TPTP Problem File: SLH0898^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Interpolation_Polynomials_HOL_Algebra/0000_Bounded_Degree_Polynomials/prob_00060_002421__16993400_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1501 ( 516 unt; 231 typ; 0 def)
% Number of atoms : 3657 (1096 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9674 ( 377 ~; 67 |; 243 &;7452 @)
% ( 0 <=>;1535 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 6 avg)
% Number of types : 39 ( 38 usr)
% Number of type conns : 1272 (1272 >; 0 *; 0 +; 0 <<)
% Number of symbols : 196 ( 193 usr; 27 con; 0-3 aty)
% Number of variables : 3717 ( 507 ^;3128 !; 82 ?;3717 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:35:31.939
%------------------------------------------------------------------------------
% Could-be-implicit typings (38)
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Int__Oint_M_062_It__Nat__Onat_Mtf__a_J_J_J_J,type,
set_set_int_nat_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Int__Oint_J_J_J,type,
set_nat_nat_int: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__a_J_J_J,type,
set_nat_nat_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Int__Oint_M_062_It__Nat__Onat_Mtf__a_J_J_J,type,
set_int_nat_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J_J,type,
set_set_nat_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Int__Oint_Mtf__a_J_J_J,type,
set_set_int_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_M_062_It__Nat__Onat_Mtf__a_J_J_J,type,
set_a_nat_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_J,type,
set_nat_a_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_M_062_It__Nat__Onat_Mtf__a_J_J_J,type,
set_o_nat_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_Itf__a_Mtf__a_J_J_J,type,
set_set_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_I_Eo_Mtf__a_J_J_J,type,
set_set_o_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Int__Oint_J_J,type,
set_nat_int: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
set_int_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Int__Oint_Mt__Int__Oint_J_J,type,
set_int_int: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__Nat__Onat_J_J,type,
set_a_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__Int__Oint_J_J,type,
set_a_int: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
set_nat_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Int__Oint_Mtf__a_J_J,type,
set_int_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
set_nat_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Int__Oint_M_Eo_J_J,type,
set_int_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_Mt__Nat__Onat_J_J,type,
set_o_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_Mt__Int__Oint_J_J,type,
set_o_int: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
set_a_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_M_Eo_J_J,type,
set_a_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_Mtf__a_J_J,type,
set_o_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_M_Eo_J_J,type,
set_o_o: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (193)
thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
archim7802044766580827645g_real: real > int ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
complete_Sup_Sup_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
inverse_inverse_real: real > real ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_Mtf__a_J_001tf__a,type,
piE_nat_a_a: set_nat_a > ( ( nat > a ) > set_a ) > set_nat_a_a ).
thf(sy_c_FuncSet_OPiE_001_Eo_001_062_It__Nat__Onat_Mtf__a_J,type,
piE_o_nat_a: set_o > ( $o > set_nat_a ) > set_o_nat_a ).
thf(sy_c_FuncSet_OPiE_001_Eo_001_Eo,type,
piE_o_o: set_o > ( $o > set_o ) > set_o_o ).
thf(sy_c_FuncSet_OPiE_001_Eo_001t__Int__Oint,type,
piE_o_int: set_o > ( $o > set_int ) > set_o_int ).
thf(sy_c_FuncSet_OPiE_001_Eo_001t__Nat__Onat,type,
piE_o_nat: set_o > ( $o > set_nat ) > set_o_nat ).
thf(sy_c_FuncSet_OPiE_001_Eo_001tf__a,type,
piE_o_a: set_o > ( $o > set_a ) > set_o_a ).
thf(sy_c_FuncSet_OPiE_001t__Int__Oint_001_062_It__Nat__Onat_Mtf__a_J,type,
piE_int_nat_a: set_int > ( int > set_nat_a ) > set_int_nat_a ).
thf(sy_c_FuncSet_OPiE_001t__Int__Oint_001_Eo,type,
piE_int_o: set_int > ( int > set_o ) > set_int_o ).
thf(sy_c_FuncSet_OPiE_001t__Int__Oint_001t__Int__Oint,type,
piE_int_int: set_int > ( int > set_int ) > set_int_int ).
thf(sy_c_FuncSet_OPiE_001t__Int__Oint_001t__Nat__Onat,type,
piE_int_nat: set_int > ( int > set_nat ) > set_int_nat ).
thf(sy_c_FuncSet_OPiE_001t__Int__Oint_001tf__a,type,
piE_int_a: set_int > ( int > set_a ) > set_int_a ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Int__Oint_J,type,
piE_nat_nat_int: set_nat > ( nat > set_nat_int ) > set_nat_nat_int ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001_062_It__Nat__Onat_Mtf__a_J,type,
piE_nat_nat_a: set_nat > ( nat > set_nat_a ) > set_nat_nat_a ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001_Eo,type,
piE_nat_o: set_nat > ( nat > set_o ) > set_nat_o ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001t__Int__Oint,type,
piE_nat_int: set_nat > ( nat > set_int ) > set_nat_int ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001t__Nat__Onat,type,
piE_nat_nat: set_nat > ( nat > set_nat ) > set_nat_nat ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001tf__a,type,
piE_nat_a: set_nat > ( nat > set_a ) > set_nat_a ).
thf(sy_c_FuncSet_OPiE_001tf__a_001_062_It__Nat__Onat_Mtf__a_J,type,
piE_a_nat_a: set_a > ( a > set_nat_a ) > set_a_nat_a ).
thf(sy_c_FuncSet_OPiE_001tf__a_001_Eo,type,
piE_a_o: set_a > ( a > set_o ) > set_a_o ).
thf(sy_c_FuncSet_OPiE_001tf__a_001t__Int__Oint,type,
piE_a_int: set_a > ( a > set_int ) > set_a_int ).
thf(sy_c_FuncSet_OPiE_001tf__a_001t__Nat__Onat,type,
piE_a_nat: set_a > ( a > set_nat ) > set_a_nat ).
thf(sy_c_FuncSet_OPiE_001tf__a_001tf__a,type,
piE_a_a: set_a > ( a > set_a ) > set_a_a ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
ring_1_of_int_real: int > real ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_ONats_001t__Int__Oint,type,
semiring_1_Nats_int: set_int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Orderings_Oord__class_Oless_001_Eo,type,
ord_less_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_I_Eo_Mtf__a_J_J,type,
ord_less_set_o_a: set_o_a > set_o_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_It__Int__Oint_M_062_It__Nat__Onat_Mtf__a_J_J_J,type,
ord_le2026899869173067772_nat_a: set_int_nat_a > set_int_nat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_It__Int__Oint_Mtf__a_J_J,type,
ord_less_set_int_a: set_int_a > set_int_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
ord_less_set_nat_a: set_nat_a > set_nat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_Itf__a_M_062_It__Nat__Onat_Mtf__a_J_J_J,type,
ord_less_set_a_nat_a: set_a_nat_a > set_a_nat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
ord_less_set_a_a: set_a_a > set_a_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_I_Eo_Mtf__a_J_M_Eo_J,type,
ord_less_eq_o_a_o: ( ( $o > a ) > $o ) > ( ( $o > a ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Int__Oint_M_062_It__Nat__Onat_Mtf__a_J_J_M_Eo_J,type,
ord_le998279727272312301at_a_o: ( ( int > nat > a ) > $o ) > ( ( int > nat > a ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Int__Oint_Mtf__a_J_M_Eo_J,type,
ord_less_eq_int_a_o: ( ( int > a ) > $o ) > ( ( int > a ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mtf__a_J_M_Eo_J,type,
ord_less_eq_nat_a_o: ( ( nat > a ) > $o ) > ( ( nat > a ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_Itf__a_M_062_It__Nat__Onat_Mtf__a_J_J_M_Eo_J,type,
ord_le6051534669759718563at_a_o: ( ( a > nat > a ) > $o ) > ( ( a > nat > a ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_Itf__a_Mtf__a_J_M_Eo_J,type,
ord_less_eq_a_a_o: ( ( a > a ) > $o ) > ( ( a > a ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_Eo_J,type,
ord_less_eq_o_o: ( $o > $o ) > ( $o > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Int__Oint_M_Eo_J,type,
ord_less_eq_int_o: ( int > $o ) > ( int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
ord_less_eq_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_J,type,
ord_le3509452538356653652at_a_a: set_nat_a_a > set_nat_a_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_Eo_M_062_It__Nat__Onat_Mtf__a_J_J_J,type,
ord_le1609037065912973128_nat_a: set_o_nat_a > set_o_nat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_Eo_Mtf__a_J_J,type,
ord_less_eq_set_o_a: set_o_a > set_o_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Int__Oint_M_062_It__Nat__Onat_Mtf__a_J_J_J,type,
ord_le6072919132162605296_nat_a: set_int_nat_a > set_int_nat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Int__Oint_Mt__Int__Oint_J_J,type,
ord_le8756421791413902349nt_int: set_int_int > set_int_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
ord_le2023132899490853297nt_nat: set_int_nat > set_int_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Int__Oint_Mtf__a_J_J,type,
ord_le943418215940126601_int_a: set_int_a > set_int_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__a_J_J_J,type,
ord_le3343314562563788876_nat_a: set_nat_nat_a > set_nat_nat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Int__Oint_J_J,type,
ord_le6569500216720880561at_int: set_nat_int > set_nat_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
ord_le871467723717165285_nat_a: set_nat_a > set_nat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__a_M_062_It__Nat__Onat_Mtf__a_J_J_J,type,
ord_le2508512696544544866_nat_a: set_a_nat_a > set_a_nat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
ord_less_eq_set_a_a: set_a_a > set_a_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
top_top_int_int_o: ( int > int ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_It__Int__Oint_Mt__Nat__Onat_J_M_Eo_J,type,
top_top_int_nat_o: ( int > nat ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_It__Nat__Onat_Mt__Int__Oint_J_M_Eo_J,type,
top_top_nat_int_o: ( nat > int ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_It__Nat__Onat_Mtf__a_J_M_Eo_J,type,
top_top_nat_a_o: ( nat > a ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Int__Oint_M_Eo_J,type,
top_top_int_o: int > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_Itf__a_M_Eo_J,type,
top_top_a_o: a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_I_Eo_Mtf__a_J_J,type,
top_top_set_o_a: set_o_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Int__Oint_Mt__Int__Oint_J_J,type,
top_top_set_int_int: set_int_int ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
top_top_set_int_nat: set_int_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Int__Oint_Mtf__a_J_J,type,
top_top_set_int_a: set_int_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Int__Oint_J_J_J,type,
top_to8065595215973171184at_int: set_nat_nat_int ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Int__Oint_J_J,type,
top_top_set_nat_int: set_nat_int ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_top_set_nat_nat: set_nat_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
top_top_set_nat_a: set_nat_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__a_Mt__Int__Oint_J_J,type,
top_top_set_a_int: set_a_int ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__a_Mt__Nat__Onat_J_J,type,
top_top_set_a_nat: set_a_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
top_top_set_a_a: set_a_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
top_top_set_o: set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
top_top_set_int: set_int ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Set_OCollect_001_062_I_Eo_Mtf__a_J,type,
collect_o_a: ( ( $o > a ) > $o ) > set_o_a ).
thf(sy_c_Set_OCollect_001_062_It__Int__Oint_M_062_It__Nat__Onat_Mtf__a_J_J,type,
collect_int_nat_a: ( ( int > nat > a ) > $o ) > set_int_nat_a ).
thf(sy_c_Set_OCollect_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
collect_int_int: ( ( int > int ) > $o ) > set_int_int ).
thf(sy_c_Set_OCollect_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
collect_int_nat: ( ( int > nat ) > $o ) > set_int_nat ).
thf(sy_c_Set_OCollect_001_062_It__Int__Oint_Mtf__a_J,type,
collect_int_a: ( ( int > a ) > $o ) > set_int_a ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Int__Oint_J,type,
collect_nat_int: ( ( nat > int ) > $o ) > set_nat_int ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mtf__a_J,type,
collect_nat_a: ( ( nat > a ) > $o ) > set_nat_a ).
thf(sy_c_Set_OCollect_001_062_Itf__a_M_062_It__Nat__Onat_Mtf__a_J_J,type,
collect_a_nat_a: ( ( a > nat > a ) > $o ) > set_a_nat_a ).
thf(sy_c_Set_OCollect_001_062_Itf__a_Mtf__a_J,type,
collect_a_a: ( ( a > a ) > $o ) > set_a_a ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mtf__a_J_001_062_It__Nat__Onat_Mtf__a_J,type,
image_nat_a_nat_a: ( ( nat > a ) > nat > a ) > set_nat_a > set_nat_a ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mtf__a_J_001_Eo,type,
image_nat_a_o: ( ( nat > a ) > $o ) > set_nat_a > set_o ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mtf__a_J_001t__Int__Oint,type,
image_nat_a_int: ( ( nat > a ) > int ) > set_nat_a > set_int ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mtf__a_J_001tf__a,type,
image_nat_a_a: ( ( nat > a ) > a ) > set_nat_a > set_a ).
thf(sy_c_Set_Oimage_001_Eo_001_Eo,type,
image_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Int__Oint,type,
image_o_int: ( $o > int ) > set_o > set_int ).
thf(sy_c_Set_Oimage_001_Eo_001t__Nat__Onat,type,
image_o_nat: ( $o > nat ) > set_o > set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001tf__a,type,
image_o_a: ( $o > a ) > set_o > set_a ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001_Eo,type,
image_int_o: ( int > $o ) > set_int > set_o ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
image_int_nat: ( int > nat ) > set_int > set_nat ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001tf__a,type,
image_int_a: ( int > a ) > set_int > set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_Eo,type,
image_nat_o: ( nat > $o ) > set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
image_nat_int: ( nat > int ) > set_nat > set_int ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001_062_It__Nat__Onat_Mtf__a_J,type,
image_a_nat_a: ( a > nat > a ) > set_a > set_nat_a ).
thf(sy_c_Set_Oimage_001tf__a_001_Eo,type,
image_a_o: ( a > $o ) > set_a > set_o ).
thf(sy_c_Set_Oimage_001tf__a_001t__Int__Oint,type,
image_a_int: ( a > int ) > set_a > set_int ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001_Eo,type,
set_or7139685690850216873Than_o: $o > $o > set_o ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Real__Oreal,type,
set_or66887138388493659n_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_I_062_I_Eo_Mtf__a_J_J,type,
set_or1317968236485584176et_o_a: set_o_a > set_o_a > set_set_o_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_I_062_It__Int__Oint_M_062_It__Nat__Onat_Mtf__a_J_J_J,type,
set_or7189319678020346337_nat_a: set_int_nat_a > set_int_nat_a > set_set_int_nat_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_I_062_It__Int__Oint_Mtf__a_J_J,type,
set_or8749074377923073530_int_a: set_int_a > set_int_a > set_set_int_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
set_or8677123885700112214_nat_a: set_nat_a > set_nat_a > set_set_nat_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
set_or5743114581318135242et_a_a: set_a_a > set_a_a > set_set_a_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_Itf__a_J,type,
set_or2348907005316661231_set_a: set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_member_001_062_I_Eo_M_Eo_J,type,
member_o_o: ( $o > $o ) > set_o_o > $o ).
thf(sy_c_member_001_062_I_Eo_Mt__Int__Oint_J,type,
member_o_int: ( $o > int ) > set_o_int > $o ).
thf(sy_c_member_001_062_I_Eo_Mt__Nat__Onat_J,type,
member_o_nat: ( $o > nat ) > set_o_nat > $o ).
thf(sy_c_member_001_062_I_Eo_Mtf__a_J,type,
member_o_a: ( $o > a ) > set_o_a > $o ).
thf(sy_c_member_001_062_It__Int__Oint_M_062_It__Nat__Onat_Mtf__a_J_J,type,
member_int_nat_a: ( int > nat > a ) > set_int_nat_a > $o ).
thf(sy_c_member_001_062_It__Int__Oint_M_Eo_J,type,
member_int_o: ( int > $o ) > set_int_o > $o ).
thf(sy_c_member_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
member_int_int: ( int > int ) > set_int_int > $o ).
thf(sy_c_member_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
member_int_nat: ( int > nat ) > set_int_nat > $o ).
thf(sy_c_member_001_062_It__Int__Oint_Mtf__a_J,type,
member_int_a: ( int > a ) > set_int_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_Eo_J,type,
member_nat_o: ( nat > $o ) > set_nat_o > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Int__Oint_J,type,
member_nat_int: ( nat > int ) > set_nat_int > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mtf__a_J,type,
member_nat_a: ( nat > a ) > set_nat_a > $o ).
thf(sy_c_member_001_062_Itf__a_M_062_It__Nat__Onat_Mtf__a_J_J,type,
member_a_nat_a: ( a > nat > a ) > set_a_nat_a > $o ).
thf(sy_c_member_001_062_Itf__a_M_Eo_J,type,
member_a_o: ( a > $o ) > set_a_o > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__Int__Oint_J,type,
member_a_int: ( a > int ) > set_a_int > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__Nat__Onat_J,type,
member_a_nat: ( a > nat ) > set_a_nat > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_I_Eo_Mtf__a_J_J,type,
member_set_o_a: set_o_a > set_set_o_a > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__Int__Oint_M_062_It__Nat__Onat_Mtf__a_J_J_J,type,
member_set_int_nat_a: set_int_nat_a > set_set_int_nat_a > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__Int__Oint_Mtf__a_J_J,type,
member_set_int_a: set_int_a > set_set_int_a > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
member_set_nat_a: set_nat_a > set_set_nat_a > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
member_set_a_a: set_a_a > set_set_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_B,type,
b: set_a ).
thf(sy_v_f____,type,
f: ( nat > a ) > nat > a ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_y,type,
y: a ).
% Relevant facts (1264)
thf(fact_0_assms,axiom,
member_a @ y @ b ).
% assms
thf(fact_1_in__image__by__witness,axiom,
! [A: set_nat,G: nat > nat,B: set_nat,F: nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( member_nat @ ( G @ X ) @ B )
& ( ( F @ ( G @ X ) )
= X ) ) )
=> ( ord_less_eq_set_nat @ A @ ( image_nat_nat @ F @ B ) ) ) ).
% in_image_by_witness
thf(fact_2_in__image__by__witness,axiom,
! [A: set_o,G: $o > a,B: set_a,F: a > $o] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ( member_a @ ( G @ X ) @ B )
& ( ( F @ ( G @ X ) )
= X ) ) )
=> ( ord_less_eq_set_o @ A @ ( image_a_o @ F @ B ) ) ) ).
% in_image_by_witness
thf(fact_3_in__image__by__witness,axiom,
! [A: set_o,G: $o > $o,B: set_o,F: $o > $o] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ( member_o @ ( G @ X ) @ B )
& ( ( F @ ( G @ X ) )
= X ) ) )
=> ( ord_less_eq_set_o @ A @ ( image_o_o @ F @ B ) ) ) ).
% in_image_by_witness
thf(fact_4_in__image__by__witness,axiom,
! [A: set_o,G: $o > int,B: set_int,F: int > $o] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ( member_int @ ( G @ X ) @ B )
& ( ( F @ ( G @ X ) )
= X ) ) )
=> ( ord_less_eq_set_o @ A @ ( image_int_o @ F @ B ) ) ) ).
% in_image_by_witness
thf(fact_5_in__image__by__witness,axiom,
! [A: set_int,G: int > nat,B: set_nat,F: nat > int] :
( ! [X: int] :
( ( member_int @ X @ A )
=> ( ( member_nat @ ( G @ X ) @ B )
& ( ( F @ ( G @ X ) )
= X ) ) )
=> ( ord_less_eq_set_int @ A @ ( image_nat_int @ F @ B ) ) ) ).
% in_image_by_witness
thf(fact_6_in__image__by__witness,axiom,
! [A: set_int,G: int > a,B: set_a,F: a > int] :
( ! [X: int] :
( ( member_int @ X @ A )
=> ( ( member_a @ ( G @ X ) @ B )
& ( ( F @ ( G @ X ) )
= X ) ) )
=> ( ord_less_eq_set_int @ A @ ( image_a_int @ F @ B ) ) ) ).
% in_image_by_witness
thf(fact_7_in__image__by__witness,axiom,
! [A: set_int,G: int > $o,B: set_o,F: $o > int] :
( ! [X: int] :
( ( member_int @ X @ A )
=> ( ( member_o @ ( G @ X ) @ B )
& ( ( F @ ( G @ X ) )
= X ) ) )
=> ( ord_less_eq_set_int @ A @ ( image_o_int @ F @ B ) ) ) ).
% in_image_by_witness
thf(fact_8_in__image__by__witness,axiom,
! [A: set_int,G: int > int,B: set_int,F: int > int] :
( ! [X: int] :
( ( member_int @ X @ A )
=> ( ( member_int @ ( G @ X ) @ B )
& ( ( F @ ( G @ X ) )
= X ) ) )
=> ( ord_less_eq_set_int @ A @ ( image_int_int @ F @ B ) ) ) ).
% in_image_by_witness
thf(fact_9_in__image__by__witness,axiom,
! [A: set_a,G: a > nat,B: set_nat,F: nat > a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ( member_nat @ ( G @ X ) @ B )
& ( ( F @ ( G @ X ) )
= X ) ) )
=> ( ord_less_eq_set_a @ A @ ( image_nat_a @ F @ B ) ) ) ).
% in_image_by_witness
thf(fact_10_in__image__by__witness,axiom,
! [A: set_a,G: a > a,B: set_a,F: a > a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ( member_a @ ( G @ X ) @ B )
& ( ( F @ ( G @ X ) )
= X ) ) )
=> ( ord_less_eq_set_a @ A @ ( image_a_a @ F @ B ) ) ) ).
% in_image_by_witness
thf(fact_11_f__def,axiom,
( f
= ( ^ [F2: nat > a,K: nat] : ( if_a @ ( ord_less_nat @ K @ n ) @ ( F2 @ K ) @ y ) ) ) ).
% f_def
thf(fact_12_ivl__subset,axiom,
! [I: real,J: real,M: real,N: real] :
( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ I @ J ) @ ( set_or66887138388493659n_real @ M @ N ) )
= ( ( ord_less_eq_real @ J @ I )
| ( ( ord_less_eq_real @ M @ I )
& ( ord_less_eq_real @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_13_ivl__subset,axiom,
! [I: nat,J: nat,M: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ J @ I )
| ( ( ord_less_eq_nat @ M @ I )
& ( ord_less_eq_nat @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_14_ivl__subset,axiom,
! [I: int,J: int,M: int,N: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ I @ J ) @ ( set_or4662586982721622107an_int @ M @ N ) )
= ( ( ord_less_eq_int @ J @ I )
| ( ( ord_less_eq_int @ M @ I )
& ( ord_less_eq_int @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_15_PiE__UNIV,axiom,
( ( piE_nat_a @ top_top_set_nat
@ ^ [I2: nat] : top_top_set_a )
= top_top_set_nat_a ) ).
% PiE_UNIV
thf(fact_16_PiE__UNIV,axiom,
( ( piE_nat_nat @ top_top_set_nat
@ ^ [I2: nat] : top_top_set_nat )
= top_top_set_nat_nat ) ).
% PiE_UNIV
thf(fact_17_PiE__UNIV,axiom,
( ( piE_nat_int @ top_top_set_nat
@ ^ [I2: nat] : top_top_set_int )
= top_top_set_nat_int ) ).
% PiE_UNIV
thf(fact_18_PiE__UNIV,axiom,
( ( piE_int_nat @ top_top_set_int
@ ^ [I2: int] : top_top_set_nat )
= top_top_set_int_nat ) ).
% PiE_UNIV
thf(fact_19_PiE__UNIV,axiom,
( ( piE_int_int @ top_top_set_int
@ ^ [I2: int] : top_top_set_int )
= top_top_set_int_int ) ).
% PiE_UNIV
thf(fact_20_PiE__UNIV,axiom,
( ( piE_int_a @ top_top_set_int
@ ^ [I2: int] : top_top_set_a )
= top_top_set_int_a ) ).
% PiE_UNIV
thf(fact_21_PiE__UNIV,axiom,
( ( piE_a_nat @ top_top_set_a
@ ^ [I2: a] : top_top_set_nat )
= top_top_set_a_nat ) ).
% PiE_UNIV
thf(fact_22_PiE__UNIV,axiom,
( ( piE_a_int @ top_top_set_a
@ ^ [I2: a] : top_top_set_int )
= top_top_set_a_int ) ).
% PiE_UNIV
thf(fact_23_PiE__UNIV,axiom,
( ( piE_a_a @ top_top_set_a
@ ^ [I2: a] : top_top_set_a )
= top_top_set_a_a ) ).
% PiE_UNIV
thf(fact_24_PiE__UNIV,axiom,
( ( piE_nat_nat_int @ top_top_set_nat
@ ^ [I2: nat] : top_top_set_nat_int )
= top_to8065595215973171184at_int ) ).
% PiE_UNIV
thf(fact_25_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_26_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_27_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_28_image__ident,axiom,
! [Y: set_a] :
( ( image_a_a
@ ^ [X2: a] : X2
@ Y )
= Y ) ).
% image_ident
thf(fact_29_image__ident,axiom,
! [Y: set_nat] :
( ( image_nat_nat
@ ^ [X2: nat] : X2
@ Y )
= Y ) ).
% image_ident
thf(fact_30_image__ident,axiom,
! [Y: set_int] :
( ( image_int_int
@ ^ [X2: int] : X2
@ Y )
= Y ) ).
% image_ident
thf(fact_31_image__ident,axiom,
! [Y: set_nat_a] :
( ( image_nat_a_nat_a
@ ^ [X2: nat > a] : X2
@ Y )
= Y ) ).
% image_ident
thf(fact_32_PiE__mono,axiom,
! [A: set_a,B: a > set_a,C: a > set_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ ( B @ X ) @ ( C @ X ) ) )
=> ( ord_less_eq_set_a_a @ ( piE_a_a @ A @ B ) @ ( piE_a_a @ A @ C ) ) ) ).
% PiE_mono
thf(fact_33_PiE__mono,axiom,
! [A: set_o,B: $o > set_a,C: $o > set_a] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_set_a @ ( B @ X ) @ ( C @ X ) ) )
=> ( ord_less_eq_set_o_a @ ( piE_o_a @ A @ B ) @ ( piE_o_a @ A @ C ) ) ) ).
% PiE_mono
thf(fact_34_PiE__mono,axiom,
! [A: set_int,B: int > set_a,C: int > set_a] :
( ! [X: int] :
( ( member_int @ X @ A )
=> ( ord_less_eq_set_a @ ( B @ X ) @ ( C @ X ) ) )
=> ( ord_le943418215940126601_int_a @ ( piE_int_a @ A @ B ) @ ( piE_int_a @ A @ C ) ) ) ).
% PiE_mono
thf(fact_35_PiE__mono,axiom,
! [A: set_nat,B: nat > set_a,C: nat > set_a] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_set_a @ ( B @ X ) @ ( C @ X ) ) )
=> ( ord_le871467723717165285_nat_a @ ( piE_nat_a @ A @ B ) @ ( piE_nat_a @ A @ C ) ) ) ).
% PiE_mono
thf(fact_36_PiE__mono,axiom,
! [A: set_a,B: a > set_nat_a,C: a > set_nat_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_le871467723717165285_nat_a @ ( B @ X ) @ ( C @ X ) ) )
=> ( ord_le2508512696544544866_nat_a @ ( piE_a_nat_a @ A @ B ) @ ( piE_a_nat_a @ A @ C ) ) ) ).
% PiE_mono
thf(fact_37_PiE__mono,axiom,
! [A: set_o,B: $o > set_nat_a,C: $o > set_nat_a] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_le871467723717165285_nat_a @ ( B @ X ) @ ( C @ X ) ) )
=> ( ord_le1609037065912973128_nat_a @ ( piE_o_nat_a @ A @ B ) @ ( piE_o_nat_a @ A @ C ) ) ) ).
% PiE_mono
thf(fact_38_PiE__mono,axiom,
! [A: set_int,B: int > set_nat_a,C: int > set_nat_a] :
( ! [X: int] :
( ( member_int @ X @ A )
=> ( ord_le871467723717165285_nat_a @ ( B @ X ) @ ( C @ X ) ) )
=> ( ord_le6072919132162605296_nat_a @ ( piE_int_nat_a @ A @ B ) @ ( piE_int_nat_a @ A @ C ) ) ) ).
% PiE_mono
thf(fact_39_PiE__mono,axiom,
! [A: set_int,B: int > set_int,C: int > set_int] :
( ! [X: int] :
( ( member_int @ X @ A )
=> ( ord_less_eq_set_int @ ( B @ X ) @ ( C @ X ) ) )
=> ( ord_le8756421791413902349nt_int @ ( piE_int_int @ A @ B ) @ ( piE_int_int @ A @ C ) ) ) ).
% PiE_mono
thf(fact_40_PiE__mono,axiom,
! [A: set_nat,B: nat > set_nat_a,C: nat > set_nat_a] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_le871467723717165285_nat_a @ ( B @ X ) @ ( C @ X ) ) )
=> ( ord_le3343314562563788876_nat_a @ ( piE_nat_nat_a @ A @ B ) @ ( piE_nat_nat_a @ A @ C ) ) ) ).
