TPTP Problem File: ITP111^2.p
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%------------------------------------------------------------------------------
% File : ITP111^2 : 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 : 0.00 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 358 ( 77 unt; 60 typ; 0 def)
% Number of atoms : 973 ( 342 equ; 0 cnn)
% Maximal formula atoms : 7 ( 3 avg)
% Number of connectives : 3496 ( 99 ~; 32 |; 63 &;2858 @)
% ( 0 <=>; 444 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 65 ( 65 >; 0 *; 0 +; 0 <<)
% Number of symbols : 58 ( 57 usr; 3 con; 0-5 aty)
% Number of variables : 874 ( 9 ^; 773 !; 39 ?; 874 :)
% ( 53 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:25:45.493
%------------------------------------------------------------------------------
% Could-be-implicit typings (4)
thf(ty_t_Extended__Real_Oereal,type,
extended_ereal: $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (56)
thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
real_V2090557954_space:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom,type,
idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield,type,
field:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Odivision__ring,type,
division_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__divide,type,
semidom_divide:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__mult,type,
semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring,type,
linordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Oboolean__algebra,type,
boolean_algebra:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot0__space,type,
topological_t0_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot1__space,type,
topological_t1_space:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring__strict,type,
linord581940658strict:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
semiri1193490041visors:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__strict,type,
linord20386208strict:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord1659791738miring:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Odiscrete__topology,type,
topolo2133971006pology:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Otopological__space,type,
topolo503727757_space:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__comm__semiring__strict,type,
linord893533164strict:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
semiri1923998003cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel,type,
euclid24285859cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__semiring__cancel,type,
euclid191655569cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit1656338222tinuum:
!>[A: $tType] : $o ).
thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity,type,
extend1396239628finity:
!>[A: $tType] : A ).
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_Extended__Real_Oereal_Orec__ereal,type,
extended_rec_ereal:
!>[A: $tType] : ( ( real > A ) > A > A > extended_ereal > A ) ).
thf(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Olsc__at,type,
lower_582600101lsc_at:
!>[A: $tType,B: $tType] : ( A > ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Rings_Odivide__class_Odivide,type,
divide_divide:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen,type,
topolo1751647064n_open:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_f,type,
f: a > extended_ereal ).
thf(sy_v_x0,type,
x0: a ).
% Relevant facts (255)
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 )
= ( extend1396239628finity @ extended_ereal ) )
=> ( ( lower_582600101lsc_at @ a @ extended_ereal @ x0 @ f )
= ( ! [C: extended_ereal] :
( ( ord_less @ extended_ereal @ C @ ( f @ x0 ) )
=> ? [T: set @ a] :
( ( topolo1751647064n_open @ a @ T )
& ( member @ a @ x0 @ T )
& ! [X: a] :
( ( member @ a @ X @ T )
=> ( ord_less @ extended_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 )
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( lower_582600101lsc_at @ a @ extended_ereal @ x0 @ f )
= ( ! [C: extended_ereal] :
( ( ord_less @ extended_ereal @ C @ ( f @ x0 ) )
=> ? [T: set @ a] :
( ( topolo1751647064n_open @ a @ T )
& ( member @ a @ x0 @ T )
& ! [X: a] :
( ( member @ a @ X @ T )
=> ( ord_less @ extended_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 )
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( ( f @ x0 )
!= ( extend1396239628finity @ extended_ereal ) )
=> ( ( lower_582600101lsc_at @ a @ extended_ereal @ x0 @ f )
= ( ! [C: extended_ereal] :
( ( ord_less @ extended_ereal @ C @ ( f @ x0 ) )
=> ? [T: set @ a] :
( ( topolo1751647064n_open @ a @ T )
& ( member @ a @ x0 @ T )
& ! [X: a] :
( ( member @ a @ X @ T )
=> ( ord_less @ extended_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,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ! [F: A > extended_ereal,X0: A] :
( ( ( F @ X0 )
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( lower_582600101lsc_at @ A @ extended_ereal @ X0 @ F ) ) ) ).
% lsc_at_MInfty
thf(fact_4_t0__space,axiom,
! [A: $tType] :
( ( topological_t0_space @ A )
=> ! [X2: A,Y: A] :
( ( X2 != Y )
=> ? [U: set @ A] :
( ( topolo1751647064n_open @ A @ U )
& ( ( member @ A @ X2 @ U )
!= ( member @ A @ Y @ U ) ) ) ) ) ).
% t0_space
thf(fact_5_t1__space,axiom,
! [A: $tType] :
( ( topological_t1_space @ A )
=> ! [X2: A,Y: A] :
( ( X2 != Y )
=> ? [U: set @ A] :
( ( topolo1751647064n_open @ A @ U )
& ( member @ A @ X2 @ U )
& ~ ( member @ A @ Y @ U ) ) ) ) ).
% t1_space
thf(fact_6_separation__t0,axiom,
! [A: $tType] :
( ( topological_t0_space @ A )
=> ! [X2: A,Y: A] :
( ( X2 != Y )
= ( ? [U2: set @ A] :
( ( topolo1751647064n_open @ A @ U2 )
& ( ( member @ A @ X2 @ U2 )
!= ( member @ A @ Y @ U2 ) ) ) ) ) ) ).
% separation_t0
thf(fact_7_separation__t1,axiom,
! [A: $tType] :
( ( topological_t1_space @ A )
=> ! [X2: A,Y: A] :
( ( X2 != Y )
= ( ? [U2: set @ A] :
( ( topolo1751647064n_open @ A @ U2 )
& ( member @ A @ X2 @ U2 )
& ~ ( member @ A @ Y @ U2 ) ) ) ) ) ).
% separation_t1
thf(fact_8_open__discrete,axiom,
! [A: $tType] :
( ( topolo2133971006pology @ A )
=> ! [A2: set @ A] : ( topolo1751647064n_open @ A @ A2 ) ) ).
% open_discrete
thf(fact_9_minf_I11_J,axiom,
! [C2: $tType,D: $tType] :
( ( ord @ C2 )
=> ! [F2: D] :
? [Z: C2] :
! [X3: C2] :
( ( ord_less @ C2 @ X3 @ Z )
=> ( F2 = F2 ) ) ) ).
% minf(11)
thf(fact_10_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z )
=> ~ ( ord_less @ A @ T2 @ X3 ) ) ) ).
% minf(7)
thf(fact_11_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z )
=> ( ord_less @ A @ X3 @ T2 ) ) ) ).
% minf(5)
thf(fact_12_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z )
=> ( X3 != T2 ) ) ) ).
% minf(4)
thf(fact_13_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z )
=> ( X3 != T2 ) ) ) ).
% minf(3)
thf(fact_14_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z2: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z2 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z2: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z2 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_15_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z2: A] :
! [X4: A] :
( ( ord_less @ A @ Z2 @ X4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z2: A] :
! [X4: A] :
( ( ord_less @ A @ Z2 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_16_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z2: A] :
! [X4: A] :
( ( ord_less @ A @ Z2 @ X4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z2: A] :
! [X4: A] :
( ( ord_less @ A @ Z2 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_17_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( X3 != T2 ) ) ) ).
