TPTP Problem File: ITP111^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP111^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Lower_Semicontinuous problem prob_214__6249176_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Lower_Semicontinuous/prob_214__6249176_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 138 ( 59 unt; 17 typ; 0 def)
% Number of atoms : 359 ( 148 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 932 ( 58 ~; 18 |; 35 &; 684 @)
% ( 0 <=>; 137 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 42 ( 42 >; 0 *; 0 +; 0 <<)
% Number of symbols : 15 ( 14 usr; 5 con; 0-2 aty)
% Number of variables : 288 ( 3 ^; 255 !; 30 ?; 288 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:40:48.195
%------------------------------------------------------------------------------
% Could-be-implicit typings (3)
thf(ty_n_t__Extended____Real__Oereal,type,
extended_ereal: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (14)
thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity_001t__Extended____Real__Oereal,type,
extend1289208545_ereal: extended_ereal ).
thf(sy_c_Extended__Real_Oereal_OMInfty,type,
extended_MInfty: extended_ereal ).
thf(sy_c_Extended__Real_Oereal_OPInfty,type,
extended_PInfty: extended_ereal ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Real__Oereal,type,
times_1966848393_ereal: extended_ereal > extended_ereal > extended_ereal ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Extended____Real__Oereal,type,
uminus1208298309_ereal: extended_ereal > extended_ereal ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Real__Oereal,type,
zero_z163181189_ereal: extended_ereal ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001tf__a_001t__Extended____Real__Oereal,type,
lower_191460856_ereal: a > ( a > extended_ereal ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Real__Oereal,type,
ord_le2001149050_ereal: extended_ereal > extended_ereal > $o ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Extended____Real__Oereal,type,
divide595620860_ereal: extended_ereal > extended_ereal > extended_ereal ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001tf__a,type,
topolo1276428101open_a: set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_f,type,
f: a > extended_ereal ).
thf(sy_v_x0,type,
x0: a ).
% Relevant facts (120)
thf(fact_0__092_060open_062f_Ax0_A_061_A_092_060infinity_062_A_092_060Longrightarrow_062_Alsc__at_Ax0_Af_A_061_A_I_092_060forall_062C_060f_Ax0_O_A_092_060exists_062T_O_Aopen_AT_A_092_060and_062_Ax0_A_092_060in_062_AT_A_092_060and_062_A_I_092_060forall_062y_092_060in_062T_O_AC_A_060_Af_Ay_J_J_092_060close_062,axiom,
( ( ( f @ x0 )
= extend1289208545_ereal )
=> ( ( lower_191460856_ereal @ x0 @ f )
= ( ! [C: extended_ereal] :
( ( ord_le2001149050_ereal @ C @ ( f @ x0 ) )
=> ? [T: set_a] :
( ( topolo1276428101open_a @ T )
& ( member_a @ x0 @ T )
& ! [X: a] :
( ( member_a @ X @ T )
=> ( ord_le2001149050_ereal @ C @ ( f @ X ) ) ) ) ) ) ) ) ).
% \<open>f x0 = \<infinity> \<Longrightarrow> lsc_at x0 f = (\<forall>C<f x0. \<exists>T. open T \<and> x0 \<in> T \<and> (\<forall>y\<in>T. C < f y))\<close>
thf(fact_1__092_060open_062f_Ax0_A_061_A_N_A_092_060infinity_062_A_092_060Longrightarrow_062_Alsc__at_Ax0_Af_A_061_A_I_092_060forall_062C_060f_Ax0_O_A_092_060exists_062T_O_Aopen_AT_A_092_060and_062_Ax0_A_092_060in_062_AT_A_092_060and_062_A_I_092_060forall_062y_092_060in_062T_O_AC_A_060_Af_Ay_J_J_092_060close_062,axiom,
( ( ( f @ x0 )
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ( lower_191460856_ereal @ x0 @ f )
= ( ! [C: extended_ereal] :
( ( ord_le2001149050_ereal @ C @ ( f @ x0 ) )
=> ? [T: set_a] :
( ( topolo1276428101open_a @ T )
& ( member_a @ x0 @ T )
& ! [X: a] :
( ( member_a @ X @ T )
=> ( ord_le2001149050_ereal @ C @ ( f @ X ) ) ) ) ) ) ) ) ).
% \<open>f x0 = - \<infinity> \<Longrightarrow> lsc_at x0 f = (\<forall>C<f x0. \<exists>T. open T \<and> x0 \<in> T \<and> (\<forall>y\<in>T. C < f y))\<close>
thf(fact_2__092_060open_062_092_060lbrakk_062f_Ax0_A_092_060noteq_062_A_N_A_092_060infinity_062_059_Af_Ax0_A_092_060noteq_062_A_092_060infinity_062_092_060rbrakk_062_A_092_060Longrightarrow_062_Alsc__at_Ax0_Af_A_061_A_I_092_060forall_062C_060f_Ax0_O_A_092_060exists_062T_O_Aopen_AT_A_092_060and_062_Ax0_A_092_060in_062_AT_A_092_060and_062_A_I_092_060forall_062y_092_060in_062T_O_AC_A_060_Af_Ay_J_J_092_060close_062,axiom,
( ( ( f @ x0 )
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ( ( f @ x0 )
!= extend1289208545_ereal )
=> ( ( lower_191460856_ereal @ x0 @ f )
= ( ! [C: extended_ereal] :
( ( ord_le2001149050_ereal @ C @ ( f @ x0 ) )
=> ? [T: set_a] :
( ( topolo1276428101open_a @ T )
& ( member_a @ x0 @ T )
& ! [X: a] :
( ( member_a @ X @ T )
=> ( ord_le2001149050_ereal @ C @ ( f @ X ) ) ) ) ) ) ) ) ) ).
