% TIMEFORMAT='%3R'; { time (exec 2>&1; /home/martin/bin/satallax -E /home/martin/.isabelle/contrib/e-2.5-1/x86_64-linux/eprover -p tstp -t 5 /home/martin/judgement-day/tptp-thf/tptp/Fundamental_Theorem_Algebra/prob_119__5368088_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:27:13.420

% Could-be-implicit typings (1)
thf(ty_n_t__Real__Oreal, type,
    real : $tType).

% Explicit typings (5)
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal, type,
    ord_less_real : real > real > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal, type,
    ord_less_eq_real : real > real > $o).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Real__Oreal, type,
    order_Greatest_real : (real > $o) > real).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Real__Oreal, type,
    order_1818878995l_real : (real > real) > $o).
thf(sy_v_P, type,
    p : real > $o).

% Relevant facts (126)
thf(fact_0_ex, axiom,
    ((?[X_1 : real]: (p @ X_1)))). % ex
thf(fact_1_bz, axiom,
    ((?[Z : real]: (![X : real]: ((p @ X) => (ord_less_real @ X @ Z)))))). % bz
thf(fact_2_minf_I7_J, axiom,
    ((![T : real]: (?[Z : real]: (![X : real]: ((ord_less_real @ X @ Z) => (~ ((ord_less_real @ T @ X))))))))). % minf(7)
thf(fact_3_minf_I5_J, axiom,
    ((![T : real]: (?[Z : real]: (![X : real]: ((ord_less_real @ X @ Z) => (ord_less_real @ X @ T))))))). % minf(5)
thf(fact_4_minf_I4_J, axiom,
    ((![T : real]: (?[Z : real]: (![X : real]: ((ord_less_real @ X @ Z) => (~ ((X = T))))))))). % minf(4)
thf(fact_5_minf_I3_J, axiom,
    ((![T : real]: (?[Z : real]: (![X : real]: ((ord_less_real @ X @ Z) => (~ ((X = T))))))))). % minf(3)
thf(fact_6_minf_I2_J, axiom,
    ((![P : real > $o, P2 : real > $o, Q : real > $o, Q2 : real > $o]: ((?[Z2 : real]: (![X2 : real]: ((ord_less_real @ X2 @ Z2) => ((P @ X2) = (P2 @ X2))))) => ((?[Z2 : real]: (![X2 : real]: ((ord_less_real @ X2 @ Z2) => ((Q @ X2) = (Q2 @ X2))))) => (?[Z : real]: (![X : real]: ((ord_less_real @ X @ Z) => ((((P @ X)) | ((Q @ X))) = (((P2 @ X)) | ((Q2 @ X)))))))))))). % minf(2)
thf(fact_7_minf_I1_J, axiom,
    ((![P : real > $o, P2 : real > $o, Q : real > $o, Q2 : real > $o]: ((?[Z2 : real]: (![X2 : real]: ((ord_less_real @ X2 @ Z2) => ((P @ X2) = (P2 @ X2))))) => ((?[Z2 : real]: (![X2 : real]: ((ord_less_real @ X2 @ Z2) => ((Q @ X2) = (Q2 @ X2))))) => (?[Z : real]: (![X : real]: ((ord_less_real @ X @ Z) => ((((P @ X)) & ((Q @ X))) = (((P2 @ X)) & ((Q2 @ X)))))))))))). % minf(1)
thf(fact_8_pinf_I7_J, axiom,
    ((![T : real]: (?[Z : real]: (![X : real]: ((ord_less_real @ Z @ X) => (ord_less_real @ T @ X))))))). % pinf(7)
thf(fact_9_pinf_I5_J, axiom,
    ((![T : real]: (?[Z : real]: (![X : real]: ((ord_less_real @ Z @ X) => (~ ((ord_less_real @ X @ T))))))))). % pinf(5)
thf(fact_10_pinf_I4_J, axiom,
    ((![T : real]: (?[Z : real]: (![X : real]: ((ord_less_real @ Z @ X) => (~ ((X = T))))))))). % pinf(4)
thf(fact_11_pinf_I3_J, axiom,
    ((![T : real]: (?[Z : real]: (![X : real]: ((ord_less_real @ Z @ X) => (~ ((X = T))))))))). % pinf(3)
thf(fact_12_pinf_I1_J, axiom,