% PiE_mono
thf(fact_41_PiE__mono,axiom,
! [A: set_nat_a,B: ( nat > a ) > set_a,C: ( nat > a ) > set_a] :
( ! [X: nat > a] :
( ( member_nat_a @ X @ A )
=> ( ord_less_eq_set_a @ ( B @ X ) @ ( C @ X ) ) )
=> ( ord_le3509452538356653652at_a_a @ ( piE_nat_a_a @ A @ B ) @ ( piE_nat_a_a @ A @ C ) ) ) ).
% PiE_mono
thf(fact_42_atLeastLessThan__subset__iff,axiom,
! [A2: nat,B2: nat,C2: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A2 @ B2 ) @ ( set_or4665077453230672383an_nat @ C2 @ D ) )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
| ( ( ord_less_eq_nat @ C2 @ A2 )
& ( ord_less_eq_nat @ B2 @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_43_atLeastLessThan__subset__iff,axiom,
! [A2: int,B2: int,C2: int,D: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ A2 @ B2 ) @ ( set_or4662586982721622107an_int @ C2 @ D ) )
=> ( ( ord_less_eq_int @ B2 @ A2 )
| ( ( ord_less_eq_int @ C2 @ A2 )
& ( ord_less_eq_int @ B2 @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_44_atLeastLessThan__subset__iff,axiom,
! [A2: real,B2: real,C2: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ A2 @ B2 ) @ ( set_or66887138388493659n_real @ C2 @ D ) )
=> ( ( ord_less_eq_real @ B2 @ A2 )
| ( ( ord_less_eq_real @ C2 @ A2 )
& ( ord_less_eq_real @ B2 @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_45_range__subsetD,axiom,
! [F: nat > $o,B: set_o,I: nat] :
( ( ord_less_eq_set_o @ ( image_nat_o @ F @ top_top_set_nat ) @ B )
=> ( member_o @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_46_range__subsetD,axiom,
! [F: nat > int,B: set_int,I: nat] :
( ( ord_less_eq_set_int @ ( image_nat_int @ F @ top_top_set_nat ) @ B )
=> ( member_int @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_47_range__subsetD,axiom,
! [F: nat > nat,B: set_nat,I: nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ top_top_set_nat ) @ B )
=> ( member_nat @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_48_range__subsetD,axiom,
! [F: int > $o,B: set_o,I: int] :
( ( ord_less_eq_set_o @ ( image_int_o @ F @ top_top_set_int ) @ B )
=> ( member_o @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_49_range__subsetD,axiom,
! [F: int > int,B: set_int,I: int] :
( ( ord_less_eq_set_int @ ( image_int_int @ F @ top_top_set_int ) @ B )
=> ( member_int @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_50_range__subsetD,axiom,
! [F: int > nat,B: set_nat,I: int] :
( ( ord_less_eq_set_nat @ ( image_int_nat @ F @ top_top_set_int ) @ B )
=> ( member_nat @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_51_range__subsetD,axiom,
! [F: a > $o,B: set_o,I: a] :
( ( ord_less_eq_set_o @ ( image_a_o @ F @ top_top_set_a ) @ B )
=> ( member_o @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_52_range__subsetD,axiom,
! [F: a > int,B: set_int,I: a] :
( ( ord_less_eq_set_int @ ( image_a_int @ F @ top_top_set_a ) @ B )
=> ( member_int @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_53_range__subsetD,axiom,
! [F: a > nat,B: set_nat,I: a] :
( ( ord_less_eq_set_nat @ ( image_a_nat @ F @ top_top_set_a ) @ B )
=> ( member_nat @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_54_range__subsetD,axiom,
! [F: nat > a,B: set_a,I: nat] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F @ top_top_set_nat ) @ B )
=> ( member_a @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_55_image__eqI,axiom,
! [B2: a,F: a > a,X3: a,A: set_a] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_a @ X3 @ A )
=> ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_56_image__eqI,axiom,
! [B2: $o,F: a > $o,X3: a,A: set_a] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_a @ X3 @ A )
=> ( member_o @ B2 @ ( image_a_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_57_image__eqI,axiom,
! [B2: int,F: a > int,X3: a,A: set_a] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_a @ X3 @ A )
=> ( member_int @ B2 @ ( image_a_int @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_58_image__eqI,axiom,
! [B2: nat,F: a > nat,X3: a,A: set_a] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_a @ X3 @ A )
=> ( member_nat @ B2 @ ( image_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_59_image__eqI,axiom,
! [B2: a,F: $o > a,X3: $o,A: set_o] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_o @ X3 @ A )
=> ( member_a @ B2 @ ( image_o_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_60_image__eqI,axiom,
! [B2: $o,F: $o > $o,X3: $o,A: set_o] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_o @ X3 @ A )
=> ( member_o @ B2 @ ( image_o_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_61_image__eqI,axiom,
! [B2: int,F: $o > int,X3: $o,A: set_o] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_o @ X3 @ A )
=> ( member_int @ B2 @ ( image_o_int @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_62_image__eqI,axiom,
! [B2: nat,F: $o > nat,X3: $o,A: set_o] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_o @ X3 @ A )
=> ( member_nat @ B2 @ ( image_o_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_63_image__eqI,axiom,
! [B2: a,F: int > a,X3: int,A: set_int] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_int @ X3 @ A )
=> ( member_a @ B2 @ ( image_int_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_64_image__eqI,axiom,
! [B2: $o,F: int > $o,X3: int,A: set_int] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_int @ X3 @ A )
=> ( member_o @ B2 @ ( image_int_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_65_UNIV__I,axiom,
! [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).
% UNIV_I
thf(fact_66_UNIV__I,axiom,
! [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% UNIV_I
thf(fact_67_UNIV__I,axiom,
! [X3: int] : ( member_int @ X3 @ top_top_set_int ) ).
% UNIV_I
thf(fact_68_UNIV__I,axiom,
! [X3: nat > int] : ( member_nat_int @ X3 @ top_top_set_nat_int ) ).
% UNIV_I
thf(fact_69_UNIV__I,axiom,
! [X3: nat > a] : ( member_nat_a @ X3 @ top_top_set_nat_a ) ).
% UNIV_I
thf(fact_70_UNIV__I,axiom,
! [X3: int > nat] : ( member_int_nat @ X3 @ top_top_set_int_nat ) ).
% UNIV_I
thf(fact_71_UNIV__I,axiom,
! [X3: int > int] : ( member_int_int @ X3 @ top_top_set_int_int ) ).
% UNIV_I
thf(fact_72_UNIV__I,axiom,
! [X3: a] : ( member_a @ X3 @ top_top_set_a ) ).
% UNIV_I
thf(fact_73_subset__antisym,axiom,
! [A: set_nat_a,B: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A @ B )
=> ( ( ord_le871467723717165285_nat_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_74_subset__antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_75_subset__antisym,axiom,
! [A: set_o_a,B: set_o_a] :
( ( ord_less_eq_set_o_a @ A @ B )
=> ( ( ord_less_eq_set_o_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_76_subset__antisym,axiom,
! [A: set_int_nat_a,B: set_int_nat_a] :
( ( ord_le6072919132162605296_nat_a @ A @ B )
=> ( ( ord_le6072919132162605296_nat_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_77_subset__antisym,axiom,
! [A: set_int_a,B: set_int_a] :
( ( ord_le943418215940126601_int_a @ A @ B )
=> ( ( ord_le943418215940126601_int_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_78_subset__antisym,axiom,
! [A: set_a_nat_a,B: set_a_nat_a] :
( ( ord_le2508512696544544866_nat_a @ A @ B )
=> ( ( ord_le2508512696544544866_nat_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_79_subset__antisym,axiom,
! [A: set_a_a,B: set_a_a] :
( ( ord_less_eq_set_a_a @ A @ B )
=> ( ( ord_less_eq_set_a_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_80_subsetI,axiom,
! [A: set_o,B: set_o] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( member_o @ X @ B ) )
=> ( ord_less_eq_set_o @ A @ B ) ) ).
% subsetI
thf(fact_81_subsetI,axiom,
! [A: set_int,B: set_int] :
( ! [X: int] :
( ( member_int @ X @ A )
=> ( member_int @ X @ B ) )
=> ( ord_less_eq_set_int @ A @ B ) ) ).
% subsetI
thf(fact_82_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ X @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_83_subsetI,axiom,
! [A: set_nat_a,B: set_nat_a] :
( ! [X: nat > a] :
( ( member_nat_a @ X @ A )
=> ( member_nat_a @ X @ B ) )
=> ( ord_le871467723717165285_nat_a @ A @ B ) ) ).
% subsetI
thf(fact_84_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_a @ X @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_85_subsetI,axiom,
! [A: set_o_a,B: set_o_a] :
( ! [X: $o > a] :
( ( member_o_a @ X @ A )
=> ( member_o_a @ X @ B ) )
=> ( ord_less_eq_set_o_a @ A @ B ) ) ).
% subsetI
thf(fact_86_subsetI,axiom,
! [A: set_int_nat_a,B: set_int_nat_a] :
( ! [X: int > nat > a] :
( ( member_int_nat_a @ X @ A )
=> ( member_int_nat_a @ X @ B ) )
=> ( ord_le6072919132162605296_nat_a @ A @ B ) ) ).
% subsetI
thf(fact_87_subsetI,axiom,
! [A: set_int_a,B: set_int_a] :
( ! [X: int > a] :
( ( member_int_a @ X @ A )
=> ( member_int_a @ X @ B ) )
=> ( ord_le943418215940126601_int_a @ A @ B ) ) ).
% subsetI
thf(fact_88_subsetI,axiom,
! [A: set_a_nat_a,B: set_a_nat_a] :
( ! [X: a > nat > a] :
( ( member_a_nat_a @ X @ A )
=> ( member_a_nat_a @ X @ B ) )
=> ( ord_le2508512696544544866_nat_a @ A @ B ) ) ).
% subsetI
thf(fact_89_subsetI,axiom,
! [A: set_a_a,B: set_a_a] :
( ! [X: a > a] :
( ( member_a_a @ X @ A )
=> ( member_a_a @ X @ B ) )
=> ( ord_less_eq_set_a_a @ A @ B ) ) ).
% subsetI
thf(fact_90_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_91_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_92_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_93_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_94_atLeastLessThan__iff,axiom,
! [I: $o,L: $o,U: $o] :
( ( member_o @ I @ ( set_or7139685690850216873Than_o @ L @ U ) )
= ( ( ord_less_eq_o @ L @ I )
& ( ord_less_o @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_95_atLeastLessThan__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_96_atLeastLessThan__iff,axiom,
! [I: int,L: int,U: int] :
( ( member_int @ I @ ( set_or4662586982721622107an_int @ L @ U ) )
= ( ( ord_less_eq_int @ L @ I )
& ( ord_less_int @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_97_atLeastLessThan__iff,axiom,
! [I: real,L: real,U: real] :
( ( member_real @ I @ ( set_or66887138388493659n_real @ L @ U ) )
= ( ( ord_less_eq_real @ L @ I )
& ( ord_less_real @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_98_atLeastLessThan__iff,axiom,
! [I: set_a,L: set_a,U: set_a] :
( ( member_set_a @ I @ ( set_or2348907005316661231_set_a @ L @ U ) )
= ( ( ord_less_eq_set_a @ L @ I )
& ( ord_less_set_a @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_99_atLeastLessThan__iff,axiom,
! [I: set_nat_a,L: set_nat_a,U: set_nat_a] :
( ( member_set_nat_a @ I @ ( set_or8677123885700112214_nat_a @ L @ U ) )
= ( ( ord_le871467723717165285_nat_a @ L @ I )
& ( ord_less_set_nat_a @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_100_atLeastLessThan__iff,axiom,
! [I: set_o_a,L: set_o_a,U: set_o_a] :
( ( member_set_o_a @ I @ ( set_or1317968236485584176et_o_a @ L @ U ) )
= ( ( ord_less_eq_set_o_a @ L @ I )
& ( ord_less_set_o_a @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_101_atLeastLessThan__iff,axiom,
! [I: set_int_a,L: set_int_a,U: set_int_a] :
( ( member_set_int_a @ I @ ( set_or8749074377923073530_int_a @ L @ U ) )
= ( ( ord_le943418215940126601_int_a @ L @ I )
& ( ord_less_set_int_a @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_102_atLeastLessThan__iff,axiom,
! [I: set_a_a,L: set_a_a,U: set_a_a] :
( ( member_set_a_a @ I @ ( set_or5743114581318135242et_a_a @ L @ U ) )
= ( ( ord_less_eq_set_a_a @ L @ I )
& ( ord_less_set_a_a @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_103_atLeastLessThan__iff,axiom,
! [I: set_int_nat_a,L: set_int_nat_a,U: set_int_nat_a] :
( ( member_set_int_nat_a @ I @ ( set_or7189319678020346337_nat_a @ L @ U ) )
= ( ( ord_le6072919132162605296_nat_a @ L @ I )
& ( ord_le2026899869173067772_nat_a @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_104_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_105_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_106_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_107_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_108_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_109_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_110_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_111_linorder__neqE__nat,axiom,
! [X3: nat,Y2: nat] :
( ( X3 != Y2 )
=> ( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ Y2 @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_112_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_113_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_114_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_115_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_116_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_117_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_118_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_119_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_120_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_121_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_122_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_123_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_124_mem__Collect__eq,axiom,
! [A2: $o,P: $o > $o] :
( ( member_o @ A2 @ ( collect_o @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_125_mem__Collect__eq,axiom,
! [A2: int,P: int > $o] :
( ( member_int @ A2 @ ( collect_int @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_126_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_127_mem__Collect__eq,axiom,
! [A2: nat > a,P: ( nat > a ) > $o] :
( ( member_nat_a @ A2 @ ( collect_nat_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_128_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_129_Collect__mem__eq,axiom,
! [A: set_o] :
( ( collect_o
@ ^ [X2: $o] : ( member_o @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_130_Collect__mem__eq,axiom,
! [A: set_int] :
( ( collect_int
@ ^ [X2: int] : ( member_int @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_131_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_132_Collect__mem__eq,axiom,
! [A: set_nat_a] :
( ( collect_nat_a
@ ^ [X2: nat > a] : ( member_nat_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_133_Collect__cong,axiom,
! [P: ( nat > a ) > $o,Q: ( nat > a ) > $o] :
( ! [X: nat > a] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_nat_a @ P )
= ( collect_nat_a @ Q ) ) ) ).
% Collect_cong
thf(fact_134_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_135_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_136_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_137_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_138_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_139_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_140_atLeastLessThan__eq__iff,axiom,
! [A2: nat,B2: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ( ( set_or4665077453230672383an_nat @ A2 @ B2 )
= ( set_or4665077453230672383an_nat @ C2 @ D ) )
= ( ( A2 = C2 )
& ( B2 = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_141_atLeastLessThan__eq__iff,axiom,
! [A2: int,B2: int,C2: int,D: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ C2 @ D )
=> ( ( ( set_or4662586982721622107an_int @ A2 @ B2 )
= ( set_or4662586982721622107an_int @ C2 @ D ) )
= ( ( A2 = C2 )
& ( B2 = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_142_atLeastLessThan__eq__iff,axiom,
! [A2: real,B2: real,C2: real,D: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ C2 @ D )
=> ( ( ( set_or66887138388493659n_real @ A2 @ B2 )
= ( set_or66887138388493659n_real @ C2 @ D ) )
= ( ( A2 = C2 )
& ( B2 = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_143_Ico__eq__Ico,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H )
= ( set_or4665077453230672383an_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_nat @ L @ H )
& ~ ( ord_less_nat @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_144_Ico__eq__Ico,axiom,
! [L: int,H: int,L2: int,H2: int] :
( ( ( set_or4662586982721622107an_int @ L @ H )
= ( set_or4662586982721622107an_int @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_int @ L @ H )
& ~ ( ord_less_int @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_145_Ico__eq__Ico,axiom,
! [L: real,H: real,L2: real,H2: real] :
( ( ( set_or66887138388493659n_real @ L @ H )
= ( set_or66887138388493659n_real @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_real @ L @ H )
& ~ ( ord_less_real @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_146_atLeastLessThan__inj_I1_J,axiom,
! [A2: nat,B2: nat,C2: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A2 @ B2 )
= ( set_or4665077453230672383an_nat @ C2 @ D ) )
=> ( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( A2 = C2 ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_147_atLeastLessThan__inj_I1_J,axiom,
! [A2: int,B2: int,C2: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A2 @ B2 )
= ( set_or4662586982721622107an_int @ C2 @ D ) )
=> ( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ C2 @ D )
=> ( A2 = C2 ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_148_atLeastLessThan__inj_I1_J,axiom,
! [A2: real,B2: real,C2: real,D: real] :
( ( ( set_or66887138388493659n_real @ A2 @ B2 )
= ( set_or66887138388493659n_real @ C2 @ D ) )
=> ( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ C2 @ D )
=> ( A2 = C2 ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_149_atLeastLessThan__inj_I2_J,axiom,
! [A2: nat,B2: nat,C2: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A2 @ B2 )
= ( set_or4665077453230672383an_nat @ C2 @ D ) )
=> ( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( B2 = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_150_atLeastLessThan__inj_I2_J,axiom,
! [A2: int,B2: int,C2: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A2 @ B2 )
= ( set_or4662586982721622107an_int @ C2 @ D ) )
=> ( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ C2 @ D )
=> ( B2 = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_151_atLeastLessThan__inj_I2_J,axiom,
! [A2: real,B2: real,C2: real,D: real] :
( ( ( set_or66887138388493659n_real @ A2 @ B2 )
= ( set_or66887138388493659n_real @ C2 @ D ) )
=> ( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ C2 @ D )
=> ( B2 = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_152_less__eq__set__def,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B3: set_o] :
( ord_less_eq_o_o
@ ^ [X2: $o] : ( member_o @ X2 @ A3 )
@ ^ [X2: $o] : ( member_o @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_153_less__eq__set__def,axiom,
( ord_less_eq_set_int
= ( ^ [A3: set_int,B3: set_int] :
( ord_less_eq_int_o
@ ^ [X2: int] : ( member_int @ X2 @ A3 )
@ ^ [X2: int] : ( member_int @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_154_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_155_less__eq__set__def,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A3: set_nat_a,B3: set_nat_a] :
( ord_less_eq_nat_a_o
@ ^ [X2: nat > a] : ( member_nat_a @ X2 @ A3 )
@ ^ [X2: nat > a] : ( member_nat_a @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_156_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ord_less_eq_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A3 )
@ ^ [X2: a] : ( member_a @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_157_less__eq__set__def,axiom,
( ord_less_eq_set_o_a
= ( ^ [A3: set_o_a,B3: set_o_a] :
( ord_less_eq_o_a_o
@ ^ [X2: $o > a] : ( member_o_a @ X2 @ A3 )
@ ^ [X2: $o > a] : ( member_o_a @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_158_less__eq__set__def,axiom,
( ord_le6072919132162605296_nat_a
= ( ^ [A3: set_int_nat_a,B3: set_int_nat_a] :
( ord_le998279727272312301at_a_o
@ ^ [X2: int > nat > a] : ( member_int_nat_a @ X2 @ A3 )
@ ^ [X2: int > nat > a] : ( member_int_nat_a @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_159_less__eq__set__def,axiom,
( ord_le943418215940126601_int_a
= ( ^ [A3: set_int_a,B3: set_int_a] :
( ord_less_eq_int_a_o
@ ^ [X2: int > a] : ( member_int_a @ X2 @ A3 )
@ ^ [X2: int > a] : ( member_int_a @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_160_less__eq__set__def,axiom,
( ord_le2508512696544544866_nat_a
= ( ^ [A3: set_a_nat_a,B3: set_a_nat_a] :
( ord_le6051534669759718563at_a_o
@ ^ [X2: a > nat > a] : ( member_a_nat_a @ X2 @ A3 )
@ ^ [X2: a > nat > a] : ( member_a_nat_a @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_161_less__eq__set__def,axiom,
( ord_less_eq_set_a_a
= ( ^ [A3: set_a_a,B3: set_a_a] :
( ord_less_eq_a_a_o
@ ^ [X2: a > a] : ( member_a_a @ X2 @ A3 )
@ ^ [X2: a > a] : ( member_a_a @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_162_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_163_zero__reorient,axiom,
! [X3: nat] :
( ( zero_zero_nat = X3 )
= ( X3 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_164_zero__reorient,axiom,
! [X3: int] :
( ( zero_zero_int = X3 )
= ( X3 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_165_zero__reorient,axiom,
! [X3: real] :
( ( zero_zero_real = X3 )
= ( X3 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_166_Collect__mono__iff,axiom,
! [P: ( nat > a ) > $o,Q: ( nat > a ) > $o] :
( ( ord_le871467723717165285_nat_a @ ( collect_nat_a @ P ) @ ( collect_nat_a @ Q ) )
= ( ! [X2: nat > a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_167_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_168_Collect__mono__iff,axiom,
! [P: ( $o > a ) > $o,Q: ( $o > a ) > $o] :
( ( ord_less_eq_set_o_a @ ( collect_o_a @ P ) @ ( collect_o_a @ Q ) )
= ( ! [X2: $o > a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_169_Collect__mono__iff,axiom,
! [P: ( int > nat > a ) > $o,Q: ( int > nat > a ) > $o] :
( ( ord_le6072919132162605296_nat_a @ ( collect_int_nat_a @ P ) @ ( collect_int_nat_a @ Q ) )
= ( ! [X2: int > nat > a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_170_Collect__mono__iff,axiom,
! [P: ( int > a ) > $o,Q: ( int > a ) > $o] :
( ( ord_le943418215940126601_int_a @ ( collect_int_a @ P ) @ ( collect_int_a @ Q ) )
= ( ! [X2: int > a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_171_Collect__mono__iff,axiom,
! [P: ( a > nat > a ) > $o,Q: ( a > nat > a ) > $o] :
( ( ord_le2508512696544544866_nat_a @ ( collect_a_nat_a @ P ) @ ( collect_a_nat_a @ Q ) )
= ( ! [X2: a > nat > a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_172_Collect__mono__iff,axiom,
! [P: ( a > a ) > $o,Q: ( a > a ) > $o] :
( ( ord_less_eq_set_a_a @ ( collect_a_a @ P ) @ ( collect_a_a @ Q ) )
= ( ! [X2: a > a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_173_set__eq__subset,axiom,
( ( ^ [Y3: set_nat_a,Z: set_nat_a] : ( Y3 = Z ) )
= ( ^ [A3: set_nat_a,B3: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A3 @ B3 )
& ( ord_le871467723717165285_nat_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_174_set__eq__subset,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_175_set__eq__subset,axiom,
( ( ^ [Y3: set_o_a,Z: set_o_a] : ( Y3 = Z ) )
= ( ^ [A3: set_o_a,B3: set_o_a] :
( ( ord_less_eq_set_o_a @ A3 @ B3 )
& ( ord_less_eq_set_o_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_176_set__eq__subset,axiom,
( ( ^ [Y3: set_int_nat_a,Z: set_int_nat_a] : ( Y3 = Z ) )
= ( ^ [A3: set_int_nat_a,B3: set_int_nat_a] :
( ( ord_le6072919132162605296_nat_a @ A3 @ B3 )
& ( ord_le6072919132162605296_nat_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_177_set__eq__subset,axiom,
( ( ^ [Y3: set_int_a,Z: set_int_a] : ( Y3 = Z ) )
= ( ^ [A3: set_int_a,B3: set_int_a] :
( ( ord_le943418215940126601_int_a @ A3 @ B3 )
& ( ord_le943418215940126601_int_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_178_set__eq__subset,axiom,
( ( ^ [Y3: set_a_nat_a,Z: set_a_nat_a] : ( Y3 = Z ) )
= ( ^ [A3: set_a_nat_a,B3: set_a_nat_a] :
( ( ord_le2508512696544544866_nat_a @ A3 @ B3 )
& ( ord_le2508512696544544866_nat_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_179_set__eq__subset,axiom,
( ( ^ [Y3: set_a_a,Z: set_a_a] : ( Y3 = Z ) )
= ( ^ [A3: set_a_a,B3: set_a_a] :
( ( ord_less_eq_set_a_a @ A3 @ B3 )
& ( ord_less_eq_set_a_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_180_subset__trans,axiom,
! [A: set_nat_a,B: set_nat_a,C: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A @ B )
=> ( ( ord_le871467723717165285_nat_a @ B @ C )
=> ( ord_le871467723717165285_nat_a @ A @ C ) ) ) ).
% subset_trans
thf(fact_181_subset__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% subset_trans
thf(fact_182_subset__trans,axiom,
! [A: set_o_a,B: set_o_a,C: set_o_a] :
( ( ord_less_eq_set_o_a @ A @ B )
=> ( ( ord_less_eq_set_o_a @ B @ C )
=> ( ord_less_eq_set_o_a @ A @ C ) ) ) ).
% subset_trans
thf(fact_183_subset__trans,axiom,
! [A: set_int_nat_a,B: set_int_nat_a,C: set_int_nat_a] :
( ( ord_le6072919132162605296_nat_a @ A @ B )
=> ( ( ord_le6072919132162605296_nat_a @ B @ C )
=> ( ord_le6072919132162605296_nat_a @ A @ C ) ) ) ).
% subset_trans
thf(fact_184_subset__trans,axiom,
! [A: set_int_a,B: set_int_a,C: set_int_a] :
( ( ord_le943418215940126601_int_a @ A @ B )
=> ( ( ord_le943418215940126601_int_a @ B @ C )
=> ( ord_le943418215940126601_int_a @ A @ C ) ) ) ).
% subset_trans
thf(fact_185_subset__trans,axiom,
! [A: set_a_nat_a,B: set_a_nat_a,C: set_a_nat_a] :
( ( ord_le2508512696544544866_nat_a @ A @ B )
=> ( ( ord_le2508512696544544866_nat_a @ B @ C )
=> ( ord_le2508512696544544866_nat_a @ A @ C ) ) ) ).
% subset_trans
thf(fact_186_subset__trans,axiom,
! [A: set_a_a,B: set_a_a,C: set_a_a] :
( ( ord_less_eq_set_a_a @ A @ B )
=> ( ( ord_less_eq_set_a_a @ B @ C )
=> ( ord_less_eq_set_a_a @ A @ C ) ) ) ).
% subset_trans
thf(fact_187_Collect__mono,axiom,
! [P: ( nat > a ) > $o,Q: ( nat > a ) > $o] :
( ! [X: nat > a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le871467723717165285_nat_a @ ( collect_nat_a @ P ) @ ( collect_nat_a @ Q ) ) ) ).
% Collect_mono
thf(fact_188_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_189_Collect__mono,axiom,
! [P: ( $o > a ) > $o,Q: ( $o > a ) > $o] :
( ! [X: $o > a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_o_a @ ( collect_o_a @ P ) @ ( collect_o_a @ Q ) ) ) ).
% Collect_mono
thf(fact_190_Collect__mono,axiom,
! [P: ( int > nat > a ) > $o,Q: ( int > nat > a ) > $o] :
( ! [X: int > nat > a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le6072919132162605296_nat_a @ ( collect_int_nat_a @ P ) @ ( collect_int_nat_a @ Q ) ) ) ).
% Collect_mono
thf(fact_191_Collect__mono,axiom,
! [P: ( int > a ) > $o,Q: ( int > a ) > $o] :
( ! [X: int > a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le943418215940126601_int_a @ ( collect_int_a @ P ) @ ( collect_int_a @ Q ) ) ) ).
% Collect_mono
thf(fact_192_Collect__mono,axiom,
! [P: ( a > nat > a ) > $o,Q: ( a > nat > a ) > $o] :
( ! [X: a > nat > a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le2508512696544544866_nat_a @ ( collect_a_nat_a @ P ) @ ( collect_a_nat_a @ Q ) ) ) ).
% Collect_mono
thf(fact_193_Collect__mono,axiom,
! [P: ( a > a ) > $o,Q: ( a > a ) > $o] :
( ! [X: a > a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_a_a @ ( collect_a_a @ P ) @ ( collect_a_a @ Q ) ) ) ).
% Collect_mono
thf(fact_194_subset__refl,axiom,
! [A: set_nat_a] : ( ord_le871467723717165285_nat_a @ A @ A ) ).
% subset_refl
thf(fact_195_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_196_subset__refl,axiom,
! [A: set_o_a] : ( ord_less_eq_set_o_a @ A @ A ) ).
% subset_refl
thf(fact_197_subset__refl,axiom,
! [A: set_int_nat_a] : ( ord_le6072919132162605296_nat_a @ A @ A ) ).
% subset_refl
thf(fact_198_subset__refl,axiom,
! [A: set_int_a] : ( ord_le943418215940126601_int_a @ A @ A ) ).
% subset_refl
thf(fact_199_subset__refl,axiom,
! [A: set_a_nat_a] : ( ord_le2508512696544544866_nat_a @ A @ A ) ).
% subset_refl
thf(fact_200_subset__refl,axiom,
! [A: set_a_a] : ( ord_less_eq_set_a_a @ A @ A ) ).