% \<open>\<lbrakk>f x0 \<noteq> - \<infinity>; f x0 \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> lsc_at x0 f = (\<forall>C<f x0. \<exists>T. open T \<and> x0 \<in> T \<and> (\<forall>y\<in>T. C < f y))\<close>
thf(fact_3_lsc__at__MInfty,axiom,
! [F: a > extended_ereal,X0: a] :
( ( ( F @ X0 )
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( lower_191460856_ereal @ X0 @ F ) ) ).
% lsc_at_MInfty
thf(fact_4_t0__space,axiom,
! [X2: a,Y: a] :
( ( X2 != Y )
=> ? [U: set_a] :
( ( topolo1276428101open_a @ U )
& ( ( member_a @ X2 @ U )
!= ( member_a @ Y @ U ) ) ) ) ).
% t0_space
thf(fact_5_t1__space,axiom,
! [X2: a,Y: a] :
( ( X2 != Y )
=> ? [U: set_a] :
( ( topolo1276428101open_a @ U )
& ( member_a @ X2 @ U )
& ~ ( member_a @ Y @ U ) ) ) ).
% t1_space
thf(fact_6_separation__t0,axiom,
! [X2: a,Y: a] :
( ( X2 != Y )
= ( ? [U2: set_a] :
( ( topolo1276428101open_a @ U2 )
& ( ( member_a @ X2 @ U2 )
!= ( member_a @ Y @ U2 ) ) ) ) ) ).
% separation_t0
thf(fact_7_separation__t1,axiom,
! [X2: a,Y: a] :
( ( X2 != Y )
= ( ? [U2: set_a] :
( ( topolo1276428101open_a @ U2 )
& ( member_a @ X2 @ U2 )
& ~ ( member_a @ Y @ U2 ) ) ) ) ).
% separation_t1
thf(fact_8_minf_I7_J,axiom,
! [T2: extended_ereal] :
? [Z: extended_ereal] :
! [X3: extended_ereal] :
( ( ord_le2001149050_ereal @ X3 @ Z )
=> ~ ( ord_le2001149050_ereal @ T2 @ X3 ) ) ).
% minf(7)
thf(fact_9_minf_I5_J,axiom,
! [T2: extended_ereal] :
? [Z: extended_ereal] :
! [X3: extended_ereal] :
( ( ord_le2001149050_ereal @ X3 @ Z )
=> ( ord_le2001149050_ereal @ X3 @ T2 ) ) ).
% minf(5)
thf(fact_10_minf_I4_J,axiom,
! [T2: extended_ereal] :
? [Z: extended_ereal] :
! [X3: extended_ereal] :
( ( ord_le2001149050_ereal @ X3 @ Z )
=> ( X3 != T2 ) ) ).
% minf(4)
thf(fact_11_minf_I3_J,axiom,
! [T2: extended_ereal] :
? [Z: extended_ereal] :
! [X3: extended_ereal] :
( ( ord_le2001149050_ereal @ X3 @ Z )
=> ( X3 != T2 ) ) ).
% minf(3)
thf(fact_12_minf_I2_J,axiom,
! [P: extended_ereal > $o,P2: extended_ereal > $o,Q: extended_ereal > $o,Q2: extended_ereal > $o] :
( ? [Z2: extended_ereal] :
! [X4: extended_ereal] :
( ( ord_le2001149050_ereal @ X4 @ Z2 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z2: extended_ereal] :
! [X4: extended_ereal] :
( ( ord_le2001149050_ereal @ X4 @ Z2 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z: extended_ereal] :
! [X3: extended_ereal] :
( ( ord_le2001149050_ereal @ X3 @ Z )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_13_pinf_I1_J,axiom,
! [P: extended_ereal > $o,P2: extended_ereal > $o,Q: extended_ereal > $o,Q2: extended_ereal > $o] :
( ? [Z2: extended_ereal] :
! [X4: extended_ereal] :
( ( ord_le2001149050_ereal @ Z2 @ X4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z2: extended_ereal] :
! [X4: extended_ereal] :
( ( ord_le2001149050_ereal @ Z2 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z: extended_ereal] :
! [X3: extended_ereal] :
( ( ord_le2001149050_ereal @ Z @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_14_pinf_I2_J,axiom,
! [P: extended_ereal > $o,P2: extended_ereal > $o,Q: extended_ereal > $o,Q2: extended_ereal > $o] :
( ? [Z2: extended_ereal] :
! [X4: extended_ereal] :
( ( ord_le2001149050_ereal @ Z2 @ X4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z2: extended_ereal] :
! [X4: extended_ereal] :
( ( ord_le2001149050_ereal @ Z2 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z: extended_ereal] :
! [X3: extended_ereal] :
( ( ord_le2001149050_ereal @ Z @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_15_pinf_I3_J,axiom,
! [T2: extended_ereal] :
? [Z: extended_ereal] :
! [X3: extended_ereal] :
( ( ord_le2001149050_ereal @ Z @ X3 )
=> ( X3 != T2 ) ) ).