    ((![P : real > $o, P2 : real > $o, Q : real > $o, Q2 : real > $o]: ((?[Z2 : real]: (![X2 : real]: ((ord_less_real @ Z2 @ X2) => ((P @ X2) = (P2 @ X2))))) => ((?[Z2 : real]: (![X2 : real]: ((ord_less_real @ Z2 @ X2) => ((Q @ X2) = (Q2 @ X2))))) => (?[Z : real]: (![X : real]: ((ord_less_real @ Z @ X) => ((((P @ X)) & ((Q @ X))) = (((P2 @ X)) & ((Q2 @ X)))))))))))). % pinf(1)
thf(fact_13_pinf_I2_J, axiom,
    ((![P : real > $o, P2 : real > $o, Q : real > $o, Q2 : real > $o]: ((?[Z2 : real]: (![X2 : real]: ((ord_less_real @ Z2 @ X2) => ((P @ X2) = (P2 @ X2))))) => ((?[Z2 : real]: (![X2 : real]: ((ord_less_real @ Z2 @ X2) => ((Q @ X2) = (Q2 @ X2))))) => (?[Z : real]: (![X : real]: ((ord_less_real @ Z @ X) => ((((P @ X)) | ((Q @ X))) = (((P2 @ X)) | ((Q2 @ X)))))))))))). % pinf(2)
thf(fact_14_ex__gt__or__lt, axiom,
    ((![A : real]: (?[B : real]: ((ord_less_real @ A @ B) | (ord_less_real @ B @ A)))))). % ex_gt_or_lt
thf(fact_15_linorder__neqE__linordered__idom, axiom,
    ((![X3 : real, Y : real]: ((~ ((X3 = Y))) => ((~ ((ord_less_real @ X3 @ Y))) => (ord_less_real @ Y @ X3)))))). % linorder_neqE_linordered_idom
thf(fact_16_dual__order_Ostrict__implies__not__eq, axiom,
    ((![B2 : real, A : real]: ((ord_less_real @ B2 @ A) => (~ ((A = B2))))))). % dual_order.strict_implies_not_eq
thf(fact_17_linordered__field__no__ub, axiom,
    ((![X : real]: (?[X_1 : real]: (ord_less_real @ X @ X_1))))). % linordered_field_no_ub
thf(fact_18_linordered__field__no__lb, axiom,
    ((![X : real]: (?[Y2 : real]: (ord_less_real @ Y2 @ X))))). % linordered_field_no_lb
thf(fact_19_order_Ostrict__implies__not__eq, axiom,
    ((![A : real, B2 : real]: ((ord_less_real @ A @ B2) => (~ ((A = B2))))))). % order.strict_implies_not_eq
thf(fact_20_not__less__iff__gr__or__eq, axiom,
    ((![X3 : real, Y : real]: ((~ ((ord_less_real @ X3 @ Y))) = (((ord_less_real @ Y @ X3)) | ((X3 = Y))))))). % not_less_iff_gr_or_eq
thf(fact_21_dual__order_Ostrict__trans, axiom,
    ((![B2 : real, A : real, C : real]: ((ord_less_real @ B2 @ A) => ((ord_less_real @ C @ B2) => (ord_less_real @ C @ A)))))). % dual_order.strict_trans
thf(fact_22_linorder__less__wlog, axiom,
    ((![P : real > real > $o, A : real, B2 : real]: ((![A2 : real, B : real]: ((ord_less_real @ A2 @ B) => (P @ A2 @ B))) => ((![A2 : real]: (P @ A2 @ A2)) => ((![A2 : real, B : real]: ((P @ B @ A2) => (P @ A2 @ B))) => (P @ A @ B2))))))). % linorder_less_wlog
thf(fact_23_ord__eq__less__subst, axiom,
    ((![A : real, F : real > real, B2 : real, C : real]: ((A = (F @ B2)) => ((ord_less_real @ B2 @ C) => ((![X2 : real, Y2 : real]: ((ord_less_real @ X2 @ Y2) => (ord_less_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_real @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_24_ord__less__eq__subst, axiom,
    ((![A : real, B2 : real, F : real > real, C : real]: ((ord_less_real @ A @ B2) => (((F @ B2) = C) => ((![X2 : real, Y2 : real]: ((ord_less_real @ X2 @ Y2) => (ord_less_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_real @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_25_order__less__subst1, axiom,
    ((![A : real, F : real > real, B2 : real, C : real]: ((ord_less_real @ A @ (F @ B2)) => ((ord_less_real @ B2 @ C) => ((![X2 : real, Y2 : real]: ((ord_less_real @ X2 @ Y2) => (ord_less_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_26_order__less__subst2, axiom,