% subset_refl
thf(fact_201_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B3: set_o] :
! [T2: $o] :
( ( member_o @ T2 @ A3 )
=> ( member_o @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_202_subset__iff,axiom,
( ord_less_eq_set_int
= ( ^ [A3: set_int,B3: set_int] :
! [T2: int] :
( ( member_int @ T2 @ A3 )
=> ( member_int @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_203_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A3 )
=> ( member_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_204_subset__iff,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A3: set_nat_a,B3: set_nat_a] :
! [T2: nat > a] :
( ( member_nat_a @ T2 @ A3 )
=> ( member_nat_a @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_205_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A3 )
=> ( member_a @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_206_subset__iff,axiom,
( ord_less_eq_set_o_a
= ( ^ [A3: set_o_a,B3: set_o_a] :
! [T2: $o > a] :
( ( member_o_a @ T2 @ A3 )
=> ( member_o_a @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_207_subset__iff,axiom,
( ord_le6072919132162605296_nat_a
= ( ^ [A3: set_int_nat_a,B3: set_int_nat_a] :
! [T2: int > nat > a] :
( ( member_int_nat_a @ T2 @ A3 )
=> ( member_int_nat_a @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_208_subset__iff,axiom,
( ord_le943418215940126601_int_a
= ( ^ [A3: set_int_a,B3: set_int_a] :
! [T2: int > a] :
( ( member_int_a @ T2 @ A3 )
=> ( member_int_a @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_209_subset__iff,axiom,
( ord_le2508512696544544866_nat_a
= ( ^ [A3: set_a_nat_a,B3: set_a_nat_a] :
! [T2: a > nat > a] :
( ( member_a_nat_a @ T2 @ A3 )
=> ( member_a_nat_a @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_210_subset__iff,axiom,
( ord_less_eq_set_a_a
= ( ^ [A3: set_a_a,B3: set_a_a] :
! [T2: a > a] :
( ( member_a_a @ T2 @ A3 )
=> ( member_a_a @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_211_equalityD2,axiom,
! [A: set_nat_a,B: set_nat_a] :
( ( A = B )
=> ( ord_le871467723717165285_nat_a @ B @ A ) ) ).
% equalityD2
thf(fact_212_equalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_213_equalityD2,axiom,
! [A: set_o_a,B: set_o_a] :
( ( A = B )
=> ( ord_less_eq_set_o_a @ B @ A ) ) ).
% equalityD2
thf(fact_214_equalityD2,axiom,
! [A: set_int_nat_a,B: set_int_nat_a] :
( ( A = B )
=> ( ord_le6072919132162605296_nat_a @ B @ A ) ) ).
% equalityD2
thf(fact_215_equalityD2,axiom,
! [A: set_int_a,B: set_int_a] :
( ( A = B )
=> ( ord_le943418215940126601_int_a @ B @ A ) ) ).
% equalityD2
thf(fact_216_equalityD2,axiom,
! [A: set_a_nat_a,B: set_a_nat_a] :
( ( A = B )
=> ( ord_le2508512696544544866_nat_a @ B @ A ) ) ).
% equalityD2
thf(fact_217_equalityD2,axiom,
! [A: set_a_a,B: set_a_a] :
( ( A = B )
=> ( ord_less_eq_set_a_a @ B @ A ) ) ).
% equalityD2
thf(fact_218_equalityD1,axiom,
! [A: set_nat_a,B: set_nat_a] :
( ( A = B )
=> ( ord_le871467723717165285_nat_a @ A @ B ) ) ).
% equalityD1
thf(fact_219_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_220_equalityD1,axiom,
! [A: set_o_a,B: set_o_a] :
( ( A = B )
=> ( ord_less_eq_set_o_a @ A @ B ) ) ).
% equalityD1
thf(fact_221_equalityD1,axiom,
! [A: set_int_nat_a,B: set_int_nat_a] :
( ( A = B )
=> ( ord_le6072919132162605296_nat_a @ A @ B ) ) ).
% equalityD1
thf(fact_222_equalityD1,axiom,
! [A: set_int_a,B: set_int_a] :
( ( A = B )
=> ( ord_le943418215940126601_int_a @ A @ B ) ) ).
% equalityD1
thf(fact_223_equalityD1,axiom,
! [A: set_a_nat_a,B: set_a_nat_a] :
( ( A = B )
=> ( ord_le2508512696544544866_nat_a @ A @ B ) ) ).
% equalityD1
thf(fact_224_equalityD1,axiom,
! [A: set_a_a,B: set_a_a] :
( ( A = B )
=> ( ord_less_eq_set_a_a @ A @ B ) ) ).
% equalityD1
thf(fact_225_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B3: set_o] :
! [X2: $o] :
( ( member_o @ X2 @ A3 )
=> ( member_o @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_226_subset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A3: set_int,B3: set_int] :
! [X2: int] :
( ( member_int @ X2 @ A3 )
=> ( member_int @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_227_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( member_nat @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_228_subset__eq,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A3: set_nat_a,B3: set_nat_a] :
! [X2: nat > a] :
( ( member_nat_a @ X2 @ A3 )
=> ( member_nat_a @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_229_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( member_a @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_230_subset__eq,axiom,
( ord_less_eq_set_o_a
= ( ^ [A3: set_o_a,B3: set_o_a] :
! [X2: $o > a] :
( ( member_o_a @ X2 @ A3 )
=> ( member_o_a @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_231_subset__eq,axiom,
( ord_le6072919132162605296_nat_a
= ( ^ [A3: set_int_nat_a,B3: set_int_nat_a] :
! [X2: int > nat > a] :
( ( member_int_nat_a @ X2 @ A3 )
=> ( member_int_nat_a @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_232_subset__eq,axiom,
( ord_le943418215940126601_int_a
= ( ^ [A3: set_int_a,B3: set_int_a] :
! [X2: int > a] :
( ( member_int_a @ X2 @ A3 )
=> ( member_int_a @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_233_subset__eq,axiom,
( ord_le2508512696544544866_nat_a
= ( ^ [A3: set_a_nat_a,B3: set_a_nat_a] :
! [X2: a > nat > a] :
( ( member_a_nat_a @ X2 @ A3 )
=> ( member_a_nat_a @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_234_subset__eq,axiom,
( ord_less_eq_set_a_a
= ( ^ [A3: set_a_a,B3: set_a_a] :
! [X2: a > a] :
( ( member_a_a @ X2 @ A3 )
=> ( member_a_a @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_235_equalityE,axiom,
! [A: set_nat_a,B: set_nat_a] :
( ( A = B )
=> ~ ( ( ord_le871467723717165285_nat_a @ A @ B )
=> ~ ( ord_le871467723717165285_nat_a @ B @ A ) ) ) ).
% equalityE
thf(fact_236_equalityE,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_237_equalityE,axiom,
! [A: set_o_a,B: set_o_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_o_a @ A @ B )
=> ~ ( ord_less_eq_set_o_a @ B @ A ) ) ) ).
% equalityE
thf(fact_238_equalityE,axiom,
! [A: set_int_nat_a,B: set_int_nat_a] :
( ( A = B )
=> ~ ( ( ord_le6072919132162605296_nat_a @ A @ B )
=> ~ ( ord_le6072919132162605296_nat_a @ B @ A ) ) ) ).
% equalityE
thf(fact_239_equalityE,axiom,
! [A: set_int_a,B: set_int_a] :
( ( A = B )
=> ~ ( ( ord_le943418215940126601_int_a @ A @ B )
=> ~ ( ord_le943418215940126601_int_a @ B @ A ) ) ) ).
% equalityE
thf(fact_240_equalityE,axiom,
! [A: set_a_nat_a,B: set_a_nat_a] :
( ( A = B )
=> ~ ( ( ord_le2508512696544544866_nat_a @ A @ B )
=> ~ ( ord_le2508512696544544866_nat_a @ B @ A ) ) ) ).
% equalityE
thf(fact_241_equalityE,axiom,
! [A: set_a_a,B: set_a_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a_a @ A @ B )
=> ~ ( ord_less_eq_set_a_a @ B @ A ) ) ) ).
% equalityE
thf(fact_242_subsetD,axiom,
! [A: set_o,B: set_o,C2: $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_o @ C2 @ A )
=> ( member_o @ C2 @ B ) ) ) ).
% subsetD
thf(fact_243_subsetD,axiom,
! [A: set_int,B: set_int,C2: int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( member_int @ C2 @ A )
=> ( member_int @ C2 @ B ) ) ) ).
% subsetD
thf(fact_244_subsetD,axiom,
! [A: set_nat,B: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_245_subsetD,axiom,
! [A: set_nat_a,B: set_nat_a,C2: nat > a] :
( ( ord_le871467723717165285_nat_a @ A @ B )
=> ( ( member_nat_a @ C2 @ A )
=> ( member_nat_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_246_subsetD,axiom,
! [A: set_a,B: set_a,C2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_247_subsetD,axiom,
! [A: set_o_a,B: set_o_a,C2: $o > a] :
( ( ord_less_eq_set_o_a @ A @ B )
=> ( ( member_o_a @ C2 @ A )
=> ( member_o_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_248_subsetD,axiom,
! [A: set_int_nat_a,B: set_int_nat_a,C2: int > nat > a] :
( ( ord_le6072919132162605296_nat_a @ A @ B )
=> ( ( member_int_nat_a @ C2 @ A )
=> ( member_int_nat_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_249_subsetD,axiom,
! [A: set_int_a,B: set_int_a,C2: int > a] :
( ( ord_le943418215940126601_int_a @ A @ B )
=> ( ( member_int_a @ C2 @ A )
=> ( member_int_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_250_subsetD,axiom,
! [A: set_a_nat_a,B: set_a_nat_a,C2: a > nat > a] :
( ( ord_le2508512696544544866_nat_a @ A @ B )
=> ( ( member_a_nat_a @ C2 @ A )
=> ( member_a_nat_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_251_subsetD,axiom,
! [A: set_a_a,B: set_a_a,C2: a > a] :
( ( ord_less_eq_set_a_a @ A @ B )
=> ( ( member_a_a @ C2 @ A )
=> ( member_a_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_252_in__mono,axiom,
! [A: set_o,B: set_o,X3: $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_o @ X3 @ A )
=> ( member_o @ X3 @ B ) ) ) ).
% in_mono
thf(fact_253_in__mono,axiom,
! [A: set_int,B: set_int,X3: int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( member_int @ X3 @ A )
=> ( member_int @ X3 @ B ) ) ) ).
% in_mono
thf(fact_254_in__mono,axiom,
! [A: set_nat,B: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_255_in__mono,axiom,
! [A: set_nat_a,B: set_nat_a,X3: nat > a] :
( ( ord_le871467723717165285_nat_a @ A @ B )
=> ( ( member_nat_a @ X3 @ A )
=> ( member_nat_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_256_in__mono,axiom,
! [A: set_a,B: set_a,X3: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X3 @ A )
=> ( member_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_257_in__mono,axiom,
! [A: set_o_a,B: set_o_a,X3: $o > a] :
( ( ord_less_eq_set_o_a @ A @ B )
=> ( ( member_o_a @ X3 @ A )
=> ( member_o_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_258_in__mono,axiom,
! [A: set_int_nat_a,B: set_int_nat_a,X3: int > nat > a] :
( ( ord_le6072919132162605296_nat_a @ A @ B )
=> ( ( member_int_nat_a @ X3 @ A )
=> ( member_int_nat_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_259_in__mono,axiom,
! [A: set_int_a,B: set_int_a,X3: int > a] :
( ( ord_le943418215940126601_int_a @ A @ B )
=> ( ( member_int_a @ X3 @ A )
=> ( member_int_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_260_in__mono,axiom,
! [A: set_a_nat_a,B: set_a_nat_a,X3: a > nat > a] :
( ( ord_le2508512696544544866_nat_a @ A @ B )
=> ( ( member_a_nat_a @ X3 @ A )
=> ( member_a_nat_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_261_in__mono,axiom,
! [A: set_a_a,B: set_a_a,X3: a > a] :
( ( ord_less_eq_set_a_a @ A @ B )
=> ( ( member_a_a @ X3 @ A )
=> ( member_a_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_262_UNIV__witness,axiom,
? [X: $o] : ( member_o @ X @ top_top_set_o ) ).
% UNIV_witness
thf(fact_263_UNIV__witness,axiom,
? [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_264_UNIV__witness,axiom,
? [X: int] : ( member_int @ X @ top_top_set_int ) ).
% UNIV_witness
thf(fact_265_UNIV__witness,axiom,
? [X: nat > int] : ( member_nat_int @ X @ top_top_set_nat_int ) ).
% UNIV_witness
thf(fact_266_UNIV__witness,axiom,
? [X: nat > a] : ( member_nat_a @ X @ top_top_set_nat_a ) ).
% UNIV_witness
thf(fact_267_UNIV__witness,axiom,
? [X: int > nat] : ( member_int_nat @ X @ top_top_set_int_nat ) ).
% UNIV_witness
thf(fact_268_UNIV__witness,axiom,
? [X: int > int] : ( member_int_int @ X @ top_top_set_int_int ) ).
% UNIV_witness
thf(fact_269_UNIV__witness,axiom,
? [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_witness
thf(fact_270_UNIV__eq__I,axiom,
! [A: set_o] :
( ! [X: $o] : ( member_o @ X @ A )
=> ( top_top_set_o = A ) ) ).
% UNIV_eq_I
thf(fact_271_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X: nat] : ( member_nat @ X @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_272_UNIV__eq__I,axiom,
! [A: set_int] :
( ! [X: int] : ( member_int @ X @ A )
=> ( top_top_set_int = A ) ) ).
% UNIV_eq_I
thf(fact_273_UNIV__eq__I,axiom,
! [A: set_nat_int] :
( ! [X: nat > int] : ( member_nat_int @ X @ A )
=> ( top_top_set_nat_int = A ) ) ).
% UNIV_eq_I
thf(fact_274_UNIV__eq__I,axiom,
! [A: set_nat_a] :
( ! [X: nat > a] : ( member_nat_a @ X @ A )
=> ( top_top_set_nat_a = A ) ) ).
% UNIV_eq_I
thf(fact_275_UNIV__eq__I,axiom,
! [A: set_int_nat] :
( ! [X: int > nat] : ( member_int_nat @ X @ A )
=> ( top_top_set_int_nat = A ) ) ).
% UNIV_eq_I
thf(fact_276_UNIV__eq__I,axiom,
! [A: set_int_int] :
( ! [X: int > int] : ( member_int_int @ X @ A )
=> ( top_top_set_int_int = A ) ) ).
% UNIV_eq_I
thf(fact_277_UNIV__eq__I,axiom,
! [A: set_a] :
( ! [X: a] : ( member_a @ X @ A )
=> ( top_top_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_278_rev__image__eqI,axiom,
! [X3: a,A: set_a,B2: a,F: a > a] :
( ( member_a @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_279_rev__image__eqI,axiom,
! [X3: a,A: set_a,B2: $o,F: a > $o] :
( ( member_a @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_o @ B2 @ ( image_a_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_280_rev__image__eqI,axiom,
! [X3: a,A: set_a,B2: int,F: a > int] :
( ( member_a @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_int @ B2 @ ( image_a_int @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_281_rev__image__eqI,axiom,
! [X3: a,A: set_a,B2: nat,F: a > nat] :
( ( member_a @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_nat @ B2 @ ( image_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_282_rev__image__eqI,axiom,
! [X3: $o,A: set_o,B2: a,F: $o > a] :
( ( member_o @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_a @ B2 @ ( image_o_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_283_rev__image__eqI,axiom,
! [X3: $o,A: set_o,B2: $o,F: $o > $o] :
( ( member_o @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_o @ B2 @ ( image_o_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_284_rev__image__eqI,axiom,
! [X3: $o,A: set_o,B2: int,F: $o > int] :
( ( member_o @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_int @ B2 @ ( image_o_int @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_285_rev__image__eqI,axiom,
! [X3: $o,A: set_o,B2: nat,F: $o > nat] :
( ( member_o @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_nat @ B2 @ ( image_o_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_286_rev__image__eqI,axiom,
! [X3: int,A: set_int,B2: a,F: int > a] :
( ( member_int @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_a @ B2 @ ( image_int_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_287_rev__image__eqI,axiom,
! [X3: int,A: set_int,B2: $o,F: int > $o] :
( ( member_int @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_o @ B2 @ ( image_int_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_288_ball__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ! [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A ) )
=> ( P @ X ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_289_ball__imageD,axiom,
! [F: nat > int,A: set_nat,P: int > $o] :
( ! [X: int] :
( ( member_int @ X @ ( image_nat_int @ F @ A ) )
=> ( P @ X ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_290_ball__imageD,axiom,
! [F: nat > a,A: set_nat,P: a > $o] :
( ! [X: a] :
( ( member_a @ X @ ( image_nat_a @ F @ A ) )
=> ( P @ X ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_291_ball__imageD,axiom,
! [F: int > int,A: set_int,P: int > $o] :
( ! [X: int] :
( ( member_int @ X @ ( image_int_int @ F @ A ) )
=> ( P @ X ) )
=> ! [X4: int] :
( ( member_int @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_292_ball__imageD,axiom,
! [F: ( nat > a ) > nat > a,A: set_nat_a,P: ( nat > a ) > $o] :
( ! [X: nat > a] :
( ( member_nat_a @ X @ ( image_nat_a_nat_a @ F @ A ) )
=> ( P @ X ) )
=> ! [X4: nat > a] :
( ( member_nat_a @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_293_ball__imageD,axiom,
! [F: int > a,A: set_int,P: a > $o] :
( ! [X: a] :
( ( member_a @ X @ ( image_int_a @ F @ A ) )
=> ( P @ X ) )
=> ! [X4: int] :
( ( member_int @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_294_ball__imageD,axiom,
! [F: a > $o,A: set_a,P: $o > $o] :
( ! [X: $o] :
( ( member_o @ X @ ( image_a_o @ F @ A ) )
=> ( P @ X ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_295_ball__imageD,axiom,
! [F: a > int,A: set_a,P: int > $o] :
( ! [X: int] :
( ( member_int @ X @ ( image_a_int @ F @ A ) )
=> ( P @ X ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_296_ball__imageD,axiom,
! [F: a > nat > a,A: set_a,P: ( nat > a ) > $o] :
( ! [X: nat > a] :
( ( member_nat_a @ X @ ( image_a_nat_a @ F @ A ) )
=> ( P @ X ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_297_ball__imageD,axiom,
! [F: a > a,A: set_a,P: a > $o] :
( ! [X: a] :
( ( member_a @ X @ ( image_a_a @ F @ A ) )
=> ( P @ X ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_298_image__cong,axiom,
! [M4: set_a,N4: set_a,F: a > $o,G: a > $o] :
( ( M4 = N4 )
=> ( ! [X: a] :
( ( member_a @ X @ N4 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_a_o @ F @ M4 )
= ( image_a_o @ G @ N4 ) ) ) ) ).
% image_cong
thf(fact_299_image__cong,axiom,
! [M4: set_a,N4: set_a,F: a > int,G: a > int] :
( ( M4 = N4 )
=> ( ! [X: a] :
( ( member_a @ X @ N4 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_a_int @ F @ M4 )
= ( image_a_int @ G @ N4 ) ) ) ) ).
% image_cong
thf(fact_300_image__cong,axiom,
! [M4: set_a,N4: set_a,F: a > nat > a,G: a > nat > a] :
( ( M4 = N4 )
=> ( ! [X: a] :
( ( member_a @ X @ N4 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_a_nat_a @ F @ M4 )
= ( image_a_nat_a @ G @ N4 ) ) ) ) ).
% image_cong
thf(fact_301_image__cong,axiom,
! [M4: set_a,N4: set_a,F: a > a,G: a > a] :
( ( M4 = N4 )
=> ( ! [X: a] :
( ( member_a @ X @ N4 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_a_a @ F @ M4 )
= ( image_a_a @ G @ N4 ) ) ) ) ).
% image_cong
thf(fact_302_image__cong,axiom,
! [M4: set_int,N4: set_int,F: int > int,G: int > int] :
( ( M4 = N4 )
=> ( ! [X: int] :
( ( member_int @ X @ N4 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_int_int @ F @ M4 )
= ( image_int_int @ G @ N4 ) ) ) ) ).
% image_cong
thf(fact_303_image__cong,axiom,
! [M4: set_int,N4: set_int,F: int > a,G: int > a] :
( ( M4 = N4 )
=> ( ! [X: int] :
( ( member_int @ X @ N4 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_int_a @ F @ M4 )
= ( image_int_a @ G @ N4 ) ) ) ) ).
% image_cong
thf(fact_304_image__cong,axiom,
! [M4: set_nat,N4: set_nat,F: nat > nat,G: nat > nat] :
( ( M4 = N4 )
=> ( ! [X: nat] :
( ( member_nat @ X @ N4 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_nat @ F @ M4 )
= ( image_nat_nat @ G @ N4 ) ) ) ) ).
% image_cong
thf(fact_305_image__cong,axiom,
! [M4: set_nat,N4: set_nat,F: nat > int,G: nat > int] :
( ( M4 = N4 )
=> ( ! [X: nat] :
( ( member_nat @ X @ N4 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_int @ F @ M4 )
= ( image_nat_int @ G @ N4 ) ) ) ) ).
% image_cong
thf(fact_306_image__cong,axiom,
! [M4: set_nat,N4: set_nat,F: nat > a,G: nat > a] :
( ( M4 = N4 )
=> ( ! [X: nat] :
( ( member_nat @ X @ N4 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_a @ F @ M4 )
= ( image_nat_a @ G @ N4 ) ) ) ) ).
% image_cong
thf(fact_307_image__cong,axiom,
! [M4: set_nat_a,N4: set_nat_a,F: ( nat > a ) > nat > a,G: ( nat > a ) > nat > a] :
( ( M4 = N4 )
=> ( ! [X: nat > a] :
( ( member_nat_a @ X @ N4 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_a_nat_a @ F @ M4 )
= ( image_nat_a_nat_a @ G @ N4 ) ) ) ) ).
% image_cong
thf(fact_308_bex__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A ) )
& ( P @ X4 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_309_bex__imageD,axiom,
! [F: nat > int,A: set_nat,P: int > $o] :
( ? [X4: int] :
( ( member_int @ X4 @ ( image_nat_int @ F @ A ) )
& ( P @ X4 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_310_bex__imageD,axiom,
! [F: nat > a,A: set_nat,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( image_nat_a @ F @ A ) )
& ( P @ X4 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_311_bex__imageD,axiom,
! [F: int > int,A: set_int,P: int > $o] :
( ? [X4: int] :
( ( member_int @ X4 @ ( image_int_int @ F @ A ) )
& ( P @ X4 ) )
=> ? [X: int] :
( ( member_int @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_312_bex__imageD,axiom,
! [F: ( nat > a ) > nat > a,A: set_nat_a,P: ( nat > a ) > $o] :
( ? [X4: nat > a] :
( ( member_nat_a @ X4 @ ( image_nat_a_nat_a @ F @ A ) )
& ( P @ X4 ) )
=> ? [X: nat > a] :
( ( member_nat_a @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_313_bex__imageD,axiom,
! [F: int > a,A: set_int,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( image_int_a @ F @ A ) )
& ( P @ X4 ) )
=> ? [X: int] :
( ( member_int @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_314_bex__imageD,axiom,
! [F: a > $o,A: set_a,P: $o > $o] :
( ? [X4: $o] :
( ( member_o @ X4 @ ( image_a_o @ F @ A ) )
& ( P @ X4 ) )
=> ? [X: a] :
( ( member_a @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_315_bex__imageD,axiom,
! [F: a > int,A: set_a,P: int > $o] :
( ? [X4: int] :
( ( member_int @ X4 @ ( image_a_int @ F @ A ) )
& ( P @ X4 ) )
=> ? [X: a] :
( ( member_a @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_316_bex__imageD,axiom,
! [F: a > nat > a,A: set_a,P: ( nat > a ) > $o] :
( ? [X4: nat > a] :
( ( member_nat_a @ X4 @ ( image_a_nat_a @ F @ A ) )
& ( P @ X4 ) )
=> ? [X: a] :
( ( member_a @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_317_bex__imageD,axiom,
! [F: a > a,A: set_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( image_a_a @ F @ A ) )
& ( P @ X4 ) )
=> ? [X: a] :
( ( member_a @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_318_image__iff,axiom,
! [Z2: a,F: nat > a,A: set_nat] :
( ( member_a @ Z2 @ ( image_nat_a @ F @ A ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_319_image__iff,axiom,
! [Z2: a,F: int > a,A: set_int] :
( ( member_a @ Z2 @ ( image_int_a @ F @ A ) )
= ( ? [X2: int] :
( ( member_int @ X2 @ A )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_320_image__iff,axiom,
! [Z2: a,F: a > a,A: set_a] :
( ( member_a @ Z2 @ ( image_a_a @ F @ A ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_321_image__iff,axiom,
! [Z2: $o,F: a > $o,A: set_a] :
( ( member_o @ Z2 @ ( image_a_o @ F @ A ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_322_image__iff,axiom,
! [Z2: int,F: nat > int,A: set_nat] :
( ( member_int @ Z2 @ ( image_nat_int @ F @ A ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_323_image__iff,axiom,
! [Z2: int,F: int > int,A: set_int] :
( ( member_int @ Z2 @ ( image_int_int @ F @ A ) )
= ( ? [X2: int] :
( ( member_int @ X2 @ A )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_324_image__iff,axiom,
! [Z2: int,F: a > int,A: set_a] :
( ( member_int @ Z2 @ ( image_a_int @ F @ A ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_325_image__iff,axiom,
! [Z2: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ Z2 @ ( image_nat_nat @ F @ A ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_326_image__iff,axiom,
! [Z2: nat > a,F: ( nat > a ) > nat > a,A: set_nat_a] :
( ( member_nat_a @ Z2 @ ( image_nat_a_nat_a @ F @ A ) )
= ( ? [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_327_image__iff,axiom,
! [Z2: nat > a,F: a > nat > a,A: set_a] :
( ( member_nat_a @ Z2 @ ( image_a_nat_a @ F @ A ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_328_imageI,axiom,
! [X3: a,A: set_a,F: a > a] :
( ( member_a @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ ( image_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_329_imageI,axiom,
! [X3: a,A: set_a,F: a > $o] :
( ( member_a @ X3 @ A )
=> ( member_o @ ( F @ X3 ) @ ( image_a_o @ F @ A ) ) ) ).
% imageI
thf(fact_330_imageI,axiom,
! [X3: a,A: set_a,F: a > int] :
( ( member_a @ X3 @ A )
=> ( member_int @ ( F @ X3 ) @ ( image_a_int @ F @ A ) ) ) ).
% imageI
thf(fact_331_imageI,axiom,
! [X3: a,A: set_a,F: a > nat] :
( ( member_a @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ ( image_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_332_imageI,axiom,
! [X3: $o,A: set_o,F: $o > a] :
( ( member_o @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ ( image_o_a @ F @ A ) ) ) ).
% imageI
thf(fact_333_imageI,axiom,
! [X3: $o,A: set_o,F: $o > $o] :
( ( member_o @ X3 @ A )
=> ( member_o @ ( F @ X3 ) @ ( image_o_o @ F @ A ) ) ) ).
% imageI
thf(fact_334_imageI,axiom,
! [X3: $o,A: set_o,F: $o > int] :
( ( member_o @ X3 @ A )
=> ( member_int @ ( F @ X3 ) @ ( image_o_int @ F @ A ) ) ) ).
% imageI
thf(fact_335_imageI,axiom,
! [X3: $o,A: set_o,F: $o > nat] :
( ( member_o @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ ( image_o_nat @ F @ A ) ) ) ).
% imageI
thf(fact_336_imageI,axiom,
! [X3: int,A: set_int,F: int > a] :
( ( member_int @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ ( image_int_a @ F @ A ) ) ) ).
% imageI
thf(fact_337_imageI,axiom,
! [X3: int,A: set_int,F: int > $o] :
( ( member_int @ X3 @ A )
=> ( member_o @ ( F @ X3 ) @ ( image_int_o @ F @ A ) ) ) ).
% imageI
thf(fact_338_bounded__Max__nat,axiom,
! [P: nat > $o,X3: nat,M4: nat] :
( ( P @ X3 )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq_nat @ X @ M4 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_339_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K3: nat,B2: nat] :
( ( P @ K3 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B2 ) )
=> ? [X: nat] :
( ( P @ X )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_340_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_341_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_342_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_343_le__trans,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K3 )
=> ( ord_less_eq_nat @ I @ K3 ) ) ) ).
% le_trans
thf(fact_344_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_345_PiE__cong,axiom,
! [I5: set_nat,A: nat > set_a,B: nat > set_a] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( A @ I3 )
= ( B @ I3 ) ) )
=> ( ( piE_nat_a @ I5 @ A )
= ( piE_nat_a @ I5 @ B ) ) ) ).
