TPTP Problem File: SLH0976^1.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 : Frequency_Moments/0080_Frequency_Moments_Preliminary_Results/prob_00042_001706__19781164_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1622 ( 731 unt; 355 typ; 0 def)
% Number of atoms : 2985 (1155 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 11432 ( 355 ~; 53 |; 244 &;9853 @)
% ( 0 <=>; 927 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 60 ( 59 usr)
% Number of type conns : 1180 (1180 >; 0 *; 0 +; 0 <<)
% Number of symbols : 298 ( 296 usr; 8 con; 0-4 aty)
% Number of variables : 3919 ( 542 ^;3264 !; 113 ?;3919 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:10:07.443
%------------------------------------------------------------------------------
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thf(sy_c_Set_Oimage_001tf__a_001t__Num__Onum,type,
image_a_num: ( a > num ) > set_a > set_num ).
thf(sy_c_Set_Oimage_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
image_7400625782589995694od_a_a: ( a > product_prod_a_a ) > set_a > set_Product_prod_a_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $o ).
thf(sy_c_member_001t__Extended____Nat__Oenat,type,
member_Extended_enat: extended_enat > set_Extended_enat > $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__Num__Onum,type,
member_num: num > set_num > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member6519385997651362275od_a_a: produc2100646355360341836od_a_a > set_Pr5009853057886717698od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
member8962352052110095674_nat_a: product_prod_nat_a > set_Pr4193341848836149977_nat_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
member7279096912039735102um_num: product_prod_num_num > set_Pr8218934625190621173um_num > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Nat__Onat_J,type,
member1268613965720311869_a_nat: produc6073246360284067238_a_nat > set_Pr8287382171553239132_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member6330455413206600464od_a_a: produc3498347346309940967od_a_a > set_Pr8600417178894128327od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J,type,
member2829916517802586983_a_a_a: produc3802892049952890430_a_a_a > set_Pr8876520727511657886_a_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
member5724188588386418708_a_nat: product_prod_a_nat > set_Pr4934435412358123699_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member3071122053849602553od_a_a: produc4044097585999906000od_a_a > set_Pr5530083903271594800od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_M,type,
m: set_a ).
% Relevant facts (1266)
thf(fact_0_assms,axiom,
finite_finite_a @ m ).
% assms
thf(fact_1_power2__commute,axiom,
! [X: real,Y: real] :
( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ ( minus_minus_real @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_2_power2__commute,axiom,
! [X: complex,Y: complex] :
( ( power_power_complex @ ( minus_minus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_complex @ ( minus_minus_complex @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_3_power2__commute,axiom,
! [X: int,Y: int] :
( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ ( minus_minus_int @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_4_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_5_semiring__norm_I83_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_6_card__2__iff_H,axiom,
! [S: set_a] :
( ( ( finite_card_a @ S )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ S )
& ? [Y2: a] :
( ( member_a @ Y2 @ S )
& ( X2 != Y2 )
& ! [Z: a] :
( ( member_a @ Z @ S )
=> ( ( Z = X2 )
| ( Z = Y2 ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_7_card__2__iff_H,axiom,
! [S: set_Product_prod_a_a] :
( ( ( finite4795055649997197647od_a_a @ S )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ? [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ S )
& ? [Y2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y2 @ S )
& ( X2 != Y2 )
& ! [Z: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z @ S )
=> ( ( Z = X2 )
| ( Z = Y2 ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_8_card__2__iff_H,axiom,
! [S: set_nat] :
( ( ( finite_card_nat @ S )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ S )
& ? [Y2: nat] :
( ( member_nat @ Y2 @ S )
& ( X2 != Y2 )
& ! [Z: nat] :
( ( member_nat @ Z @ S )
=> ( ( Z = X2 )
| ( Z = Y2 ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_9_card__2__iff_H,axiom,
! [S: set_complex] :
( ( ( finite_card_complex @ S )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ? [X2: complex] :
( ( member_complex @ X2 @ S )
& ? [Y2: complex] :
( ( member_complex @ Y2 @ S )
& ( X2 != Y2 )
& ! [Z: complex] :
( ( member_complex @ Z @ S )
=> ( ( Z = X2 )
| ( Z = Y2 ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_10_image__ident,axiom,
! [Y3: set_nat] :
( ( image_nat_nat
@ ^ [X2: nat] : X2
@ Y3 )
= Y3 ) ).
% image_ident
thf(fact_11_image__ident,axiom,
! [Y3: set_Product_prod_a_a] :
( ( image_4636654165204879301od_a_a
@ ^ [X2: product_prod_a_a] : X2
@ Y3 )
= Y3 ) ).
% image_ident
thf(fact_12__092_060open_062card_A_IM_A_092_060times_062_AM_J_A_N_Acard_A_123_Ix_M_Ay_J_O_A_Ix_M_Ay_J_A_092_060in_062_AM_A_092_060times_062_AM_A_092_060and_062_Ax_A_061_Ay_125_A_061_A_Icard_AM_J_092_060_094sup_0622_A_N_Acard_A_I_I_092_060lambda_062x_O_A_Ix_M_Ax_J_J_A_096_AM_J_092_060close_062,axiom,
( ( minus_minus_nat
@ ( finite4795055649997197647od_a_a
@ ( product_Sigma_a_a @ m
@ ^ [Uu: a] : m ) )
@ ( finite4795055649997197647od_a_a
@ ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X2: a,Y2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 )
@ ( product_Sigma_a_a @ m
@ ^ [Uu: a] : m ) )
& ( X2 = Y2 ) ) ) ) ) )
= ( minus_minus_nat @ ( power_power_nat @ ( finite_card_a @ m ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
@ ( finite4795055649997197647od_a_a
@ ( image_7400625782589995694od_a_a
@ ^ [X2: a] : ( product_Pair_a_a @ X2 @ X2 )
@ m ) ) ) ) ).