    ((![A : real, B2 : real, F : real > real, C : real]: ((ord_less_real @ A @ B2) => ((ord_less_real @ (F @ B2) @ C) => ((![X2 : real, Y2 : real]: ((ord_less_real @ X2 @ Y2) => (ord_less_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_27_lt__ex, axiom,
    ((![X3 : real]: (?[Y2 : real]: (ord_less_real @ Y2 @ X3))))). % lt_ex
thf(fact_28_gt__ex, axiom,
    ((![X3 : real]: (?[X_1 : real]: (ord_less_real @ X3 @ X_1))))). % gt_ex
thf(fact_29_neqE, axiom,
    ((![X3 : real, Y : real]: ((~ ((X3 = Y))) => ((~ ((ord_less_real @ X3 @ Y))) => (ord_less_real @ Y @ X3)))))). % neqE
thf(fact_30_neq__iff, axiom,
    ((![X3 : real, Y : real]: ((~ ((X3 = Y))) = (((ord_less_real @ X3 @ Y)) | ((ord_less_real @ Y @ X3))))))). % neq_iff
thf(fact_31_order_Oasym, axiom,
    ((![A : real, B2 : real]: ((ord_less_real @ A @ B2) => (~ ((ord_less_real @ B2 @ A))))))). % order.asym
thf(fact_32_dense, axiom,
    ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (?[Z : real]: ((ord_less_real @ X3 @ Z) & (ord_less_real @ Z @ Y))))))). % dense
thf(fact_33_less__imp__neq, axiom,
    ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (~ ((X3 = Y))))))). % less_imp_neq
thf(fact_34_less__asym, axiom,
    ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (~ ((ord_less_real @ Y @ X3))))))). % less_asym
thf(fact_35_less__asym_H, axiom,
    ((![A : real, B2 : real]: ((ord_less_real @ A @ B2) => (~ ((ord_less_real @ B2 @ A))))))). % less_asym'
thf(fact_36_less__trans, axiom,
    ((![X3 : real, Y : real, Z3 : real]: ((ord_less_real @ X3 @ Y) => ((ord_less_real @ Y @ Z3) => (ord_less_real @ X3 @ Z3)))))). % less_trans
thf(fact_37_less__linear, axiom,
    ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) | ((X3 = Y) | (ord_less_real @ Y @ X3)))))). % less_linear
thf(fact_38_less__irrefl, axiom,
    ((![X3 : real]: (~ ((ord_less_real @ X3 @ X3)))))). % less_irrefl
thf(fact_39_ord__eq__less__trans, axiom,
    ((![A : real, B2 : real, C : real]: ((A = B2) => ((ord_less_real @ B2 @ C) => (ord_less_real @ A @ C)))))). % ord_eq_less_trans
thf(fact_40_ord__less__eq__trans, axiom,
    ((![A : real, B2 : real, C : real]: ((ord_less_real @ A @ B2) => ((B2 = C) => (ord_less_real @ A @ C)))))). % ord_less_eq_trans
thf(fact_41_dual__order_Oasym, axiom,
    ((![B2 : real, A : real]: ((ord_less_real @ B2 @ A) => (~ ((ord_less_real @ A @ B2))))))). % dual_order.asym
thf(fact_42_less__imp__not__eq, axiom,
    ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (~ ((X3 = Y))))))). % less_imp_not_eq
thf(fact_43_less__not__sym, axiom,
    ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (~ ((ord_less_real @ Y @ X3))))))). % less_not_sym
thf(fact_44_antisym__conv3, axiom,
    ((![Y : real, X3 : real]: ((~ ((ord_less_real @ Y @ X3))) => ((~ ((ord_less_real @ X3 @ Y))) = (X3 = Y)))))). % antisym_conv3
thf(fact_45_less__imp__not__eq2, axiom,
    ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (~ ((Y = X3))))))). % less_imp_not_eq2
thf(fact_46_less__imp__triv, axiom,
    ((![X3 : real, Y : real, P : $o]: ((ord_less_real @ X3 @ Y) => ((ord_less_real @ Y @ X3) => P))))). % less_imp_triv