% PiE_cong
thf(fact_346_PiE__cong,axiom,
! [I5: set_int,A: int > set_int,B: int > set_int] :
( ! [I3: int] :
( ( member_int @ I3 @ I5 )
=> ( ( A @ I3 )
= ( B @ I3 ) ) )
=> ( ( piE_int_int @ I5 @ A )
= ( piE_int_int @ I5 @ B ) ) ) ).
% PiE_cong
thf(fact_347_PiE__cong,axiom,
! [I5: set_int,A: int > set_nat_a,B: int > set_nat_a] :
( ! [I3: int] :
( ( member_int @ I3 @ I5 )
=> ( ( A @ I3 )
= ( B @ I3 ) ) )
=> ( ( piE_int_nat_a @ I5 @ A )
= ( piE_int_nat_a @ I5 @ B ) ) ) ).
% PiE_cong
thf(fact_348_PiE__cong,axiom,
! [I5: set_int,A: int > set_a,B: int > set_a] :
( ! [I3: int] :
( ( member_int @ I3 @ I5 )
=> ( ( A @ I3 )
= ( B @ I3 ) ) )
=> ( ( piE_int_a @ I5 @ A )
= ( piE_int_a @ I5 @ B ) ) ) ).
% PiE_cong
thf(fact_349_PiE__cong,axiom,
! [I5: set_a,A: a > set_nat_a,B: a > set_nat_a] :
( ! [I3: a] :
( ( member_a @ I3 @ I5 )
=> ( ( A @ I3 )
= ( B @ I3 ) ) )
=> ( ( piE_a_nat_a @ I5 @ A )
= ( piE_a_nat_a @ I5 @ B ) ) ) ).
% PiE_cong
thf(fact_350_PiE__cong,axiom,
! [I5: set_a,A: a > set_a,B: a > set_a] :
( ! [I3: a] :
( ( member_a @ I3 @ I5 )
=> ( ( A @ I3 )
= ( B @ I3 ) ) )
=> ( ( piE_a_a @ I5 @ A )
= ( piE_a_a @ I5 @ B ) ) ) ).
% PiE_cong
thf(fact_351_PiE__mem,axiom,
! [F: a > $o,S2: set_a,T3: a > set_o,X3: a] :
( ( member_a_o @ F @ ( piE_a_o @ S2 @ T3 ) )
=> ( ( member_a @ X3 @ S2 )
=> ( member_o @ ( F @ X3 ) @ ( T3 @ X3 ) ) ) ) ).
% PiE_mem
thf(fact_352_PiE__mem,axiom,
! [F: a > int,S2: set_a,T3: a > set_int,X3: a] :
( ( member_a_int @ F @ ( piE_a_int @ S2 @ T3 ) )
=> ( ( member_a @ X3 @ S2 )
=> ( member_int @ ( F @ X3 ) @ ( T3 @ X3 ) ) ) ) ).
% PiE_mem
thf(fact_353_PiE__mem,axiom,
! [F: a > nat,S2: set_a,T3: a > set_nat,X3: a] :
( ( member_a_nat @ F @ ( piE_a_nat @ S2 @ T3 ) )
=> ( ( member_a @ X3 @ S2 )
=> ( member_nat @ ( F @ X3 ) @ ( T3 @ X3 ) ) ) ) ).
% PiE_mem
thf(fact_354_PiE__mem,axiom,
! [F: $o > a,S2: set_o,T3: $o > set_a,X3: $o] :
( ( member_o_a @ F @ ( piE_o_a @ S2 @ T3 ) )
=> ( ( member_o @ X3 @ S2 )
=> ( member_a @ ( F @ X3 ) @ ( T3 @ X3 ) ) ) ) ).
% PiE_mem
thf(fact_355_PiE__mem,axiom,
! [F: $o > $o,S2: set_o,T3: $o > set_o,X3: $o] :
( ( member_o_o @ F @ ( piE_o_o @ S2 @ T3 ) )
=> ( ( member_o @ X3 @ S2 )
=> ( member_o @ ( F @ X3 ) @ ( T3 @ X3 ) ) ) ) ).
% PiE_mem
thf(fact_356_PiE__mem,axiom,
! [F: $o > int,S2: set_o,T3: $o > set_int,X3: $o] :
( ( member_o_int @ F @ ( piE_o_int @ S2 @ T3 ) )
=> ( ( member_o @ X3 @ S2 )
=> ( member_int @ ( F @ X3 ) @ ( T3 @ X3 ) ) ) ) ).
% PiE_mem
thf(fact_357_PiE__mem,axiom,
! [F: $o > nat,S2: set_o,T3: $o > set_nat,X3: $o] :
( ( member_o_nat @ F @ ( piE_o_nat @ S2 @ T3 ) )
=> ( ( member_o @ X3 @ S2 )
=> ( member_nat @ ( F @ X3 ) @ ( T3 @ X3 ) ) ) ) ).
% PiE_mem
thf(fact_358_PiE__mem,axiom,
! [F: int > $o,S2: set_int,T3: int > set_o,X3: int] :
( ( member_int_o @ F @ ( piE_int_o @ S2 @ T3 ) )
=> ( ( member_int @ X3 @ S2 )
=> ( member_o @ ( F @ X3 ) @ ( T3 @ X3 ) ) ) ) ).
% PiE_mem
thf(fact_359_PiE__mem,axiom,
! [F: int > nat,S2: set_int,T3: int > set_nat,X3: int] :
( ( member_int_nat @ F @ ( piE_int_nat @ S2 @ T3 ) )
=> ( ( member_int @ X3 @ S2 )
=> ( member_nat @ ( F @ X3 ) @ ( T3 @ X3 ) ) ) ) ).
% PiE_mem
thf(fact_360_PiE__mem,axiom,
! [F: nat > $o,S2: set_nat,T3: nat > set_o,X3: nat] :
( ( member_nat_o @ F @ ( piE_nat_o @ S2 @ T3 ) )
=> ( ( member_nat @ X3 @ S2 )
=> ( member_o @ ( F @ X3 ) @ ( T3 @ X3 ) ) ) ) ).
% PiE_mem
thf(fact_361_PiE__ext,axiom,
! [X3: nat > a,K3: set_nat,S: nat > set_a,Y2: nat > a] :
( ( member_nat_a @ X3 @ ( piE_nat_a @ K3 @ S ) )
=> ( ( member_nat_a @ Y2 @ ( piE_nat_a @ K3 @ S ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ K3 )
=> ( ( X3 @ I3 )
= ( Y2 @ I3 ) ) )
=> ( X3 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_362_PiE__ext,axiom,
! [X3: int > int,K3: set_int,S: int > set_int,Y2: int > int] :
( ( member_int_int @ X3 @ ( piE_int_int @ K3 @ S ) )
=> ( ( member_int_int @ Y2 @ ( piE_int_int @ K3 @ S ) )
=> ( ! [I3: int] :
( ( member_int @ I3 @ K3 )
=> ( ( X3 @ I3 )
= ( Y2 @ I3 ) ) )
=> ( X3 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_363_PiE__ext,axiom,
! [X3: int > nat > a,K3: set_int,S: int > set_nat_a,Y2: int > nat > a] :
( ( member_int_nat_a @ X3 @ ( piE_int_nat_a @ K3 @ S ) )
=> ( ( member_int_nat_a @ Y2 @ ( piE_int_nat_a @ K3 @ S ) )
=> ( ! [I3: int] :
( ( member_int @ I3 @ K3 )
=> ( ( X3 @ I3 )
= ( Y2 @ I3 ) ) )
=> ( X3 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_364_PiE__ext,axiom,
! [X3: int > a,K3: set_int,S: int > set_a,Y2: int > a] :
( ( member_int_a @ X3 @ ( piE_int_a @ K3 @ S ) )
=> ( ( member_int_a @ Y2 @ ( piE_int_a @ K3 @ S ) )
=> ( ! [I3: int] :
( ( member_int @ I3 @ K3 )
=> ( ( X3 @ I3 )
= ( Y2 @ I3 ) ) )
=> ( X3 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_365_PiE__ext,axiom,
! [X3: a > nat > a,K3: set_a,S: a > set_nat_a,Y2: a > nat > a] :
( ( member_a_nat_a @ X3 @ ( piE_a_nat_a @ K3 @ S ) )
=> ( ( member_a_nat_a @ Y2 @ ( piE_a_nat_a @ K3 @ S ) )
=> ( ! [I3: a] :
( ( member_a @ I3 @ K3 )
=> ( ( X3 @ I3 )
= ( Y2 @ I3 ) ) )
=> ( X3 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_366_PiE__ext,axiom,
! [X3: a > a,K3: set_a,S: a > set_a,Y2: a > a] :
( ( member_a_a @ X3 @ ( piE_a_a @ K3 @ S ) )
=> ( ( member_a_a @ Y2 @ ( piE_a_a @ K3 @ S ) )
=> ( ! [I3: a] :
( ( member_a @ I3 @ K3 )
=> ( ( X3 @ I3 )
= ( Y2 @ I3 ) ) )
=> ( X3 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_367_Collect__subset,axiom,
! [A: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_368_Collect__subset,axiom,
! [A: set_int,P: int > $o] :
( ord_less_eq_set_int
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_369_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_370_Collect__subset,axiom,
! [A: set_nat_a,P: ( nat > a ) > $o] :
( ord_le871467723717165285_nat_a
@ ( collect_nat_a
@ ^ [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_371_Collect__subset,axiom,
! [A: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_372_Collect__subset,axiom,
! [A: set_o_a,P: ( $o > a ) > $o] :
( ord_less_eq_set_o_a
@ ( collect_o_a
@ ^ [X2: $o > a] :
( ( member_o_a @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_373_Collect__subset,axiom,
! [A: set_int_nat_a,P: ( int > nat > a ) > $o] :
( ord_le6072919132162605296_nat_a
@ ( collect_int_nat_a
@ ^ [X2: int > nat > a] :
( ( member_int_nat_a @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_374_Collect__subset,axiom,
! [A: set_int_a,P: ( int > a ) > $o] :
( ord_le943418215940126601_int_a
@ ( collect_int_a
@ ^ [X2: int > a] :
( ( member_int_a @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_375_Collect__subset,axiom,
! [A: set_a_nat_a,P: ( a > nat > a ) > $o] :
( ord_le2508512696544544866_nat_a
@ ( collect_a_nat_a
@ ^ [X2: a > nat > a] :
( ( member_a_nat_a @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_376_Collect__subset,axiom,
! [A: set_a_a,P: ( a > a ) > $o] :
( ord_less_eq_set_a_a
@ ( collect_a_a
@ ^ [X2: a > a] :
( ( member_a_a @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_377_UNIV__def,axiom,
( top_top_set_nat
= ( collect_nat
@ ^ [X2: nat] : $true ) ) ).
% UNIV_def
thf(fact_378_UNIV__def,axiom,
( top_top_set_int
= ( collect_int
@ ^ [X2: int] : $true ) ) ).
% UNIV_def
thf(fact_379_UNIV__def,axiom,
( top_top_set_nat_int
= ( collect_nat_int
@ ^ [X2: nat > int] : $true ) ) ).
% UNIV_def
thf(fact_380_UNIV__def,axiom,
( top_top_set_nat_a
= ( collect_nat_a
@ ^ [X2: nat > a] : $true ) ) ).
% UNIV_def
thf(fact_381_UNIV__def,axiom,
( top_top_set_int_nat
= ( collect_int_nat
@ ^ [X2: int > nat] : $true ) ) ).
% UNIV_def
thf(fact_382_UNIV__def,axiom,
( top_top_set_int_int
= ( collect_int_int
@ ^ [X2: int > int] : $true ) ) ).
% UNIV_def
thf(fact_383_UNIV__def,axiom,
( top_top_set_a
= ( collect_a
@ ^ [X2: a] : $true ) ) ).
% UNIV_def
thf(fact_384_Compr__image__eq,axiom,
! [F: a > a,A: set_a,P: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ ( image_a_a @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_a_a @ F
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_385_Compr__image__eq,axiom,
! [F: $o > a,A: set_o,P: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ ( image_o_a @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_o_a @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_386_Compr__image__eq,axiom,
! [F: int > a,A: set_int,P: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ ( image_int_a @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_int_a @ F
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_387_Compr__image__eq,axiom,
! [F: nat > a,A: set_nat,P: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ ( image_nat_a @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_nat_a @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_388_Compr__image__eq,axiom,
! [F: a > $o,A: set_a,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_a_o @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_a_o @ F
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_389_Compr__image__eq,axiom,
! [F: $o > $o,A: set_o,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_o_o @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_o_o @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_390_Compr__image__eq,axiom,
! [F: int > $o,A: set_int,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_int_o @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_int_o @ F
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_391_Compr__image__eq,axiom,
! [F: nat > $o,A: set_nat,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_nat_o @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_nat_o @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_392_Compr__image__eq,axiom,
! [F: a > int,A: set_a,P: int > $o] :
( ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ ( image_a_int @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_a_int @ F
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_393_Compr__image__eq,axiom,
! [F: $o > int,A: set_o,P: int > $o] :
( ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ ( image_o_int @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_o_int @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_394_image__image,axiom,
! [F: int > nat,G: nat > int,A: set_nat] :
( ( image_int_nat @ F @ ( image_nat_int @ G @ A ) )
= ( image_nat_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_395_image__image,axiom,
! [F: a > nat,G: nat > a,A: set_nat] :
( ( image_a_nat @ F @ ( image_nat_a @ G @ A ) )
= ( image_nat_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_396_image__image,axiom,
! [F: $o > $o,G: a > $o,A: set_a] :
( ( image_o_o @ F @ ( image_a_o @ G @ A ) )
= ( image_a_o
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_397_image__image,axiom,
! [F: $o > int,G: a > $o,A: set_a] :
( ( image_o_int @ F @ ( image_a_o @ G @ A ) )
= ( image_a_int
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_398_image__image,axiom,
! [F: $o > a,G: a > $o,A: set_a] :
( ( image_o_a @ F @ ( image_a_o @ G @ A ) )
= ( image_a_a
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_399_image__image,axiom,
! [F: int > $o,G: a > int,A: set_a] :
( ( image_int_o @ F @ ( image_a_int @ G @ A ) )
= ( image_a_o
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_400_image__image,axiom,
! [F: nat > nat,G: nat > nat,A: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G @ A ) )
= ( image_nat_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_401_image__image,axiom,
! [F: nat > int,G: int > nat,A: set_int] :
( ( image_nat_int @ F @ ( image_int_nat @ G @ A ) )
= ( image_int_int
@ ^ [X2: int] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_402_image__image,axiom,
! [F: nat > int,G: a > nat,A: set_a] :
( ( image_nat_int @ F @ ( image_a_nat @ G @ A ) )
= ( image_a_int
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_403_image__image,axiom,
! [F: nat > int,G: nat > nat,A: set_nat] :
( ( image_nat_int @ F @ ( image_nat_nat @ G @ A ) )
= ( image_nat_int
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_404_imageE,axiom,
! [B2: a,F: a > a,A: set_a] :
( ( member_a @ B2 @ ( image_a_a @ F @ A ) )
=> ~ ! [X: a] :
( ( B2
= ( F @ X ) )
=> ~ ( member_a @ X @ A ) ) ) ).
% imageE
thf(fact_405_imageE,axiom,
! [B2: a,F: $o > a,A: set_o] :
( ( member_a @ B2 @ ( image_o_a @ F @ A ) )
=> ~ ! [X: $o] :
( ( B2
= ( F @ X ) )
=> ~ ( member_o @ X @ A ) ) ) ).
% imageE
thf(fact_406_imageE,axiom,
! [B2: a,F: int > a,A: set_int] :
( ( member_a @ B2 @ ( image_int_a @ F @ A ) )
=> ~ ! [X: int] :
( ( B2
= ( F @ X ) )
=> ~ ( member_int @ X @ A ) ) ) ).
% imageE
thf(fact_407_imageE,axiom,
! [B2: a,F: nat > a,A: set_nat] :
( ( member_a @ B2 @ ( image_nat_a @ F @ A ) )
=> ~ ! [X: nat] :
( ( B2
= ( F @ X ) )
=> ~ ( member_nat @ X @ A ) ) ) ).
% imageE
thf(fact_408_imageE,axiom,
! [B2: $o,F: a > $o,A: set_a] :
( ( member_o @ B2 @ ( image_a_o @ F @ A ) )
=> ~ ! [X: a] :
( ( B2
= ( F @ X ) )
=> ~ ( member_a @ X @ A ) ) ) ).
% imageE
thf(fact_409_imageE,axiom,
! [B2: $o,F: $o > $o,A: set_o] :
( ( member_o @ B2 @ ( image_o_o @ F @ A ) )
=> ~ ! [X: $o] :
( ( B2
= ( F @ X ) )
=> ~ ( member_o @ X @ A ) ) ) ).
% imageE
thf(fact_410_imageE,axiom,
! [B2: $o,F: int > $o,A: set_int] :
( ( member_o @ B2 @ ( image_int_o @ F @ A ) )
=> ~ ! [X: int] :
( ( B2
= ( F @ X ) )
=> ~ ( member_int @ X @ A ) ) ) ).
% imageE
thf(fact_411_imageE,axiom,
! [B2: $o,F: nat > $o,A: set_nat] :
( ( member_o @ B2 @ ( image_nat_o @ F @ A ) )
=> ~ ! [X: nat] :
( ( B2
= ( F @ X ) )
=> ~ ( member_nat @ X @ A ) ) ) ).
% imageE
thf(fact_412_imageE,axiom,
! [B2: int,F: a > int,A: set_a] :
( ( member_int @ B2 @ ( image_a_int @ F @ A ) )
=> ~ ! [X: a] :
( ( B2
= ( F @ X ) )
=> ~ ( member_a @ X @ A ) ) ) ).
% imageE
thf(fact_413_imageE,axiom,
! [B2: int,F: $o > int,A: set_o] :
( ( member_int @ B2 @ ( image_o_int @ F @ A ) )
=> ~ ! [X: $o] :
( ( B2
= ( F @ X ) )
=> ~ ( member_o @ X @ A ) ) ) ).
% imageE
thf(fact_414_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_415_subset__UNIV,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_416_subset__UNIV,axiom,
! [A: set_int] : ( ord_less_eq_set_int @ A @ top_top_set_int ) ).
% subset_UNIV
thf(fact_417_subset__UNIV,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% subset_UNIV
thf(fact_418_subset__UNIV,axiom,
! [A: set_nat_int] : ( ord_le6569500216720880561at_int @ A @ top_top_set_nat_int ) ).
% subset_UNIV
thf(fact_419_subset__UNIV,axiom,
! [A: set_int_nat] : ( ord_le2023132899490853297nt_nat @ A @ top_top_set_int_nat ) ).
% subset_UNIV
thf(fact_420_subset__UNIV,axiom,
! [A: set_int_int] : ( ord_le8756421791413902349nt_int @ A @ top_top_set_int_int ) ).
% subset_UNIV
thf(fact_421_subset__UNIV,axiom,
! [A: set_nat_a] : ( ord_le871467723717165285_nat_a @ A @ top_top_set_nat_a ) ).
% subset_UNIV
thf(fact_422_subset__UNIV,axiom,
! [A: set_o_a] : ( ord_less_eq_set_o_a @ A @ top_top_set_o_a ) ).
% subset_UNIV
thf(fact_423_subset__UNIV,axiom,
! [A: set_int_a] : ( ord_le943418215940126601_int_a @ A @ top_top_set_int_a ) ).
% subset_UNIV
thf(fact_424_subset__UNIV,axiom,
! [A: set_a_a] : ( ord_less_eq_set_a_a @ A @ top_top_set_a_a ) ).
% subset_UNIV
thf(fact_425_subset__image__iff,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_426_subset__image__iff,axiom,
! [B: set_int,F: nat > int,A: set_nat] :
( ( ord_less_eq_set_int @ B @ ( image_nat_int @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_int @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_427_subset__image__iff,axiom,
! [B: set_int,F: int > int,A: set_int] :
( ( ord_less_eq_set_int @ B @ ( image_int_int @ F @ A ) )
= ( ? [AA: set_int] :
( ( ord_less_eq_set_int @ AA @ A )
& ( B
= ( image_int_int @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_428_subset__image__iff,axiom,
! [B: set_o,F: a > $o,A: set_a] :
( ( ord_less_eq_set_o @ B @ ( image_a_o @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_o @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_429_subset__image__iff,axiom,
! [B: set_int,F: a > int,A: set_a] :
( ( ord_less_eq_set_int @ B @ ( image_a_int @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_int @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_430_subset__image__iff,axiom,
! [B: set_a,F: nat > a,A: set_nat] :
( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_431_subset__image__iff,axiom,
! [B: set_a,F: int > a,A: set_int] :
( ( ord_less_eq_set_a @ B @ ( image_int_a @ F @ A ) )
= ( ? [AA: set_int] :
( ( ord_less_eq_set_int @ AA @ A )
& ( B
= ( image_int_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_432_subset__image__iff,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_433_subset__image__iff,axiom,
! [B: set_nat_a,F: a > nat > a,A: set_a] :
( ( ord_le871467723717165285_nat_a @ B @ ( image_a_nat_a @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_nat_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_434_subset__image__iff,axiom,
! [B: set_a,F: ( nat > a ) > a,A: set_nat_a] :
( ( ord_less_eq_set_a @ B @ ( image_nat_a_a @ F @ A ) )
= ( ? [AA: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ AA @ A )
& ( B
= ( image_nat_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_435_image__subset__iff,axiom,
! [F: a > $o,A: set_a,B: set_o] :
( ( ord_less_eq_set_o @ ( image_a_o @ F @ A ) @ B )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_o @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_436_image__subset__iff,axiom,
! [F: nat > int,A: set_nat,B: set_int] :
( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A ) @ B )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_int @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_437_image__subset__iff,axiom,
! [F: int > int,A: set_int,B: set_int] :
( ( ord_less_eq_set_int @ ( image_int_int @ F @ A ) @ B )
= ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( member_int @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_438_image__subset__iff,axiom,
! [F: a > int,A: set_a,B: set_int] :
( ( ord_less_eq_set_int @ ( image_a_int @ F @ A ) @ B )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_int @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_439_image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_440_image__subset__iff,axiom,
! [F: ( nat > a ) > nat > a,A: set_nat_a,B: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ ( image_nat_a_nat_a @ F @ A ) @ B )
= ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
=> ( member_nat_a @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_441_image__subset__iff,axiom,
! [F: a > nat > a,A: set_a,B: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ ( image_a_nat_a @ F @ A ) @ B )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_nat_a @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_442_image__subset__iff,axiom,
! [F: nat > a,A: set_nat,B: set_a] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ B )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_443_image__subset__iff,axiom,
! [F: int > a,A: set_int,B: set_a] :
( ( ord_less_eq_set_a @ ( image_int_a @ F @ A ) @ B )
= ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_444_image__subset__iff,axiom,
! [F: a > a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_445_subset__imageE,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B
!= ( image_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_446_subset__imageE,axiom,
! [B: set_int,F: nat > int,A: set_nat] :
( ( ord_less_eq_set_int @ B @ ( image_nat_int @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B
!= ( image_nat_int @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_447_subset__imageE,axiom,
! [B: set_int,F: int > int,A: set_int] :
( ( ord_less_eq_set_int @ B @ ( image_int_int @ F @ A ) )
=> ~ ! [C3: set_int] :
( ( ord_less_eq_set_int @ C3 @ A )
=> ( B
!= ( image_int_int @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_448_subset__imageE,axiom,
! [B: set_o,F: a > $o,A: set_a] :
( ( ord_less_eq_set_o @ B @ ( image_a_o @ F @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B
!= ( image_a_o @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_449_subset__imageE,axiom,
! [B: set_int,F: a > int,A: set_a] :
( ( ord_less_eq_set_int @ B @ ( image_a_int @ F @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B
!= ( image_a_int @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_450_subset__imageE,axiom,
! [B: set_a,F: nat > a,A: set_nat] :
( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B
!= ( image_nat_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_451_subset__imageE,axiom,
! [B: set_a,F: int > a,A: set_int] :
( ( ord_less_eq_set_a @ B @ ( image_int_a @ F @ A ) )
=> ~ ! [C3: set_int] :
( ( ord_less_eq_set_int @ C3 @ A )
=> ( B
!= ( image_int_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_452_subset__imageE,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B
!= ( image_a_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_453_subset__imageE,axiom,
! [B: set_nat_a,F: a > nat > a,A: set_a] :
( ( ord_le871467723717165285_nat_a @ B @ ( image_a_nat_a @ F @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B
!= ( image_a_nat_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_454_subset__imageE,axiom,
! [B: set_a,F: ( nat > a ) > a,A: set_nat_a] :
( ( ord_less_eq_set_a @ B @ ( image_nat_a_a @ F @ A ) )
=> ~ ! [C3: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ C3 @ A )
=> ( B
!= ( image_nat_a_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_455_image__subsetI,axiom,
! [A: set_a,F: a > $o,B: set_o] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_o @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_a_o @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_456_image__subsetI,axiom,
! [A: set_a,F: a > int,B: set_int] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_int @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_int @ ( image_a_int @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_457_image__subsetI,axiom,
! [A: set_a,F: a > nat,B: set_nat] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_458_image__subsetI,axiom,
! [A: set_o,F: $o > $o,B: set_o] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( member_o @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_459_image__subsetI,axiom,
! [A: set_o,F: $o > int,B: set_int] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( member_int @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_int @ ( image_o_int @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_460_image__subsetI,axiom,
! [A: set_o,F: $o > nat,B: set_nat] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_o_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_461_image__subsetI,axiom,
! [A: set_int,F: int > $o,B: set_o] :
( ! [X: int] :
( ( member_int @ X @ A )
=> ( member_o @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_int_o @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_462_image__subsetI,axiom,
! [A: set_int,F: int > int,B: set_int] :
( ! [X: int] :
( ( member_int @ X @ A )
=> ( member_int @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_int @ ( image_int_int @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_463_image__subsetI,axiom,
! [A: set_int,F: int > nat,B: set_nat] :
( ! [X: int] :
( ( member_int @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_464_image__subsetI,axiom,
! [A: set_nat,F: nat > $o,B: set_o] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_o @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_nat_o @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_465_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_466_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > int] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_int @ ( image_nat_int @ F @ A ) @ ( image_nat_int @ F @ B ) ) ) ).
% image_mono
thf(fact_467_image__mono,axiom,
! [A: set_int,B: set_int,F: int > int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ord_less_eq_set_int @ ( image_int_int @ F @ A ) @ ( image_int_int @ F @ B ) ) ) ).
% image_mono
thf(fact_468_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > a] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ ( image_nat_a @ F @ B ) ) ) ).
% image_mono
thf(fact_469_image__mono,axiom,
! [A: set_int,B: set_int,F: int > a] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ord_less_eq_set_a @ ( image_int_a @ F @ A ) @ ( image_int_a @ F @ B ) ) ) ).
% image_mono
thf(fact_470_image__mono,axiom,
! [A: set_a,B: set_a,F: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_o @ ( image_a_o @ F @ A ) @ ( image_a_o @ F @ B ) ) ) ).
% image_mono
thf(fact_471_image__mono,axiom,
! [A: set_a,B: set_a,F: a > int] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_int @ ( image_a_int @ F @ A ) @ ( image_a_int @ F @ B ) ) ) ).
% image_mono
thf(fact_472_image__mono,axiom,
! [A: set_a,B: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) ) ).
% image_mono
thf(fact_473_image__mono,axiom,
! [A: set_nat_a,B: set_nat_a,F: ( nat > a ) > a] :
( ( ord_le871467723717165285_nat_a @ A @ B )
=> ( ord_less_eq_set_a @ ( image_nat_a_a @ F @ A ) @ ( image_nat_a_a @ F @ B ) ) ) ).
% image_mono
thf(fact_474_image__mono,axiom,
! [A: set_a,B: set_a,F: a > nat > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_le871467723717165285_nat_a @ ( image_a_nat_a @ F @ A ) @ ( image_a_nat_a @ F @ B ) ) ) ).
% image_mono
thf(fact_475_range__eqI,axiom,
! [B2: a,F: nat > a,X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ( member_a @ B2 @ ( image_nat_a @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_476_range__eqI,axiom,
! [B2: $o,F: nat > $o,X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ( member_o @ B2 @ ( image_nat_o @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_477_range__eqI,axiom,
! [B2: int,F: nat > int,X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ( member_int @ B2 @ ( image_nat_int @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_478_range__eqI,axiom,
! [B2: nat,F: nat > nat,X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_479_range__eqI,axiom,
! [B2: a,F: int > a,X3: int] :
( ( B2
= ( F @ X3 ) )
=> ( member_a @ B2 @ ( image_int_a @ F @ top_top_set_int ) ) ) ).
% range_eqI
thf(fact_480_range__eqI,axiom,
! [B2: $o,F: int > $o,X3: int] :
( ( B2
= ( F @ X3 ) )
=> ( member_o @ B2 @ ( image_int_o @ F @ top_top_set_int ) ) ) ).
% range_eqI
thf(fact_481_range__eqI,axiom,
! [B2: int,F: int > int,X3: int] :
( ( B2
= ( F @ X3 ) )
=> ( member_int @ B2 @ ( image_int_int @ F @ top_top_set_int ) ) ) ).