% pinf(3)
thf(fact_18_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( X3 != T2 ) ) ) ).
% pinf(4)
thf(fact_19_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ~ ( ord_less @ A @ X3 @ T2 ) ) ) ).
% pinf(5)
thf(fact_20_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( ord_less @ A @ T2 @ X3 ) ) ) ).
% pinf(7)
thf(fact_21_pinf_I11_J,axiom,
! [C2: $tType,D: $tType] :
( ( ord @ C2 )
=> ! [F2: D] :
? [Z: C2] :
! [X3: C2] :
( ( ord_less @ C2 @ Z @ X3 )
=> ( F2 = F2 ) ) ) ).
% pinf(11)
thf(fact_22_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z2: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z2 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z2: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z2 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_23_ereal__MInfty__lessI,axiom,
! [A3: extended_ereal] :
( ( A3
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ord_less @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ A3 ) ) ).
% ereal_MInfty_lessI
thf(fact_24_ereal__infty__less_I2_J,axiom,
! [X2: extended_ereal] :
( ( ord_less @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ X2 )
= ( X2
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% ereal_infty_less(2)
thf(fact_25_ereal__minus__less__minus,axiom,
! [A3: extended_ereal,B2: extended_ereal] :
( ( ord_less @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A3 ) @ ( uminus_uminus @ extended_ereal @ B2 ) )
= ( ord_less @ extended_ereal @ B2 @ A3 ) ) ).
% ereal_minus_less_minus
thf(fact_26_ereal__less__PInfty,axiom,
! [A3: extended_ereal] :
( ( A3
!= ( extend1396239628finity @ extended_ereal ) )
=> ( ord_less @ extended_ereal @ A3 @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% ereal_less_PInfty
thf(fact_27_ereal__infty__less_I1_J,axiom,
! [X2: extended_ereal] :
( ( ord_less @ extended_ereal @ X2 @ ( extend1396239628finity @ extended_ereal ) )
= ( X2
!= ( extend1396239628finity @ extended_ereal ) ) ) ).
% ereal_infty_less(1)
thf(fact_28_neg__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% neg_less_iff_less
thf(fact_29_less__ereal_Osimps_I3_J,axiom,
! [A3: extended_ereal] :
~ ( ord_less @ extended_ereal @ A3 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% less_ereal.simps(3)
thf(fact_30_less__ereal_Osimps_I6_J,axiom,
ord_less @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extend1396239628finity @ extended_ereal ) ).
% less_ereal.simps(6)
thf(fact_31_ereal__uminus__eq__iff,axiom,
! [A3: extended_ereal,B2: extended_ereal] :
( ( ( uminus_uminus @ extended_ereal @ A3 )
= ( uminus_uminus @ extended_ereal @ B2 ) )
= ( A3 = B2 ) ) ).
% ereal_uminus_eq_iff
thf(fact_32_ereal__uminus__uminus,axiom,
! [A3: extended_ereal] :
( ( uminus_uminus @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A3 ) )
= A3 ) ).
% ereal_uminus_uminus
thf(fact_33_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( uminus_uminus @ A @ A3 )
= ( uminus_uminus @ A @ B2 ) )
= ( A3 = B2 ) ) ) ).
% neg_equal_iff_equal
thf(fact_34_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A3 ) )
= A3 ) ) ).
% add.inverse_inverse
thf(fact_35_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( uminus_uminus @ A @ A3 )
= B2 )
= ( ( uminus_uminus @ A @ B2 )
= A3 ) ) ) ).
% minus_equation_iff
thf(fact_36_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( uminus_uminus @ A @ B2 ) )
= ( B2
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% equation_minus_iff
thf(fact_37_ereal__uminus__eq__reorder,axiom,
! [A3: extended_ereal,B2: extended_ereal] :
( ( ( uminus_uminus @ extended_ereal @ A3 )
= B2 )
= ( A3
= ( uminus_uminus @ extended_ereal @ B2 ) ) ) ).
% ereal_uminus_eq_reorder
thf(fact_38_less__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less @ A @ B2 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% less_minus_iff
thf(fact_39_minus__less__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ A3 ) ) ) ).
% minus_less_iff
thf(fact_40_MInfty__neq__PInfty_I1_J,axiom,
( ( extend1396239628finity @ extended_ereal )
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% MInfty_neq_PInfty(1)
thf(fact_41_less__ereal_Osimps_I2_J,axiom,
! [A3: extended_ereal] :
~ ( ord_less @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ A3 ) ).
% less_ereal.simps(2)
thf(fact_42_ereal__less__uminus__reorder,axiom,
! [A3: extended_ereal,B2: extended_ereal] :
( ( ord_less @ extended_ereal @ A3 @ ( uminus_uminus @ extended_ereal @ B2 ) )
= ( ord_less @ extended_ereal @ B2 @ ( uminus_uminus @ extended_ereal @ A3 ) ) ) ).
% ereal_less_uminus_reorder
thf(fact_43_ereal__uminus__less__reorder,axiom,
! [A3: extended_ereal,B2: extended_ereal] :
( ( ord_less @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A3 ) @ B2 )
= ( ord_less @ extended_ereal @ ( uminus_uminus @ extended_ereal @ B2 ) @ A3 ) ) ).
% ereal_uminus_less_reorder
thf(fact_44_compl__eq__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X2: A,Y: A] :
( ( ( uminus_uminus @ A @ X2 )
= ( uminus_uminus @ A @ Y ) )
= ( X2 = Y ) ) ) ).
% compl_eq_compl_iff
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X4: A] :
( ( F @ X4 )
= ( G @ X4 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_double__compl,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X2: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X2 ) )
= X2 ) ) ).
% double_compl
thf(fact_50_uminus__apply,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A4: A > B,X: A] : ( uminus_uminus @ B @ ( A4 @ X ) ) ) ) ) ).
% uminus_apply
thf(fact_51_verit__minus__simplify_I4_J,axiom,
! [B: $tType] :
( ( group_add @ B )
=> ! [B2: B] :
( ( uminus_uminus @ B @ ( uminus_uminus @ B @ B2 ) )
= B2 ) ) ).
% verit_minus_simplify(4)
thf(fact_52_verit__negate__coefficient_I2_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_53_compl__less__swap1,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [Y: A,X2: A] :
( ( ord_less @ A @ Y @ ( uminus_uminus @ A @ X2 ) )
=> ( ord_less @ A @ X2 @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_less_swap1
thf(fact_54_compl__less__swap2,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [Y: A,X2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ Y ) @ X2 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ X2 ) @ Y ) ) ) ).
% compl_less_swap2
thf(fact_55_compl__less__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ X2 ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less @ A @ Y @ X2 ) ) ) ).