thf(fact_47_linorder__cases, axiom,
    ((![X3 : real, Y : real]: ((~ ((ord_less_real @ X3 @ Y))) => ((~ ((X3 = Y))) => (ord_less_real @ Y @ X3)))))). % linorder_cases
thf(fact_48_dual__order_Oirrefl, axiom,
    ((![A : real]: (~ ((ord_less_real @ A @ A)))))). % dual_order.irrefl
thf(fact_49_order_Ostrict__trans, axiom,
    ((![A : real, B2 : real, C : real]: ((ord_less_real @ A @ B2) => ((ord_less_real @ B2 @ C) => (ord_less_real @ A @ C)))))). % order.strict_trans
thf(fact_50_less__imp__not__less, axiom,
    ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (~ ((ord_less_real @ Y @ X3))))))). % less_imp_not_less
thf(fact_51_verit__comp__simplify1_I1_J, axiom,
    ((![A : real]: (~ ((ord_less_real @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_52_strict__monoD, axiom,
    ((![F : real > real, X3 : real, Y : real]: ((order_1818878995l_real @ F) => ((ord_less_real @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y))))))). % strict_monoD
thf(fact_53_strict__monoI, axiom,
    ((![F : real > real]: ((![X2 : real, Y2 : real]: ((ord_less_real @ X2 @ Y2) => (ord_less_real @ (F @ X2) @ (F @ Y2)))) => (order_1818878995l_real @ F))))). % strict_monoI
thf(fact_54_strict__mono__def, axiom,
    ((order_1818878995l_real = (^[F2 : real > real]: (![X4 : real]: (![Y3 : real]: (((ord_less_real @ X4 @ Y3)) => ((ord_less_real @ (F2 @ X4) @ (F2 @ Y3)))))))))). % strict_mono_def
thf(fact_55_strict__mono__less, axiom,
    ((![F : real > real, X3 : real, Y : real]: ((order_1818878995l_real @ F) => ((ord_less_real @ (F @ X3) @ (F @ Y)) = (ord_less_real @ X3 @ Y)))))). % strict_mono_less
thf(fact_56_strict__mono__less__eq, axiom,
    ((![F : real > real, X3 : real, Y : real]: ((order_1818878995l_real @ F) => ((ord_less_eq_real @ (F @ X3) @ (F @ Y)) = (ord_less_eq_real @ X3 @ Y)))))). % strict_mono_less_eq
thf(fact_57_order__refl, axiom,
    ((![X3 : real]: (ord_less_eq_real @ X3 @ X3)))). % order_refl
thf(fact_58_verit__comp__simplify1_I3_J, axiom,
    ((![B3 : real, A3 : real]: ((~ ((ord_less_eq_real @ B3 @ A3))) = (ord_less_real @ A3 @ B3))))). % verit_comp_simplify1(3)
thf(fact_59_verit__la__disequality, axiom,
    ((![A : real, B2 : real]: ((A = B2) | ((~ ((ord_less_eq_real @ A @ B2))) | (~ ((ord_less_eq_real @ B2 @ A)))))))). % verit_la_disequality
thf(fact_60_pinf_I6_J, axiom,
    ((![T : real]: (?[Z : real]: (![X : real]: ((ord_less_real @ Z @ X) => (~ ((ord_less_eq_real @ X @ T))))))))). % pinf(6)
thf(fact_61_pinf_I8_J, axiom,
    ((![T : real]: (?[Z : real]: (![X : real]: ((ord_less_real @ Z @ X) => (ord_less_eq_real @ T @ X))))))). % pinf(8)
thf(fact_62_minf_I6_J, axiom,
    ((![T : real]: (?[Z : real]: (![X : real]: ((ord_less_real @ X @ Z) => (ord_less_eq_real @ X @ T))))))). % minf(6)
thf(fact_63_minf_I8_J, axiom,
    ((![T : real]: (?[Z : real]: (![X : real]: ((ord_less_real @ X @ Z) => (~ ((ord_less_eq_real @ T @ X))))))))). % minf(8)
thf(fact_64_complete__interval, axiom,
    ((![A : real, B2 : real, P : real > $o]: ((ord_less_real @ A @ B2) => ((P @ A) => ((~ ((P @ B2))) => (?[C2 : real]: ((ord_less_eq_real @ A @ C2) & ((ord_less_eq_real @ C2 @ B2) & ((![X : real]: (((ord_less_eq_real @ A @ X) & (ord_less_real @ X @ C2)) => (P @ X))) & (![D : real]: ((![X2 : real]: (((ord_less_eq_real @ A @ X2) & (ord_less_real @ X2 @ D)) => (P @ X2))) => (ord_less_eq_real @ D @ C2))))))))))))). % complete_interval