% range_eqI
thf(fact_482_range__eqI,axiom,
! [B2: nat,F: int > nat,X3: int] :
( ( B2
= ( F @ X3 ) )
=> ( member_nat @ B2 @ ( image_int_nat @ F @ top_top_set_int ) ) ) ).
% range_eqI
thf(fact_483_range__eqI,axiom,
! [B2: a,F: a > a,X3: a] :
( ( B2
= ( F @ X3 ) )
=> ( member_a @ B2 @ ( image_a_a @ F @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_484_range__eqI,axiom,
! [B2: $o,F: a > $o,X3: a] :
( ( B2
= ( F @ X3 ) )
=> ( member_o @ B2 @ ( image_a_o @ F @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_485_rangeI,axiom,
! [F: nat > a,X3: nat] : ( member_a @ ( F @ X3 ) @ ( image_nat_a @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_486_rangeI,axiom,
! [F: nat > $o,X3: nat] : ( member_o @ ( F @ X3 ) @ ( image_nat_o @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_487_rangeI,axiom,
! [F: nat > int,X3: nat] : ( member_int @ ( F @ X3 ) @ ( image_nat_int @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_488_rangeI,axiom,
! [F: nat > nat,X3: nat] : ( member_nat @ ( F @ X3 ) @ ( image_nat_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_489_rangeI,axiom,
! [F: int > a,X3: int] : ( member_a @ ( F @ X3 ) @ ( image_int_a @ F @ top_top_set_int ) ) ).
% rangeI
thf(fact_490_rangeI,axiom,
! [F: int > $o,X3: int] : ( member_o @ ( F @ X3 ) @ ( image_int_o @ F @ top_top_set_int ) ) ).
% rangeI
thf(fact_491_rangeI,axiom,
! [F: int > int,X3: int] : ( member_int @ ( F @ X3 ) @ ( image_int_int @ F @ top_top_set_int ) ) ).
% rangeI
thf(fact_492_rangeI,axiom,
! [F: int > nat,X3: int] : ( member_nat @ ( F @ X3 ) @ ( image_int_nat @ F @ top_top_set_int ) ) ).
% rangeI
thf(fact_493_rangeI,axiom,
! [F: a > a,X3: a] : ( member_a @ ( F @ X3 ) @ ( image_a_a @ F @ top_top_set_a ) ) ).
% rangeI
thf(fact_494_rangeI,axiom,
! [F: a > $o,X3: a] : ( member_o @ ( F @ X3 ) @ ( image_a_o @ F @ top_top_set_a ) ) ).
% rangeI
thf(fact_495_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_496_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_497_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_498_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_499_range__composition,axiom,
! [F: a > $o,G: nat > a] :
( ( image_nat_o
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ top_top_set_nat )
= ( image_a_o @ F @ ( image_nat_a @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_500_range__composition,axiom,
! [F: int > nat,G: nat > int] :
( ( image_nat_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ top_top_set_nat )
= ( image_int_nat @ F @ ( image_nat_int @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_501_range__composition,axiom,
! [F: a > nat,G: nat > a] :
( ( image_nat_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ top_top_set_nat )
= ( image_a_nat @ F @ ( image_nat_a @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_502_range__composition,axiom,
! [F: nat > nat,G: nat > nat] :
( ( image_nat_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ top_top_set_nat )
= ( image_nat_nat @ F @ ( image_nat_nat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_503_range__composition,axiom,
! [F: nat > int,G: nat > nat] :
( ( image_nat_int
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ top_top_set_nat )
= ( image_nat_int @ F @ ( image_nat_nat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_504_range__composition,axiom,
! [F: int > int,G: nat > int] :
( ( image_nat_int
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ top_top_set_nat )
= ( image_int_int @ F @ ( image_nat_int @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_505_range__composition,axiom,
! [F: a > int,G: nat > a] :
( ( image_nat_int
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ top_top_set_nat )
= ( image_a_int @ F @ ( image_nat_a @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_506_range__composition,axiom,
! [F: nat > a,G: nat > nat] :
( ( image_nat_a
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ top_top_set_nat )
= ( image_nat_a @ F @ ( image_nat_nat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_507_range__composition,axiom,
! [F: int > a,G: nat > int] :
( ( image_nat_a
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ top_top_set_nat )
= ( image_int_a @ F @ ( image_nat_int @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_508_range__composition,axiom,
! [F: a > a,G: nat > a] :
( ( image_nat_a
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ top_top_set_nat )
= ( image_a_a @ F @ ( image_nat_a @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_509_rangeE,axiom,
! [B2: a,F: nat > a] :
( ( member_a @ B2 @ ( image_nat_a @ F @ top_top_set_nat ) )
=> ~ ! [X: nat] :
( B2
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_510_rangeE,axiom,
! [B2: $o,F: nat > $o] :
( ( member_o @ B2 @ ( image_nat_o @ F @ top_top_set_nat ) )
=> ~ ! [X: nat] :
( B2
= ( ~ ( F @ X ) ) ) ) ).
% rangeE
thf(fact_511_rangeE,axiom,
! [B2: int,F: nat > int] :
( ( member_int @ B2 @ ( image_nat_int @ F @ top_top_set_nat ) )
=> ~ ! [X: nat] :
( B2
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_512_rangeE,axiom,
! [B2: nat,F: nat > nat] :
( ( member_nat @ B2 @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ~ ! [X: nat] :
( B2
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_513_rangeE,axiom,
! [B2: a,F: int > a] :
( ( member_a @ B2 @ ( image_int_a @ F @ top_top_set_int ) )
=> ~ ! [X: int] :
( B2
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_514_rangeE,axiom,
! [B2: $o,F: int > $o] :
( ( member_o @ B2 @ ( image_int_o @ F @ top_top_set_int ) )
=> ~ ! [X: int] :
( B2
= ( ~ ( F @ X ) ) ) ) ).
% rangeE
thf(fact_515_rangeE,axiom,
! [B2: int,F: int > int] :
( ( member_int @ B2 @ ( image_int_int @ F @ top_top_set_int ) )
=> ~ ! [X: int] :
( B2
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_516_rangeE,axiom,
! [B2: nat,F: int > nat] :
( ( member_nat @ B2 @ ( image_int_nat @ F @ top_top_set_int ) )
=> ~ ! [X: int] :
( B2
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_517_rangeE,axiom,
! [B2: a,F: a > a] :
( ( member_a @ B2 @ ( image_a_a @ F @ top_top_set_a ) )
=> ~ ! [X: a] :
( B2
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_518_rangeE,axiom,
! [B2: $o,F: a > $o] :
( ( member_o @ B2 @ ( image_a_o @ F @ top_top_set_a ) )
=> ~ ! [X: a] :
( B2
= ( ~ ( F @ X ) ) ) ) ).
% rangeE
thf(fact_519_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( P @ M3 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less_eq
thf(fact_520_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M3: nat] :
( ( ord_less_nat @ M3 @ N )
& ( P @ M3 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_521_iso__tuple__UNIV__I,axiom,
! [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).
% iso_tuple_UNIV_I
thf(fact_522_iso__tuple__UNIV__I,axiom,
! [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_523_iso__tuple__UNIV__I,axiom,
! [X3: int] : ( member_int @ X3 @ top_top_set_int ) ).
% iso_tuple_UNIV_I
thf(fact_524_iso__tuple__UNIV__I,axiom,
! [X3: nat > int] : ( member_nat_int @ X3 @ top_top_set_nat_int ) ).
% iso_tuple_UNIV_I
thf(fact_525_iso__tuple__UNIV__I,axiom,
! [X3: nat > a] : ( member_nat_a @ X3 @ top_top_set_nat_a ) ).
% iso_tuple_UNIV_I
thf(fact_526_iso__tuple__UNIV__I,axiom,
! [X3: int > nat] : ( member_int_nat @ X3 @ top_top_set_int_nat ) ).
% iso_tuple_UNIV_I
thf(fact_527_iso__tuple__UNIV__I,axiom,
! [X3: int > int] : ( member_int_int @ X3 @ top_top_set_int_int ) ).
% iso_tuple_UNIV_I
thf(fact_528_iso__tuple__UNIV__I,axiom,
! [X3: a] : ( member_a @ X3 @ top_top_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_529_dual__order_Orefl,axiom,
! [A2: set_nat_a] : ( ord_le871467723717165285_nat_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_530_dual__order_Orefl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_531_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_532_dual__order_Orefl,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_533_dual__order_Orefl,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_534_dual__order_Orefl,axiom,
! [A2: set_o_a] : ( ord_less_eq_set_o_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_535_dual__order_Orefl,axiom,
! [A2: set_int_nat_a] : ( ord_le6072919132162605296_nat_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_536_dual__order_Orefl,axiom,
! [A2: set_int_a] : ( ord_le943418215940126601_int_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_537_dual__order_Orefl,axiom,
! [A2: set_a_nat_a] : ( ord_le2508512696544544866_nat_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_538_dual__order_Orefl,axiom,
! [A2: set_a_a] : ( ord_less_eq_set_a_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_539_order__refl,axiom,
! [X3: set_nat_a] : ( ord_le871467723717165285_nat_a @ X3 @ X3 ) ).
% order_refl
thf(fact_540_order__refl,axiom,
! [X3: set_a] : ( ord_less_eq_set_a @ X3 @ X3 ) ).
% order_refl
thf(fact_541_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_542_order__refl,axiom,
! [X3: int] : ( ord_less_eq_int @ X3 @ X3 ) ).
% order_refl
thf(fact_543_order__refl,axiom,
! [X3: real] : ( ord_less_eq_real @ X3 @ X3 ) ).
% order_refl
thf(fact_544_order__refl,axiom,
! [X3: set_o_a] : ( ord_less_eq_set_o_a @ X3 @ X3 ) ).
% order_refl
thf(fact_545_order__refl,axiom,
! [X3: set_int_nat_a] : ( ord_le6072919132162605296_nat_a @ X3 @ X3 ) ).
% order_refl
thf(fact_546_order__refl,axiom,
! [X3: set_int_a] : ( ord_le943418215940126601_int_a @ X3 @ X3 ) ).
% order_refl
thf(fact_547_order__refl,axiom,
! [X3: set_a_nat_a] : ( ord_le2508512696544544866_nat_a @ X3 @ X3 ) ).
% order_refl
thf(fact_548_order__refl,axiom,
! [X3: set_a_a] : ( ord_less_eq_set_a_a @ X3 @ X3 ) ).
% order_refl
thf(fact_549_image__Collect__subsetI,axiom,
! [P: a > $o,F: a > $o,B: set_o] :
( ! [X: a] :
( ( P @ X )
=> ( member_o @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_a_o @ F @ ( collect_a @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_550_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > int,B: set_int] :
( ! [X: nat] :
( ( P @ X )
=> ( member_int @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_int @ ( image_nat_int @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_551_image__Collect__subsetI,axiom,
! [P: int > $o,F: int > int,B: set_int] :
( ! [X: int] :
( ( P @ X )
=> ( member_int @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_int @ ( image_int_int @ F @ ( collect_int @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_552_image__Collect__subsetI,axiom,
! [P: a > $o,F: a > int,B: set_int] :
( ! [X: a] :
( ( P @ X )
=> ( member_int @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_int @ ( image_a_int @ F @ ( collect_a @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_553_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > nat,B: set_nat] :
( ! [X: nat] :
( ( P @ X )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_554_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > a,B: set_a] :
( ! [X: nat] :
( ( P @ X )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_555_image__Collect__subsetI,axiom,
! [P: int > $o,F: int > a,B: set_a] :
( ! [X: int] :
( ( P @ X )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_int_a @ F @ ( collect_int @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_556_image__Collect__subsetI,axiom,
! [P: a > $o,F: a > a,B: set_a] :
( ! [X: a] :
( ( P @ X )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ ( collect_a @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_557_image__Collect__subsetI,axiom,
! [P: ( nat > a ) > $o,F: ( nat > a ) > $o,B: set_o] :
( ! [X: nat > a] :
( ( P @ X )
=> ( member_o @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_nat_a_o @ F @ ( collect_nat_a @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_558_image__Collect__subsetI,axiom,
! [P: ( nat > a ) > $o,F: ( nat > a ) > int,B: set_int] :
( ! [X: nat > a] :
( ( P @ X )
=> ( member_int @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_int @ ( image_nat_a_int @ F @ ( collect_nat_a @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_559_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_560_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y6: nat] :
? [X2: nat] :
( Y6
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_561_surj__def,axiom,
! [F: nat > int] :
( ( ( image_nat_int @ F @ top_top_set_nat )
= top_top_set_int )
= ( ! [Y6: int] :
? [X2: nat] :
( Y6
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_562_surj__def,axiom,
! [F: nat > a] :
( ( ( image_nat_a @ F @ top_top_set_nat )
= top_top_set_a )
= ( ! [Y6: a] :
? [X2: nat] :
( Y6
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_563_surj__def,axiom,
! [F: int > nat] :
( ( ( image_int_nat @ F @ top_top_set_int )
= top_top_set_nat )
= ( ! [Y6: nat] :
? [X2: int] :
( Y6
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_564_surj__def,axiom,
! [F: int > int] :
( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
= ( ! [Y6: int] :
? [X2: int] :
( Y6
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_565_surj__def,axiom,
! [F: int > a] :
( ( ( image_int_a @ F @ top_top_set_int )
= top_top_set_a )
= ( ! [Y6: a] :
? [X2: int] :
( Y6
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_566_surj__def,axiom,
! [F: a > $o] :
( ( ( image_a_o @ F @ top_top_set_a )
= top_top_set_o )
= ( ! [Y6: $o] :
? [X2: a] :
( Y6
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_567_surj__def,axiom,
! [F: a > nat] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
= ( ! [Y6: nat] :
? [X2: a] :
( Y6
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_568_surj__def,axiom,
! [F: a > int] :
( ( ( image_a_int @ F @ top_top_set_a )
= top_top_set_int )
= ( ! [Y6: int] :
? [X2: a] :
( Y6
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_569_surj__def,axiom,
! [F: a > a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
= ( ! [Y6: a] :
? [X2: a] :
( Y6
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_570_surjI,axiom,
! [G: nat > nat,F: nat > nat] :
( ! [X: nat] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_571_surjI,axiom,
! [G: nat > int,F: int > nat] :
( ! [X: int] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_nat_int @ G @ top_top_set_nat )
= top_top_set_int ) ) ).
% surjI
thf(fact_572_surjI,axiom,
! [G: nat > a,F: a > nat] :
( ! [X: a] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_nat_a @ G @ top_top_set_nat )
= top_top_set_a ) ) ).
% surjI
thf(fact_573_surjI,axiom,
! [G: int > nat,F: nat > int] :
( ! [X: nat] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_int_nat @ G @ top_top_set_int )
= top_top_set_nat ) ) ).
% surjI
thf(fact_574_surjI,axiom,
! [G: int > int,F: int > int] :
( ! [X: int] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_int_int @ G @ top_top_set_int )
= top_top_set_int ) ) ).
% surjI
thf(fact_575_surjI,axiom,
! [G: int > a,F: a > int] :
( ! [X: a] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_int_a @ G @ top_top_set_int )
= top_top_set_a ) ) ).
% surjI
thf(fact_576_surjI,axiom,
! [G: a > $o,F: $o > a] :
( ! [X: $o] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_a_o @ G @ top_top_set_a )
= top_top_set_o ) ) ).
% surjI
thf(fact_577_surjI,axiom,
! [G: a > nat,F: nat > a] :
( ! [X: nat] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_a_nat @ G @ top_top_set_a )
= top_top_set_nat ) ) ).
% surjI
thf(fact_578_surjI,axiom,
! [G: a > int,F: int > a] :
( ! [X: int] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_a_int @ G @ top_top_set_a )
= top_top_set_int ) ) ).
% surjI
thf(fact_579_surjI,axiom,
! [G: a > a,F: a > a] :
( ! [X: a] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_a_a @ G @ top_top_set_a )
= top_top_set_a ) ) ).
% surjI
thf(fact_580_surjE,axiom,
! [F: nat > nat,Y2: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X: nat] :
( Y2
!= ( F @ X ) ) ) ).
% surjE
thf(fact_581_surjE,axiom,
! [F: nat > int,Y2: int] :
( ( ( image_nat_int @ F @ top_top_set_nat )
= top_top_set_int )
=> ~ ! [X: nat] :
( Y2
!= ( F @ X ) ) ) ).
% surjE
thf(fact_582_surjE,axiom,
! [F: nat > a,Y2: a] :
( ( ( image_nat_a @ F @ top_top_set_nat )
= top_top_set_a )
=> ~ ! [X: nat] :
( Y2
!= ( F @ X ) ) ) ).
% surjE
thf(fact_583_surjE,axiom,
! [F: int > nat,Y2: nat] :
( ( ( image_int_nat @ F @ top_top_set_int )
= top_top_set_nat )
=> ~ ! [X: int] :
( Y2
!= ( F @ X ) ) ) ).
% surjE
thf(fact_584_surjE,axiom,
! [F: int > int,Y2: int] :
( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
=> ~ ! [X: int] :
( Y2
!= ( F @ X ) ) ) ).
% surjE
thf(fact_585_surjE,axiom,
! [F: int > a,Y2: a] :
( ( ( image_int_a @ F @ top_top_set_int )
= top_top_set_a )
=> ~ ! [X: int] :
( Y2
!= ( F @ X ) ) ) ).
% surjE
thf(fact_586_surjE,axiom,
! [F: a > $o,Y2: $o] :
( ( ( image_a_o @ F @ top_top_set_a )
= top_top_set_o )
=> ~ ! [X: a] :
( Y2
= ( ~ ( F @ X ) ) ) ) ).
% surjE
thf(fact_587_surjE,axiom,
! [F: a > nat,Y2: nat] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
=> ~ ! [X: a] :
( Y2
!= ( F @ X ) ) ) ).
% surjE
thf(fact_588_surjE,axiom,
! [F: a > int,Y2: int] :
( ( ( image_a_int @ F @ top_top_set_a )
= top_top_set_int )
=> ~ ! [X: a] :
( Y2
!= ( F @ X ) ) ) ).
% surjE
thf(fact_589_surjE,axiom,
! [F: a > a,Y2: a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ~ ! [X: a] :
( Y2
!= ( F @ X ) ) ) ).
% surjE
thf(fact_590_psubsetI,axiom,
! [A: set_nat_a,B: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat_a @ A @ B ) ) ) ).
% psubsetI
thf(fact_591_psubsetI,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% psubsetI
thf(fact_592_psubsetI,axiom,
! [A: set_o_a,B: set_o_a] :
( ( ord_less_eq_set_o_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_o_a @ A @ B ) ) ) ).
% psubsetI
thf(fact_593_psubsetI,axiom,
! [A: set_int_nat_a,B: set_int_nat_a] :
( ( ord_le6072919132162605296_nat_a @ A @ B )
=> ( ( A != B )
=> ( ord_le2026899869173067772_nat_a @ A @ B ) ) ) ).
% psubsetI
thf(fact_594_psubsetI,axiom,
! [A: set_int_a,B: set_int_a] :
( ( ord_le943418215940126601_int_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_int_a @ A @ B ) ) ) ).
% psubsetI
thf(fact_595_psubsetI,axiom,
! [A: set_a_nat_a,B: set_a_nat_a] :
( ( ord_le2508512696544544866_nat_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a_nat_a @ A @ B ) ) ) ).
% psubsetI
thf(fact_596_psubsetI,axiom,
! [A: set_a_a,B: set_a_a] :
( ( ord_less_eq_set_a_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a_a @ A @ B ) ) ) ).
% psubsetI
thf(fact_597_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_598_top__set__def,axiom,
( top_top_set_int
= ( collect_int @ top_top_int_o ) ) ).
% top_set_def
thf(fact_599_top__set__def,axiom,
( top_top_set_nat_int
= ( collect_nat_int @ top_top_nat_int_o ) ) ).
% top_set_def
thf(fact_600_top__set__def,axiom,
( top_top_set_nat_a
= ( collect_nat_a @ top_top_nat_a_o ) ) ).
% top_set_def
thf(fact_601_top__set__def,axiom,
( top_top_set_int_nat
= ( collect_int_nat @ top_top_int_nat_o ) ) ).
% top_set_def
thf(fact_602_top__set__def,axiom,
( top_top_set_int_int
= ( collect_int_int @ top_top_int_int_o ) ) ).
% top_set_def
thf(fact_603_top__set__def,axiom,
( top_top_set_a
= ( collect_a @ top_top_a_o ) ) ).
% top_set_def
thf(fact_604_psubsetE,axiom,
! [A: set_nat_a,B: set_nat_a] :
( ( ord_less_set_nat_a @ A @ B )
=> ~ ( ( ord_le871467723717165285_nat_a @ A @ B )
=> ( ord_le871467723717165285_nat_a @ B @ A ) ) ) ).
% psubsetE
thf(fact_605_psubsetE,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ B @ A ) ) ) ).
% psubsetE
thf(fact_606_psubsetE,axiom,
! [A: set_o_a,B: set_o_a] :
( ( ord_less_set_o_a @ A @ B )
=> ~ ( ( ord_less_eq_set_o_a @ A @ B )
=> ( ord_less_eq_set_o_a @ B @ A ) ) ) ).
% psubsetE
thf(fact_607_psubsetE,axiom,
! [A: set_int_nat_a,B: set_int_nat_a] :
( ( ord_le2026899869173067772_nat_a @ A @ B )
=> ~ ( ( ord_le6072919132162605296_nat_a @ A @ B )
=> ( ord_le6072919132162605296_nat_a @ B @ A ) ) ) ).
% psubsetE
thf(fact_608_psubsetE,axiom,
! [A: set_int_a,B: set_int_a] :
( ( ord_less_set_int_a @ A @ B )
=> ~ ( ( ord_le943418215940126601_int_a @ A @ B )
=> ( ord_le943418215940126601_int_a @ B @ A ) ) ) ).
% psubsetE
thf(fact_609_psubsetE,axiom,
! [A: set_a_nat_a,B: set_a_nat_a] :
( ( ord_less_set_a_nat_a @ A @ B )
=> ~ ( ( ord_le2508512696544544866_nat_a @ A @ B )
=> ( ord_le2508512696544544866_nat_a @ B @ A ) ) ) ).
% psubsetE
thf(fact_610_psubsetE,axiom,
! [A: set_a_a,B: set_a_a] :
( ( ord_less_set_a_a @ A @ B )
=> ~ ( ( ord_less_eq_set_a_a @ A @ B )
=> ( ord_less_eq_set_a_a @ B @ A ) ) ) ).
% psubsetE
thf(fact_611_psubset__eq,axiom,
( ord_less_set_nat_a
= ( ^ [A3: set_nat_a,B3: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_612_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_613_psubset__eq,axiom,
( ord_less_set_o_a
= ( ^ [A3: set_o_a,B3: set_o_a] :
( ( ord_less_eq_set_o_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_614_psubset__eq,axiom,
( ord_le2026899869173067772_nat_a
= ( ^ [A3: set_int_nat_a,B3: set_int_nat_a] :
( ( ord_le6072919132162605296_nat_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_615_psubset__eq,axiom,
( ord_less_set_int_a
= ( ^ [A3: set_int_a,B3: set_int_a] :
( ( ord_le943418215940126601_int_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_616_psubset__eq,axiom,
( ord_less_set_a_nat_a
= ( ^ [A3: set_a_nat_a,B3: set_a_nat_a] :
( ( ord_le2508512696544544866_nat_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_617_psubset__eq,axiom,
( ord_less_set_a_a
= ( ^ [A3: set_a_a,B3: set_a_a] :
( ( ord_less_eq_set_a_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_618_psubset__imp__subset,axiom,
! [A: set_nat_a,B: set_nat_a] :
( ( ord_less_set_nat_a @ A @ B )
=> ( ord_le871467723717165285_nat_a @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_619_psubset__imp__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_620_psubset__imp__subset,axiom,
! [A: set_o_a,B: set_o_a] :
( ( ord_less_set_o_a @ A @ B )
=> ( ord_less_eq_set_o_a @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_621_psubset__imp__subset,axiom,
! [A: set_int_nat_a,B: set_int_nat_a] :
( ( ord_le2026899869173067772_nat_a @ A @ B )
=> ( ord_le6072919132162605296_nat_a @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_622_psubset__imp__subset,axiom,
! [A: set_int_a,B: set_int_a] :
( ( ord_less_set_int_a @ A @ B )
=> ( ord_le943418215940126601_int_a @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_623_psubset__imp__subset,axiom,
! [A: set_a_nat_a,B: set_a_nat_a] :
( ( ord_less_set_a_nat_a @ A @ B )
=> ( ord_le2508512696544544866_nat_a @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_624_psubset__imp__subset,axiom,
! [A: set_a_a,B: set_a_a] :
( ( ord_less_set_a_a @ A @ B )
=> ( ord_less_eq_set_a_a @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_625_psubset__subset__trans,axiom,
! [A: set_nat_a,B: set_nat_a,C: set_nat_a] :
( ( ord_less_set_nat_a @ A @ B )
=> ( ( ord_le871467723717165285_nat_a @ B @ C )
=> ( ord_less_set_nat_a @ A @ C ) ) ) ).
% psubset_subset_trans
thf(fact_626_psubset__subset__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% psubset_subset_trans
thf(fact_627_psubset__subset__trans,axiom,
! [A: set_o_a,B: set_o_a,C: set_o_a] :
( ( ord_less_set_o_a @ A @ B )
=> ( ( ord_less_eq_set_o_a @ B @ C )
=> ( ord_less_set_o_a @ A @ C ) ) ) ).
% psubset_subset_trans
thf(fact_628_psubset__subset__trans,axiom,
! [A: set_int_nat_a,B: set_int_nat_a,C: set_int_nat_a] :
( ( ord_le2026899869173067772_nat_a @ A @ B )
=> ( ( ord_le6072919132162605296_nat_a @ B @ C )
=> ( ord_le2026899869173067772_nat_a @ A @ C ) ) ) ).
% psubset_subset_trans
thf(fact_629_psubset__subset__trans,axiom,
! [A: set_int_a,B: set_int_a,C: set_int_a] :
( ( ord_less_set_int_a @ A @ B )
=> ( ( ord_le943418215940126601_int_a @ B @ C )
=> ( ord_less_set_int_a @ A @ C ) ) ) ).
% psubset_subset_trans
thf(fact_630_psubset__subset__trans,axiom,
! [A: set_a_nat_a,B: set_a_nat_a,C: set_a_nat_a] :
( ( ord_less_set_a_nat_a @ A @ B )
=> ( ( ord_le2508512696544544866_nat_a @ B @ C )
=> ( ord_less_set_a_nat_a @ A @ C ) ) ) ).
% psubset_subset_trans
thf(fact_631_psubset__subset__trans,axiom,
! [A: set_a_a,B: set_a_a,C: set_a_a] :
( ( ord_less_set_a_a @ A @ B )
=> ( ( ord_less_eq_set_a_a @ B @ C )
=> ( ord_less_set_a_a @ A @ C ) ) ) ).
% psubset_subset_trans
thf(fact_632_subset__not__subset__eq,axiom,
( ord_less_set_nat_a
= ( ^ [A3: set_nat_a,B3: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A3 @ B3 )
& ~ ( ord_le871467723717165285_nat_a @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_633_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ~ ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_634_subset__not__subset__eq,axiom,
( ord_less_set_o_a
= ( ^ [A3: set_o_a,B3: set_o_a] :
( ( ord_less_eq_set_o_a @ A3 @ B3 )
& ~ ( ord_less_eq_set_o_a @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_635_subset__not__subset__eq,axiom,
( ord_le2026899869173067772_nat_a
= ( ^ [A3: set_int_nat_a,B3: set_int_nat_a] :
( ( ord_le6072919132162605296_nat_a @ A3 @ B3 )
& ~ ( ord_le6072919132162605296_nat_a @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_636_subset__not__subset__eq,axiom,
( ord_less_set_int_a
= ( ^ [A3: set_int_a,B3: set_int_a] :
( ( ord_le943418215940126601_int_a @ A3 @ B3 )
& ~ ( ord_le943418215940126601_int_a @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_637_subset__not__subset__eq,axiom,
( ord_less_set_a_nat_a
= ( ^ [A3: set_a_nat_a,B3: set_a_nat_a] :
( ( ord_le2508512696544544866_nat_a @ A3 @ B3 )
& ~ ( ord_le2508512696544544866_nat_a @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_638_subset__not__subset__eq,axiom,
( ord_less_set_a_a
= ( ^ [A3: set_a_a,B3: set_a_a] :
( ( ord_less_eq_set_a_a @ A3 @ B3 )
& ~ ( ord_less_eq_set_a_a @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_639_subset__psubset__trans,axiom,
! [A: set_nat_a,B: set_nat_a,C: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A @ B )
=> ( ( ord_less_set_nat_a @ B @ C )
=> ( ord_less_set_nat_a @ A @ C ) ) ) ).