% \<open>card (M \<times> M) - card {(x, y). (x, y) \<in> M \<times> M \<and> x = y} = (card M)\<^sup>2 - card ((\<lambda>x. (x, x)) ` M)\<close>
thf(fact_13_image__eqI,axiom,
! [B: a,F: product_prod_a_a > a,X: product_prod_a_a,A: set_Product_prod_a_a] :
( ( B
= ( F @ X ) )
=> ( ( member1426531477525435216od_a_a @ X @ A )
=> ( member_a @ B @ ( image_3437945252899457948_a_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_14_image__eqI,axiom,
! [B: a,F: nat > a,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_a @ B @ ( image_nat_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_15_image__eqI,axiom,
! [B: nat,F: a > nat,X: a,A: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A )
=> ( member_nat @ B @ ( image_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_16_image__eqI,axiom,
! [B: a,F: a > a,X: a,A: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A )
=> ( member_a @ B @ ( image_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_17_image__eqI,axiom,
! [B: product_prod_a_a,F: a > product_prod_a_a,X: a,A: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A )
=> ( member1426531477525435216od_a_a @ B @ ( image_7400625782589995694od_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_18_image__eqI,axiom,
! [B: product_prod_a_a,F: product_prod_a_a > product_prod_a_a,X: product_prod_a_a,A: set_Product_prod_a_a] :
( ( B
= ( F @ X ) )
=> ( ( member1426531477525435216od_a_a @ X @ A )
=> ( member1426531477525435216od_a_a @ B @ ( image_4636654165204879301od_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_19_image__eqI,axiom,
! [B: nat,F: product_prod_a_a > nat,X: product_prod_a_a,A: set_Product_prod_a_a] :
( ( B
= ( F @ X ) )
=> ( ( member1426531477525435216od_a_a @ X @ A )
=> ( member_nat @ B @ ( image_9053670898913107890_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_20_image__eqI,axiom,
! [B: product_prod_a_a,F: nat > product_prod_a_a,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member1426531477525435216od_a_a @ B @ ( image_372941888232738320od_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_21_image__eqI,axiom,
! [B: nat,F: nat > nat,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_22_prod_Oinject,axiom,
! [X1: a,X22: a,Y1: a,Y22: a] :
( ( ( product_Pair_a_a @ X1 @ X22 )
= ( product_Pair_a_a @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_23_prod_Oinject,axiom,
! [X1: num,X22: num,Y1: num,Y22: num] :
( ( ( product_Pair_num_num @ X1 @ X22 )
= ( product_Pair_num_num @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_24_old_Oprod_Oinject,axiom,
! [A2: a,B: a,A3: a,B2: a] :
( ( ( product_Pair_a_a @ A2 @ B )
= ( product_Pair_a_a @ A3 @ B2 ) )
= ( ( A2 = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_25_old_Oprod_Oinject,axiom,
! [A2: num,B: num,A3: num,B2: num] :
( ( ( product_Pair_num_num @ A2 @ B )
= ( product_Pair_num_num @ A3 @ B2 ) )
= ( ( A2 = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_26_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_27_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_28_split__part,axiom,
! [P: $o,Q: a > a > $o] :
( ( produc6436628058953941356_a_a_o
@ ^ [A4: a,B3: a] :
( P
& ( Q @ A4 @ B3 ) ) )
= ( ^ [Ab: product_prod_a_a] :
( P
& ( produc6436628058953941356_a_a_o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_29__092_060open_062card_A_IM_A_092_060times_062_AM_A_N_A_123_Ix_M_Ay_J_O_A_Ix_M_Ay_J_A_092_060in_062_AM_A_092_060times_062_AM_A_092_060and_062_Ax_A_061_Ay_125_J_A_061_Acard_A_IM_A_092_060times_062_AM_J_A_N_Acard_A_123_Ix_M_Ay_J_O_A_Ix_M_Ay_J_A_092_060in_062_AM_A_092_060times_062_AM_A_092_060and_062_Ax_A_061_Ay_125_092_060close_062,axiom,
( ( finite4795055649997197647od_a_a
@ ( minus_6817036919807184750od_a_a
@ ( product_Sigma_a_a @ m
@ ^ [Uu: a] : m )
@ ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X2: a,Y2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 )
@ ( product_Sigma_a_a @ m
@ ^ [Uu: a] : m ) )
& ( X2 = Y2 ) ) ) ) ) )
= ( minus_minus_nat
@ ( finite4795055649997197647od_a_a
@ ( product_Sigma_a_a @ m
@ ^ [Uu: a] : m ) )
@ ( finite4795055649997197647od_a_a
@ ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X2: a,Y2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 )
@ ( product_Sigma_a_a @ m
@ ^ [Uu: a] : m ) )
& ( X2 = Y2 ) ) ) ) ) ) ) ).