thf(fact_65_order_Onot__eq__order__implies__strict, axiom,
    ((![A : real, B2 : real]: ((~ ((A = B2))) => ((ord_less_eq_real @ A @ B2) => (ord_less_real @ A @ B2)))))). % order.not_eq_order_implies_strict
thf(fact_66_dual__order_Ostrict__implies__order, axiom,
    ((![B2 : real, A : real]: ((ord_less_real @ B2 @ A) => (ord_less_eq_real @ B2 @ A))))). % dual_order.strict_implies_order
thf(fact_67_dual__order_Ostrict__iff__order, axiom,
    ((ord_less_real = (^[B4 : real]: (^[A4 : real]: (((ord_less_eq_real @ B4 @ A4)) & ((~ ((A4 = B4)))))))))). % dual_order.strict_iff_order
thf(fact_68_dual__order_Oorder__iff__strict, axiom,
    ((ord_less_eq_real = (^[B4 : real]: (^[A4 : real]: (((ord_less_real @ B4 @ A4)) | ((A4 = B4)))))))). % dual_order.order_iff_strict
thf(fact_69_order_Ostrict__implies__order, axiom,
    ((![A : real, B2 : real]: ((ord_less_real @ A @ B2) => (ord_less_eq_real @ A @ B2))))). % order.strict_implies_order
thf(fact_70_dense__le__bounded, axiom,
    ((![X3 : real, Y : real, Z3 : real]: ((ord_less_real @ X3 @ Y) => ((![W : real]: ((ord_less_real @ X3 @ W) => ((ord_less_real @ W @ Y) => (ord_less_eq_real @ W @ Z3)))) => (ord_less_eq_real @ Y @ Z3)))))). % dense_le_bounded
thf(fact_71_dense__ge__bounded, axiom,
    ((![Z3 : real, X3 : real, Y : real]: ((ord_less_real @ Z3 @ X3) => ((![W : real]: ((ord_less_real @ Z3 @ W) => ((ord_less_real @ W @ X3) => (ord_less_eq_real @ Y @ W)))) => (ord_less_eq_real @ Y @ Z3)))))). % dense_ge_bounded
thf(fact_72_dual__order_Ostrict__trans2, axiom,
    ((![B2 : real, A : real, C : real]: ((ord_less_real @ B2 @ A) => ((ord_less_eq_real @ C @ B2) => (ord_less_real @ C @ A)))))). % dual_order.strict_trans2
thf(fact_73_dual__order_Ostrict__trans1, axiom,
    ((![B2 : real, A : real, C : real]: ((ord_less_eq_real @ B2 @ A) => ((ord_less_real @ C @ B2) => (ord_less_real @ C @ A)))))). % dual_order.strict_trans1
thf(fact_74_order_Ostrict__iff__order, axiom,
    ((ord_less_real = (^[A4 : real]: (^[B4 : real]: (((ord_less_eq_real @ A4 @ B4)) & ((~ ((A4 = B4)))))))))). % order.strict_iff_order
thf(fact_75_order_Oorder__iff__strict, axiom,
    ((ord_less_eq_real = (^[A4 : real]: (^[B4 : real]: (((ord_less_real @ A4 @ B4)) | ((A4 = B4)))))))). % order.order_iff_strict
thf(fact_76_order_Ostrict__trans2, axiom,
    ((![A : real, B2 : real, C : real]: ((ord_less_real @ A @ B2) => ((ord_less_eq_real @ B2 @ C) => (ord_less_real @ A @ C)))))). % order.strict_trans2
thf(fact_77_order_Ostrict__trans1, axiom,
    ((![A : real, B2 : real, C : real]: ((ord_less_eq_real @ A @ B2) => ((ord_less_real @ B2 @ C) => (ord_less_real @ A @ C)))))). % order.strict_trans1
thf(fact_78_not__le__imp__less, axiom,
    ((![Y : real, X3 : real]: ((~ ((ord_less_eq_real @ Y @ X3))) => (ord_less_real @ X3 @ Y))))). % not_le_imp_less
thf(fact_79_less__le__not__le, axiom,
    ((ord_less_real = (^[X4 : real]: (^[Y3 : real]: (((ord_less_eq_real @ X4 @ Y3)) & ((~ ((ord_less_eq_real @ Y3 @ X4)))))))))). % less_le_not_le