% pinf(3)
thf(fact_16_pinf_I4_J,axiom,
! [T2: extended_ereal] :
? [Z: extended_ereal] :
! [X3: extended_ereal] :
( ( ord_le2001149050_ereal @ Z @ X3 )
=> ( X3 != T2 ) ) ).
% pinf(4)
thf(fact_17_pinf_I5_J,axiom,
! [T2: extended_ereal] :
? [Z: extended_ereal] :
! [X3: extended_ereal] :
( ( ord_le2001149050_ereal @ Z @ X3 )
=> ~ ( ord_le2001149050_ereal @ X3 @ T2 ) ) ).
% pinf(5)
thf(fact_18_pinf_I7_J,axiom,
! [T2: extended_ereal] :
? [Z: extended_ereal] :
! [X3: extended_ereal] :
( ( ord_le2001149050_ereal @ Z @ X3 )
=> ( ord_le2001149050_ereal @ T2 @ X3 ) ) ).
% pinf(7)
thf(fact_19_minf_I1_J,axiom,
! [P: extended_ereal > $o,P2: extended_ereal > $o,Q: extended_ereal > $o,Q2: extended_ereal > $o] :
( ? [Z2: extended_ereal] :
! [X4: extended_ereal] :
( ( ord_le2001149050_ereal @ X4 @ Z2 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z2: extended_ereal] :
! [X4: extended_ereal] :
( ( ord_le2001149050_ereal @ X4 @ Z2 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z: extended_ereal] :
! [X3: extended_ereal] :
( ( ord_le2001149050_ereal @ X3 @ Z )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_20_ereal__MInfty__lessI,axiom,
! [A: extended_ereal] :
( ( A
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ord_le2001149050_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ A ) ) ).
% ereal_MInfty_lessI
thf(fact_21_ereal__infty__less_I2_J,axiom,
! [X2: extended_ereal] :
( ( ord_le2001149050_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ X2 )
= ( X2
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ).
% ereal_infty_less(2)
thf(fact_22_ereal__minus__less__minus,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le2001149050_ereal @ ( uminus1208298309_ereal @ A ) @ ( uminus1208298309_ereal @ B ) )
= ( ord_le2001149050_ereal @ B @ A ) ) ).
% ereal_minus_less_minus
thf(fact_23_ereal__less__PInfty,axiom,
! [A: extended_ereal] :
( ( A != extend1289208545_ereal )
=> ( ord_le2001149050_ereal @ A @ extend1289208545_ereal ) ) ).
% ereal_less_PInfty
thf(fact_24_ereal__infty__less_I1_J,axiom,
! [X2: extended_ereal] :
( ( ord_le2001149050_ereal @ X2 @ extend1289208545_ereal )
= ( X2 != extend1289208545_ereal ) ) ).
% ereal_infty_less(1)
thf(fact_25_less__ereal_Osimps_I3_J,axiom,
! [A: extended_ereal] :
~ ( ord_le2001149050_ereal @ A @ ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% less_ereal.simps(3)
thf(fact_26_less__ereal_Osimps_I6_J,axiom,
ord_le2001149050_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ extend1289208545_ereal ).
% less_ereal.simps(6)
thf(fact_27_ereal__uminus__eq__iff,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( uminus1208298309_ereal @ A )
= ( uminus1208298309_ereal @ B ) )
= ( A = B ) ) ).
% ereal_uminus_eq_iff
thf(fact_28_ereal__uminus__uminus,axiom,
! [A: extended_ereal] :
( ( uminus1208298309_ereal @ ( uminus1208298309_ereal @ A ) )
= A ) ).
% ereal_uminus_uminus
thf(fact_29_ereal__uminus__eq__reorder,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( uminus1208298309_ereal @ A )
= B )
= ( A
= ( uminus1208298309_ereal @ B ) ) ) ).
% ereal_uminus_eq_reorder
thf(fact_30_MInfty__neq__PInfty_I1_J,axiom,
( extend1289208545_ereal
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% MInfty_neq_PInfty(1)
thf(fact_31_less__ereal_Osimps_I2_J,axiom,
! [A: extended_ereal] :
~ ( ord_le2001149050_ereal @ extend1289208545_ereal @ A ) ).
% less_ereal.simps(2)
thf(fact_32_ereal__less__uminus__reorder,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le2001149050_ereal @ A @ ( uminus1208298309_ereal @ B ) )
= ( ord_le2001149050_ereal @ B @ ( uminus1208298309_ereal @ A ) ) ) ).
% ereal_less_uminus_reorder
thf(fact_33_ereal__uminus__less__reorder,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le2001149050_ereal @ ( uminus1208298309_ereal @ A ) @ B )
= ( ord_le2001149050_ereal @ ( uminus1208298309_ereal @ B ) @ A ) ) ).