% compl_less_compl_iff
thf(fact_56_MInfty__eq__minfinity,axiom,
( extended_MInfty
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% MInfty_eq_minfinity
thf(fact_57_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% verit_comp_simplify1(1)
thf(fact_58_verit__negate__coefficient_I3_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( A3 = B2 )
=> ( ( uminus_uminus @ A @ A3 )
= ( uminus_uminus @ A @ B2 ) ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_59_fun__Compl__def,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A4: A > B,X: A] : ( uminus_uminus @ B @ ( A4 @ X ) ) ) ) ) ).
% fun_Compl_def
thf(fact_60_uminus__ereal_Osimps_I2_J,axiom,
( ( uminus_uminus @ extended_ereal @ extended_PInfty )
= extended_MInfty ) ).
% uminus_ereal.simps(2)
thf(fact_61_uminus__ereal_Osimps_I3_J,axiom,
( ( uminus_uminus @ extended_ereal @ extended_MInfty )
= extended_PInfty ) ).
% uminus_ereal.simps(3)
thf(fact_62_ereal__less_I6_J,axiom,
ord_less @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( zero_zero @ extended_ereal ) ).
% ereal_less(6)
thf(fact_63_ereal_Osimps_I13_J,axiom,
! [A: $tType,F1: real > A,F22: A,F3: A] :
( ( extended_rec_ereal @ A @ F1 @ F22 @ F3 @ extended_MInfty )
= F3 ) ).
% ereal.simps(13)
thf(fact_64_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A )
=> ! [A3: A] :
? [B3: A] :
( ( ord_less @ A @ A3 @ B3 )
| ( ord_less @ A @ B3 @ A3 ) ) ) ).
% ex_gt_or_lt
thf(fact_65_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y: A] :
( ( X2 != Y )
=> ( ~ ( ord_less @ A @ X2 @ Y )
=> ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_66_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( A3 != B2 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_67_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_68_neg__equal__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A3: A] :
( ( ( uminus_uminus @ A @ A3 )
= A3 )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_zero
thf(fact_69_equal__neg__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A3: A] :
( ( A3
= ( uminus_uminus @ A @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% equal_neg_zero
thf(fact_70_neg__equal__0__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( ( uminus_uminus @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_0_iff_equal
thf(fact_71_neg__0__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( uminus_uminus @ A @ A3 ) )
= ( ( zero_zero @ A )
= A3 ) ) ) ).
% neg_0_equal_iff_equal
thf(fact_72_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_73_ereal__uminus__zero__iff,axiom,
! [A3: extended_ereal] :
( ( ( uminus_uminus @ extended_ereal @ A3 )
= ( zero_zero @ extended_ereal ) )
= ( A3
= ( zero_zero @ extended_ereal ) ) ) ).
% ereal_uminus_zero_iff
thf(fact_74_ereal__uminus__zero,axiom,
( ( uminus_uminus @ extended_ereal @ ( zero_zero @ extended_ereal ) )
= ( zero_zero @ extended_ereal ) ) ).
% ereal_uminus_zero
thf(fact_75_less__neg__neg,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% less_neg_neg
thf(fact_76_neg__less__pos,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_pos
thf(fact_77_neg__0__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_less_iff_less
thf(fact_78_neg__less__0__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_0_iff_less
thf(fact_79_neg__0__less__iff__less__erea,axiom,
! [A3: extended_ereal] :
( ( ord_less @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( uminus_uminus @ extended_ereal @ A3 ) )
= ( ord_less @ extended_ereal @ A3 @ ( zero_zero @ extended_ereal ) ) ) ).
% neg_0_less_iff_less_erea
thf(fact_80_ereal_Osimps_I12_J,axiom,
! [A: $tType,F1: real > A,F22: A,F3: A] :
( ( extended_rec_ereal @ A @ F1 @ F22 @ F3 @ extended_PInfty )
= F22 ) ).
% ereal.simps(12)
thf(fact_81_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X2: A] :
( ( ( zero_zero @ A )
= X2 )
= ( X2
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_82_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( N
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% gr_zeroI
thf(fact_83_not__less__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_84_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [M: A,N: A] :
( ( ord_less @ A @ M @ N )
=> ( N
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_85_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N )
= ( N
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_86_Infty__neq__0_I1_J,axiom,
( ( extend1396239628finity @ extended_ereal )
!= ( zero_zero @ extended_ereal ) ) ).
% Infty_neq_0(1)
thf(fact_87_infinity__ereal__def,axiom,
( ( extend1396239628finity @ extended_ereal )
= extended_PInfty ) ).
% infinity_ereal_def
thf(fact_88_ereal_Odistinct_I5_J,axiom,
extended_PInfty != extended_MInfty ).
% ereal.distinct(5)
thf(fact_89_Infty__neq__0_I3_J,axiom,
( ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) )
!= ( zero_zero @ extended_ereal ) ) ).
% Infty_neq_0(3)
thf(fact_90_ereal__less_I5_J,axiom,
ord_less @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( extend1396239628finity @ extended_ereal ) ).
% ereal_less(5)
thf(fact_91_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F: B > A,B2: B,C3: B] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C3 )
=> ( ! [X4: B,Y2: B] :
( ( ord_less @ B @ X4 @ Y2 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_92_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B2: A,F: A > B,C3: B] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C3 )
=> ( ! [X4: A,Y2: A] :
( ( ord_less @ A @ X4 @ Y2 )
=> ( ord_less @ B @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ B @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_93_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B2: B,C3: B] :
( ( ord_less @ A @ A3 @ ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C3 )
=> ( ! [X4: B,Y2: B] :
( ( ord_less @ B @ X4 @ Y2 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_94_order__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 )
& ( order @ A ) )
=> ! [A3: A,B2: A,F: A > C2,C3: C2] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ C2 @ ( F @ B2 ) @ C3 )
=> ( ! [X4: A,Y2: A] :
( ( ord_less @ A @ X4 @ Y2 )
=> ( ord_less @ C2 @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ C2 @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% order_less_subst2
thf(fact_95_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X2: A] :
? [Y2: A] : ( ord_less @ A @ Y2 @ X2 ) ) ).
% lt_ex
thf(fact_96_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X2: A] :
? [X_1: A] : ( ord_less @ A @ X2 @ X_1 ) ) ).
% gt_ex
thf(fact_97_neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( X2 != Y )
=> ( ~ ( ord_less @ A @ X2 @ Y )
=> ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% neqE
thf(fact_98_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( X2 != Y )
= ( ( ord_less @ A @ X2 @ Y )
| ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% neq_iff
thf(fact_99_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% order.asym
thf(fact_100_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ? [Z: A] :
( ( ord_less @ A @ X2 @ Z )
& ( ord_less @ A @ Z @ Y ) ) ) ) ).
% dense
thf(fact_101_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( X2 != Y ) ) ) ).