thf(fact_80_le__imp__less__or__eq, axiom,
    ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => ((ord_less_real @ X3 @ Y) | (X3 = Y)))))). % le_imp_less_or_eq
thf(fact_81_le__less__linear, axiom,
    ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) | (ord_less_real @ Y @ X3))))). % le_less_linear
thf(fact_82_dense__le, axiom,
    ((![Y : real, Z3 : real]: ((![X2 : real]: ((ord_less_real @ X2 @ Y) => (ord_less_eq_real @ X2 @ Z3))) => (ord_less_eq_real @ Y @ Z3))))). % dense_le
thf(fact_83_dense__ge, axiom,
    ((![Z3 : real, Y : real]: ((![X2 : real]: ((ord_less_real @ Z3 @ X2) => (ord_less_eq_real @ Y @ X2))) => (ord_less_eq_real @ Y @ Z3))))). % dense_ge
thf(fact_84_less__le__trans, axiom,
    ((![X3 : real, Y : real, Z3 : real]: ((ord_less_real @ X3 @ Y) => ((ord_less_eq_real @ Y @ Z3) => (ord_less_real @ X3 @ Z3)))))). % less_le_trans
thf(fact_85_le__less__trans, axiom,
    ((![X3 : real, Y : real, Z3 : real]: ((ord_less_eq_real @ X3 @ Y) => ((ord_less_real @ Y @ Z3) => (ord_less_real @ X3 @ Z3)))))). % le_less_trans
thf(fact_86_less__imp__le, axiom,
    ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_eq_real @ X3 @ Y))))). % less_imp_le
thf(fact_87_antisym__conv2, axiom,
    ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => ((~ ((ord_less_real @ X3 @ Y))) = (X3 = Y)))))). % antisym_conv2
thf(fact_88_antisym__conv1, axiom,
    ((![X3 : real, Y : real]: ((~ ((ord_less_real @ X3 @ Y))) => ((ord_less_eq_real @ X3 @ Y) = (X3 = Y)))))). % antisym_conv1
thf(fact_89_le__neq__trans, axiom,
    ((![A : real, B2 : real]: ((ord_less_eq_real @ A @ B2) => ((~ ((A = B2))) => (ord_less_real @ A @ B2)))))). % le_neq_trans
thf(fact_90_not__less, axiom,
    ((![X3 : real, Y : real]: ((~ ((ord_less_real @ X3 @ Y))) = (ord_less_eq_real @ Y @ X3))))). % not_less
thf(fact_91_not__le, axiom,
    ((![X3 : real, Y : real]: ((~ ((ord_less_eq_real @ X3 @ Y))) = (ord_less_real @ Y @ X3))))). % not_le
thf(fact_92_order__less__le__subst2, axiom,
    ((![A : real, B2 : real, F : real > real, C : real]: ((ord_less_real @ A @ B2) => ((ord_less_eq_real @ (F @ B2) @ C) => ((![X2 : real, Y2 : real]: ((ord_less_real @ X2 @ Y2) => (ord_less_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_le_subst2
thf(fact_93_order__less__le__subst1, axiom,
    ((![A : real, F : real > real, B2 : real, C : real]: ((ord_less_real @ A @ (F @ B2)) => ((ord_less_eq_real @ B2 @ C) => ((![X2 : real, Y2 : real]: ((ord_less_eq_real @ X2 @ Y2) => (ord_less_eq_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_le_subst1
thf(fact_94_order__le__less__subst2, axiom,
    ((![A : real, B2 : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B2) => ((ord_less_real @ (F @ B2) @ C) => ((![X2 : real, Y2 : real]: ((ord_less_eq_real @ X2 @ Y2) => (ord_less_eq_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_real @ (F @ A) @ C))))))). % order_le_less_subst2
thf(fact_95_order__le__less__subst1, axiom,
    ((![A : real, F : real > real, B2 : real, C : real]: ((ord_less_eq_real @ A @ (F @ B2)) => ((ord_less_real @ B2 @ C) => ((![X2 : real, Y2 : real]: ((ord_less_real @ X2 @ Y2) => (ord_less_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_real @ A @ (F @ C)))))))). % order_le_less_subst1