% subset_psubset_trans
thf(fact_640_subset__psubset__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% subset_psubset_trans
thf(fact_641_subset__psubset__trans,axiom,
! [A: set_o_a,B: set_o_a,C: set_o_a] :
( ( ord_less_eq_set_o_a @ A @ B )
=> ( ( ord_less_set_o_a @ B @ C )
=> ( ord_less_set_o_a @ A @ C ) ) ) ).
% subset_psubset_trans
thf(fact_642_subset__psubset__trans,axiom,
! [A: set_int_nat_a,B: set_int_nat_a,C: set_int_nat_a] :
( ( ord_le6072919132162605296_nat_a @ A @ B )
=> ( ( ord_le2026899869173067772_nat_a @ B @ C )
=> ( ord_le2026899869173067772_nat_a @ A @ C ) ) ) ).
% subset_psubset_trans
thf(fact_643_subset__psubset__trans,axiom,
! [A: set_int_a,B: set_int_a,C: set_int_a] :
( ( ord_le943418215940126601_int_a @ A @ B )
=> ( ( ord_less_set_int_a @ B @ C )
=> ( ord_less_set_int_a @ A @ C ) ) ) ).
% subset_psubset_trans
thf(fact_644_subset__psubset__trans,axiom,
! [A: set_a_nat_a,B: set_a_nat_a,C: set_a_nat_a] :
( ( ord_le2508512696544544866_nat_a @ A @ B )
=> ( ( ord_less_set_a_nat_a @ B @ C )
=> ( ord_less_set_a_nat_a @ A @ C ) ) ) ).
% subset_psubset_trans
thf(fact_645_subset__psubset__trans,axiom,
! [A: set_a_a,B: set_a_a,C: set_a_a] :
( ( ord_less_eq_set_a_a @ A @ B )
=> ( ( ord_less_set_a_a @ B @ C )
=> ( ord_less_set_a_a @ A @ C ) ) ) ).
% subset_psubset_trans
thf(fact_646_subset__iff__psubset__eq,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A3: set_nat_a,B3: set_nat_a] :
( ( ord_less_set_nat_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_647_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_648_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_o_a
= ( ^ [A3: set_o_a,B3: set_o_a] :
( ( ord_less_set_o_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_649_subset__iff__psubset__eq,axiom,
( ord_le6072919132162605296_nat_a
= ( ^ [A3: set_int_nat_a,B3: set_int_nat_a] :
( ( ord_le2026899869173067772_nat_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_650_subset__iff__psubset__eq,axiom,
( ord_le943418215940126601_int_a
= ( ^ [A3: set_int_a,B3: set_int_a] :
( ( ord_less_set_int_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_651_subset__iff__psubset__eq,axiom,
( ord_le2508512696544544866_nat_a
= ( ^ [A3: set_a_nat_a,B3: set_a_nat_a] :
( ( ord_less_set_a_nat_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_652_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a_a
= ( ^ [A3: set_a_a,B3: set_a_a] :
( ( ord_less_set_a_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_653_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_654_nle__le,axiom,
! [A2: int,B2: int] :
( ( ~ ( ord_less_eq_int @ A2 @ B2 ) )
= ( ( ord_less_eq_int @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_655_nle__le,axiom,
! [A2: real,B2: real] :
( ( ~ ( ord_less_eq_real @ A2 @ B2 ) )
= ( ( ord_less_eq_real @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_656_le__cases3,axiom,
! [X3: nat,Y2: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_657_le__cases3,axiom,
! [X3: int,Y2: int,Z2: int] :
( ( ( ord_less_eq_int @ X3 @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y2 @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X3 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ X3 ) )
=> ( ( ( ord_less_eq_int @ Y2 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_658_le__cases3,axiom,
! [X3: real,Y2: real,Z2: real] :
( ( ( ord_less_eq_real @ X3 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X3 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ X3 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_659_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat_a,Z: set_nat_a] : ( Y3 = Z ) )
= ( ^ [X2: set_nat_a,Y6: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ X2 @ Y6 )
& ( ord_le871467723717165285_nat_a @ Y6 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_660_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [X2: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y6 )
& ( ord_less_eq_set_a @ Y6 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_661_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [X2: nat,Y6: nat] :
( ( ord_less_eq_nat @ X2 @ Y6 )
& ( ord_less_eq_nat @ Y6 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_662_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [X2: int,Y6: int] :
( ( ord_less_eq_int @ X2 @ Y6 )
& ( ord_less_eq_int @ Y6 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_663_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [X2: real,Y6: real] :
( ( ord_less_eq_real @ X2 @ Y6 )
& ( ord_less_eq_real @ Y6 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_664_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_o_a,Z: set_o_a] : ( Y3 = Z ) )
= ( ^ [X2: set_o_a,Y6: set_o_a] :
( ( ord_less_eq_set_o_a @ X2 @ Y6 )
& ( ord_less_eq_set_o_a @ Y6 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_665_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_int_nat_a,Z: set_int_nat_a] : ( Y3 = Z ) )
= ( ^ [X2: set_int_nat_a,Y6: set_int_nat_a] :
( ( ord_le6072919132162605296_nat_a @ X2 @ Y6 )
& ( ord_le6072919132162605296_nat_a @ Y6 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_666_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_int_a,Z: set_int_a] : ( Y3 = Z ) )
= ( ^ [X2: set_int_a,Y6: set_int_a] :
( ( ord_le943418215940126601_int_a @ X2 @ Y6 )
& ( ord_le943418215940126601_int_a @ Y6 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_667_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a_nat_a,Z: set_a_nat_a] : ( Y3 = Z ) )
= ( ^ [X2: set_a_nat_a,Y6: set_a_nat_a] :
( ( ord_le2508512696544544866_nat_a @ X2 @ Y6 )
& ( ord_le2508512696544544866_nat_a @ Y6 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_668_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a_a,Z: set_a_a] : ( Y3 = Z ) )
= ( ^ [X2: set_a_a,Y6: set_a_a] :
( ( ord_less_eq_set_a_a @ X2 @ Y6 )
& ( ord_less_eq_set_a_a @ Y6 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_669_ord__eq__le__trans,axiom,
! [A2: set_nat_a,B2: set_nat_a,C2: set_nat_a] :
( ( A2 = B2 )
=> ( ( ord_le871467723717165285_nat_a @ B2 @ C2 )
=> ( ord_le871467723717165285_nat_a @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_670_ord__eq__le__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_671_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_672_ord__eq__le__trans,axiom,
! [A2: int,B2: int,C2: int] :
( ( A2 = B2 )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ord_less_eq_int @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_673_ord__eq__le__trans,axiom,
! [A2: real,B2: real,C2: real] :
( ( A2 = B2 )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ord_less_eq_real @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_674_ord__eq__le__trans,axiom,
! [A2: set_o_a,B2: set_o_a,C2: set_o_a] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_o_a @ B2 @ C2 )
=> ( ord_less_eq_set_o_a @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_675_ord__eq__le__trans,axiom,
! [A2: set_int_nat_a,B2: set_int_nat_a,C2: set_int_nat_a] :
( ( A2 = B2 )
=> ( ( ord_le6072919132162605296_nat_a @ B2 @ C2 )
=> ( ord_le6072919132162605296_nat_a @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_676_ord__eq__le__trans,axiom,
! [A2: set_int_a,B2: set_int_a,C2: set_int_a] :
( ( A2 = B2 )
=> ( ( ord_le943418215940126601_int_a @ B2 @ C2 )
=> ( ord_le943418215940126601_int_a @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_677_ord__eq__le__trans,axiom,
! [A2: set_a_nat_a,B2: set_a_nat_a,C2: set_a_nat_a] :
( ( A2 = B2 )
=> ( ( ord_le2508512696544544866_nat_a @ B2 @ C2 )
=> ( ord_le2508512696544544866_nat_a @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_678_ord__eq__le__trans,axiom,
! [A2: set_a_a,B2: set_a_a,C2: set_a_a] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_a_a @ B2 @ C2 )
=> ( ord_less_eq_set_a_a @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_679_ord__le__eq__trans,axiom,
! [A2: set_nat_a,B2: set_nat_a,C2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le871467723717165285_nat_a @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_680_ord__le__eq__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_681_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_682_ord__le__eq__trans,axiom,
! [A2: int,B2: int,C2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_int @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_683_ord__le__eq__trans,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_real @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_684_ord__le__eq__trans,axiom,
! [A2: set_o_a,B2: set_o_a,C2: set_o_a] :
( ( ord_less_eq_set_o_a @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_set_o_a @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_685_ord__le__eq__trans,axiom,
! [A2: set_int_nat_a,B2: set_int_nat_a,C2: set_int_nat_a] :
( ( ord_le6072919132162605296_nat_a @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le6072919132162605296_nat_a @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_686_ord__le__eq__trans,axiom,
! [A2: set_int_a,B2: set_int_a,C2: set_int_a] :
( ( ord_le943418215940126601_int_a @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le943418215940126601_int_a @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_687_ord__le__eq__trans,axiom,
! [A2: set_a_nat_a,B2: set_a_nat_a,C2: set_a_nat_a] :
( ( ord_le2508512696544544866_nat_a @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le2508512696544544866_nat_a @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_688_ord__le__eq__trans,axiom,
! [A2: set_a_a,B2: set_a_a,C2: set_a_a] :
( ( ord_less_eq_set_a_a @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_set_a_a @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_689_order__antisym,axiom,
! [X3: set_nat_a,Y2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ X3 @ Y2 )
=> ( ( ord_le871467723717165285_nat_a @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_690_order__antisym,axiom,
! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_691_order__antisym,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_692_order__antisym,axiom,
! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_693_order__antisym,axiom,
! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_694_order__antisym,axiom,
! [X3: set_o_a,Y2: set_o_a] :
( ( ord_less_eq_set_o_a @ X3 @ Y2 )
=> ( ( ord_less_eq_set_o_a @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_695_order__antisym,axiom,
! [X3: set_int_nat_a,Y2: set_int_nat_a] :
( ( ord_le6072919132162605296_nat_a @ X3 @ Y2 )
=> ( ( ord_le6072919132162605296_nat_a @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_696_order__antisym,axiom,
! [X3: set_int_a,Y2: set_int_a] :
( ( ord_le943418215940126601_int_a @ X3 @ Y2 )
=> ( ( ord_le943418215940126601_int_a @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_697_order__antisym,axiom,
! [X3: set_a_nat_a,Y2: set_a_nat_a] :
( ( ord_le2508512696544544866_nat_a @ X3 @ Y2 )
=> ( ( ord_le2508512696544544866_nat_a @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_698_order__antisym,axiom,
! [X3: set_a_a,Y2: set_a_a] :
( ( ord_less_eq_set_a_a @ X3 @ Y2 )
=> ( ( ord_less_eq_set_a_a @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_699_order_Otrans,axiom,
! [A2: set_nat_a,B2: set_nat_a,C2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A2 @ B2 )
=> ( ( ord_le871467723717165285_nat_a @ B2 @ C2 )
=> ( ord_le871467723717165285_nat_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_700_order_Otrans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_701_order_Otrans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_702_order_Otrans,axiom,
! [A2: int,B2: int,C2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ord_less_eq_int @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_703_order_Otrans,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ord_less_eq_real @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_704_order_Otrans,axiom,
! [A2: set_o_a,B2: set_o_a,C2: set_o_a] :
( ( ord_less_eq_set_o_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_o_a @ B2 @ C2 )
=> ( ord_less_eq_set_o_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_705_order_Otrans,axiom,
! [A2: set_int_nat_a,B2: set_int_nat_a,C2: set_int_nat_a] :
( ( ord_le6072919132162605296_nat_a @ A2 @ B2 )
=> ( ( ord_le6072919132162605296_nat_a @ B2 @ C2 )
=> ( ord_le6072919132162605296_nat_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_706_order_Otrans,axiom,
! [A2: set_int_a,B2: set_int_a,C2: set_int_a] :
( ( ord_le943418215940126601_int_a @ A2 @ B2 )
=> ( ( ord_le943418215940126601_int_a @ B2 @ C2 )
=> ( ord_le943418215940126601_int_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_707_order_Otrans,axiom,
! [A2: set_a_nat_a,B2: set_a_nat_a,C2: set_a_nat_a] :
( ( ord_le2508512696544544866_nat_a @ A2 @ B2 )
=> ( ( ord_le2508512696544544866_nat_a @ B2 @ C2 )
=> ( ord_le2508512696544544866_nat_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_708_order_Otrans,axiom,
! [A2: set_a_a,B2: set_a_a,C2: set_a_a] :
( ( ord_less_eq_set_a_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a_a @ B2 @ C2 )
=> ( ord_less_eq_set_a_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_709_order__trans,axiom,
! [X3: set_nat_a,Y2: set_nat_a,Z2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ X3 @ Y2 )
=> ( ( ord_le871467723717165285_nat_a @ Y2 @ Z2 )
=> ( ord_le871467723717165285_nat_a @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_710_order__trans,axiom,
! [X3: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ Z2 )
=> ( ord_less_eq_set_a @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_711_order__trans,axiom,
! [X3: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_712_order__trans,axiom,
! [X3: int,Y2: int,Z2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z2 )
=> ( ord_less_eq_int @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_713_order__trans,axiom,
! [X3: real,Y2: real,Z2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ( ord_less_eq_real @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_714_order__trans,axiom,
! [X3: set_o_a,Y2: set_o_a,Z2: set_o_a] :
( ( ord_less_eq_set_o_a @ X3 @ Y2 )
=> ( ( ord_less_eq_set_o_a @ Y2 @ Z2 )
=> ( ord_less_eq_set_o_a @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_715_order__trans,axiom,
! [X3: set_int_nat_a,Y2: set_int_nat_a,Z2: set_int_nat_a] :
( ( ord_le6072919132162605296_nat_a @ X3 @ Y2 )
=> ( ( ord_le6072919132162605296_nat_a @ Y2 @ Z2 )
=> ( ord_le6072919132162605296_nat_a @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_716_order__trans,axiom,
! [X3: set_int_a,Y2: set_int_a,Z2: set_int_a] :
( ( ord_le943418215940126601_int_a @ X3 @ Y2 )
=> ( ( ord_le943418215940126601_int_a @ Y2 @ Z2 )
=> ( ord_le943418215940126601_int_a @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_717_order__trans,axiom,
! [X3: set_a_nat_a,Y2: set_a_nat_a,Z2: set_a_nat_a] :
( ( ord_le2508512696544544866_nat_a @ X3 @ Y2 )
=> ( ( ord_le2508512696544544866_nat_a @ Y2 @ Z2 )
=> ( ord_le2508512696544544866_nat_a @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_718_order__trans,axiom,
! [X3: set_a_a,Y2: set_a_a,Z2: set_a_a] :
( ( ord_less_eq_set_a_a @ X3 @ Y2 )
=> ( ( ord_less_eq_set_a_a @ Y2 @ Z2 )
=> ( ord_less_eq_set_a_a @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_719_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_720_linorder__wlog,axiom,
! [P: int > int > $o,A2: int,B2: int] :
( ! [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: int,B4: int] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_721_linorder__wlog,axiom,
! [P: real > real > $o,A2: real,B2: real] :
( ! [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: real,B4: real] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_722_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_nat_a,Z: set_nat_a] : ( Y3 = Z ) )
= ( ^ [A5: set_nat_a,B5: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ B5 @ A5 )
& ( ord_le871467723717165285_nat_a @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_723_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A5 )
& ( ord_less_eq_set_a @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_724_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_725_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A5: int,B5: int] :
( ( ord_less_eq_int @ B5 @ A5 )
& ( ord_less_eq_int @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_726_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [A5: real,B5: real] :
( ( ord_less_eq_real @ B5 @ A5 )
& ( ord_less_eq_real @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_727_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_o_a,Z: set_o_a] : ( Y3 = Z ) )
= ( ^ [A5: set_o_a,B5: set_o_a] :
( ( ord_less_eq_set_o_a @ B5 @ A5 )
& ( ord_less_eq_set_o_a @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_728_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_int_nat_a,Z: set_int_nat_a] : ( Y3 = Z ) )
= ( ^ [A5: set_int_nat_a,B5: set_int_nat_a] :
( ( ord_le6072919132162605296_nat_a @ B5 @ A5 )
& ( ord_le6072919132162605296_nat_a @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_729_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_int_a,Z: set_int_a] : ( Y3 = Z ) )
= ( ^ [A5: set_int_a,B5: set_int_a] :
( ( ord_le943418215940126601_int_a @ B5 @ A5 )
& ( ord_le943418215940126601_int_a @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_730_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_a_nat_a,Z: set_a_nat_a] : ( Y3 = Z ) )
= ( ^ [A5: set_a_nat_a,B5: set_a_nat_a] :
( ( ord_le2508512696544544866_nat_a @ B5 @ A5 )
& ( ord_le2508512696544544866_nat_a @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_731_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_a_a,Z: set_a_a] : ( Y3 = Z ) )
= ( ^ [A5: set_a_a,B5: set_a_a] :
( ( ord_less_eq_set_a_a @ B5 @ A5 )
& ( ord_less_eq_set_a_a @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_732_dual__order_Oantisym,axiom,
! [B2: set_nat_a,A2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ B2 @ A2 )
=> ( ( ord_le871467723717165285_nat_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_733_dual__order_Oantisym,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_734_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_735_dual__order_Oantisym,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_736_dual__order_Oantisym,axiom,
! [B2: real,A2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_737_dual__order_Oantisym,axiom,
! [B2: set_o_a,A2: set_o_a] :
( ( ord_less_eq_set_o_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_o_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_738_dual__order_Oantisym,axiom,
! [B2: set_int_nat_a,A2: set_int_nat_a] :
( ( ord_le6072919132162605296_nat_a @ B2 @ A2 )
=> ( ( ord_le6072919132162605296_nat_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_739_dual__order_Oantisym,axiom,
! [B2: set_int_a,A2: set_int_a] :
( ( ord_le943418215940126601_int_a @ B2 @ A2 )
=> ( ( ord_le943418215940126601_int_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_740_dual__order_Oantisym,axiom,
! [B2: set_a_nat_a,A2: set_a_nat_a] :
( ( ord_le2508512696544544866_nat_a @ B2 @ A2 )
=> ( ( ord_le2508512696544544866_nat_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_741_dual__order_Oantisym,axiom,
! [B2: set_a_a,A2: set_a_a] :
( ( ord_less_eq_set_a_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_742_dual__order_Otrans,axiom,
! [B2: set_nat_a,A2: set_nat_a,C2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ B2 @ A2 )
=> ( ( ord_le871467723717165285_nat_a @ C2 @ B2 )
=> ( ord_le871467723717165285_nat_a @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_743_dual__order_Otrans,axiom,
! [B2: set_a,A2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C2 @ B2 )
=> ( ord_less_eq_set_a @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_744_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_745_dual__order_Otrans,axiom,
! [B2: int,A2: int,C2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C2 @ B2 )
=> ( ord_less_eq_int @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_746_dual__order_Otrans,axiom,
! [B2: real,A2: real,C2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ C2 @ B2 )
=> ( ord_less_eq_real @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_747_dual__order_Otrans,axiom,
! [B2: set_o_a,A2: set_o_a,C2: set_o_a] :
( ( ord_less_eq_set_o_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_o_a @ C2 @ B2 )
=> ( ord_less_eq_set_o_a @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_748_dual__order_Otrans,axiom,
! [B2: set_int_nat_a,A2: set_int_nat_a,C2: set_int_nat_a] :
( ( ord_le6072919132162605296_nat_a @ B2 @ A2 )
=> ( ( ord_le6072919132162605296_nat_a @ C2 @ B2 )
=> ( ord_le6072919132162605296_nat_a @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_749_dual__order_Otrans,axiom,
! [B2: set_int_a,A2: set_int_a,C2: set_int_a] :
( ( ord_le943418215940126601_int_a @ B2 @ A2 )
=> ( ( ord_le943418215940126601_int_a @ C2 @ B2 )
=> ( ord_le943418215940126601_int_a @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_750_dual__order_Otrans,axiom,
! [B2: set_a_nat_a,A2: set_a_nat_a,C2: set_a_nat_a] :
( ( ord_le2508512696544544866_nat_a @ B2 @ A2 )
=> ( ( ord_le2508512696544544866_nat_a @ C2 @ B2 )
=> ( ord_le2508512696544544866_nat_a @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_751_dual__order_Otrans,axiom,
! [B2: set_a_a,A2: set_a_a,C2: set_a_a] :
( ( ord_less_eq_set_a_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a_a @ C2 @ B2 )
=> ( ord_less_eq_set_a_a @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_752_antisym,axiom,
! [A2: set_nat_a,B2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A2 @ B2 )
=> ( ( ord_le871467723717165285_nat_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_753_antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_754_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_755_antisym,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_756_antisym,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_757_antisym,axiom,
! [A2: set_o_a,B2: set_o_a] :
( ( ord_less_eq_set_o_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_o_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_758_antisym,axiom,
! [A2: set_int_nat_a,B2: set_int_nat_a] :
( ( ord_le6072919132162605296_nat_a @ A2 @ B2 )
=> ( ( ord_le6072919132162605296_nat_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_759_antisym,axiom,
! [A2: set_int_a,B2: set_int_a] :
( ( ord_le943418215940126601_int_a @ A2 @ B2 )
=> ( ( ord_le943418215940126601_int_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_760_antisym,axiom,
! [A2: set_a_nat_a,B2: set_a_nat_a] :
( ( ord_le2508512696544544866_nat_a @ A2 @ B2 )
=> ( ( ord_le2508512696544544866_nat_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_761_antisym,axiom,
! [A2: set_a_a,B2: set_a_a] :
( ( ord_less_eq_set_a_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_762_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat_a,Z: set_nat_a] : ( Y3 = Z ) )
= ( ^ [A5: set_nat_a,B5: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A5 @ B5 )
& ( ord_le871467723717165285_nat_a @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_763_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_764_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_765_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
& ( ord_less_eq_int @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_766_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [A5: real,B5: real] :
( ( ord_less_eq_real @ A5 @ B5 )
& ( ord_less_eq_real @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_767_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_o_a,Z: set_o_a] : ( Y3 = Z ) )
= ( ^ [A5: set_o_a,B5: set_o_a] :
( ( ord_less_eq_set_o_a @ A5 @ B5 )
& ( ord_less_eq_set_o_a @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_768_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_int_nat_a,Z: set_int_nat_a] : ( Y3 = Z ) )
= ( ^ [A5: set_int_nat_a,B5: set_int_nat_a] :
( ( ord_le6072919132162605296_nat_a @ A5 @ B5 )
& ( ord_le6072919132162605296_nat_a @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_769_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_int_a,Z: set_int_a] : ( Y3 = Z ) )
= ( ^ [A5: set_int_a,B5: set_int_a] :
( ( ord_le943418215940126601_int_a @ A5 @ B5 )
& ( ord_le943418215940126601_int_a @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_770_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a_nat_a,Z: set_a_nat_a] : ( Y3 = Z ) )
= ( ^ [A5: set_a_nat_a,B5: set_a_nat_a] :
( ( ord_le2508512696544544866_nat_a @ A5 @ B5 )
& ( ord_le2508512696544544866_nat_a @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_771_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a_a,Z: set_a_a] : ( Y3 = Z ) )
= ( ^ [A5: set_a_a,B5: set_a_a] :
( ( ord_less_eq_set_a_a @ A5 @ B5 )
& ( ord_less_eq_set_a_a @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_772_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_773_order__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C2: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_774_order__subst1,axiom,
! [A2: nat,F: real > nat,B2: real,C2: real] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_775_order__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C2: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_776_order__subst1,axiom,
! [A2: int,F: int > int,B2: int,C2: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_777_order__subst1,axiom,
! [A2: int,F: real > int,B2: real,C2: real] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_778_order__subst1,axiom,
! [A2: real,F: nat > real,B2: nat,C2: nat] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_779_order__subst1,axiom,
! [A2: real,F: int > real,B2: int,C2: int] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_780_order__subst1,axiom,
! [A2: real,F: real > real,B2: real,C2: real] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_781_order__subst1,axiom,
! [A2: set_a,F: nat > set_a,B2: nat,C2: nat] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_782_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_783_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_784_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_785_order__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_786_order__subst2,axiom,
! [A2: int,B2: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C2 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_787_order__subst2,axiom,
! [A2: int,B2: int,F: int > real,C2: real] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C2 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_788_order__subst2,axiom,
! [A2: real,B2: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_789_order__subst2,axiom,
! [A2: real,B2: real,F: real > int,C2: int] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C2 )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_790_order__subst2,axiom,
! [A2: real,B2: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C2 )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_791_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_792_order__eq__refl,axiom,
! [X3: set_nat_a,Y2: set_nat_a] :
( ( X3 = Y2 )
=> ( ord_le871467723717165285_nat_a @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_793_order__eq__refl,axiom,
! [X3: set_a,Y2: set_a] :
( ( X3 = Y2 )
=> ( ord_less_eq_set_a @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_794_order__eq__refl,axiom,
! [X3: nat,Y2: nat] :
( ( X3 = Y2 )
=> ( ord_less_eq_nat @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_795_order__eq__refl,axiom,
! [X3: int,Y2: int] :
( ( X3 = Y2 )
=> ( ord_less_eq_int @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_796_order__eq__refl,axiom,
! [X3: real,Y2: real] :
( ( X3 = Y2 )
=> ( ord_less_eq_real @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_797_order__eq__refl,axiom,
! [X3: set_o_a,Y2: set_o_a] :
( ( X3 = Y2 )
=> ( ord_less_eq_set_o_a @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_798_order__eq__refl,axiom,
! [X3: set_int_nat_a,Y2: set_int_nat_a] :
( ( X3 = Y2 )
=> ( ord_le6072919132162605296_nat_a @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_799_order__eq__refl,axiom,
! [X3: set_int_a,Y2: set_int_a] :
( ( X3 = Y2 )
=> ( ord_le943418215940126601_int_a @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_800_order__eq__refl,axiom,
! [X3: set_a_nat_a,Y2: set_a_nat_a] :
( ( X3 = Y2 )
=> ( ord_le2508512696544544866_nat_a @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_801_order__eq__refl,axiom,
! [X3: set_a_a,Y2: set_a_a] :
( ( X3 = Y2 )
=> ( ord_less_eq_set_a_a @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_802_linorder__linear,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% linorder_linear
thf(fact_803_linorder__linear,axiom,
! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
| ( ord_less_eq_int @ Y2 @ X3 ) ) ).
% linorder_linear
thf(fact_804_linorder__linear,axiom,
! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
| ( ord_less_eq_real @ Y2 @ X3 ) ) ).
% linorder_linear
thf(fact_805_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_806_ord__eq__le__subst,axiom,
! [A2: int,F: nat > int,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_807_ord__eq__le__subst,axiom,
! [A2: real,F: nat > real,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_808_ord__eq__le__subst,axiom,
! [A2: nat,F: int > nat,B2: int,C2: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_809_ord__eq__le__subst,axiom,
! [A2: int,F: int > int,B2: int,C2: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_810_ord__eq__le__subst,axiom,
! [A2: real,F: int > real,B2: int,C2: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_811_ord__eq__le__subst,axiom,
! [A2: nat,F: real > nat,B2: real,C2: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_812_ord__eq__le__subst,axiom,
! [A2: int,F: real > int,B2: real,C2: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_813_ord__eq__le__subst,axiom,
! [A2: real,F: real > real,B2: real,C2: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_814_ord__eq__le__subst,axiom,
! [A2: nat,F: set_a > nat,B2: set_a,C2: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_815_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_816_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_817_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_818_ord__le__eq__subst,axiom,
! [A2: int,B2: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_819_ord__le__eq__subst,axiom,
! [A2: int,B2: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_820_ord__le__eq__subst,axiom,
! [A2: int,B2: int,F: int > real,C2: real] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_821_ord__le__eq__subst,axiom,
! [A2: real,B2: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_822_ord__le__eq__subst,axiom,
! [A2: real,B2: real,F: real > int,C2: int] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_823_ord__le__eq__subst,axiom,
! [A2: real,B2: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_824_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_825_linorder__le__cases,axiom,
! [X3: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% linorder_le_cases
thf(fact_826_linorder__le__cases,axiom,
! [X3: int,Y2: int] :
( ~ ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X3 ) ) ).
% linorder_le_cases
thf(fact_827_linorder__le__cases,axiom,
! [X3: real,Y2: real] :
( ~ ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X3 ) ) ).