% less_imp_neq
thf(fact_102_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% less_asym
thf(fact_103_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% less_asym'
thf(fact_104_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A,Z3: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ( ord_less @ A @ Y @ Z3 )
=> ( ord_less @ A @ X2 @ Z3 ) ) ) ) ).
% less_trans
thf(fact_105_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
| ( X2 = Y )
| ( ord_less @ A @ Y @ X2 ) ) ) ).
% less_linear
thf(fact_106_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] :
~ ( ord_less @ A @ X2 @ X2 ) ) ).
% less_irrefl
thf(fact_107_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( A3 = B2 )
=> ( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_108_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( B2 = C3 )
=> ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_109_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ~ ( ord_less @ A @ A3 @ B2 ) ) ) ).
% dual_order.asym
thf(fact_110_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( X2 != Y ) ) ) ).
% less_imp_not_eq
thf(fact_111_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% less_not_sym
thf(fact_112_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A] :
( ! [X4: A] :
( ! [Y3: A] :
( ( ord_less @ A @ Y3 @ X4 )
=> ( P @ Y3 ) )
=> ( P @ X4 ) )
=> ( P @ A3 ) ) ) ).
% less_induct
thf(fact_113_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X2: A] :
( ~ ( ord_less @ A @ Y @ X2 )
=> ( ( ~ ( ord_less @ A @ X2 @ Y ) )
= ( X2 = Y ) ) ) ) ).
% antisym_conv3
thf(fact_114_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( Y != X2 ) ) ) ).
% less_imp_not_eq2
thf(fact_115_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A,P: $o] :
( ( ord_less @ A @ X2 @ Y )
=> ( ( ord_less @ A @ Y @ X2 )
=> P ) ) ) ).
% less_imp_triv
thf(fact_116_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ~ ( ord_less @ A @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% linorder_cases
thf(fact_117_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% dual_order.irrefl
thf(fact_118_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% order.strict_trans
thf(fact_119_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% less_imp_not_less
thf(fact_120_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P3: A > $o] :
? [X5: A] : ( P3 @ X5 ) )
= ( ^ [P4: A > $o] :
? [N2: A] :
( ( P4 @ N2 )
& ! [M2: A] :
( ( ord_less @ A @ M2 @ N2 )
=> ~ ( P4 @ M2 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_121_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B2: A] :
( ! [A5: A,B3: A] :
( ( ord_less @ A @ A5 @ B3 )
=> ( P @ A5 @ B3 ) )
=> ( ! [A5: A] : ( P @ A5 @ A5 )
=> ( ! [A5: A,B3: A] :
( ( P @ B3 @ A5 )
=> ( P @ A5 @ B3 ) )
=> ( P @ A3 @ B2 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_122_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C3 @ B2 )
=> ( ord_less @ A @ C3 @ A3 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_123_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ~ ( ord_less @ A @ X2 @ Y ) )
= ( ( ord_less @ A @ Y @ X2 )
| ( X2 = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_124_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( A3 != B2 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_125_field__lbound__gt__zero,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D1: A,D2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D1 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
=> ? [E: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E )
& ( ord_less @ A @ E @ D1 )
& ( ord_less @ A @ E @ D2 ) ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_126_less__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(3)
thf(fact_127_ereal__mult__eq__PInfty,axiom,
! [A3: extended_ereal,B2: extended_ereal] :
( ( ( times_times @ extended_ereal @ A3 @ B2 )
= ( extend1396239628finity @ extended_ereal ) )
= ( ( ( A3
= ( extend1396239628finity @ extended_ereal ) )
& ( ord_less @ extended_ereal @ ( zero_zero @ extended_ereal ) @ B2 ) )
| ( ( ord_less @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A3 )
& ( B2
= ( extend1396239628finity @ extended_ereal ) ) )
| ( ( A3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
& ( ord_less @ extended_ereal @ B2 @ ( zero_zero @ extended_ereal ) ) )
| ( ( ord_less @ extended_ereal @ A3 @ ( zero_zero @ extended_ereal ) )
& ( B2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ).
% ereal_mult_eq_PInfty
thf(fact_128_ereal__mult__eq__MInfty,axiom,
! [A3: extended_ereal,B2: extended_ereal] :
( ( ( times_times @ extended_ereal @ A3 @ B2 )
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
= ( ( ( A3
= ( extend1396239628finity @ extended_ereal ) )
& ( ord_less @ extended_ereal @ B2 @ ( zero_zero @ extended_ereal ) ) )
| ( ( ord_less @ extended_ereal @ A3 @ ( zero_zero @ extended_ereal ) )
& ( B2
= ( extend1396239628finity @ extended_ereal ) ) )
| ( ( A3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
& ( ord_less @ extended_ereal @ ( zero_zero @ extended_ereal ) @ B2 ) )
| ( ( ord_less @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A3 )
& ( B2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ).
% ereal_mult_eq_MInfty
thf(fact_129_ereal__mult__infty,axiom,
! [A3: extended_ereal] :
( ( ( A3
= ( zero_zero @ extended_ereal ) )
=> ( ( times_times @ extended_ereal @ A3 @ ( extend1396239628finity @ extended_ereal ) )
= ( zero_zero @ extended_ereal ) ) )
& ( ( A3
!= ( zero_zero @ extended_ereal ) )
=> ( ( ( ord_less @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A3 )
=> ( ( times_times @ extended_ereal @ A3 @ ( extend1396239628finity @ extended_ereal ) )
= ( extend1396239628finity @ extended_ereal ) ) )
& ( ~ ( ord_less @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A3 )
=> ( ( times_times @ extended_ereal @ A3 @ ( extend1396239628finity @ extended_ereal ) )
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ) ).
% ereal_mult_infty
thf(fact_130_ereal__infty__mult,axiom,
! [A3: extended_ereal] :
( ( ( A3
= ( zero_zero @ extended_ereal ) )
=> ( ( times_times @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ A3 )
= ( zero_zero @ extended_ereal ) ) )
& ( ( A3
!= ( zero_zero @ extended_ereal ) )
=> ( ( ( ord_less @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A3 )
=> ( ( times_times @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ A3 )
= ( extend1396239628finity @ extended_ereal ) ) )
& ( ~ ( ord_less @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A3 )
=> ( ( times_times @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ A3 )
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ) ).
% ereal_infty_mult
thf(fact_131_ereal__divide__Infty_I2_J,axiom,
! [X2: extended_ereal] :
( ( divide_divide @ extended_ereal @ X2 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
= ( zero_zero @ extended_ereal ) ) ).
% ereal_divide_Infty(2)
thf(fact_132_mult__zero__left,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% mult_zero_left
thf(fact_133_mult__zero__right,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% mult_zero_right
thf(fact_134_mult__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% mult_eq_0_iff
thf(fact_135_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ( times_times @ A @ C3 @ A3 )
= ( times_times @ A @ C3 @ B2 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( A3 = B2 ) ) ) ) ).