% ereal_uminus_less_reorder
thf(fact_34_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_35_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_36_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_37_MInfty__eq__minfinity,axiom,
( extended_MInfty
= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% MInfty_eq_minfinity
thf(fact_38_verit__comp__simplify1_I1_J,axiom,
! [A: extended_ereal] :
~ ( ord_le2001149050_ereal @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_39_uminus__ereal_Osimps_I2_J,axiom,
( ( uminus1208298309_ereal @ extended_PInfty )
= extended_MInfty ) ).
% uminus_ereal.simps(2)
thf(fact_40_uminus__ereal_Osimps_I3_J,axiom,
( ( uminus1208298309_ereal @ extended_MInfty )
= extended_PInfty ) ).
% uminus_ereal.simps(3)
thf(fact_41_ereal__less_I6_J,axiom,
ord_le2001149050_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ zero_z163181189_ereal ).
% ereal_less(6)
thf(fact_42_ex__gt__or__lt,axiom,
! [A: extended_ereal] :
? [B2: extended_ereal] :
( ( ord_le2001149050_ereal @ A @ B2 )
| ( ord_le2001149050_ereal @ B2 @ A ) ) ).
% ex_gt_or_lt
thf(fact_43_dual__order_Ostrict__implies__not__eq,axiom,
! [B: extended_ereal,A: extended_ereal] :
( ( ord_le2001149050_ereal @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_44_ereal__uminus__zero__iff,axiom,
! [A: extended_ereal] :
( ( ( uminus1208298309_ereal @ A )
= zero_z163181189_ereal )
= ( A = zero_z163181189_ereal ) ) ).
% ereal_uminus_zero_iff
thf(fact_45_ereal__uminus__zero,axiom,
( ( uminus1208298309_ereal @ zero_z163181189_ereal )
= zero_z163181189_ereal ) ).
% ereal_uminus_zero
thf(fact_46_neg__0__less__iff__less__erea,axiom,
! [A: extended_ereal] :
( ( ord_le2001149050_ereal @ zero_z163181189_ereal @ ( uminus1208298309_ereal @ A ) )
= ( ord_le2001149050_ereal @ A @ zero_z163181189_ereal ) ) ).
% neg_0_less_iff_less_erea
thf(fact_47_zero__reorient,axiom,
! [X2: extended_ereal] :
( ( zero_z163181189_ereal = X2 )
= ( X2 = zero_z163181189_ereal ) ) ).
% zero_reorient
thf(fact_48_Infty__neq__0_I1_J,axiom,
extend1289208545_ereal != zero_z163181189_ereal ).
% Infty_neq_0(1)
thf(fact_49_infinity__ereal__def,axiom,
extend1289208545_ereal = extended_PInfty ).
% infinity_ereal_def
thf(fact_50_ereal_Odistinct_I5_J,axiom,
extended_PInfty != extended_MInfty ).
% ereal.distinct(5)
thf(fact_51_Infty__neq__0_I3_J,axiom,
( ( uminus1208298309_ereal @ extend1289208545_ereal )
!= zero_z163181189_ereal ) ).
% Infty_neq_0(3)
thf(fact_52_ereal__less_I5_J,axiom,
ord_le2001149050_ereal @ zero_z163181189_ereal @ extend1289208545_ereal ).
% ereal_less(5)
thf(fact_53_ord__eq__less__subst,axiom,
! [A: extended_ereal,F: extended_ereal > extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2001149050_ereal @ B @ C2 )
=> ( ! [X4: extended_ereal,Y2: extended_ereal] :
( ( ord_le2001149050_ereal @ X4 @ Y2 )
=> ( ord_le2001149050_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le2001149050_ereal @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_54_ord__less__eq__subst,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > extended_ereal,C2: extended_ereal] :
( ( ord_le2001149050_ereal @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: extended_ereal,Y2: extended_ereal] :
( ( ord_le2001149050_ereal @ X4 @ Y2 )
=> ( ord_le2001149050_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le2001149050_ereal @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_55_order__less__subst1,axiom,
! [A: extended_ereal,F: extended_ereal > extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( ord_le2001149050_ereal @ A @ ( F @ B ) )
=> ( ( ord_le2001149050_ereal @ B @ C2 )
=> ( ! [X4: extended_ereal,Y2: extended_ereal] :
( ( ord_le2001149050_ereal @ X4 @ Y2 )
=> ( ord_le2001149050_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le2001149050_ereal @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_56_order__less__subst2,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > extended_ereal,C2: extended_ereal] :
( ( ord_le2001149050_ereal @ A @ B )
=> ( ( ord_le2001149050_ereal @ ( F @ B ) @ C2 )
=> ( ! [X4: extended_ereal,Y2: extended_ereal] :
( ( ord_le2001149050_ereal @ X4 @ Y2 )
=> ( ord_le2001149050_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le2001149050_ereal @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_57_neqE,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( X2 != Y )
=> ( ~ ( ord_le2001149050_ereal @ X2 @ Y )
=> ( ord_le2001149050_ereal @ Y @ X2 ) ) ) ).
% neqE
thf(fact_58_neq__iff,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( X2 != Y )
= ( ( ord_le2001149050_ereal @ X2 @ Y )
| ( ord_le2001149050_ereal @ Y @ X2 ) ) ) ).
% neq_iff
thf(fact_59_order_Oasym,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le2001149050_ereal @ A @ B )
=> ~ ( ord_le2001149050_ereal @ B @ A ) ) ).