% \<open>card (M \<times> M - {(x, y). (x, y) \<in> M \<times> M \<and> x = y}) = card (M \<times> M) - card {(x, y). (x, y) \<in> M \<times> M \<and> x = y}\<close>
thf(fact_30_mem__Sigma__iff,axiom,
! [A2: product_prod_a_a,B: product_prod_a_a,A: set_Product_prod_a_a,B4: product_prod_a_a > set_Product_prod_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ B ) @ ( produc5899993699339346696od_a_a @ A @ B4 ) )
= ( ( member1426531477525435216od_a_a @ A2 @ A )
& ( member1426531477525435216od_a_a @ B @ ( B4 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_31_mem__Sigma__iff,axiom,
! [A2: product_prod_a_a,B: nat,A: set_Product_prod_a_a,B4: product_prod_a_a > set_nat] :
( ( member1268613965720311869_a_nat @ ( produc6483765539767234976_a_nat @ A2 @ B ) @ ( produc1049071135499013807_a_nat @ A @ B4 ) )
= ( ( member1426531477525435216od_a_a @ A2 @ A )
& ( member_nat @ B @ ( B4 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_32_mem__Sigma__iff,axiom,
! [A2: product_prod_a_a,B: a,A: set_Product_prod_a_a,B4: product_prod_a_a > set_a] :
( ( member2829916517802586983_a_a_a @ ( produc5692536848587512110_a_a_a @ A2 @ B ) @ ( produc2379640491490746847_a_a_a @ A @ B4 ) )
= ( ( member1426531477525435216od_a_a @ A2 @ A )
& ( member_a @ B @ ( B4 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_33_mem__Sigma__iff,axiom,
! [A2: nat,B: product_prod_a_a,A: set_nat,B4: nat > set_Product_prod_a_a] :
( ( member6519385997651362275od_a_a @ ( produc7026408565941641214od_a_a @ A2 @ B ) @ ( produc1591714161673420045od_a_a @ A @ B4 ) )
= ( ( member_nat @ A2 @ A )
& ( member1426531477525435216od_a_a @ B @ ( B4 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_34_mem__Sigma__iff,axiom,
! [A2: nat,B: nat,A: set_nat,B4: nat > set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( produc457027306803732586at_nat @ A @ B4 ) )
= ( ( member_nat @ A2 @ A )
& ( member_nat @ B @ ( B4 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_35_mem__Sigma__iff,axiom,
! [A2: nat,B: a,A: set_nat,B4: nat > set_a] :
( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A2 @ B ) @ ( product_Sigma_nat_a @ A @ B4 ) )
= ( ( member_nat @ A2 @ A )
& ( member_a @ B @ ( B4 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_36_mem__Sigma__iff,axiom,
! [A2: a,B: product_prod_a_a,A: set_a,B4: a > set_Product_prod_a_a] :
( ( member3071122053849602553od_a_a @ ( produc431845341423274048od_a_a @ A2 @ B ) @ ( produc6342321021181284593od_a_a @ A @ B4 ) )
= ( ( member_a @ A2 @ A )
& ( member1426531477525435216od_a_a @ B @ ( B4 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_37_mem__Sigma__iff,axiom,
! [A2: a,B: nat,A: set_a,B4: a > set_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ B ) @ ( product_Sigma_a_nat @ A @ B4 ) )
= ( ( member_a @ A2 @ A )
& ( member_nat @ B @ ( B4 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_38_mem__Sigma__iff,axiom,
! [A2: num,B: num,A: set_num,B4: num > set_num] :
( ( member7279096912039735102um_num @ ( product_Pair_num_num @ A2 @ B ) @ ( produc4368061533121756414um_num @ A @ B4 ) )
= ( ( member_num @ A2 @ A )
& ( member_num @ B @ ( B4 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_39_mem__Sigma__iff,axiom,
! [A2: a,B: a,A: set_a,B4: a > set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ ( product_Sigma_a_a @ A @ B4 ) )
= ( ( member_a @ A2 @ A )
& ( member_a @ B @ ( B4 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_40_SigmaI,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a,B: product_prod_a_a,B4: product_prod_a_a > set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ A )
=> ( ( member1426531477525435216od_a_a @ B @ ( B4 @ A2 ) )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ B ) @ ( produc5899993699339346696od_a_a @ A @ B4 ) ) ) ) ).
% SigmaI
thf(fact_41_SigmaI,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a,B: nat,B4: product_prod_a_a > set_nat] :
( ( member1426531477525435216od_a_a @ A2 @ A )
=> ( ( member_nat @ B @ ( B4 @ A2 ) )
=> ( member1268613965720311869_a_nat @ ( produc6483765539767234976_a_nat @ A2 @ B ) @ ( produc1049071135499013807_a_nat @ A @ B4 ) ) ) ) ).
% SigmaI
thf(fact_42_SigmaI,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a,B: a,B4: product_prod_a_a > set_a] :
( ( member1426531477525435216od_a_a @ A2 @ A )
=> ( ( member_a @ B @ ( B4 @ A2 ) )
=> ( member2829916517802586983_a_a_a @ ( produc5692536848587512110_a_a_a @ A2 @ B ) @ ( produc2379640491490746847_a_a_a @ A @ B4 ) ) ) ) ).
% SigmaI
thf(fact_43_SigmaI,axiom,
! [A2: nat,A: set_nat,B: product_prod_a_a,B4: nat > set_Product_prod_a_a] :
( ( member_nat @ A2 @ A )
=> ( ( member1426531477525435216od_a_a @ B @ ( B4 @ A2 ) )
=> ( member6519385997651362275od_a_a @ ( produc7026408565941641214od_a_a @ A2 @ B ) @ ( produc1591714161673420045od_a_a @ A @ B4 ) ) ) ) ).
% SigmaI
thf(fact_44_SigmaI,axiom,
! [A2: nat,A: set_nat,B: nat,B4: nat > set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( member_nat @ B @ ( B4 @ A2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( produc457027306803732586at_nat @ A @ B4 ) ) ) ) ).
% SigmaI
thf(fact_45_SigmaI,axiom,
! [A2: nat,A: set_nat,B: a,B4: nat > set_a] :
( ( member_nat @ A2 @ A )
=> ( ( member_a @ B @ ( B4 @ A2 ) )
=> ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A2 @ B ) @ ( product_Sigma_nat_a @ A @ B4 ) ) ) ) ).
% SigmaI
thf(fact_46_SigmaI,axiom,
! [A2: a,A: set_a,B: product_prod_a_a,B4: a > set_Product_prod_a_a] :
( ( member_a @ A2 @ A )
=> ( ( member1426531477525435216od_a_a @ B @ ( B4 @ A2 ) )
=> ( member3071122053849602553od_a_a @ ( produc431845341423274048od_a_a @ A2 @ B ) @ ( produc6342321021181284593od_a_a @ A @ B4 ) ) ) ) ).