% linorder_le_cases
thf(fact_828_order__antisym__conv,axiom,
! [Y2: set_nat_a,X3: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ Y2 @ X3 )
=> ( ( ord_le871467723717165285_nat_a @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_829_order__antisym__conv,axiom,
! [Y2: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X3 )
=> ( ( ord_less_eq_set_a @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_830_order__antisym__conv,axiom,
! [Y2: nat,X3: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_831_order__antisym__conv,axiom,
! [Y2: int,X3: int] :
( ( ord_less_eq_int @ Y2 @ X3 )
=> ( ( ord_less_eq_int @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_832_order__antisym__conv,axiom,
! [Y2: real,X3: real] :
( ( ord_less_eq_real @ Y2 @ X3 )
=> ( ( ord_less_eq_real @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_833_order__antisym__conv,axiom,
! [Y2: set_o_a,X3: set_o_a] :
( ( ord_less_eq_set_o_a @ Y2 @ X3 )
=> ( ( ord_less_eq_set_o_a @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_834_order__antisym__conv,axiom,
! [Y2: set_int_nat_a,X3: set_int_nat_a] :
( ( ord_le6072919132162605296_nat_a @ Y2 @ X3 )
=> ( ( ord_le6072919132162605296_nat_a @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_835_order__antisym__conv,axiom,
! [Y2: set_int_a,X3: set_int_a] :
( ( ord_le943418215940126601_int_a @ Y2 @ X3 )
=> ( ( ord_le943418215940126601_int_a @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_836_order__antisym__conv,axiom,
! [Y2: set_a_nat_a,X3: set_a_nat_a] :
( ( ord_le2508512696544544866_nat_a @ Y2 @ X3 )
=> ( ( ord_le2508512696544544866_nat_a @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_837_order__antisym__conv,axiom,
! [Y2: set_a_a,X3: set_a_a] :
( ( ord_less_eq_set_a_a @ Y2 @ X3 )
=> ( ( ord_less_eq_set_a_a @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_838_order__less__imp__not__less,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_839_order__less__imp__not__less,axiom,
! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_840_order__less__imp__not__less,axiom,
! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_841_order__less__imp__not__eq2,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( Y2 != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_842_order__less__imp__not__eq2,axiom,
! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( Y2 != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_843_order__less__imp__not__eq2,axiom,
! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( Y2 != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_844_order__less__imp__not__eq,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_845_order__less__imp__not__eq,axiom,
! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_846_order__less__imp__not__eq,axiom,
! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_847_linorder__less__linear,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
| ( X3 = Y2 )
| ( ord_less_nat @ Y2 @ X3 ) ) ).
% linorder_less_linear
thf(fact_848_linorder__less__linear,axiom,
! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
| ( X3 = Y2 )
| ( ord_less_int @ Y2 @ X3 ) ) ).
% linorder_less_linear
thf(fact_849_linorder__less__linear,axiom,
! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
| ( X3 = Y2 )
| ( ord_less_real @ Y2 @ X3 ) ) ).
% linorder_less_linear
thf(fact_850_order__less__imp__triv,axiom,
! [X3: nat,Y2: nat,P: $o] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_851_order__less__imp__triv,axiom,
! [X3: int,Y2: int,P: $o] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ( ord_less_int @ Y2 @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_852_order__less__imp__triv,axiom,
! [X3: real,Y2: real,P: $o] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ( ord_less_real @ Y2 @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_853_order__less__not__sym,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X3 ) ) ).
% order_less_not_sym
thf(fact_854_order__less__not__sym,axiom,
! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X3 ) ) ).
% order_less_not_sym
thf(fact_855_order__less__not__sym,axiom,
! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X3 ) ) ).
% order_less_not_sym
thf(fact_856_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_857_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_858_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_859_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_860_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > int,C2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C2 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_861_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > real,C2: real] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C2 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_862_order__less__subst2,axiom,
! [A2: real,B2: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_863_order__less__subst2,axiom,
! [A2: real,B2: real,F: real > int,C2: int] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C2 )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_864_order__less__subst2,axiom,
! [A2: real,B2: real,F: real > real,C2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C2 )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_865_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_866_order__less__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C2: int] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C2 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_867_order__less__subst1,axiom,
! [A2: nat,F: real > nat,B2: real,C2: real] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_868_order__less__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C2: nat] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_869_order__less__subst1,axiom,
! [A2: int,F: int > int,B2: int,C2: int] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C2 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_870_order__less__subst1,axiom,
! [A2: int,F: real > int,B2: real,C2: real] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_871_order__less__subst1,axiom,
! [A2: real,F: nat > real,B2: nat,C2: nat] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_872_order__less__subst1,axiom,
! [A2: real,F: int > real,B2: int,C2: int] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C2 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_873_order__less__subst1,axiom,
! [A2: real,F: real > real,B2: real,C2: real] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_874_order__less__irrefl,axiom,
! [X3: nat] :
~ ( ord_less_nat @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_875_order__less__irrefl,axiom,
! [X3: int] :
~ ( ord_less_int @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_876_order__less__irrefl,axiom,
! [X3: real] :
~ ( ord_less_real @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_877_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_878_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_879_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_880_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_881_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > int,C2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_882_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > real,C2: real] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_883_ord__less__eq__subst,axiom,
! [A2: real,B2: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_884_ord__less__eq__subst,axiom,
! [A2: real,B2: real,F: real > int,C2: int] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_885_ord__less__eq__subst,axiom,
! [A2: real,B2: real,F: real > real,C2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_886_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_887_ord__eq__less__subst,axiom,
! [A2: int,F: nat > int,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_888_ord__eq__less__subst,axiom,
! [A2: real,F: nat > real,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_889_ord__eq__less__subst,axiom,
! [A2: nat,F: int > nat,B2: int,C2: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C2 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_890_ord__eq__less__subst,axiom,
! [A2: int,F: int > int,B2: int,C2: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C2 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_891_ord__eq__less__subst,axiom,
! [A2: real,F: int > real,B2: int,C2: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C2 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_892_ord__eq__less__subst,axiom,
! [A2: nat,F: real > nat,B2: real,C2: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_893_ord__eq__less__subst,axiom,
! [A2: int,F: real > int,B2: real,C2: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_894_ord__eq__less__subst,axiom,
! [A2: real,F: real > real,B2: real,C2: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_895_order__less__trans,axiom,
! [X3: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X3 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_896_order__less__trans,axiom,
! [X3: int,Y2: int,Z2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z2 )
=> ( ord_less_int @ X3 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_897_order__less__trans,axiom,
! [X3: real,Y2: real,Z2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z2 )
=> ( ord_less_real @ X3 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_898_order__less__asym_H,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_899_order__less__asym_H,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ~ ( ord_less_int @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_900_order__less__asym_H,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ~ ( ord_less_real @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_901_linorder__neq__iff,axiom,
! [X3: nat,Y2: nat] :
( ( X3 != Y2 )
= ( ( ord_less_nat @ X3 @ Y2 )
| ( ord_less_nat @ Y2 @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_902_linorder__neq__iff,axiom,
! [X3: int,Y2: int] :
( ( X3 != Y2 )
= ( ( ord_less_int @ X3 @ Y2 )
| ( ord_less_int @ Y2 @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_903_linorder__neq__iff,axiom,
! [X3: real,Y2: real] :
( ( X3 != Y2 )
= ( ( ord_less_real @ X3 @ Y2 )
| ( ord_less_real @ Y2 @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_904_order__less__asym,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X3 ) ) ).
% order_less_asym
thf(fact_905_order__less__asym,axiom,
! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X3 ) ) ).
% order_less_asym
thf(fact_906_order__less__asym,axiom,
! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X3 ) ) ).
% order_less_asym
thf(fact_907_linorder__neqE,axiom,
! [X3: nat,Y2: nat] :
( ( X3 != Y2 )
=> ( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ Y2 @ X3 ) ) ) ).
% linorder_neqE
thf(fact_908_linorder__neqE,axiom,
! [X3: int,Y2: int] :
( ( X3 != Y2 )
=> ( ~ ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ Y2 @ X3 ) ) ) ).
% linorder_neqE
thf(fact_909_linorder__neqE,axiom,
! [X3: real,Y2: real] :
( ( X3 != Y2 )
=> ( ~ ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ Y2 @ X3 ) ) ) ).
% linorder_neqE
thf(fact_910_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_911_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_912_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: real,A2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_913_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_914_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_915_order_Ostrict__implies__not__eq,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_916_dual__order_Ostrict__trans,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C2 @ B2 )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_917_dual__order_Ostrict__trans,axiom,
! [B2: int,A2: int,C2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_int @ C2 @ B2 )
=> ( ord_less_int @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_918_dual__order_Ostrict__trans,axiom,
! [B2: real,A2: real,C2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( ( ord_less_real @ C2 @ B2 )
=> ( ord_less_real @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_919_not__less__iff__gr__or__eq,axiom,
! [X3: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X3 )
| ( X3 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_920_not__less__iff__gr__or__eq,axiom,
! [X3: int,Y2: int] :
( ( ~ ( ord_less_int @ X3 @ Y2 ) )
= ( ( ord_less_int @ Y2 @ X3 )
| ( X3 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_921_not__less__iff__gr__or__eq,axiom,
! [X3: real,Y2: real] :
( ( ~ ( ord_less_real @ X3 @ Y2 ) )
= ( ( ord_less_real @ Y2 @ X3 )
| ( X3 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_922_order_Ostrict__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_923_order_Ostrict__trans,axiom,
! [A2: int,B2: int,C2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ B2 @ C2 )
=> ( ord_less_int @ A2 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_924_order_Ostrict__trans,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ord_less_real @ A2 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_925_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_926_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: int,B2: int] :
( ! [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B4: int] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_927_linorder__less__wlog,axiom,
! [P: real > real > $o,A2: real,B2: real] :
( ! [A4: real,B4: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: real] : ( P @ A4 @ A4 )
=> ( ! [A4: real,B4: real] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_928_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N3: nat] :
( ( P3 @ N3 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ~ ( P3 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_929_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_930_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_931_dual__order_Oirrefl,axiom,
! [A2: real] :
~ ( ord_less_real @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_932_dual__order_Oasym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_933_dual__order_Oasym,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ~ ( ord_less_int @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_934_dual__order_Oasym,axiom,
! [B2: real,A2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ~ ( ord_less_real @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_935_linorder__cases,axiom,
! [X3: nat,Y2: nat] :
( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( ( X3 != Y2 )
=> ( ord_less_nat @ Y2 @ X3 ) ) ) ).
% linorder_cases
thf(fact_936_linorder__cases,axiom,
! [X3: int,Y2: int] :
( ~ ( ord_less_int @ X3 @ Y2 )
=> ( ( X3 != Y2 )
=> ( ord_less_int @ Y2 @ X3 ) ) ) ).
% linorder_cases
thf(fact_937_linorder__cases,axiom,
! [X3: real,Y2: real] :
( ~ ( ord_less_real @ X3 @ Y2 )
=> ( ( X3 != Y2 )
=> ( ord_less_real @ Y2 @ X3 ) ) ) ).
% linorder_cases
thf(fact_938_antisym__conv3,axiom,
! [Y2: nat,X3: nat] :
( ~ ( ord_less_nat @ Y2 @ X3 )
=> ( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_939_antisym__conv3,axiom,
! [Y2: int,X3: int] :
( ~ ( ord_less_int @ Y2 @ X3 )
=> ( ( ~ ( ord_less_int @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_940_antisym__conv3,axiom,
! [Y2: real,X3: real] :
( ~ ( ord_less_real @ Y2 @ X3 )
=> ( ( ~ ( ord_less_real @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_941_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_942_ord__less__eq__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_943_ord__less__eq__trans,axiom,
! [A2: int,B2: int,C2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_int @ A2 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_944_ord__less__eq__trans,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_real @ A2 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_945_ord__eq__less__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( A2 = B2 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_946_ord__eq__less__trans,axiom,
! [A2: int,B2: int,C2: int] :
( ( A2 = B2 )
=> ( ( ord_less_int @ B2 @ C2 )
=> ( ord_less_int @ A2 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_947_ord__eq__less__trans,axiom,
! [A2: real,B2: real,C2: real] :
( ( A2 = B2 )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ord_less_real @ A2 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_948_order_Oasym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_949_order_Oasym,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ~ ( ord_less_int @ B2 @ A2 ) ) ).
% order.asym
thf(fact_950_order_Oasym,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ~ ( ord_less_real @ B2 @ A2 ) ) ).
% order.asym
thf(fact_951_less__imp__neq,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% less_imp_neq
thf(fact_952_less__imp__neq,axiom,
! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% less_imp_neq
thf(fact_953_less__imp__neq,axiom,
! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% less_imp_neq
thf(fact_954_dense,axiom,
! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ? [Z3: real] :
( ( ord_less_real @ X3 @ Z3 )
& ( ord_less_real @ Z3 @ Y2 ) ) ) ).
% dense
thf(fact_955_gt__ex,axiom,
! [X3: nat] :
? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).
% gt_ex
thf(fact_956_gt__ex,axiom,
! [X3: int] :
? [X_1: int] : ( ord_less_int @ X3 @ X_1 ) ).
% gt_ex
thf(fact_957_gt__ex,axiom,
! [X3: real] :
? [X_1: real] : ( ord_less_real @ X3 @ X_1 ) ).
% gt_ex
thf(fact_958_lt__ex,axiom,
! [X3: int] :
? [Y4: int] : ( ord_less_int @ Y4 @ X3 ) ).
% lt_ex
thf(fact_959_lt__ex,axiom,
! [X3: real] :
? [Y4: real] : ( ord_less_real @ Y4 @ X3 ) ).
% lt_ex
thf(fact_960_prop__restrict,axiom,
! [X3: $o,Z4: set_o,X6: set_o,P: $o > $o] :
( ( member_o @ X3 @ Z4 )
=> ( ( ord_less_eq_set_o @ Z4
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ X6 )
& ( P @ X2 ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_961_prop__restrict,axiom,
! [X3: int,Z4: set_int,X6: set_int,P: int > $o] :
( ( member_int @ X3 @ Z4 )
=> ( ( ord_less_eq_set_int @ Z4
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ X6 )
& ( P @ X2 ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_962_prop__restrict,axiom,
! [X3: nat,Z4: set_nat,X6: set_nat,P: nat > $o] :
( ( member_nat @ X3 @ Z4 )
=> ( ( ord_less_eq_set_nat @ Z4
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X6 )
& ( P @ X2 ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_963_prop__restrict,axiom,
! [X3: nat > a,Z4: set_nat_a,X6: set_nat_a,P: ( nat > a ) > $o] :
( ( member_nat_a @ X3 @ Z4 )
=> ( ( ord_le871467723717165285_nat_a @ Z4
@ ( collect_nat_a
@ ^ [X2: nat > a] :
( ( member_nat_a @ X2 @ X6 )
& ( P @ X2 ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_964_prop__restrict,axiom,
! [X3: a,Z4: set_a,X6: set_a,P: a > $o] :
( ( member_a @ X3 @ Z4 )
=> ( ( ord_less_eq_set_a @ Z4
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ X6 )
& ( P @ X2 ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_965_prop__restrict,axiom,
! [X3: $o > a,Z4: set_o_a,X6: set_o_a,P: ( $o > a ) > $o] :
( ( member_o_a @ X3 @ Z4 )
=> ( ( ord_less_eq_set_o_a @ Z4
@ ( collect_o_a
@ ^ [X2: $o > a] :
( ( member_o_a @ X2 @ X6 )
& ( P @ X2 ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_966_prop__restrict,axiom,
! [X3: int > nat > a,Z4: set_int_nat_a,X6: set_int_nat_a,P: ( int > nat > a ) > $o] :
( ( member_int_nat_a @ X3 @ Z4 )
=> ( ( ord_le6072919132162605296_nat_a @ Z4
@ ( collect_int_nat_a
@ ^ [X2: int > nat > a] :
( ( member_int_nat_a @ X2 @ X6 )
& ( P @ X2 ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_967_prop__restrict,axiom,
! [X3: int > a,Z4: set_int_a,X6: set_int_a,P: ( int > a ) > $o] :
( ( member_int_a @ X3 @ Z4 )
=> ( ( ord_le943418215940126601_int_a @ Z4
@ ( collect_int_a
@ ^ [X2: int > a] :
( ( member_int_a @ X2 @ X6 )
& ( P @ X2 ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_968_prop__restrict,axiom,
! [X3: a > nat > a,Z4: set_a_nat_a,X6: set_a_nat_a,P: ( a > nat > a ) > $o] :
( ( member_a_nat_a @ X3 @ Z4 )
=> ( ( ord_le2508512696544544866_nat_a @ Z4
@ ( collect_a_nat_a
@ ^ [X2: a > nat > a] :
( ( member_a_nat_a @ X2 @ X6 )
& ( P @ X2 ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_969_prop__restrict,axiom,
! [X3: a > a,Z4: set_a_a,X6: set_a_a,P: ( a > a ) > $o] :
( ( member_a_a @ X3 @ Z4 )
=> ( ( ord_less_eq_set_a_a @ Z4
@ ( collect_a_a
@ ^ [X2: a > a] :
( ( member_a_a @ X2 @ X6 )
& ( P @ X2 ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_970_Collect__restrict,axiom,
! [X6: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ X6 )
& ( P @ X2 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_971_Collect__restrict,axiom,
! [X6: set_int,P: int > $o] :
( ord_less_eq_set_int
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ X6 )
& ( P @ X2 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_972_Collect__restrict,axiom,
! [X6: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X6 )
& ( P @ X2 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_973_Collect__restrict,axiom,
! [X6: set_nat_a,P: ( nat > a ) > $o] :
( ord_le871467723717165285_nat_a
@ ( collect_nat_a
@ ^ [X2: nat > a] :
( ( member_nat_a @ X2 @ X6 )
& ( P @ X2 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_974_Collect__restrict,axiom,
! [X6: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ X6 )
& ( P @ X2 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_975_Collect__restrict,axiom,
! [X6: set_o_a,P: ( $o > a ) > $o] :
( ord_less_eq_set_o_a
@ ( collect_o_a
@ ^ [X2: $o > a] :
( ( member_o_a @ X2 @ X6 )
& ( P @ X2 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_976_Collect__restrict,axiom,
! [X6: set_int_nat_a,P: ( int > nat > a ) > $o] :
( ord_le6072919132162605296_nat_a
@ ( collect_int_nat_a
@ ^ [X2: int > nat > a] :
( ( member_int_nat_a @ X2 @ X6 )
& ( P @ X2 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_977_Collect__restrict,axiom,
! [X6: set_int_a,P: ( int > a ) > $o] :
( ord_le943418215940126601_int_a
@ ( collect_int_a
@ ^ [X2: int > a] :
( ( member_int_a @ X2 @ X6 )
& ( P @ X2 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_978_Collect__restrict,axiom,
! [X6: set_a_nat_a,P: ( a > nat > a ) > $o] :
( ord_le2508512696544544866_nat_a
@ ( collect_a_nat_a
@ ^ [X2: a > nat > a] :
( ( member_a_nat_a @ X2 @ X6 )
& ( P @ X2 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_979_Collect__restrict,axiom,
! [X6: set_a_a,P: ( a > a ) > $o] :
( ord_less_eq_set_a_a
@ ( collect_a_a
@ ^ [X2: a > a] :
( ( member_a_a @ X2 @ X6 )
& ( P @ X2 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_980_leD,axiom,
! [Y2: set_nat_a,X3: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ Y2 @ X3 )
=> ~ ( ord_less_set_nat_a @ X3 @ Y2 ) ) ).
% leD
thf(fact_981_leD,axiom,
! [Y2: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X3 )
=> ~ ( ord_less_set_a @ X3 @ Y2 ) ) ).
% leD
thf(fact_982_leD,axiom,
! [Y2: nat,X3: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ~ ( ord_less_nat @ X3 @ Y2 ) ) ).
% leD
thf(fact_983_leD,axiom,
! [Y2: int,X3: int] :
( ( ord_less_eq_int @ Y2 @ X3 )
=> ~ ( ord_less_int @ X3 @ Y2 ) ) ).
% leD
thf(fact_984_leD,axiom,
! [Y2: real,X3: real] :
( ( ord_less_eq_real @ Y2 @ X3 )
=> ~ ( ord_less_real @ X3 @ Y2 ) ) ).
% leD
thf(fact_985_leD,axiom,
! [Y2: set_o_a,X3: set_o_a] :
( ( ord_less_eq_set_o_a @ Y2 @ X3 )
=> ~ ( ord_less_set_o_a @ X3 @ Y2 ) ) ).
% leD
thf(fact_986_leD,axiom,
! [Y2: set_int_nat_a,X3: set_int_nat_a] :
( ( ord_le6072919132162605296_nat_a @ Y2 @ X3 )
=> ~ ( ord_le2026899869173067772_nat_a @ X3 @ Y2 ) ) ).
% leD
thf(fact_987_leD,axiom,
! [Y2: set_int_a,X3: set_int_a] :
( ( ord_le943418215940126601_int_a @ Y2 @ X3 )
=> ~ ( ord_less_set_int_a @ X3 @ Y2 ) ) ).
% leD
thf(fact_988_leD,axiom,
! [Y2: set_a_nat_a,X3: set_a_nat_a] :
( ( ord_le2508512696544544866_nat_a @ Y2 @ X3 )
=> ~ ( ord_less_set_a_nat_a @ X3 @ Y2 ) ) ).
% leD
thf(fact_989_leD,axiom,
! [Y2: set_a_a,X3: set_a_a] :
( ( ord_less_eq_set_a_a @ Y2 @ X3 )
=> ~ ( ord_less_set_a_a @ X3 @ Y2 ) ) ).
% leD
thf(fact_990_leI,axiom,
! [X3: nat,Y2: nat] :
( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% leI
thf(fact_991_leI,axiom,
! [X3: int,Y2: int] :
( ~ ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X3 ) ) ).
% leI
thf(fact_992_leI,axiom,
! [X3: real,Y2: real] :
( ~ ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X3 ) ) ).
% leI
thf(fact_993_nless__le,axiom,
! [A2: set_nat_a,B2: set_nat_a] :
( ( ~ ( ord_less_set_nat_a @ A2 @ B2 ) )
= ( ~ ( ord_le871467723717165285_nat_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_994_nless__le,axiom,
! [A2: set_a,B2: set_a] :
( ( ~ ( ord_less_set_a @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_set_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_995_nless__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_996_nless__le,axiom,
! [A2: int,B2: int] :
( ( ~ ( ord_less_int @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_int @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_997_nless__le,axiom,
! [A2: real,B2: real] :
( ( ~ ( ord_less_real @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_real @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_998_nless__le,axiom,
! [A2: set_o_a,B2: set_o_a] :
( ( ~ ( ord_less_set_o_a @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_set_o_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_999_nless__le,axiom,
! [A2: set_int_nat_a,B2: set_int_nat_a] :
( ( ~ ( ord_le2026899869173067772_nat_a @ A2 @ B2 ) )
= ( ~ ( ord_le6072919132162605296_nat_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_1000_nless__le,axiom,
! [A2: set_int_a,B2: set_int_a] :
( ( ~ ( ord_less_set_int_a @ A2 @ B2 ) )
= ( ~ ( ord_le943418215940126601_int_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_1001_nless__le,axiom,
! [A2: set_a_nat_a,B2: set_a_nat_a] :
( ( ~ ( ord_less_set_a_nat_a @ A2 @ B2 ) )
= ( ~ ( ord_le2508512696544544866_nat_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_1002_nless__le,axiom,
! [A2: set_a_a,B2: set_a_a] :
( ( ~ ( ord_less_set_a_a @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_set_a_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_1003_antisym__conv1,axiom,
! [X3: set_nat_a,Y2: set_nat_a] :
( ~ ( ord_less_set_nat_a @ X3 @ Y2 )
=> ( ( ord_le871467723717165285_nat_a @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_1004_antisym__conv1,axiom,
! [X3: set_a,Y2: set_a] :
( ~ ( ord_less_set_a @ X3 @ Y2 )
=> ( ( ord_less_eq_set_a @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_1005_antisym__conv1,axiom,
! [X3: nat,Y2: nat] :
( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_1006_antisym__conv1,axiom,
! [X3: int,Y2: int] :
( ~ ( ord_less_int @ X3 @ Y2 )
=> ( ( ord_less_eq_int @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_1007_antisym__conv1,axiom,
! [X3: real,Y2: real] :
( ~ ( ord_less_real @ X3 @ Y2 )
=> ( ( ord_less_eq_real @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_1008_antisym__conv1,axiom,
! [X3: set_o_a,Y2: set_o_a] :
( ~ ( ord_less_set_o_a @ X3 @ Y2 )
=> ( ( ord_less_eq_set_o_a @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_1009_antisym__conv1,axiom,
! [X3: set_int_nat_a,Y2: set_int_nat_a] :
( ~ ( ord_le2026899869173067772_nat_a @ X3 @ Y2 )
=> ( ( ord_le6072919132162605296_nat_a @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_1010_antisym__conv1,axiom,
! [X3: set_int_a,Y2: set_int_a] :
( ~ ( ord_less_set_int_a @ X3 @ Y2 )
=> ( ( ord_le943418215940126601_int_a @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_1011_antisym__conv1,axiom,
! [X3: set_a_nat_a,Y2: set_a_nat_a] :
( ~ ( ord_less_set_a_nat_a @ X3 @ Y2 )
=> ( ( ord_le2508512696544544866_nat_a @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_1012_antisym__conv1,axiom,
! [X3: set_a_a,Y2: set_a_a] :
( ~ ( ord_less_set_a_a @ X3 @ Y2 )
=> ( ( ord_less_eq_set_a_a @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_1013_antisym__conv2,axiom,
! [X3: set_nat_a,Y2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ X3 @ Y2 )
=> ( ( ~ ( ord_less_set_nat_a @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_1014_antisym__conv2,axiom,
! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ( ~ ( ord_less_set_a @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_1015_antisym__conv2,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_1016_antisym__conv2,axiom,
! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ( ~ ( ord_less_int @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_1017_antisym__conv2,axiom,
! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ( ~ ( ord_less_real @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_1018_antisym__conv2,axiom,
! [X3: set_o_a,Y2: set_o_a] :
( ( ord_less_eq_set_o_a @ X3 @ Y2 )
=> ( ( ~ ( ord_less_set_o_a @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_1019_antisym__conv2,axiom,
! [X3: set_int_nat_a,Y2: set_int_nat_a] :
( ( ord_le6072919132162605296_nat_a @ X3 @ Y2 )
=> ( ( ~ ( ord_le2026899869173067772_nat_a @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_1020_antisym__conv2,axiom,
! [X3: set_int_a,Y2: set_int_a] :
( ( ord_le943418215940126601_int_a @ X3 @ Y2 )
=> ( ( ~ ( ord_less_set_int_a @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_1021_antisym__conv2,axiom,
! [X3: set_a_nat_a,Y2: set_a_nat_a] :
( ( ord_le2508512696544544866_nat_a @ X3 @ Y2 )
=> ( ( ~ ( ord_less_set_a_nat_a @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_1022_antisym__conv2,axiom,
! [X3: set_a_a,Y2: set_a_a] :
( ( ord_less_eq_set_a_a @ X3 @ Y2 )
=> ( ( ~ ( ord_less_set_a_a @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_1023_dense__ge,axiom,
! [Z2: real,Y2: real] :
( ! [X: real] :
( ( ord_less_real @ Z2 @ X )
=> ( ord_less_eq_real @ Y2 @ X ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ).
% dense_ge
thf(fact_1024_dense__le,axiom,
! [Y2: real,Z2: real] :
( ! [X: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ord_less_eq_real @ X @ Z2 ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ).