% order.asym
thf(fact_60_dense,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( ord_le2001149050_ereal @ X2 @ Y )
=> ? [Z: extended_ereal] :
( ( ord_le2001149050_ereal @ X2 @ Z )
& ( ord_le2001149050_ereal @ Z @ Y ) ) ) ).
% dense
thf(fact_61_less__imp__neq,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( ord_le2001149050_ereal @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_62_less__asym,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( ord_le2001149050_ereal @ X2 @ Y )
=> ~ ( ord_le2001149050_ereal @ Y @ X2 ) ) ).
% less_asym
thf(fact_63_less__asym_H,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le2001149050_ereal @ A @ B )
=> ~ ( ord_le2001149050_ereal @ B @ A ) ) ).
% less_asym'
thf(fact_64_less__trans,axiom,
! [X2: extended_ereal,Y: extended_ereal,Z3: extended_ereal] :
( ( ord_le2001149050_ereal @ X2 @ Y )
=> ( ( ord_le2001149050_ereal @ Y @ Z3 )
=> ( ord_le2001149050_ereal @ X2 @ Z3 ) ) ) ).
% less_trans
thf(fact_65_less__linear,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( ord_le2001149050_ereal @ X2 @ Y )
| ( X2 = Y )
| ( ord_le2001149050_ereal @ Y @ X2 ) ) ).
% less_linear
thf(fact_66_less__irrefl,axiom,
! [X2: extended_ereal] :
~ ( ord_le2001149050_ereal @ X2 @ X2 ) ).
% less_irrefl
thf(fact_67_ord__eq__less__trans,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( A = B )
=> ( ( ord_le2001149050_ereal @ B @ C2 )
=> ( ord_le2001149050_ereal @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_68_ord__less__eq__trans,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( ord_le2001149050_ereal @ A @ B )
=> ( ( B = C2 )
=> ( ord_le2001149050_ereal @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_69_dual__order_Oasym,axiom,
! [B: extended_ereal,A: extended_ereal] :
( ( ord_le2001149050_ereal @ B @ A )
=> ~ ( ord_le2001149050_ereal @ A @ B ) ) ).
% dual_order.asym
thf(fact_70_less__imp__not__eq,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( ord_le2001149050_ereal @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_not_eq
thf(fact_71_less__not__sym,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( ord_le2001149050_ereal @ X2 @ Y )
=> ~ ( ord_le2001149050_ereal @ Y @ X2 ) ) ).
% less_not_sym
thf(fact_72_antisym__conv3,axiom,
! [Y: extended_ereal,X2: extended_ereal] :
( ~ ( ord_le2001149050_ereal @ Y @ X2 )
=> ( ( ~ ( ord_le2001149050_ereal @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_73_less__imp__not__eq2,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( ord_le2001149050_ereal @ X2 @ Y )
=> ( Y != X2 ) ) ).
% less_imp_not_eq2
thf(fact_74_less__imp__triv,axiom,
! [X2: extended_ereal,Y: extended_ereal,P: $o] :
( ( ord_le2001149050_ereal @ X2 @ Y )
=> ( ( ord_le2001149050_ereal @ Y @ X2 )
=> P ) ) ).
% less_imp_triv
thf(fact_75_linorder__cases,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ~ ( ord_le2001149050_ereal @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_le2001149050_ereal @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_76_dual__order_Oirrefl,axiom,
! [A: extended_ereal] :
~ ( ord_le2001149050_ereal @ A @ A ) ).
% dual_order.irrefl
thf(fact_77_order_Ostrict__trans,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( ord_le2001149050_ereal @ A @ B )
=> ( ( ord_le2001149050_ereal @ B @ C2 )
=> ( ord_le2001149050_ereal @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_78_less__imp__not__less,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( ord_le2001149050_ereal @ X2 @ Y )
=> ~ ( ord_le2001149050_ereal @ Y @ X2 ) ) ).
% less_imp_not_less
thf(fact_79_linorder__less__wlog,axiom,
! [P: extended_ereal > extended_ereal > $o,A: extended_ereal,B: extended_ereal] :
( ! [A3: extended_ereal,B2: extended_ereal] :
( ( ord_le2001149050_ereal @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: extended_ereal] : ( P @ A3 @ A3 )
=> ( ! [A3: extended_ereal,B2: extended_ereal] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_80_dual__order_Ostrict__trans,axiom,
! [B: extended_ereal,A: extended_ereal,C2: extended_ereal] :
( ( ord_le2001149050_ereal @ B @ A )
=> ( ( ord_le2001149050_ereal @ C2 @ B )
=> ( ord_le2001149050_ereal @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_81_not__less__iff__gr__or__eq,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( ~ ( ord_le2001149050_ereal @ X2 @ Y ) )
= ( ( ord_le2001149050_ereal @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_82_order_Ostrict__implies__not__eq,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le2001149050_ereal @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_83_ereal__mult__eq__PInfty,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( times_1966848393_ereal @ A @ B )
= extend1289208545_ereal )
= ( ( ( A = extend1289208545_ereal )
& ( ord_le2001149050_ereal @ zero_z163181189_ereal @ B ) )
| ( ( ord_le2001149050_ereal @ zero_z163181189_ereal @ A )
& ( B = extend1289208545_ereal ) )
| ( ( A
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
& ( ord_le2001149050_ereal @ B @ zero_z163181189_ereal ) )
| ( ( ord_le2001149050_ereal @ A @ zero_z163181189_ereal )
& ( B
= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ).