% mult_cancel_left
thf(fact_136_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ( times_times @ A @ A3 @ C3 )
= ( times_times @ A @ B2 @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( A3 = B2 ) ) ) ) ).
% mult_cancel_right
thf(fact_137_div__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% div_0
thf(fact_138_div__by__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% div_by_0
thf(fact_139_mult__minus__right,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% mult_minus_right
thf(fact_140_minus__mult__minus,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
= ( times_times @ A @ A3 @ B2 ) ) ) ).
% minus_mult_minus
thf(fact_141_mult__minus__left,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( uminus_uminus @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% mult_minus_left
thf(fact_142_ereal__mult__minus__left,axiom,
! [A3: extended_ereal,B2: extended_ereal] :
( ( times_times @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A3 ) @ B2 )
= ( uminus_uminus @ extended_ereal @ ( times_times @ extended_ereal @ A3 @ B2 ) ) ) ).
% ereal_mult_minus_left
thf(fact_143_ereal__mult__minus__right,axiom,
! [A3: extended_ereal,B2: extended_ereal] :
( ( times_times @ extended_ereal @ A3 @ ( uminus_uminus @ extended_ereal @ B2 ) )
= ( uminus_uminus @ extended_ereal @ ( times_times @ extended_ereal @ A3 @ B2 ) ) ) ).
% ereal_mult_minus_right
thf(fact_144_ereal__zero__times,axiom,
! [A3: extended_ereal,B2: extended_ereal] :
( ( ( times_times @ extended_ereal @ A3 @ B2 )
= ( zero_zero @ extended_ereal ) )
= ( ( A3
= ( zero_zero @ extended_ereal ) )
| ( B2
= ( zero_zero @ extended_ereal ) ) ) ) ).
% ereal_zero_times
thf(fact_145_ereal__zero__mult,axiom,
! [A3: extended_ereal] :
( ( times_times @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A3 )
= ( zero_zero @ extended_ereal ) ) ).
% ereal_zero_mult
thf(fact_146_ereal__mult__zero,axiom,
! [A3: extended_ereal] :
( ( times_times @ extended_ereal @ A3 @ ( zero_zero @ extended_ereal ) )
= ( zero_zero @ extended_ereal ) ) ).
% ereal_mult_zero
thf(fact_147_ereal__uminus__divide,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( divide_divide @ extended_ereal @ ( uminus_uminus @ extended_ereal @ X2 ) @ Y )
= ( uminus_uminus @ extended_ereal @ ( divide_divide @ extended_ereal @ X2 @ Y ) ) ) ).
% ereal_uminus_divide
thf(fact_148_ereal__divide__zero__left,axiom,
! [A3: extended_ereal] :
( ( divide_divide @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A3 )
= ( zero_zero @ extended_ereal ) ) ).
% ereal_divide_zero_left
thf(fact_149_ereal__times__divide__eq__left,axiom,
! [B2: extended_ereal,C3: extended_ereal,A3: extended_ereal] :
( ( times_times @ extended_ereal @ ( divide_divide @ extended_ereal @ B2 @ C3 ) @ A3 )
= ( divide_divide @ extended_ereal @ ( times_times @ extended_ereal @ B2 @ A3 ) @ C3 ) ) ).
% ereal_times_divide_eq_left
thf(fact_150_nonzero__mult__div__cancel__right,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_151_nonzero__mult__div__cancel__left,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ A3 )
= B2 ) ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_152_ereal__divide__Infty_I1_J,axiom,
! [X2: extended_ereal] :
( ( divide_divide @ extended_ereal @ X2 @ ( extend1396239628finity @ extended_ereal ) )
= ( zero_zero @ extended_ereal ) ) ).
% ereal_divide_Infty(1)
thf(fact_153_mult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( times_times @ A @ B2 @ ( times_times @ A @ A3 @ C3 ) )
= ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% mult.left_commute
thf(fact_154_mult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A6: A,B4: A] : ( times_times @ A @ B4 @ A6 ) ) ) ) ).
% mult.commute
thf(fact_155_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 )
= ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% mult.assoc
thf(fact_156_ereal__times__divide__eq,axiom,
! [A3: extended_ereal,B2: extended_ereal,C3: extended_ereal] :
( ( times_times @ extended_ereal @ A3 @ ( divide_divide @ extended_ereal @ B2 @ C3 ) )
= ( divide_divide @ extended_ereal @ ( times_times @ extended_ereal @ A3 @ B2 ) @ C3 ) ) ).
% ereal_times_divide_eq
thf(fact_157_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 )
= ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_158_mult__not__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ B2 )
!= ( zero_zero @ A ) )
=> ( ( A3
!= ( zero_zero @ A ) )
& ( B2
!= ( zero_zero @ A ) ) ) ) ) ).
% mult_not_zero
thf(fact_159_divisors__zero,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
=> ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divisors_zero
thf(fact_160_no__zero__divisors,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ B2 )
!= ( zero_zero @ A ) ) ) ) ) ).
% no_zero_divisors
thf(fact_161_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C3 @ A3 )
= ( times_times @ A @ C3 @ B2 ) )
= ( A3 = B2 ) ) ) ) ).
% mult_left_cancel
thf(fact_162_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A3 @ C3 )
= ( times_times @ A @ B2 @ C3 ) )
= ( A3 = B2 ) ) ) ) ).
% mult_right_cancel
thf(fact_163_square__eq__iff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ A3 )
= ( times_times @ A @ B2 @ B2 ) )
= ( ( A3 = B2 )
| ( A3
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% square_eq_iff
thf(fact_164_minus__mult__commute,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( times_times @ A @ A3 @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_mult_commute
thf(fact_165_times__ereal_Osimps_I6_J,axiom,
( ( times_times @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ ( extend1396239628finity @ extended_ereal ) )
= ( extend1396239628finity @ extended_ereal ) ) ).
% times_ereal.simps(6)
thf(fact_166_ereal__right__mult__cong,axiom,
! [C3: extended_ereal,D3: extended_ereal,A3: extended_ereal,B2: extended_ereal] :
( ( C3 = D3 )
=> ( ( ( D3
!= ( zero_zero @ extended_ereal ) )
=> ( A3 = B2 ) )
=> ( ( times_times @ extended_ereal @ C3 @ A3 )
= ( times_times @ extended_ereal @ D3 @ B2 ) ) ) ) ).
% ereal_right_mult_cong
thf(fact_167_ereal__left__mult__cong,axiom,
! [C3: extended_ereal,D3: extended_ereal,A3: extended_ereal,B2: extended_ereal] :
( ( C3 = D3 )
=> ( ( ( D3
!= ( zero_zero @ extended_ereal ) )
=> ( A3 = B2 ) )
=> ( ( times_times @ extended_ereal @ A3 @ C3 )
= ( times_times @ extended_ereal @ B2 @ D3 ) ) ) ) ).