thf(fact_96_less__le, axiom,
    ((ord_less_real = (^[X4 : real]: (^[Y3 : real]: (((ord_less_eq_real @ X4 @ Y3)) & ((~ ((X4 = Y3)))))))))). % less_le
thf(fact_97_le__less, axiom,
    ((ord_less_eq_real = (^[X4 : real]: (^[Y3 : real]: (((ord_less_real @ X4 @ Y3)) | ((X4 = Y3)))))))). % le_less
thf(fact_98_leI, axiom,
    ((![X3 : real, Y : real]: ((~ ((ord_less_real @ X3 @ Y))) => (ord_less_eq_real @ Y @ X3))))). % leI
thf(fact_99_leD, axiom,
    ((![Y : real, X3 : real]: ((ord_less_eq_real @ Y @ X3) => (~ ((ord_less_real @ X3 @ Y))))))). % leD
thf(fact_100_order__subst1, axiom,
    ((![A : real, F : real > real, B2 : real, C : real]: ((ord_less_eq_real @ A @ (F @ B2)) => ((ord_less_eq_real @ B2 @ C) => ((![X2 : real, Y2 : real]: ((ord_less_eq_real @ X2 @ Y2) => (ord_less_eq_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % order_subst1
thf(fact_101_order__subst2, axiom,
    ((![A : real, B2 : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B2) => ((ord_less_eq_real @ (F @ B2) @ C) => ((![X2 : real, Y2 : real]: ((ord_less_eq_real @ X2 @ Y2) => (ord_less_eq_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % order_subst2
thf(fact_102_ord__eq__le__subst, axiom,
    ((![A : real, F : real > real, B2 : real, C : real]: ((A = (F @ B2)) => ((ord_less_eq_real @ B2 @ C) => ((![X2 : real, Y2 : real]: ((ord_less_eq_real @ X2 @ Y2) => (ord_less_eq_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_103_ord__le__eq__subst, axiom,
    ((![A : real, B2 : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B2) => (((F @ B2) = C) => ((![X2 : real, Y2 : real]: ((ord_less_eq_real @ X2 @ Y2) => (ord_less_eq_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_104_eq__iff, axiom,
    (((^[Y4 : real]: (^[Z4 : real]: (Y4 = Z4))) = (^[X4 : real]: (^[Y3 : real]: (((ord_less_eq_real @ X4 @ Y3)) & ((ord_less_eq_real @ Y3 @ X4)))))))). % eq_iff
thf(fact_105_antisym, axiom,
    ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => ((ord_less_eq_real @ Y @ X3) => (X3 = Y)))))). % antisym
thf(fact_106_linear, axiom,
    ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) | (ord_less_eq_real @ Y @ X3))))). % linear
thf(fact_107_eq__refl, axiom,
    ((![X3 : real, Y : real]: ((X3 = Y) => (ord_less_eq_real @ X3 @ Y))))). % eq_refl
thf(fact_108_le__cases, axiom,
    ((![X3 : real, Y : real]: ((~ ((ord_less_eq_real @ X3 @ Y))) => (ord_less_eq_real @ Y @ X3))))). % le_cases
thf(fact_109_order_Otrans, axiom,
    ((![A : real, B2 : real, C : real]: ((ord_less_eq_real @ A @ B2) => ((ord_less_eq_real @ B2 @ C) => (ord_less_eq_real @ A @ C)))))). % order.trans
thf(fact_110_le__cases3, axiom,
    ((![X3 : real, Y : real, Z3 : real]: (((ord_less_eq_real @ X3 @ Y) => (~ ((ord_less_eq_real @ Y @ Z3)))) => (((ord_less_eq_real @ Y @ X3) => (~ ((ord_less_eq_real @ X3 @ Z3)))) => (((ord_less_eq_real @ X3 @ Z3) => (~ ((ord_less_eq_real @ Z3 @ Y)))) => (((ord_less_eq_real @ Z3 @ Y) => (~ ((ord_less_eq_real @ Y @ X3)))) => (((ord_less_eq_real @ Y @ Z3) => (~ ((ord_less_eq_real @ Z3 @ X3)))) => (~ (((ord_less_eq_real @ Z3 @ X3) => (~ ((ord_less_eq_real @ X3 @ Y)))))))))))))). % le_cases3