% ereal_mult_eq_PInfty
thf(fact_84_ereal__mult__eq__MInfty,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( times_1966848393_ereal @ A @ B )
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
= ( ( ( A = extend1289208545_ereal )
& ( ord_le2001149050_ereal @ B @ zero_z163181189_ereal ) )
| ( ( ord_le2001149050_ereal @ A @ zero_z163181189_ereal )
& ( B = extend1289208545_ereal ) )
| ( ( A
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
& ( ord_le2001149050_ereal @ zero_z163181189_ereal @ B ) )
| ( ( ord_le2001149050_ereal @ zero_z163181189_ereal @ A )
& ( B
= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ).
% ereal_mult_eq_MInfty
thf(fact_85_ereal__mult__infty,axiom,
! [A: extended_ereal] :
( ( ( A = zero_z163181189_ereal )
=> ( ( times_1966848393_ereal @ A @ extend1289208545_ereal )
= zero_z163181189_ereal ) )
& ( ( A != zero_z163181189_ereal )
=> ( ( ( ord_le2001149050_ereal @ zero_z163181189_ereal @ A )
=> ( ( times_1966848393_ereal @ A @ extend1289208545_ereal )
= extend1289208545_ereal ) )
& ( ~ ( ord_le2001149050_ereal @ zero_z163181189_ereal @ A )
=> ( ( times_1966848393_ereal @ A @ extend1289208545_ereal )
= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ) ).
% ereal_mult_infty
thf(fact_86_ereal__infty__mult,axiom,
! [A: extended_ereal] :
( ( ( A = zero_z163181189_ereal )
=> ( ( times_1966848393_ereal @ extend1289208545_ereal @ A )
= zero_z163181189_ereal ) )
& ( ( A != zero_z163181189_ereal )
=> ( ( ( ord_le2001149050_ereal @ zero_z163181189_ereal @ A )
=> ( ( times_1966848393_ereal @ extend1289208545_ereal @ A )
= extend1289208545_ereal ) )
& ( ~ ( ord_le2001149050_ereal @ zero_z163181189_ereal @ A )
=> ( ( times_1966848393_ereal @ extend1289208545_ereal @ A )
= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ) ).
% ereal_infty_mult
thf(fact_87_ereal__divide__Infty_I2_J,axiom,
! [X2: extended_ereal] :
( ( divide595620860_ereal @ X2 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
= zero_z163181189_ereal ) ).
% ereal_divide_Infty(2)
thf(fact_88_ereal__mult__minus__left,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( times_1966848393_ereal @ ( uminus1208298309_ereal @ A ) @ B )
= ( uminus1208298309_ereal @ ( times_1966848393_ereal @ A @ B ) ) ) ).
% ereal_mult_minus_left
thf(fact_89_ereal__mult__minus__right,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( times_1966848393_ereal @ A @ ( uminus1208298309_ereal @ B ) )
= ( uminus1208298309_ereal @ ( times_1966848393_ereal @ A @ B ) ) ) ).
% ereal_mult_minus_right
thf(fact_90_ereal__zero__times,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( times_1966848393_ereal @ A @ B )
= zero_z163181189_ereal )
= ( ( A = zero_z163181189_ereal )
| ( B = zero_z163181189_ereal ) ) ) ).
% ereal_zero_times
thf(fact_91_ereal__zero__mult,axiom,
! [A: extended_ereal] :
( ( times_1966848393_ereal @ zero_z163181189_ereal @ A )
= zero_z163181189_ereal ) ).
% ereal_zero_mult
thf(fact_92_ereal__mult__zero,axiom,
! [A: extended_ereal] :
( ( times_1966848393_ereal @ A @ zero_z163181189_ereal )
= zero_z163181189_ereal ) ).
% ereal_mult_zero
thf(fact_93_ereal__uminus__divide,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( divide595620860_ereal @ ( uminus1208298309_ereal @ X2 ) @ Y )
= ( uminus1208298309_ereal @ ( divide595620860_ereal @ X2 @ Y ) ) ) ).
% ereal_uminus_divide
thf(fact_94_ereal__divide__zero__left,axiom,
! [A: extended_ereal] :
( ( divide595620860_ereal @ zero_z163181189_ereal @ A )
= zero_z163181189_ereal ) ).
% ereal_divide_zero_left
thf(fact_95_ereal__times__divide__eq__left,axiom,
! [B: extended_ereal,C2: extended_ereal,A: extended_ereal] :
( ( times_1966848393_ereal @ ( divide595620860_ereal @ B @ C2 ) @ A )
= ( divide595620860_ereal @ ( times_1966848393_ereal @ B @ A ) @ C2 ) ) ).
% ereal_times_divide_eq_left
thf(fact_96_ereal__divide__Infty_I1_J,axiom,
! [X2: extended_ereal] :
( ( divide595620860_ereal @ X2 @ extend1289208545_ereal )
= zero_z163181189_ereal ) ).