% ereal_left_mult_cong
thf(fact_168_ereal__mult__divide,axiom,
! [B2: extended_ereal,A3: extended_ereal] :
( ( ord_less @ extended_ereal @ ( zero_zero @ extended_ereal ) @ B2 )
=> ( ( ord_less @ extended_ereal @ B2 @ ( extend1396239628finity @ extended_ereal ) )
=> ( ( times_times @ extended_ereal @ B2 @ ( divide_divide @ extended_ereal @ A3 @ B2 ) )
= A3 ) ) ) ).
% ereal_mult_divide
thf(fact_169_ereal__divide__less__iff,axiom,
! [C3: extended_ereal,A3: extended_ereal,B2: extended_ereal] :
( ( ord_less @ extended_ereal @ ( zero_zero @ extended_ereal ) @ C3 )
=> ( ( ord_less @ extended_ereal @ C3 @ ( extend1396239628finity @ extended_ereal ) )
=> ( ( ord_less @ extended_ereal @ ( divide_divide @ extended_ereal @ A3 @ C3 ) @ B2 )
= ( ord_less @ extended_ereal @ A3 @ ( times_times @ extended_ereal @ B2 @ C3 ) ) ) ) ) ).
% ereal_divide_less_iff
thf(fact_170_ereal__divide__less__pos,axiom,
! [X2: extended_ereal,Y: extended_ereal,Z3: extended_ereal] :
( ( ord_less @ extended_ereal @ ( zero_zero @ extended_ereal ) @ X2 )
=> ( ( X2
!= ( extend1396239628finity @ extended_ereal ) )
=> ( ( ord_less @ extended_ereal @ ( divide_divide @ extended_ereal @ Y @ X2 ) @ Z3 )
= ( ord_less @ extended_ereal @ Y @ ( times_times @ extended_ereal @ X2 @ Z3 ) ) ) ) ) ).
% ereal_divide_less_pos
thf(fact_171_ereal__less__divide__iff,axiom,
! [C3: extended_ereal,A3: extended_ereal,B2: extended_ereal] :
( ( ord_less @ extended_ereal @ ( zero_zero @ extended_ereal ) @ C3 )
=> ( ( ord_less @ extended_ereal @ C3 @ ( extend1396239628finity @ extended_ereal ) )
=> ( ( ord_less @ extended_ereal @ A3 @ ( divide_divide @ extended_ereal @ B2 @ C3 ) )
= ( ord_less @ extended_ereal @ ( times_times @ extended_ereal @ A3 @ C3 ) @ B2 ) ) ) ) ).
% ereal_less_divide_iff
thf(fact_172_ereal__less__divide__pos,axiom,
! [X2: extended_ereal,Y: extended_ereal,Z3: extended_ereal] :
( ( ord_less @ extended_ereal @ ( zero_zero @ extended_ereal ) @ X2 )
=> ( ( X2
!= ( extend1396239628finity @ extended_ereal ) )
=> ( ( ord_less @ extended_ereal @ Y @ ( divide_divide @ extended_ereal @ Z3 @ X2 ) )
= ( ord_less @ extended_ereal @ ( times_times @ extended_ereal @ X2 @ Y ) @ Z3 ) ) ) ) ).
% ereal_less_divide_pos
thf(fact_173_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord893533164strict @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_174_mult__less__cancel__right__disj,axiom,
! [A: $tType] :
( ( linord581940658strict @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
& ( ord_less @ A @ A3 @ B2 ) )
| ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_175_mult__strict__right__mono,axiom,
! [A: $tType] :
( ( linord20386208strict @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ) ).
% mult_strict_right_mono
thf(fact_176_mult__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linord581940658strict @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_177_mult__less__cancel__left__disj,axiom,
! [A: $tType] :
( ( linord581940658strict @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
& ( ord_less @ A @ A3 @ B2 ) )
| ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_178_mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord20386208strict @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% mult_strict_left_mono
thf(fact_179_mult__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linord581940658strict @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_180_mult__less__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord581940658strict @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_181_mult__less__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord581940658strict @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_182_zero__less__mult__pos2,axiom,
! [A: $tType] :
( ( linord20386208strict @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ B2 @ A3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% zero_less_mult_pos2
thf(fact_183_zero__less__mult__pos,axiom,
! [A: $tType] :
( ( linord20386208strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% zero_less_mult_pos
thf(fact_184_zero__less__mult__iff,axiom,
! [A: $tType] :
( ( linord581940658strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_mult_iff
thf(fact_185_mult__pos__neg2,axiom,
! [A: $tType] :
( ( linord20386208strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ B2 @ A3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg2
thf(fact_186_mult__pos__pos,axiom,
! [A: $tType] :
( ( linord20386208strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_pos_pos
thf(fact_187_mult__pos__neg,axiom,
! [A: $tType] :
( ( linord20386208strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg
thf(fact_188_mult__neg__pos,axiom,
! [A: $tType] :
( ( linord20386208strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_neg_pos
thf(fact_189_mult__less__0__iff,axiom,
! [A: $tType] :
( ( linord581940658strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% mult_less_0_iff
thf(fact_190_not__square__less__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( times_times @ A @ A3 @ A3 ) @ ( zero_zero @ A ) ) ) ).
% not_square_less_zero
thf(fact_191_mult__neg__neg,axiom,
! [A: $tType] :
( ( linord581940658strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_neg_neg
thf(fact_192_times__ereal_Osimps_I9_J,axiom,
( ( times_times @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
= ( extend1396239628finity @ extended_ereal ) ) ).
% times_ereal.simps(9)
thf(fact_193_times__ereal_Osimps_I8_J,axiom,
( ( times_times @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% times_ereal.simps(8)
thf(fact_194_times__ereal_Osimps_I7_J,axiom,
( ( times_times @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extend1396239628finity @ extended_ereal ) )
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% times_ereal.simps(7)
thf(fact_195_ereal__mult__less__0__iff,axiom,
! [A3: extended_ereal,B2: extended_ereal] :
( ( ord_less @ extended_ereal @ ( times_times @ extended_ereal @ A3 @ B2 ) @ ( zero_zero @ extended_ereal ) )
= ( ( ( ord_less @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A3 )
& ( ord_less @ extended_ereal @ B2 @ ( zero_zero @ extended_ereal ) ) )
| ( ( ord_less @ extended_ereal @ A3 @ ( zero_zero @ extended_ereal ) )
& ( ord_less @ extended_ereal @ ( zero_zero @ extended_ereal ) @ B2 ) ) ) ) ).
% ereal_mult_less_0_iff
thf(fact_196_ereal__zero__less__0__iff,axiom,
! [A3: extended_ereal,B2: extended_ereal] :
( ( ord_less @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( times_times @ extended_ereal @ A3 @ B2 ) )
= ( ( ( ord_less @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A3 )
& ( ord_less @ extended_ereal @ ( zero_zero @ extended_ereal ) @ B2 ) )
| ( ( ord_less @ extended_ereal @ A3 @ ( zero_zero @ extended_ereal ) )
& ( ord_less @ extended_ereal @ B2 @ ( zero_zero @ extended_ereal ) ) ) ) ) ).