% dense_le
thf(fact_1025_less__le__not__le,axiom,
( ord_less_set_nat_a
= ( ^ [X2: set_nat_a,Y6: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ X2 @ Y6 )
& ~ ( ord_le871467723717165285_nat_a @ Y6 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1026_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X2: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y6 )
& ~ ( ord_less_eq_set_a @ Y6 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1027_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y6: nat] :
( ( ord_less_eq_nat @ X2 @ Y6 )
& ~ ( ord_less_eq_nat @ Y6 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1028_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y6: int] :
( ( ord_less_eq_int @ X2 @ Y6 )
& ~ ( ord_less_eq_int @ Y6 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1029_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y6: real] :
( ( ord_less_eq_real @ X2 @ Y6 )
& ~ ( ord_less_eq_real @ Y6 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1030_less__le__not__le,axiom,
( ord_less_set_o_a
= ( ^ [X2: set_o_a,Y6: set_o_a] :
( ( ord_less_eq_set_o_a @ X2 @ Y6 )
& ~ ( ord_less_eq_set_o_a @ Y6 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1031_less__le__not__le,axiom,
( ord_le2026899869173067772_nat_a
= ( ^ [X2: set_int_nat_a,Y6: set_int_nat_a] :
( ( ord_le6072919132162605296_nat_a @ X2 @ Y6 )
& ~ ( ord_le6072919132162605296_nat_a @ Y6 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1032_less__le__not__le,axiom,
( ord_less_set_int_a
= ( ^ [X2: set_int_a,Y6: set_int_a] :
( ( ord_le943418215940126601_int_a @ X2 @ Y6 )
& ~ ( ord_le943418215940126601_int_a @ Y6 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1033_less__le__not__le,axiom,
( ord_less_set_a_nat_a
= ( ^ [X2: set_a_nat_a,Y6: set_a_nat_a] :
( ( ord_le2508512696544544866_nat_a @ X2 @ Y6 )
& ~ ( ord_le2508512696544544866_nat_a @ Y6 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1034_less__le__not__le,axiom,
( ord_less_set_a_a
= ( ^ [X2: set_a_a,Y6: set_a_a] :
( ( ord_less_eq_set_a_a @ X2 @ Y6 )
& ~ ( ord_less_eq_set_a_a @ Y6 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1035_not__le__imp__less,axiom,
! [Y2: nat,X3: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ord_less_nat @ X3 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_1036_not__le__imp__less,axiom,
! [Y2: int,X3: int] :
( ~ ( ord_less_eq_int @ Y2 @ X3 )
=> ( ord_less_int @ X3 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_1037_not__le__imp__less,axiom,
! [Y2: real,X3: real] :
( ~ ( ord_less_eq_real @ Y2 @ X3 )
=> ( ord_less_real @ X3 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_1038_order_Oorder__iff__strict,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A5: set_nat_a,B5: set_nat_a] :
( ( ord_less_set_nat_a @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1039_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_set_a @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1040_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1041_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B5: int] :
( ( ord_less_int @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1042_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B5: real] :
( ( ord_less_real @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1043_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_o_a
= ( ^ [A5: set_o_a,B5: set_o_a] :
( ( ord_less_set_o_a @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1044_order_Oorder__iff__strict,axiom,
( ord_le6072919132162605296_nat_a
= ( ^ [A5: set_int_nat_a,B5: set_int_nat_a] :
( ( ord_le2026899869173067772_nat_a @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1045_order_Oorder__iff__strict,axiom,
( ord_le943418215940126601_int_a
= ( ^ [A5: set_int_a,B5: set_int_a] :
( ( ord_less_set_int_a @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1046_order_Oorder__iff__strict,axiom,
( ord_le2508512696544544866_nat_a
= ( ^ [A5: set_a_nat_a,B5: set_a_nat_a] :
( ( ord_less_set_a_nat_a @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1047_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a_a
= ( ^ [A5: set_a_a,B5: set_a_a] :
( ( ord_less_set_a_a @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1048_order_Ostrict__iff__order,axiom,
( ord_less_set_nat_a
= ( ^ [A5: set_nat_a,B5: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1049_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1050_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1051_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1052_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A5: real,B5: real] :
( ( ord_less_eq_real @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1053_order_Ostrict__iff__order,axiom,
( ord_less_set_o_a
= ( ^ [A5: set_o_a,B5: set_o_a] :
( ( ord_less_eq_set_o_a @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1054_order_Ostrict__iff__order,axiom,
( ord_le2026899869173067772_nat_a
= ( ^ [A5: set_int_nat_a,B5: set_int_nat_a] :
( ( ord_le6072919132162605296_nat_a @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1055_order_Ostrict__iff__order,axiom,
( ord_less_set_int_a
= ( ^ [A5: set_int_a,B5: set_int_a] :
( ( ord_le943418215940126601_int_a @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1056_order_Ostrict__iff__order,axiom,
( ord_less_set_a_nat_a
= ( ^ [A5: set_a_nat_a,B5: set_a_nat_a] :
( ( ord_le2508512696544544866_nat_a @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1057_order_Ostrict__iff__order,axiom,
( ord_less_set_a_a
= ( ^ [A5: set_a_a,B5: set_a_a] :
( ( ord_less_eq_set_a_a @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1058_order_Ostrict__trans1,axiom,
! [A2: set_nat_a,B2: set_nat_a,C2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A2 @ B2 )
=> ( ( ord_less_set_nat_a @ B2 @ C2 )
=> ( ord_less_set_nat_a @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1059_order_Ostrict__trans1,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C2 )
=> ( ord_less_set_a @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1060_order_Ostrict__trans1,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1061_order_Ostrict__trans1,axiom,
! [A2: int,B2: int,C2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ B2 @ C2 )
=> ( ord_less_int @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1062_order_Ostrict__trans1,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ord_less_real @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1063_order_Ostrict__trans1,axiom,
! [A2: set_o_a,B2: set_o_a,C2: set_o_a] :
( ( ord_less_eq_set_o_a @ A2 @ B2 )
=> ( ( ord_less_set_o_a @ B2 @ C2 )
=> ( ord_less_set_o_a @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1064_order_Ostrict__trans1,axiom,
! [A2: set_int_nat_a,B2: set_int_nat_a,C2: set_int_nat_a] :
( ( ord_le6072919132162605296_nat_a @ A2 @ B2 )
=> ( ( ord_le2026899869173067772_nat_a @ B2 @ C2 )
=> ( ord_le2026899869173067772_nat_a @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1065_order_Ostrict__trans1,axiom,
! [A2: set_int_a,B2: set_int_a,C2: set_int_a] :
( ( ord_le943418215940126601_int_a @ A2 @ B2 )
=> ( ( ord_less_set_int_a @ B2 @ C2 )
=> ( ord_less_set_int_a @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1066_order_Ostrict__trans1,axiom,
! [A2: set_a_nat_a,B2: set_a_nat_a,C2: set_a_nat_a] :
( ( ord_le2508512696544544866_nat_a @ A2 @ B2 )
=> ( ( ord_less_set_a_nat_a @ B2 @ C2 )
=> ( ord_less_set_a_nat_a @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1067_order_Ostrict__trans1,axiom,
! [A2: set_a_a,B2: set_a_a,C2: set_a_a] :
( ( ord_less_eq_set_a_a @ A2 @ B2 )
=> ( ( ord_less_set_a_a @ B2 @ C2 )
=> ( ord_less_set_a_a @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1068_order_Ostrict__trans2,axiom,
! [A2: set_nat_a,B2: set_nat_a,C2: set_nat_a] :
( ( ord_less_set_nat_a @ A2 @ B2 )
=> ( ( ord_le871467723717165285_nat_a @ B2 @ C2 )
=> ( ord_less_set_nat_a @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1069_order_Ostrict__trans2,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_set_a @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1070_order_Ostrict__trans2,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1071_order_Ostrict__trans2,axiom,
! [A2: int,B2: int,C2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ord_less_int @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1072_order_Ostrict__trans2,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ord_less_real @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1073_order_Ostrict__trans2,axiom,
! [A2: set_o_a,B2: set_o_a,C2: set_o_a] :
( ( ord_less_set_o_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_o_a @ B2 @ C2 )
=> ( ord_less_set_o_a @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1074_order_Ostrict__trans2,axiom,
! [A2: set_int_nat_a,B2: set_int_nat_a,C2: set_int_nat_a] :
( ( ord_le2026899869173067772_nat_a @ A2 @ B2 )
=> ( ( ord_le6072919132162605296_nat_a @ B2 @ C2 )
=> ( ord_le2026899869173067772_nat_a @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1075_order_Ostrict__trans2,axiom,
! [A2: set_int_a,B2: set_int_a,C2: set_int_a] :
( ( ord_less_set_int_a @ A2 @ B2 )
=> ( ( ord_le943418215940126601_int_a @ B2 @ C2 )
=> ( ord_less_set_int_a @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1076_order_Ostrict__trans2,axiom,
! [A2: set_a_nat_a,B2: set_a_nat_a,C2: set_a_nat_a] :
( ( ord_less_set_a_nat_a @ A2 @ B2 )
=> ( ( ord_le2508512696544544866_nat_a @ B2 @ C2 )
=> ( ord_less_set_a_nat_a @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1077_order_Ostrict__trans2,axiom,
! [A2: set_a_a,B2: set_a_a,C2: set_a_a] :
( ( ord_less_set_a_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a_a @ B2 @ C2 )
=> ( ord_less_set_a_a @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1078_order_Ostrict__iff__not,axiom,
( ord_less_set_nat_a
= ( ^ [A5: set_nat_a,B5: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A5 @ B5 )
& ~ ( ord_le871467723717165285_nat_a @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1079_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ~ ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1080_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ~ ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1081_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
& ~ ( ord_less_eq_int @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1082_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A5: real,B5: real] :
( ( ord_less_eq_real @ A5 @ B5 )
& ~ ( ord_less_eq_real @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1083_order_Ostrict__iff__not,axiom,
( ord_less_set_o_a
= ( ^ [A5: set_o_a,B5: set_o_a] :
( ( ord_less_eq_set_o_a @ A5 @ B5 )
& ~ ( ord_less_eq_set_o_a @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1084_order_Ostrict__iff__not,axiom,
( ord_le2026899869173067772_nat_a
= ( ^ [A5: set_int_nat_a,B5: set_int_nat_a] :
( ( ord_le6072919132162605296_nat_a @ A5 @ B5 )
& ~ ( ord_le6072919132162605296_nat_a @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1085_order_Ostrict__iff__not,axiom,
( ord_less_set_int_a
= ( ^ [A5: set_int_a,B5: set_int_a] :
( ( ord_le943418215940126601_int_a @ A5 @ B5 )
& ~ ( ord_le943418215940126601_int_a @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1086_order_Ostrict__iff__not,axiom,
( ord_less_set_a_nat_a
= ( ^ [A5: set_a_nat_a,B5: set_a_nat_a] :
( ( ord_le2508512696544544866_nat_a @ A5 @ B5 )
& ~ ( ord_le2508512696544544866_nat_a @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1087_order_Ostrict__iff__not,axiom,
( ord_less_set_a_a
= ( ^ [A5: set_a_a,B5: set_a_a] :
( ( ord_less_eq_set_a_a @ A5 @ B5 )
& ~ ( ord_less_eq_set_a_a @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1088_dense__ge__bounded,axiom,
! [Z2: real,X3: real,Y2: real] :
( ( ord_less_real @ Z2 @ X3 )
=> ( ! [W: real] :
( ( ord_less_real @ Z2 @ W )
=> ( ( ord_less_real @ W @ X3 )
=> ( ord_less_eq_real @ Y2 @ W ) ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_1089_dense__le__bounded,axiom,
! [X3: real,Y2: real,Z2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ! [W: real] :
( ( ord_less_real @ X3 @ W )
=> ( ( ord_less_real @ W @ Y2 )
=> ( ord_less_eq_real @ W @ Z2 ) ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_1090_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_nat @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1091_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B5: int,A5: int] :
( ( ord_less_int @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1092_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B5: real,A5: real] :
( ( ord_less_real @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1093_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_o_a
= ( ^ [B5: set_o_a,A5: set_o_a] :
( ( ord_less_set_o_a @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1094_dual__order_Oorder__iff__strict,axiom,
( ord_le6072919132162605296_nat_a
= ( ^ [B5: set_int_nat_a,A5: set_int_nat_a] :
( ( ord_le2026899869173067772_nat_a @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1095_dual__order_Oorder__iff__strict,axiom,
( ord_le943418215940126601_int_a
= ( ^ [B5: set_int_a,A5: set_int_a] :
( ( ord_less_set_int_a @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1096_dual__order_Oorder__iff__strict,axiom,
( ord_le2508512696544544866_nat_a
= ( ^ [B5: set_a_nat_a,A5: set_a_nat_a] :
( ( ord_less_set_a_nat_a @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1097_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a_a
= ( ^ [B5: set_a_a,A5: set_a_a] :
( ( ord_less_set_a_a @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1098_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X2: nat,Y6: nat] : ( ord_less_eq_nat @ Y6 @ X2 )
@ ^ [X2: nat,Y6: nat] : ( ord_less_nat @ Y6 @ X2 )
@ zero_zero_nat ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_1099_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1100_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1101_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1102_image__int__atLeastLessThan,axiom,
! [A2: nat,B2: nat] :
( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A2 @ B2 ) )
= ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% image_int_atLeastLessThan
thf(fact_1103_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_less_as_int
thf(fact_1104_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1105_pos__int__cases,axiom,
! [K3: int] :
( ( ord_less_int @ zero_zero_int @ K3 )
=> ~ ! [N2: nat] :
( ( K3
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_1106_zero__less__imp__eq__int,axiom,
! [K3: int] :
( ( ord_less_int @ zero_zero_int @ K3 )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K3
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1107_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1108_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1109_nat__add__left__cancel__less,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K3 @ M ) @ ( plus_plus_nat @ K3 @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1110_nat__add__left__cancel__le,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K3 @ M ) @ ( plus_plus_nat @ K3 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1111_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1112_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_1113_int__if,axiom,
! [P: $o,A2: nat,B2: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B2 ) )
= ( semiri1314217659103216013at_int @ A2 ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B2 ) )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% int_if
thf(fact_1114_nat__int__comparison_I1_J,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A5: nat,B5: nat] :
( ( semiri1314217659103216013at_int @ A5 )
= ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1115_zero__le__imp__eq__int,axiom,
! [K3: int] :
( ( ord_less_eq_int @ zero_zero_int @ K3 )
=> ? [N2: nat] :
( K3
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1116_nonneg__int__cases,axiom,
! [K3: int] :
( ( ord_less_eq_int @ zero_zero_int @ K3 )
=> ~ ! [N2: nat] :
( K3
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_1117_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1118_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1119_less__add__eq__less,axiom,
! [K3: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K3 @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K3 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1120_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1121_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1122_add__less__mono1,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ K3 ) ) ) ).
% add_less_mono1
thf(fact_1123_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1124_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1125_add__less__mono,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K3 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1126_add__lessD1,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K3 )
=> ( ord_less_nat @ I @ K3 ) ) ).
% add_lessD1
thf(fact_1127_add__leE,axiom,
! [M: nat,K3: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K3 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K3 @ N ) ) ) ).
% add_leE
thf(fact_1128_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1129_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1130_add__leD1,axiom,
! [M: nat,K3: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K3 ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1131_add__leD2,axiom,
! [M: nat,K3: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K3 ) @ N )
=> ( ord_less_eq_nat @ K3 @ N ) ) ).
% add_leD2
thf(fact_1132_le__Suc__ex,axiom,
! [K3: nat,L: nat] :
( ( ord_less_eq_nat @ K3 @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K3 @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1133_add__le__mono,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K3 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1134_add__le__mono1,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ K3 ) ) ) ).
% add_le_mono1
thf(fact_1135_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1136_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1137_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
? [K: nat] :
( N3
= ( plus_plus_nat @ M3 @ K ) ) ) ) ).
% nat_le_iff_add
thf(fact_1138_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1139_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K3: nat] :
( ! [M5: nat,N2: nat] :
( ( ord_less_nat @ M5 @ N2 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K3 ) @ ( F @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1140_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_1141_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
= ( P @ B4 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_1142_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_1143_int__ops_I5_J,axiom,
! [A2: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A2 @ B2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% int_ops(5)
thf(fact_1144_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_1145_zadd__int__left,axiom,
! [M: nat,N: nat,Z2: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_1146_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z5: int] :
? [N3: nat] :
( Z5
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1147_neg__int__cases,axiom,
! [K3: int] :
( ( ord_less_int @ K3 @ zero_zero_int )
=> ~ ! [N2: nat] :
( ( K3
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% neg_int_cases
thf(fact_1148_nat__less__iff,axiom,
! [W3: int,M: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W3 )
=> ( ( ord_less_nat @ ( nat2 @ W3 ) @ M )
= ( ord_less_int @ W3 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_1149_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% nat_int
thf(fact_1150_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_1151_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_1152_nat__le__0,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ Z2 @ zero_zero_int )
=> ( ( nat2 @ Z2 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_1153_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_1154_zless__nat__conj,axiom,
! [W3: int,Z2: int] :
( ( ord_less_nat @ ( nat2 @ W3 ) @ ( nat2 @ Z2 ) )
= ( ( ord_less_int @ zero_zero_int @ Z2 )
& ( ord_less_int @ W3 @ Z2 ) ) ) ).
% zless_nat_conj
thf(fact_1155_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_1156_int__nat__eq,axiom,
! [Z2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= Z2 ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_1157_zero__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% zero_less_nat_eq
thf(fact_1158_int__cases2,axiom,
! [Z2: int] :
( ! [N2: nat] :
( Z2
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% int_cases2
thf(fact_1159_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_1160_nat__mono,axiom,
! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y2 ) ) ) ).
% nat_mono
thf(fact_1161_nat__int__add,axiom,
! [A2: nat,B2: nat] :
( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) )
= ( plus_plus_nat @ A2 @ B2 ) ) ).
% nat_int_add
thf(fact_1162_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_1163_nat__plus__as__int,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B5: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_1164_nat__mono__iff,axiom,
! [Z2: int,W3: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_nat @ ( nat2 @ W3 ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W3 @ Z2 ) ) ) ).
% nat_mono_iff
thf(fact_1165_zless__nat__eq__int__zless,axiom,
! [M: nat,Z2: int] :
( ( ord_less_nat @ M @ ( nat2 @ Z2 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z2 ) ) ).
% zless_nat_eq_int_zless
thf(fact_1166_nat__le__iff,axiom,
! [X3: int,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X3 ) @ N )
= ( ord_less_eq_int @ X3 @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_le_iff
thf(fact_1167_int__eq__iff,axiom,
! [M: nat,Z2: int] :
( ( ( semiri1314217659103216013at_int @ M )
= Z2 )
= ( ( M
= ( nat2 @ Z2 ) )
& ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ) ).
% int_eq_iff
thf(fact_1168_nat__0__le,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= Z2 ) ) ).
% nat_0_le
thf(fact_1169_int__cases4,axiom,
! [M: int] :
( ! [N2: nat] :
( M
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% int_cases4
thf(fact_1170_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1171_nonpos__int__cases,axiom,
! [K3: int] :
( ( ord_less_eq_int @ K3 @ zero_zero_int )
=> ~ ! [N2: nat] :
( K3
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1172_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1173_nat__less__eq__zless,axiom,
! [W3: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ W3 )
=> ( ( ord_less_nat @ ( nat2 @ W3 ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W3 @ Z2 ) ) ) ).
% nat_less_eq_zless
thf(fact_1174_nat__eq__iff2,axiom,
! [M: nat,W3: int] :
( ( M
= ( nat2 @ W3 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W3 )
=> ( W3
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W3 )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_1175_nat__eq__iff,axiom,
! [W3: int,M: nat] :
( ( ( nat2 @ W3 )
= M )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W3 )
=> ( W3
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W3 )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_1176_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N3: nat] :
( ( I
= ( semiri1314217659103216013at_int @ N3 ) )
=> ( P @ N3 ) )
& ( ( ord_less_int @ I @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_1177_nat__le__eq__zle,axiom,
! [W3: int,Z2: int] :
( ( ( ord_less_int @ zero_zero_int @ W3 )
| ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W3 ) @ ( nat2 @ Z2 ) )
= ( ord_less_eq_int @ W3 @ Z2 ) ) ) ).
% nat_le_eq_zle
thf(fact_1178_le__nat__iff,axiom,
! [K3: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K3 )
=> ( ( ord_less_eq_nat @ N @ ( nat2 @ K3 ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K3 ) ) ) ).
% le_nat_iff
thf(fact_1179_int__cases3,axiom,
! [K3: int] :
( ( K3 != zero_zero_int )
=> ( ! [N2: nat] :
( ( K3
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
=> ~ ! [N2: nat] :
( ( K3
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% int_cases3
thf(fact_1180_image__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
= ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).
% image_atLeastZeroLessThan_int
thf(fact_1181_UN__lessThan__UNIV,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
= top_top_set_nat ) ).
% UN_lessThan_UNIV
thf(fact_1182_lessThan__atLeast0,axiom,
( set_ord_lessThan_nat
= ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% lessThan_atLeast0
thf(fact_1183_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1184_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1185_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1186_diff__diff__left,axiom,
! [I: nat,J: nat,K3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K3 )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K3 ) ) ) ).
% diff_diff_left
thf(fact_1187_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1188_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1189_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1190_Nat_Oadd__diff__assoc,axiom,
! [K3: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K3 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1191_Nat_Oadd__diff__assoc2,axiom,
! [K3: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K3 ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K3 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1192_Nat_Odiff__diff__right,axiom,
! [K3: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K3 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1193_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1194_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1195_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1196_less__imp__diff__less,axiom,
! [J: nat,K3: nat,N: nat] :
( ( ord_less_nat @ J @ K3 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K3 ) ) ).
% less_imp_diff_less
thf(fact_1197_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1198_le__diff__iff_H,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A2 ) @ ( minus_minus_nat @ C2 @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1199_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1200_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1201_Nat_Odiff__diff__eq,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K3 @ M )
=> ( ( ord_less_eq_nat @ K3 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K3 ) @ ( minus_minus_nat @ N @ K3 ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1202_le__diff__iff,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K3 @ M )
=> ( ( ord_less_eq_nat @ K3 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K3 ) @ ( minus_minus_nat @ N @ K3 ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1203_eq__diff__iff,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K3 @ M )
=> ( ( ord_less_eq_nat @ K3 @ N )
=> ( ( ( minus_minus_nat @ M @ K3 )
= ( minus_minus_nat @ N @ K3 ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1204_int__diff__cases,axiom,
! [Z2: int] :
~ ! [M5: nat,N2: nat] :
( Z2
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_1205_diff__commute,axiom,
! [I: nat,J: nat,K3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K3 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K3 ) @ J ) ) ).
% diff_commute
thf(fact_1206_int__ops_I6_J,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B2 ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B2 ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% int_ops(6)
thf(fact_1207_nat__minus__as__int,axiom,
( minus_minus_nat
= ( ^ [A5: nat,B5: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ) ).
% nat_minus_as_int
thf(fact_1208_int__minus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M ) )
= ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).
% int_minus
thf(fact_1209_Nat_Odiff__cancel,axiom,
! [K3: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K3 @ M ) @ ( plus_plus_nat @ K3 @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1210_diff__cancel2,axiom,
! [M: nat,K3: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K3 ) @ ( plus_plus_nat @ N @ K3 ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1211_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1212_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1213_real__arch__inverse,axiom,
! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
= ( ? [N3: nat] :
( ( N3 != zero_zero_nat )
& ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ E ) ) ) ) ).
% real_arch_inverse
thf(fact_1214_forall__pos__mono,axiom,
! [P: real > $o,E: real] :
( ! [D2: real,E2: real] :
( ( ord_less_real @ D2 @ E2 )
=> ( ( P @ D2 )
=> ( P @ E2 ) ) )
=> ( ! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E )
=> ( P @ E ) ) ) ) ).
% forall_pos_mono
thf(fact_1215_zdiff__int__split,axiom,
! [P: int > $o,X3: nat,Y2: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X3 @ Y2 ) ) )
= ( ( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( semiri1314217659103216013at_int @ Y2 ) ) ) )
& ( ( ord_less_nat @ X3 @ Y2 )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1216_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1217_less__diff__iff,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K3 @ M )
=> ( ( ord_less_eq_nat @ K3 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K3 ) @ ( minus_minus_nat @ N @ K3 ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1218_diff__less__mono,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C2 ) @ ( minus_minus_nat @ B2 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1219_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1220_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1221_less__diff__conv,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K3 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ J ) ) ).
% less_diff_conv
thf(fact_1222_le__diff__conv,axiom,
! [J: nat,K3: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K3 ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K3 ) ) ) ).
% le_diff_conv
thf(fact_1223_Nat_Ole__diff__conv2,axiom,
! [K3: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K3 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1224_Nat_Odiff__add__assoc,axiom,
! [K3: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K3 )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K3 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1225_Nat_Odiff__add__assoc2,axiom,
! [K3: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K3 )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K3 ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1226_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K3 )
= ( J
= ( plus_plus_nat @ K3 @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1227_diff__nat__eq__if,axiom,
! [Z6: int,Z2: int] :
( ( ( ord_less_int @ Z6 @ zero_zero_int )
=> ( ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z6 ) )
= ( nat2 @ Z2 ) ) )
& ( ~ ( ord_less_int @ Z6 @ zero_zero_int )
=> ( ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z6 ) )
= ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z2 @ Z6 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z2 @ Z6 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_1228_nat__diff__split,axiom,
! [P: nat > $o,A2: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B2 ) )
= ( ( ( ord_less_nat @ A2 @ B2 )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1229_nat__diff__split__asm,axiom,
! [P: nat > $o,A2: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B2 ) )
= ( ~ ( ( ( ord_less_nat @ A2 @ B2 )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1230_less__diff__conv2,axiom,
! [K3: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K3 ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K3 ) ) ) ) ).
% less_diff_conv2
thf(fact_1231_image__add__int__atLeastLessThan,axiom,
! [L: int,U: int] :
( ( image_int_int
@ ^ [X2: int] : ( plus_plus_int @ X2 @ L )
@ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
= ( set_or4662586982721622107an_int @ L @ U ) ) ).
% image_add_int_atLeastLessThan
thf(fact_1232_real__add__minus__iff,axiom,
! [X3: real,A2: real] :
( ( ( plus_plus_real @ X3 @ ( uminus_uminus_real @ A2 ) )
= zero_zero_real )
= ( X3 = A2 ) ) ).
% real_add_minus_iff
thf(fact_1233_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1234_real__0__le__add__iff,axiom,
! [X3: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X3 @ Y2 ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ X3 ) @ Y2 ) ) ).
% real_0_le_add_iff
thf(fact_1235_real__add__le__0__iff,axiom,
! [X3: real,Y2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X3 @ Y2 ) @ zero_zero_real )
= ( ord_less_eq_real @ Y2 @ ( uminus_uminus_real @ X3 ) ) ) ).
% real_add_le_0_iff
thf(fact_1236_real__add__less__0__iff,axiom,
! [X3: real,Y2: real] :
( ( ord_less_real @ ( plus_plus_real @ X3 @ Y2 ) @ zero_zero_real )
= ( ord_less_real @ Y2 @ ( uminus_uminus_real @ X3 ) ) ) ).
% real_add_less_0_iff
thf(fact_1237_real__0__less__add__iff,axiom,
! [X3: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X3 @ Y2 ) )
= ( ord_less_real @ ( uminus_uminus_real @ X3 ) @ Y2 ) ) ).
% real_0_less_add_iff
thf(fact_1238_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1239_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_1240_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N3: nat,M3: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M3 ) ) ) ) ).
% nat_less_real_le
thf(fact_1241_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N3: nat,M3: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M3 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_1242_int__less__real__le,axiom,
( ord_less_int
= ( ^ [N3: int,M3: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N3 ) @ one_one_real ) @ ( ring_1_of_int_real @ M3 ) ) ) ) ).
% int_less_real_le
thf(fact_1243_int__le__real__less,axiom,
( ord_less_eq_int
= ( ^ [N3: int,M3: int] : ( ord_less_real @ ( ring_1_of_int_real @ N3 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M3 ) @ one_one_real ) ) ) ) ).
% int_le_real_less
thf(fact_1244_minus__real__def,axiom,
( minus_minus_real
= ( ^ [X2: real,Y6: real] : ( plus_plus_real @ X2 @ ( uminus_uminus_real @ Y6 ) ) ) ) ).
% minus_real_def
thf(fact_1245_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K3: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K3 )
=> ( ( ord_less_eq_int @ K3 @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K3 ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1246_range__abs__Nats,axiom,
( ( image_int_int @ abs_abs_int @ top_top_set_int )
= semiring_1_Nats_int ) ).
% range_abs_Nats
thf(fact_1247_nat__ceiling__le__eq,axiom,
! [X3: real,A2: nat] :
( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X3 ) ) @ A2 )
= ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ A2 ) ) ) ).
% nat_ceiling_le_eq
thf(fact_1248_real__nat__ceiling__ge,axiom,
! [X3: real] : ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X3 ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_1249_int__in__range__abs,axiom,
! [N: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N ) @ ( image_int_int @ abs_abs_int @ top_top_set_int ) ) ).
% int_in_range_abs
thf(fact_1250_nat__abs__triangle__ineq,axiom,
! [K3: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K3 @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).
% nat_abs_triangle_ineq
thf(fact_1251_nat__abs__int__diff,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) )
= ( minus_minus_nat @ B2 @ A2 ) ) )
& ( ~ ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) )
= ( minus_minus_nat @ A2 @ B2 ) ) ) ) ).
% nat_abs_int_diff
thf(fact_1252_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K3: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K3 )
=> ( ( ord_less_eq_int @ K3 @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K3 ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1253_decr__lemma,axiom,
! [D: int,X3: int,Z2: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ ( minus_minus_int @ X3 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ Z2 ) ) @ one_one_int ) @ D ) ) @ Z2 ) ) ).
% decr_lemma
thf(fact_1254_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1255_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1256_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1257_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1258_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_1259_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_1260_Suc__diff__diff,axiom,
! [M: nat,N: nat,K3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K3 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K3 ) ) ).
% Suc_diff_diff
thf(fact_1261_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_1262_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_1263_image__Suc__atLeastLessThan,axiom,
! [I: nat,J: nat] :
( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
= ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastLessThan
% Helper facts (5)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X3: a,Y2: a] :
( ( if_a @ $false @ X3 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X3: a,Y2: a] :
( ( if_a @ $true @ X3 @ Y2 )
= X3 ) ).
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y2: nat] :
( ( if_nat @ $false @ X3 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y2: nat] :
( ( if_nat @ $true @ X3 @ Y2 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ord_le871467723717165285_nat_a
@ ( collect_nat_a
@ ^ [F2: nat > a] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F2 @ top_top_set_nat ) @ b )
& ! [X2: nat] :
( ( ord_less_eq_nat @ n @ X2 )
=> ( ( F2 @ X2 )
= y ) ) ) )
@ ( image_nat_a_nat_a @ f
@ ( piE_nat_a @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ n )
@ ^ [I2: nat] : b ) ) ) ).
%------------------------------------------------------------------------------