% ereal_divide_Infty(1)
thf(fact_97_mult_Oleft__commute,axiom,
! [B: extended_ereal,A: extended_ereal,C2: extended_ereal] :
( ( times_1966848393_ereal @ B @ ( times_1966848393_ereal @ A @ C2 ) )
= ( times_1966848393_ereal @ A @ ( times_1966848393_ereal @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_98_mult_Ocommute,axiom,
( times_1966848393_ereal
= ( ^ [A4: extended_ereal,B3: extended_ereal] : ( times_1966848393_ereal @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_99_mult_Oassoc,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( times_1966848393_ereal @ ( times_1966848393_ereal @ A @ B ) @ C2 )
= ( times_1966848393_ereal @ A @ ( times_1966848393_ereal @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_100_ereal__times__divide__eq,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( times_1966848393_ereal @ A @ ( divide595620860_ereal @ B @ C2 ) )
= ( divide595620860_ereal @ ( times_1966848393_ereal @ A @ B ) @ C2 ) ) ).
% ereal_times_divide_eq
thf(fact_101_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( times_1966848393_ereal @ ( times_1966848393_ereal @ A @ B ) @ C2 )
= ( times_1966848393_ereal @ A @ ( times_1966848393_ereal @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_102_times__ereal_Osimps_I6_J,axiom,
( ( times_1966848393_ereal @ extend1289208545_ereal @ extend1289208545_ereal )
= extend1289208545_ereal ) ).
% times_ereal.simps(6)
thf(fact_103_ereal__right__mult__cong,axiom,
! [C2: extended_ereal,D: extended_ereal,A: extended_ereal,B: extended_ereal] :
( ( C2 = D )
=> ( ( ( D != zero_z163181189_ereal )
=> ( A = B ) )
=> ( ( times_1966848393_ereal @ C2 @ A )
= ( times_1966848393_ereal @ D @ B ) ) ) ) ).
% ereal_right_mult_cong
thf(fact_104_ereal__left__mult__cong,axiom,
! [C2: extended_ereal,D: extended_ereal,A: extended_ereal,B: extended_ereal] :
( ( C2 = D )
=> ( ( ( D != zero_z163181189_ereal )
=> ( A = B ) )
=> ( ( times_1966848393_ereal @ A @ C2 )
= ( times_1966848393_ereal @ B @ D ) ) ) ) ).
% ereal_left_mult_cong
thf(fact_105_ereal__mult__divide,axiom,
! [B: extended_ereal,A: extended_ereal] :
( ( ord_le2001149050_ereal @ zero_z163181189_ereal @ B )
=> ( ( ord_le2001149050_ereal @ B @ extend1289208545_ereal )
=> ( ( times_1966848393_ereal @ B @ ( divide595620860_ereal @ A @ B ) )
= A ) ) ) ).
% ereal_mult_divide
thf(fact_106_ereal__divide__less__iff,axiom,
! [C2: extended_ereal,A: extended_ereal,B: extended_ereal] :
( ( ord_le2001149050_ereal @ zero_z163181189_ereal @ C2 )
=> ( ( ord_le2001149050_ereal @ C2 @ extend1289208545_ereal )
=> ( ( ord_le2001149050_ereal @ ( divide595620860_ereal @ A @ C2 ) @ B )
= ( ord_le2001149050_ereal @ A @ ( times_1966848393_ereal @ B @ C2 ) ) ) ) ) ).
% ereal_divide_less_iff
thf(fact_107_ereal__divide__less__pos,axiom,
! [X2: extended_ereal,Y: extended_ereal,Z3: extended_ereal] :
( ( ord_le2001149050_ereal @ zero_z163181189_ereal @ X2 )
=> ( ( X2 != extend1289208545_ereal )
=> ( ( ord_le2001149050_ereal @ ( divide595620860_ereal @ Y @ X2 ) @ Z3 )
= ( ord_le2001149050_ereal @ Y @ ( times_1966848393_ereal @ X2 @ Z3 ) ) ) ) ) ).
% ereal_divide_less_pos
thf(fact_108_ereal__less__divide__iff,axiom,
! [C2: extended_ereal,A: extended_ereal,B: extended_ereal] :
( ( ord_le2001149050_ereal @ zero_z163181189_ereal @ C2 )
=> ( ( ord_le2001149050_ereal @ C2 @ extend1289208545_ereal )
=> ( ( ord_le2001149050_ereal @ A @ ( divide595620860_ereal @ B @ C2 ) )
= ( ord_le2001149050_ereal @ ( times_1966848393_ereal @ A @ C2 ) @ B ) ) ) ) ).
% ereal_less_divide_iff
thf(fact_109_ereal__less__divide__pos,axiom,
! [X2: extended_ereal,Y: extended_ereal,Z3: extended_ereal] :
( ( ord_le2001149050_ereal @ zero_z163181189_ereal @ X2 )
=> ( ( X2 != extend1289208545_ereal )
=> ( ( ord_le2001149050_ereal @ Y @ ( divide595620860_ereal @ Z3 @ X2 ) )
= ( ord_le2001149050_ereal @ ( times_1966848393_ereal @ X2 @ Y ) @ Z3 ) ) ) ) ).