% SigmaI
thf(fact_47_SigmaI,axiom,
! [A2: a,A: set_a,B: nat,B4: a > set_nat] :
( ( member_a @ A2 @ A )
=> ( ( member_nat @ B @ ( B4 @ A2 ) )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ B ) @ ( product_Sigma_a_nat @ A @ B4 ) ) ) ) ).
% SigmaI
thf(fact_48_SigmaI,axiom,
! [A2: num,A: set_num,B: num,B4: num > set_num] :
( ( member_num @ A2 @ A )
=> ( ( member_num @ B @ ( B4 @ A2 ) )
=> ( member7279096912039735102um_num @ ( product_Pair_num_num @ A2 @ B ) @ ( produc4368061533121756414um_num @ A @ B4 ) ) ) ) ).
% SigmaI
thf(fact_49_SigmaI,axiom,
! [A2: a,A: set_a,B: a,B4: a > set_a] :
( ( member_a @ A2 @ A )
=> ( ( member_a @ B @ ( B4 @ A2 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ ( product_Sigma_a_a @ A @ B4 ) ) ) ) ).
% SigmaI
thf(fact_50_pair__imageI,axiom,
! [A2: a,B: a,A: set_Product_prod_a_a,F: a > a > product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ A )
=> ( member1426531477525435216od_a_a @ ( F @ A2 @ B ) @ ( image_4636654165204879301od_a_a @ ( produc408267641121961211od_a_a @ F ) @ A ) ) ) ).
% pair_imageI
thf(fact_51_pair__imageI,axiom,
! [A2: a,B: a,A: set_Product_prod_a_a,F: a > a > nat] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ A )
=> ( member_nat @ ( F @ A2 @ B ) @ ( image_9053670898913107890_a_nat @ ( produc3852632504931109628_a_nat @ F ) @ A ) ) ) ).
% pair_imageI
thf(fact_52_pair__imageI,axiom,
! [A2: a,B: a,A: set_Product_prod_a_a,F: a > a > a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ A )
=> ( member_a @ ( F @ A2 @ B ) @ ( image_3437945252899457948_a_a_a @ ( produc8815886927560695506_a_a_a @ F ) @ A ) ) ) ).
% pair_imageI
thf(fact_53_pair__imageI,axiom,
! [A2: num,B: num,A: set_Pr8218934625190621173um_num,F: num > num > product_prod_a_a] :
( ( member7279096912039735102um_num @ ( product_Pair_num_num @ A2 @ B ) @ A )
=> ( member1426531477525435216od_a_a @ ( F @ A2 @ B ) @ ( image_3038167635258416343od_a_a @ ( produc3351508447645921989od_a_a @ F ) @ A ) ) ) ).
% pair_imageI
thf(fact_54_pair__imageI,axiom,
! [A2: num,B: num,A: set_Pr8218934625190621173um_num,F: num > num > nat] :
( ( member7279096912039735102um_num @ ( product_Pair_num_num @ A2 @ B ) @ A )
=> ( member_nat @ ( F @ A2 @ B ) @ ( image_2445279336714102816um_nat @ ( produc2914010905598588082um_nat @ F ) @ A ) ) ) ).
% pair_imageI
thf(fact_55_pair__imageI,axiom,
! [A2: num,B: num,A: set_Pr8218934625190621173um_num,F: num > num > a] :
( ( member7279096912039735102um_num @ ( product_Pair_num_num @ A2 @ B ) @ A )
=> ( member_a @ ( F @ A2 @ B ) @ ( image_7071692275713983150_num_a @ ( produc4784766916121835164_num_a @ F ) @ A ) ) ) ).
% pair_imageI
thf(fact_56_pair__imageI,axiom,
! [A2: a,B: a,A: set_Product_prod_a_a,F: a > a > $o] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ A )
=> ( member_o @ ( F @ A2 @ B ) @ ( image_9022731552424948534_a_a_o @ ( produc6436628058953941356_a_a_o @ F ) @ A ) ) ) ).
% pair_imageI
thf(fact_57_case__prod__conv,axiom,
! [F: a > a > $o,A2: a,B: a] :
( ( produc6436628058953941356_a_a_o @ F @ ( product_Pair_a_a @ A2 @ B ) )
= ( F @ A2 @ B ) ) ).
% case_prod_conv
thf(fact_58_case__prodI2,axiom,
! [P2: product_prod_num_num,C: num > num > $o] :
( ! [A5: num,B5: num] :
( ( P2
= ( product_Pair_num_num @ A5 @ B5 ) )
=> ( C @ A5 @ B5 ) )
=> ( produc5703948589228662326_num_o @ C @ P2 ) ) ).
% case_prodI2
thf(fact_59_case__prodI2,axiom,
! [P2: product_prod_a_a,C: a > a > $o] :
( ! [A5: a,B5: a] :
( ( P2
= ( product_Pair_a_a @ A5 @ B5 ) )
=> ( C @ A5 @ B5 ) )
=> ( produc6436628058953941356_a_a_o @ C @ P2 ) ) ).
% case_prodI2
thf(fact_60_case__prodI,axiom,
! [F: num > num > $o,A2: num,B: num] :
( ( F @ A2 @ B )
=> ( produc5703948589228662326_num_o @ F @ ( product_Pair_num_num @ A2 @ B ) ) ) ).
% case_prodI
thf(fact_61_case__prodI,axiom,
! [F: a > a > $o,A2: a,B: a] :
( ( F @ A2 @ B )
=> ( produc6436628058953941356_a_a_o @ F @ ( product_Pair_a_a @ A2 @ B ) ) ) ).