% ereal_zero_less_0_iff
thf(fact_197_ereal__mult__mono__strict,axiom,
! [B2: extended_ereal,C3: extended_ereal,A3: extended_ereal,D3: extended_ereal] :
( ( ord_less @ extended_ereal @ ( zero_zero @ extended_ereal ) @ B2 )
=> ( ( ord_less @ extended_ereal @ ( zero_zero @ extended_ereal ) @ C3 )
=> ( ( ord_less @ extended_ereal @ A3 @ B2 )
=> ( ( ord_less @ extended_ereal @ C3 @ D3 )
=> ( ord_less @ extended_ereal @ ( times_times @ extended_ereal @ A3 @ C3 ) @ ( times_times @ extended_ereal @ B2 @ D3 ) ) ) ) ) ) ).
% ereal_mult_mono_strict
thf(fact_198_ereal__mult__mono__strict_H,axiom,
! [A3: extended_ereal,C3: extended_ereal,B2: extended_ereal,D3: extended_ereal] :
( ( ord_less @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A3 )
=> ( ( ord_less @ extended_ereal @ ( zero_zero @ extended_ereal ) @ C3 )
=> ( ( ord_less @ extended_ereal @ A3 @ B2 )
=> ( ( ord_less @ extended_ereal @ C3 @ D3 )
=> ( ord_less @ extended_ereal @ ( times_times @ extended_ereal @ A3 @ C3 ) @ ( times_times @ extended_ereal @ B2 @ D3 ) ) ) ) ) ) ).
% ereal_mult_mono_strict'
thf(fact_199_ereal__mult__less__right,axiom,
! [B2: extended_ereal,A3: extended_ereal,C3: extended_ereal] :
( ( ord_less @ extended_ereal @ ( times_times @ extended_ereal @ B2 @ A3 ) @ ( times_times @ extended_ereal @ C3 @ A3 ) )
=> ( ( ord_less @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A3 )
=> ( ( ord_less @ extended_ereal @ A3 @ ( extend1396239628finity @ extended_ereal ) )
=> ( ord_less @ extended_ereal @ B2 @ C3 ) ) ) ) ).
% ereal_mult_less_right
thf(fact_200_ereal__mult__strict__left__mono,axiom,
! [A3: extended_ereal,B2: extended_ereal,C3: extended_ereal] :
( ( ord_less @ extended_ereal @ A3 @ B2 )
=> ( ( ord_less @ extended_ereal @ ( zero_zero @ extended_ereal ) @ C3 )
=> ( ( ord_less @ extended_ereal @ C3 @ ( extend1396239628finity @ extended_ereal ) )
=> ( ord_less @ extended_ereal @ ( times_times @ extended_ereal @ C3 @ A3 ) @ ( times_times @ extended_ereal @ C3 @ B2 ) ) ) ) ) ).
% ereal_mult_strict_left_mono
thf(fact_201_ereal__mult__strict__right__mono,axiom,
! [A3: extended_ereal,B2: extended_ereal,C3: extended_ereal] :
( ( ord_less @ extended_ereal @ A3 @ B2 )
=> ( ( ord_less @ extended_ereal @ ( zero_zero @ extended_ereal ) @ C3 )
=> ( ( ord_less @ extended_ereal @ C3 @ ( extend1396239628finity @ extended_ereal ) )
=> ( ord_less @ extended_ereal @ ( times_times @ extended_ereal @ A3 @ C3 ) @ ( times_times @ extended_ereal @ B2 @ C3 ) ) ) ) ) ).
% ereal_mult_strict_right_mono
thf(fact_202_div__mult__mult1__if,axiom,
! [A: $tType] :
( ( euclid191655569cancel @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ( C3
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_mult1_if
thf(fact_203_div__mult__mult2,axiom,
! [A: $tType] :
( ( euclid191655569cancel @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% div_mult_mult2
thf(fact_204_div__mult__mult1,axiom,
! [A: $tType] :
( ( euclid191655569cancel @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% div_mult_mult1
thf(fact_205_nonzero__mult__divide__mult__cancel__right2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_206_divide__eq__0__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( ( divide_divide @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_eq_0_iff
thf(fact_207_divide__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ( divide_divide @ A @ C3 @ A3 )
= ( divide_divide @ A @ C3 @ B2 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( A3 = B2 ) ) ) ) ).
% divide_cancel_left
thf(fact_208_divide__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ( divide_divide @ A @ A3 @ C3 )
= ( divide_divide @ A @ B2 @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( A3 = B2 ) ) ) ) ).
% divide_cancel_right
thf(fact_209_division__ring__divide__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% division_ring_divide_zero
thf(fact_210_times__divide__eq__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% times_divide_eq_right
thf(fact_211_divide__divide__eq__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( divide_divide @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 ) ) ) ).
% divide_divide_eq_right
thf(fact_212_divide__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C3 )
= ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% divide_divide_eq_left
thf(fact_213_times__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( times_times @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A3 )
= ( divide_divide @ A @ ( times_times @ A @ B2 @ A3 ) @ C3 ) ) ) ).
% times_divide_eq_left
thf(fact_214_div__minus__minus,axiom,
! [A: $tType] :
( ( euclid24285859cancel @ A )
=> ! [A3: A,B2: A] :
( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ).
% div_minus_minus
thf(fact_215_mult__divide__mult__cancel__left__if,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ( C3
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_216_nonzero__mult__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_217_nonzero__mult__divide__mult__cancel__left2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_218_nonzero__mult__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_219_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A] :
? [Y2: A] : ( ord_less @ A @ Y2 @ X3 ) ) ).
% linordered_field_no_lb
thf(fact_220_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A] :
? [X_1: A] : ( ord_less @ A @ X3 @ X_1 ) ) ).
% linordered_field_no_ub
thf(fact_221_divide__divide__eq__left_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C3 )
= ( divide_divide @ A @ A3 @ ( times_times @ A @ C3 @ B2 ) ) ) ) ).
% divide_divide_eq_left'
thf(fact_222_divide__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X2: A,Y: A,Z3: A,W: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ X2 @ Y ) @ ( divide_divide @ A @ Z3 @ W ) )
= ( divide_divide @ A @ ( times_times @ A @ X2 @ W ) @ ( times_times @ A @ Y @ Z3 ) ) ) ) ).
% divide_divide_times_eq
thf(fact_223_times__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X2: A,Y: A,Z3: A,W: A] :
( ( times_times @ A @ ( divide_divide @ A @ X2 @ Y ) @ ( divide_divide @ A @ Z3 @ W ) )
= ( divide_divide @ A @ ( times_times @ A @ X2 @ Z3 ) @ ( times_times @ A @ Y @ W ) ) ) ) ).