thf(fact_111_antisym__conv, axiom,
    ((![Y : real, X3 : real]: ((ord_less_eq_real @ Y @ X3) => ((ord_less_eq_real @ X3 @ Y) = (X3 = Y)))))). % antisym_conv
thf(fact_112_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y4 : real]: (^[Z4 : real]: (Y4 = Z4))) = (^[A4 : real]: (^[B4 : real]: (((ord_less_eq_real @ A4 @ B4)) & ((ord_less_eq_real @ B4 @ A4)))))))). % order_class.order.eq_iff
thf(fact_113_ord__eq__le__trans, axiom,
    ((![A : real, B2 : real, C : real]: ((A = B2) => ((ord_less_eq_real @ B2 @ C) => (ord_less_eq_real @ A @ C)))))). % ord_eq_le_trans
thf(fact_114_ord__le__eq__trans, axiom,
    ((![A : real, B2 : real, C : real]: ((ord_less_eq_real @ A @ B2) => ((B2 = C) => (ord_less_eq_real @ A @ C)))))). % ord_le_eq_trans
thf(fact_115_order__class_Oorder_Oantisym, axiom,
    ((![A : real, B2 : real]: ((ord_less_eq_real @ A @ B2) => ((ord_less_eq_real @ B2 @ A) => (A = B2)))))). % order_class.order.antisym
thf(fact_116_order__trans, axiom,
    ((![X3 : real, Y : real, Z3 : real]: ((ord_less_eq_real @ X3 @ Y) => ((ord_less_eq_real @ Y @ Z3) => (ord_less_eq_real @ X3 @ Z3)))))). % order_trans
thf(fact_117_dual__order_Orefl, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ A)))). % dual_order.refl
thf(fact_118_linorder__wlog, axiom,
    ((![P : real > real > $o, A : real, B2 : real]: ((![A2 : real, B : real]: ((ord_less_eq_real @ A2 @ B) => (P @ A2 @ B))) => ((![A2 : real, B : real]: ((P @ B @ A2) => (P @ A2 @ B))) => (P @ A @ B2)))))). % linorder_wlog
thf(fact_119_dual__order_Otrans, axiom,
    ((![B2 : real, A : real, C : real]: ((ord_less_eq_real @ B2 @ A) => ((ord_less_eq_real @ C @ B2) => (ord_less_eq_real @ C @ A)))))). % dual_order.trans
thf(fact_120_dual__order_Oeq__iff, axiom,
    (((^[Y4 : real]: (^[Z4 : real]: (Y4 = Z4))) = (^[A4 : real]: (^[B4 : real]: (((ord_less_eq_real @ B4 @ A4)) & ((ord_less_eq_real @ A4 @ B4)))))))). % dual_order.eq_iff
thf(fact_121_dual__order_Oantisym, axiom,
    ((![B2 : real, A : real]: ((ord_less_eq_real @ B2 @ A) => ((ord_less_eq_real @ A @ B2) => (A = B2)))))). % dual_order.antisym
thf(fact_122_strict__mono__leD, axiom,
    ((![R : real > real, M : real, N : real]: ((order_1818878995l_real @ R) => ((ord_less_eq_real @ M @ N) => (ord_less_eq_real @ (R @ M) @ (R @ N))))))). % strict_mono_leD
thf(fact_123_GreatestI2__order, axiom,
    ((![P : real > $o, X3 : real, Q : real > $o]: ((P @ X3) => ((![Y2 : real]: ((P @ Y2) => (ord_less_eq_real @ Y2 @ X3))) => ((![X2 : real]: ((P @ X2) => ((![Y5 : real]: ((P @ Y5) => (ord_less_eq_real @ Y5 @ X2))) => (Q @ X2)))) => (Q @ (order_Greatest_real @ P)))))))). % GreatestI2_order
thf(fact_124_Greatest__equality, axiom,
    ((![P : real > $o, X3 : real]: ((P @ X3) => ((![Y2 : real]: ((P @ Y2) => (ord_less_eq_real @ Y2 @ X3))) => ((order_Greatest_real @ P) = X3)))))). % Greatest_equality
thf(fact_125_less__eq__real__def, axiom,
    ((ord_less_eq_real = (^[X4 : real]: (^[Y3 : real]: (((ord_less_real @ X4 @ Y3)) | ((X4 = Y3)))))))). % less_eq_real_def

% Conjectures (1)
thf(conj_0, conjecture,
    ((?[S : real]: (![Y2 : real]: ((?[X4 : real]: (((p @ X4)) & ((ord_less_real @ Y2 @ X4)))) = (ord_less_real @ Y2 @ S)))))).