% case_prodI
thf(fact_62_Collect__case__prod,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [A4: nat,B3: nat] :
( ( P @ A4 )
& ( Q @ B3 ) ) ) )
= ( produc457027306803732586at_nat @ ( collect_nat @ P )
@ ^ [Uu: nat] : ( collect_nat @ Q ) ) ) ).
% Collect_case_prod
thf(fact_63_Collect__case__prod,axiom,
! [P: nat > $o,Q: int > $o] :
( ( collec8437875479827664712at_int
@ ( produc4260733461647839854_int_o
@ ^ [A4: nat,B3: int] :
( ( P @ A4 )
& ( Q @ B3 ) ) ) )
= ( produc454536836294682310at_int @ ( collect_nat @ P )
@ ^ [Uu: nat] : ( collect_int @ Q ) ) ) ).
% Collect_case_prod
thf(fact_64_Collect__case__prod,axiom,
! [P: nat > $o,Q: complex > $o] :
( ( collec84115854367371978omplex
@ ( produc373301242307897708plex_o
@ ^ [A4: nat,B3: complex] :
( ( P @ A4 )
& ( Q @ B3 ) ) ) )
= ( produc7489356884331829576omplex @ ( collect_nat @ P )
@ ^ [Uu: nat] : ( collect_complex @ Q ) ) ) ).
% Collect_case_prod
thf(fact_65_Collect__case__prod,axiom,
! [P: int > $o,Q: nat > $o] :
( ( collec4391708174383140168nt_nat
@ ( produc6768351840121078382_nat_o
@ ^ [A4: int,B3: nat] :
( ( P @ A4 )
& ( Q @ B3 ) ) ) )
= ( produc1456381018704787142nt_nat @ ( collect_int @ P )
@ ^ [Uu: int] : ( collect_nat @ Q ) ) ) ).
% Collect_case_prod
thf(fact_66_Collect__case__prod,axiom,
! [P: int > $o,Q: int > $o] :
( ( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [A4: int,B3: int] :
( ( P @ A4 )
& ( Q @ B3 ) ) ) )
= ( produc1453890548195736866nt_int @ ( collect_int @ P )
@ ^ [Uu: int] : ( collect_int @ Q ) ) ) ).
% Collect_case_prod
thf(fact_67_Collect__case__prod,axiom,
! [P: int > $o,Q: complex > $o] :
( ( collec1059651319573549350omplex
@ ( produc1928087437784968208plex_o
@ ^ [A4: int,B3: complex] :
( ( P @ A4 )
& ( Q @ B3 ) ) ) )
= ( produc8464892349538006948omplex @ ( collect_int @ P )
@ ^ [Uu: int] : ( collect_complex @ Q ) ) ) ).
% Collect_case_prod
thf(fact_68_Collect__case__prod,axiom,
! [P: complex > $o,Q: nat > $o] :
( ( collec6244025470361543626ex_nat
@ ( produc3852264011746918252_nat_o
@ ^ [A4: complex,B3: nat] :
( ( P @ A4 )
& ( Q @ B3 ) ) ) )
= ( produc1885768171911791432ex_nat @ ( collect_complex @ P )
@ ^ [Uu: complex] : ( collect_nat @ Q ) ) ) ).
% Collect_case_prod
thf(fact_69_Collect__case__prod,axiom,
! [P: complex > $o,Q: int > $o] :
( ( collec2066174450852346918ex_int
@ ( produc2031221666314230288_int_o
@ ^ [A4: complex,B3: int] :
( ( P @ A4 )
& ( Q @ B3 ) ) ) )
= ( produc1883277701402741156ex_int @ ( collect_complex @ P )
@ ^ [Uu: complex] : ( collect_int @ Q ) ) ) ).
% Collect_case_prod
thf(fact_70_Collect__case__prod,axiom,
! [P: complex > $o,Q: complex > $o] :
( ( collec8663557070575231912omplex
@ ( produc6771430404735790350plex_o
@ ^ [A4: complex,B3: complex] :
( ( P @ A4 )
& ( Q @ B3 ) ) ) )
= ( produc4778231967749636134omplex @ ( collect_complex @ P )
@ ^ [Uu: complex] : ( collect_complex @ Q ) ) ) ).
% Collect_case_prod
thf(fact_71_Collect__case__prod,axiom,
! [P: a > $o,Q: a > $o] :
( ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [A4: a,B3: a] :
( ( P @ A4 )
& ( Q @ B3 ) ) ) )
= ( product_Sigma_a_a @ ( collect_a @ P )
@ ^ [Uu: a] : ( collect_a @ Q ) ) ) ).
% Collect_case_prod
thf(fact_72_prod_Ocase__distrib,axiom,
! [H: $o > $o,F: a > a > $o,Prod: product_prod_a_a] :
( ( H @ ( produc6436628058953941356_a_a_o @ F @ Prod ) )
= ( produc6436628058953941356_a_a_o
@ ^ [X12: a,X23: a] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_73_prod_Odisc__eq__case,axiom,
! [Prod: product_prod_a_a] :
( produc6436628058953941356_a_a_o
@ ^ [Uu: a,Uv: a] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_74_Sigma__cong,axiom,
! [A: set_a,B4: set_a,C2: a > set_a,D: a > set_a] :
( ( A = B4 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B4 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( product_Sigma_a_a @ A @ C2 )
= ( product_Sigma_a_a @ B4 @ D ) ) ) ) ).
% Sigma_cong
thf(fact_75_Times__eq__cancel2,axiom,
! [X: a,C2: set_a,A: set_a,B4: set_a] :
( ( member_a @ X @ C2 )
=> ( ( ( product_Sigma_a_a @ A
@ ^ [Uu: a] : C2 )
= ( product_Sigma_a_a @ B4
@ ^ [Uu: a] : C2 ) )
= ( A = B4 ) ) ) ).