% times_divide_times_eq
thf(fact_224_div__minus__right,axiom,
! [A: $tType] :
( ( euclid24285859cancel @ A )
=> ! [A3: A,B2: A] :
( ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% div_minus_right
thf(fact_225_minus__divide__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% minus_divide_left
thf(fact_226_minus__divide__divide,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ).
% minus_divide_divide
thf(fact_227_minus__divide__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_divide_right
thf(fact_228_divide__neg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X2 @ Y ) ) ) ) ) ).
% divide_neg_neg
thf(fact_229_divide__neg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ ( divide_divide @ A @ X2 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_neg_pos
thf(fact_230_divide__pos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X2 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_pos_neg
thf(fact_231_divide__pos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X2 @ Y ) ) ) ) ) ).
% divide_pos_pos
thf(fact_232_divide__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% divide_less_0_iff
thf(fact_233_divide__less__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A3 @ B2 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A3 ) )
& ( C3
!= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_cancel
thf(fact_234_zero__less__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_divide_iff
thf(fact_235_divide__strict__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) ) ) ) ) ).
% divide_strict_right_mono
thf(fact_236_divide__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_237_frac__eq__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z3: A,X2: A,W: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ X2 @ Y )
= ( divide_divide @ A @ W @ Z3 ) )
= ( ( times_times @ A @ X2 @ Z3 )
= ( times_times @ A @ W @ Y ) ) ) ) ) ) ).
% frac_eq_eq
thf(fact_238_divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ( divide_divide @ A @ B2 @ C3 )
= A3 )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A3 @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq
thf(fact_239_eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( A3
= ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ C3 )
= B2 ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq
thf(fact_240_divide__eq__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( B2
= ( times_times @ A @ A3 @ C3 ) )
=> ( ( divide_divide @ A @ B2 @ C3 )
= A3 ) ) ) ) ).
% divide_eq_imp
thf(fact_241_eq__divide__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A3 @ C3 )
= B2 )
=> ( A3
= ( divide_divide @ A @ B2 @ C3 ) ) ) ) ) ).
% eq_divide_imp
thf(fact_242_nonzero__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ B2 @ C3 )
= A3 )
= ( B2
= ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_243_nonzero__eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( A3
= ( divide_divide @ A @ B2 @ C3 ) )
= ( ( times_times @ A @ A3 @ C3 )
= B2 ) ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_244_nonzero__minus__divide__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_245_nonzero__minus__divide__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_246_divide__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C3 @ A3 ) @ ( divide_divide @ A @ C3 @ B2 ) ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_247_divide__strict__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C3 @ A3 ) @ ( divide_divide @ A @ C3 @ B2 ) ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_248_mult__imp__less__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z3: A,X2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less @ A @ ( times_times @ A @ Z3 @ Y ) @ X2 )
=> ( ord_less @ A @ Z3 @ ( divide_divide @ A @ X2 @ Y ) ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_249_mult__imp__div__pos__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X2: A,Z3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less @ A @ X2 @ ( times_times @ A @ Z3 @ Y ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X2 @ Y ) @ Z3 ) ) ) ) ).
% mult_imp_div_pos_less
thf(fact_250_pos__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 ) ) ) ) ).
% pos_less_divide_eq
thf(fact_251_pos__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A3 )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% pos_divide_less_eq
thf(fact_252_neg__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% neg_less_divide_eq
thf(fact_253_neg__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 ) ) ) ) ).
% neg_divide_less_eq
thf(fact_254_less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq
% Subclasses (4)
thf(subcl_Real__Vector__Spaces_Ometric__space___HOL_Otype,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ( type @ A ) ) ).
thf(subcl_Real__Vector__Spaces_Ometric__space___Topological__Spaces_Ot0__space,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ( topological_t0_space @ A ) ) ).
thf(subcl_Real__Vector__Spaces_Ometric__space___Topological__Spaces_Ot1__space,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ( topological_t1_space @ A ) ) ).
thf(subcl_Real__Vector__Spaces_Ometric__space___Topological__Spaces_Otopological__space,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ( topolo503727757_space @ A ) ) ).
% Type constructors (37)
thf(tcon_fun___Topological__Spaces_Otopological__space,axiom,
! [A7: $tType,A8: $tType] :
( ( topolo503727757_space @ A8 )
=> ( topolo503727757_space @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
! [A7: $tType,A8: $tType] :
( ( boolean_algebra @ A8 )
=> ( boolean_algebra @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 )
=> ( preorder @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 )
=> ( order @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 )
=> ( ord @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A7: $tType,A8: $tType] :
( ( uminus @ A8 )
=> ( uminus @ ( A7 > A8 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Oboolean__algebra_1,axiom,
! [A7: $tType] : ( boolean_algebra @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Groups_Oab__semigroup__mult,axiom,
! [A7: $tType] :
( ( ab_semigroup_mult @ A7 )
=> ( ab_semigroup_mult @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Groups_Osemigroup__mult,axiom,
! [A7: $tType] :
( ( semigroup_mult @ A7 )
=> ( semigroup_mult @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_2,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_3,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_4,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_5,axiom,
! [A7: $tType] : ( uminus @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Groups_Ozero,axiom,
! [A7: $tType] :
( ( zero @ A7 )
=> ( zero @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_6,axiom,
topolo503727757_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology,axiom,
topolo2133971006pology @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot1__space,axiom,
topological_t1_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot0__space,axiom,
topological_t0_space @ $o ).
thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_7,axiom,
boolean_algebra @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_8,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_11,axiom,
uminus @ $o ).
thf(tcon_Extended__Real_Oereal___Conditionally__Complete__Lattices_Olinear__continuum,axiom,
condit1656338222tinuum @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Topological__Spaces_Otopological__space_12,axiom,
topolo503727757_space @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Topological__Spaces_Ot1__space_13,axiom,
topological_t1_space @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Topological__Spaces_Ot0__space_14,axiom,
topological_t0_space @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Groups_Oab__semigroup__mult_15,axiom,
ab_semigroup_mult @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Orderings_Odense__order,axiom,
dense_order @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Groups_Osemigroup__mult_16,axiom,
semigroup_mult @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Orderings_Opreorder_17,axiom,
preorder @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Orderings_Olinorder_18,axiom,
linorder @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Orderings_Oorder_19,axiom,
order @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Orderings_Oord_20,axiom,
ord @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Groups_Ouminus_21,axiom,
uminus @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Groups_Ozero_22,axiom,
zero @ extended_ereal ).
% Free types (1)
thf(tfree_0,hypothesis,
real_V2090557954_space @ a ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( lower_582600101lsc_at @ a @ extended_ereal @ x0 @ f )
!= ( ~ ! [C: extended_ereal] :
( ( ord_less @ extended_ereal @ C @ ( f @ x0 ) )
=> ? [T: set @ a] :
( ( topolo1751647064n_open @ a @ T )
& ( member @ a @ x0 @ T )
& ! [X: a] :
( ( member @ a @ X @ T )
=> ( ord_less @ extended_ereal @ C @ ( f @ X ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------