% ereal_less_divide_pos
thf(fact_110_times__ereal_Osimps_I9_J,axiom,
( ( times_1966848393_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
= extend1289208545_ereal ) ).
% times_ereal.simps(9)
thf(fact_111_times__ereal_Osimps_I8_J,axiom,
( ( times_1966848393_ereal @ extend1289208545_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% times_ereal.simps(8)
thf(fact_112_times__ereal_Osimps_I7_J,axiom,
( ( times_1966848393_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ extend1289208545_ereal )
= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% times_ereal.simps(7)
thf(fact_113_ereal__mult__less__0__iff,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le2001149050_ereal @ ( times_1966848393_ereal @ A @ B ) @ zero_z163181189_ereal )
= ( ( ( ord_le2001149050_ereal @ zero_z163181189_ereal @ A )
& ( ord_le2001149050_ereal @ B @ zero_z163181189_ereal ) )
| ( ( ord_le2001149050_ereal @ A @ zero_z163181189_ereal )
& ( ord_le2001149050_ereal @ zero_z163181189_ereal @ B ) ) ) ) ).
% ereal_mult_less_0_iff
thf(fact_114_ereal__zero__less__0__iff,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le2001149050_ereal @ zero_z163181189_ereal @ ( times_1966848393_ereal @ A @ B ) )
= ( ( ( ord_le2001149050_ereal @ zero_z163181189_ereal @ A )
& ( ord_le2001149050_ereal @ zero_z163181189_ereal @ B ) )
| ( ( ord_le2001149050_ereal @ A @ zero_z163181189_ereal )
& ( ord_le2001149050_ereal @ B @ zero_z163181189_ereal ) ) ) ) ).
% ereal_zero_less_0_iff
thf(fact_115_ereal__mult__mono__strict,axiom,
! [B: extended_ereal,C2: extended_ereal,A: extended_ereal,D: extended_ereal] :
( ( ord_le2001149050_ereal @ zero_z163181189_ereal @ B )
=> ( ( ord_le2001149050_ereal @ zero_z163181189_ereal @ C2 )
=> ( ( ord_le2001149050_ereal @ A @ B )
=> ( ( ord_le2001149050_ereal @ C2 @ D )
=> ( ord_le2001149050_ereal @ ( times_1966848393_ereal @ A @ C2 ) @ ( times_1966848393_ereal @ B @ D ) ) ) ) ) ) ).
% ereal_mult_mono_strict
thf(fact_116_ereal__mult__mono__strict_H,axiom,
! [A: extended_ereal,C2: extended_ereal,B: extended_ereal,D: extended_ereal] :
( ( ord_le2001149050_ereal @ zero_z163181189_ereal @ A )
=> ( ( ord_le2001149050_ereal @ zero_z163181189_ereal @ C2 )
=> ( ( ord_le2001149050_ereal @ A @ B )
=> ( ( ord_le2001149050_ereal @ C2 @ D )
=> ( ord_le2001149050_ereal @ ( times_1966848393_ereal @ A @ C2 ) @ ( times_1966848393_ereal @ B @ D ) ) ) ) ) ) ).
% ereal_mult_mono_strict'
thf(fact_117_ereal__mult__less__right,axiom,
! [B: extended_ereal,A: extended_ereal,C2: extended_ereal] :
( ( ord_le2001149050_ereal @ ( times_1966848393_ereal @ B @ A ) @ ( times_1966848393_ereal @ C2 @ A ) )
=> ( ( ord_le2001149050_ereal @ zero_z163181189_ereal @ A )
=> ( ( ord_le2001149050_ereal @ A @ extend1289208545_ereal )
=> ( ord_le2001149050_ereal @ B @ C2 ) ) ) ) ).
% ereal_mult_less_right
thf(fact_118_ereal__mult__strict__left__mono,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( ord_le2001149050_ereal @ A @ B )
=> ( ( ord_le2001149050_ereal @ zero_z163181189_ereal @ C2 )
=> ( ( ord_le2001149050_ereal @ C2 @ extend1289208545_ereal )
=> ( ord_le2001149050_ereal @ ( times_1966848393_ereal @ C2 @ A ) @ ( times_1966848393_ereal @ C2 @ B ) ) ) ) ) ).
% ereal_mult_strict_left_mono
thf(fact_119_ereal__mult__strict__right__mono,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( ord_le2001149050_ereal @ A @ B )
=> ( ( ord_le2001149050_ereal @ zero_z163181189_ereal @ C2 )
=> ( ( ord_le2001149050_ereal @ C2 @ extend1289208545_ereal )
=> ( ord_le2001149050_ereal @ ( times_1966848393_ereal @ A @ C2 ) @ ( times_1966848393_ereal @ B @ C2 ) ) ) ) ) ).
% ereal_mult_strict_right_mono
% Conjectures (1)
thf(conj_0,conjecture,
( ( lower_191460856_ereal @ x0 @ f )
!= ( ~ ! [C: extended_ereal] :
( ( ord_le2001149050_ereal @ C @ ( f @ x0 ) )
=> ? [T: set_a] :
( ( topolo1276428101open_a @ T )
& ( member_a @ x0 @ T )
& ! [X: a] :
( ( member_a @ X @ T )
=> ( ord_le2001149050_ereal @ C @ ( f @ X ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------