% Times_eq_cancel2
thf(fact_76_Sigma__Diff__distrib1,axiom,
! [I: set_a,J: set_a,C2: a > set_a] :
( ( product_Sigma_a_a @ ( minus_minus_set_a @ I @ J ) @ C2 )
= ( minus_6817036919807184750od_a_a @ ( product_Sigma_a_a @ I @ C2 ) @ ( product_Sigma_a_a @ J @ C2 ) ) ) ).
% Sigma_Diff_distrib1
thf(fact_77_Sigma__Diff__distrib2,axiom,
! [I: set_a,A: a > set_a,B4: a > set_a] :
( ( product_Sigma_a_a @ I
@ ^ [I2: a] : ( minus_minus_set_a @ ( A @ I2 ) @ ( B4 @ I2 ) ) )
= ( minus_6817036919807184750od_a_a @ ( product_Sigma_a_a @ I @ A ) @ ( product_Sigma_a_a @ I @ B4 ) ) ) ).
% Sigma_Diff_distrib2
thf(fact_78_Times__Diff__distrib1,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( product_Sigma_a_a @ ( minus_minus_set_a @ A @ B4 )
@ ^ [Uu: a] : C2 )
= ( minus_6817036919807184750od_a_a
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : C2 )
@ ( product_Sigma_a_a @ B4
@ ^ [Uu: a] : C2 ) ) ) ).
% Times_Diff_distrib1
thf(fact_79_Collect__case__prod__Sigma,axiom,
! [P: nat > $o,Q: nat > nat > $o] :
( ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X2: nat,Y2: nat] :
( ( P @ X2 )
& ( Q @ X2 @ Y2 ) ) ) )
= ( produc457027306803732586at_nat @ ( collect_nat @ P )
@ ^ [X2: nat] : ( collect_nat @ ( Q @ X2 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_80_Collect__case__prod__Sigma,axiom,
! [P: nat > $o,Q: nat > int > $o] :
( ( collec8437875479827664712at_int
@ ( produc4260733461647839854_int_o
@ ^ [X2: nat,Y2: int] :
( ( P @ X2 )
& ( Q @ X2 @ Y2 ) ) ) )
= ( produc454536836294682310at_int @ ( collect_nat @ P )
@ ^ [X2: nat] : ( collect_int @ ( Q @ X2 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_81_Collect__case__prod__Sigma,axiom,
! [P: nat > $o,Q: nat > complex > $o] :
( ( collec84115854367371978omplex
@ ( produc373301242307897708plex_o
@ ^ [X2: nat,Y2: complex] :
( ( P @ X2 )
& ( Q @ X2 @ Y2 ) ) ) )
= ( produc7489356884331829576omplex @ ( collect_nat @ P )
@ ^ [X2: nat] : ( collect_complex @ ( Q @ X2 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_82_Collect__case__prod__Sigma,axiom,
! [P: int > $o,Q: int > nat > $o] :
( ( collec4391708174383140168nt_nat
@ ( produc6768351840121078382_nat_o
@ ^ [X2: int,Y2: nat] :
( ( P @ X2 )
& ( Q @ X2 @ Y2 ) ) ) )
= ( produc1456381018704787142nt_nat @ ( collect_int @ P )
@ ^ [X2: int] : ( collect_nat @ ( Q @ X2 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_83_Collect__case__prod__Sigma,axiom,
! [P: int > $o,Q: int > int > $o] :
( ( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [X2: int,Y2: int] :
( ( P @ X2 )
& ( Q @ X2 @ Y2 ) ) ) )
= ( produc1453890548195736866nt_int @ ( collect_int @ P )
@ ^ [X2: int] : ( collect_int @ ( Q @ X2 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_84_Collect__case__prod__Sigma,axiom,
! [P: int > $o,Q: int > complex > $o] :
( ( collec1059651319573549350omplex
@ ( produc1928087437784968208plex_o
@ ^ [X2: int,Y2: complex] :
( ( P @ X2 )
& ( Q @ X2 @ Y2 ) ) ) )
= ( produc8464892349538006948omplex @ ( collect_int @ P )
@ ^ [X2: int] : ( collect_complex @ ( Q @ X2 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_85_Collect__case__prod__Sigma,axiom,
! [P: complex > $o,Q: complex > nat > $o] :
( ( collec6244025470361543626ex_nat
@ ( produc3852264011746918252_nat_o
@ ^ [X2: complex,Y2: nat] :
( ( P @ X2 )
& ( Q @ X2 @ Y2 ) ) ) )
= ( produc1885768171911791432ex_nat @ ( collect_complex @ P )
@ ^ [X2: complex] : ( collect_nat @ ( Q @ X2 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_86_Collect__case__prod__Sigma,axiom,
! [P: complex > $o,Q: complex > int > $o] :
( ( collec2066174450852346918ex_int
@ ( produc2031221666314230288_int_o
@ ^ [X2: complex,Y2: int] :
( ( P @ X2 )
& ( Q @ X2 @ Y2 ) ) ) )
= ( produc1883277701402741156ex_int @ ( collect_complex @ P )
@ ^ [X2: complex] : ( collect_int @ ( Q @ X2 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_87_Collect__case__prod__Sigma,axiom,
! [P: complex > $o,Q: complex > complex > $o] :
( ( collec8663557070575231912omplex
@ ( produc6771430404735790350plex_o
@ ^ [X2: complex,Y2: complex] :
( ( P @ X2 )
& ( Q @ X2 @ Y2 ) ) ) )
= ( produc4778231967749636134omplex @ ( collect_complex @ P )
@ ^ [X2: complex] : ( collect_complex @ ( Q @ X2 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_88_Collect__case__prod__Sigma,axiom,
! [P: a > $o,Q: a > a > $o] :
( ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X2: a,Y2: a] :
( ( P @ X2 )
& ( Q @ X2 @ Y2 ) ) ) )
= ( product_Sigma_a_a @ ( collect_a @ P )
@ ^ [X2: a] : ( collect_a @ ( Q @ X2 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_89_SigmaE2,axiom,
! [A2: product_prod_a_a,B: product_prod_a_a,A: set_Product_prod_a_a,B4: product_prod_a_a > set_Product_prod_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ B ) @ ( produc5899993699339346696od_a_a @ A @ B4 ) )
=> ~ ( ( member1426531477525435216od_a_a @ A2 @ A )
=> ~ ( member1426531477525435216od_a_a @ B @ ( B4 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_90_SigmaE2,axiom,
! [A2: product_prod_a_a,B: nat,A: set_Product_prod_a_a,B4: product_prod_a_a > set_nat] :
( ( member1268613965720311869_a_nat @ ( produc6483765539767234976_a_nat @ A2 @ B ) @ ( produc1049071135499013807_a_nat @ A @ B4 ) )
=> ~ ( ( member1426531477525435216od_a_a @ A2 @ A )
=> ~ ( member_nat @ B @ ( B4 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_91_SigmaE2,axiom,
! [A2: product_prod_a_a,B: a,A: set_Product_prod_a_a,B4: product_prod_a_a > set_a] :
( ( member2829916517802586983_a_a_a @ ( produc5692536848587512110_a_a_a @ A2 @ B ) @ ( produc2379640491490746847_a_a_a @ A @ B4 ) )
=> ~ ( ( member1426531477525435216od_a_a @ A2 @ A )
=> ~ ( member_a @ B @ ( B4 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_92_SigmaE2,axiom,
! [A2: nat,B: product_prod_a_a,A: set_nat,B4: nat > set_Product_prod_a_a] :
( ( member6519385997651362275od_a_a @ ( produc7026408565941641214od_a_a @ A2 @ B ) @ ( produc1591714161673420045od_a_a @ A @ B4 ) )
=> ~ ( ( member_nat @ A2 @ A )
=> ~ ( member1426531477525435216od_a_a @ B @ ( B4 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_93_SigmaE2,axiom,
! [A2: nat,B: nat,A: set_nat,B4: nat > set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( produc457027306803732586at_nat @ A @ B4 ) )
=> ~ ( ( member_nat @ A2 @ A )
=> ~ ( member_nat @ B @ ( B4 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_94_SigmaE2,axiom,
! [A2: nat,B: a,A: set_nat,B4: nat > set_a] :
( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A2 @ B ) @ ( product_Sigma_nat_a @ A @ B4 ) )
=> ~ ( ( member_nat @ A2 @ A )
=> ~ ( member_a @ B @ ( B4 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_95_SigmaE2,axiom,
! [A2: a,B: product_prod_a_a,A: set_a,B4: a > set_Product_prod_a_a] :
( ( member3071122053849602553od_a_a @ ( produc431845341423274048od_a_a @ A2 @ B ) @ ( produc6342321021181284593od_a_a @ A @ B4 ) )
=> ~ ( ( member_a @ A2 @ A )
=> ~ ( member1426531477525435216od_a_a @ B @ ( B4 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_96_SigmaE2,axiom,
! [A2: a,B: nat,A: set_a,B4: a > set_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ B ) @ ( product_Sigma_a_nat @ A @ B4 ) )
=> ~ ( ( member_a @ A2 @ A )
=> ~ ( member_nat @ B @ ( B4 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_97_SigmaE2,axiom,
! [A2: num,B: num,A: set_num,B4: num > set_num] :
( ( member7279096912039735102um_num @ ( product_Pair_num_num @ A2 @ B ) @ ( produc4368061533121756414um_num @ A @ B4 ) )
=> ~ ( ( member_num @ A2 @ A )
=> ~ ( member_num @ B @ ( B4 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_98_SigmaE2,axiom,
! [A2: a,B: a,A: set_a,B4: a > set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ ( product_Sigma_a_a @ A @ B4 ) )
=> ~ ( ( member_a @ A2 @ A )
=> ~ ( member_a @ B @ ( B4 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_99_SigmaD2,axiom,
! [A2: num,B: num,A: set_num,B4: num > set_num] :
( ( member7279096912039735102um_num @ ( product_Pair_num_num @ A2 @ B ) @ ( produc4368061533121756414um_num @ A @ B4 ) )
=> ( member_num @ B @ ( B4 @ A2 ) ) ) ).
% SigmaD2
thf(fact_100_SigmaD2,axiom,
! [A2: a,B: a,A: set_a,B4: a > set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ ( product_Sigma_a_a @ A @ B4 ) )
=> ( member_a @ B @ ( B4 @ A2 ) ) ) ).
% SigmaD2
thf(fact_101_SigmaD1,axiom,
! [A2: num,B: num,A: set_num,B4: num > set_num] :
( ( member7279096912039735102um_num @ ( product_Pair_num_num @ A2 @ B ) @ ( produc4368061533121756414um_num @ A @ B4 ) )
=> ( member_num @ A2 @ A ) ) ).
% SigmaD1
thf(fact_102_SigmaD1,axiom,
! [A2: a,B: a,A: set_a,B4: a > set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ ( product_Sigma_a_a @ A @ B4 ) )
=> ( member_a @ A2 @ A ) ) ).
% SigmaD1
thf(fact_103_SigmaE,axiom,
! [C: produc3498347346309940967od_a_a,A: set_Product_prod_a_a,B4: product_prod_a_a > set_Product_prod_a_a] :
( ( member6330455413206600464od_a_a @ C @ ( produc5899993699339346696od_a_a @ A @ B4 ) )
=> ~ ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A )
=> ! [Y4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y4 @ ( B4 @ X3 ) )
=> ( C
!= ( produc7886510207707329367od_a_a @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_104_SigmaE,axiom,
! [C: produc6073246360284067238_a_nat,A: set_Product_prod_a_a,B4: product_prod_a_a > set_nat] :
( ( member1268613965720311869_a_nat @ C @ ( produc1049071135499013807_a_nat @ A @ B4 ) )
=> ~ ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ ( B4 @ X3 ) )
=> ( C
!= ( produc6483765539767234976_a_nat @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_105_SigmaE,axiom,
! [C: produc3802892049952890430_a_a_a,A: set_Product_prod_a_a,B4: product_prod_a_a > set_a] :
( ( member2829916517802586983_a_a_a @ C @ ( produc2379640491490746847_a_a_a @ A @ B4 ) )
=> ~ ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A )
=> ! [Y4: a] :
( ( member_a @ Y4 @ ( B4 @ X3 ) )
=> ( C
!= ( produc5692536848587512110_a_a_a @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_106_SigmaE,axiom,
! [C: produc2100646355360341836od_a_a,A: set_nat,B4: nat > set_Product_prod_a_a] :
( ( member6519385997651362275od_a_a @ C @ ( produc1591714161673420045od_a_a @ A @ B4 ) )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ! [Y4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y4 @ ( B4 @ X3 ) )
=> ( C
!= ( produc7026408565941641214od_a_a @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_107_SigmaE,axiom,
! [C: product_prod_nat_nat,A: set_nat,B4: nat > set_nat] :
( ( member8440522571783428010at_nat @ C @ ( produc457027306803732586at_nat @ A @ B4 ) )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ ( B4 @ X3 ) )
=> ( C
!= ( product_Pair_nat_nat @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_108_SigmaE,axiom,
! [C: product_prod_nat_a,A: set_nat,B4: nat > set_a] :
( ( member8962352052110095674_nat_a @ C @ ( product_Sigma_nat_a @ A @ B4 ) )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ! [Y4: a] :
( ( member_a @ Y4 @ ( B4 @ X3 ) )
=> ( C
!= ( product_Pair_nat_a @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_109_SigmaE,axiom,
! [C: produc4044097585999906000od_a_a,A: set_a,B4: a > set_Product_prod_a_a] :
( ( member3071122053849602553od_a_a @ C @ ( produc6342321021181284593od_a_a @ A @ B4 ) )
=> ~ ! [X3: a] :
( ( member_a @ X3 @ A )
=> ! [Y4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y4 @ ( B4 @ X3 ) )
=> ( C
!= ( produc431845341423274048od_a_a @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_110_SigmaE,axiom,
! [C: product_prod_a_nat,A: set_a,B4: a > set_nat] :
( ( member5724188588386418708_a_nat @ C @ ( product_Sigma_a_nat @ A @ B4 ) )
=> ~ ! [X3: a] :
( ( member_a @ X3 @ A )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ ( B4 @ X3 ) )
=> ( C
!= ( product_Pair_a_nat @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_111_SigmaE,axiom,
! [C: product_prod_num_num,A: set_num,B4: num > set_num] :
( ( member7279096912039735102um_num @ C @ ( produc4368061533121756414um_num @ A @ B4 ) )
=> ~ ! [X3: num] :
( ( member_num @ X3 @ A )
=> ! [Y4: num] :
( ( member_num @ Y4 @ ( B4 @ X3 ) )
=> ( C
!= ( product_Pair_num_num @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_112_SigmaE,axiom,
! [C: product_prod_a_a,A: set_a,B4: a > set_a] :
( ( member1426531477525435216od_a_a @ C @ ( product_Sigma_a_a @ A @ B4 ) )
=> ~ ! [X3: a] :
( ( member_a @ X3 @ A )
=> ! [Y4: a] :
( ( member_a @ Y4 @ ( B4 @ X3 ) )
=> ( C
!= ( product_Pair_a_a @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_113_old_Oprod_Ocase,axiom,
! [F: a > a > $o,X1: a,X22: a] :
( ( produc6436628058953941356_a_a_o @ F @ ( product_Pair_a_a @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_114_case__prodE,axiom,
! [C: num > num > $o,P2: product_prod_num_num] :
( ( produc5703948589228662326_num_o @ C @ P2 )
=> ~ ! [X3: num,Y4: num] :
( ( P2
= ( product_Pair_num_num @ X3 @ Y4 ) )
=> ~ ( C @ X3 @ Y4 ) ) ) ).
% case_prodE
thf(fact_115_case__prodE,axiom,
! [C: a > a > $o,P2: product_prod_a_a] :
( ( produc6436628058953941356_a_a_o @ C @ P2 )
=> ~ ! [X3: a,Y4: a] :
( ( P2
= ( product_Pair_a_a @ X3 @ Y4 ) )
=> ~ ( C @ X3 @ Y4 ) ) ) ).
% case_prodE
thf(fact_116_case__prodD,axiom,
! [F: num > num > $o,A2: num,B: num] :
( ( produc5703948589228662326_num_o @ F @ ( product_Pair_num_num @ A2 @ B ) )
=> ( F @ A2 @ B ) ) ).
% case_prodD
thf(fact_117_case__prodD,axiom,
! [F: a > a > $o,A2: a,B: a] :
( ( produc6436628058953941356_a_a_o @ F @ ( product_Pair_a_a @ A2 @ B ) )
=> ( F @ A2 @ B ) ) ).
% case_prodD
thf(fact_118_cond__case__prod__eta,axiom,
! [F: a > a > $o,G: product_prod_a_a > $o] :
( ! [X3: a,Y4: a] :
( ( F @ X3 @ Y4 )
= ( G @ ( product_Pair_a_a @ X3 @ Y4 ) ) )
=> ( ( produc6436628058953941356_a_a_o @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_119_case__prod__eta,axiom,
! [F: product_prod_a_a > $o] :
( ( produc6436628058953941356_a_a_o
@ ^ [X2: a,Y2: a] : ( F @ ( product_Pair_a_a @ X2 @ Y2 ) ) )
= F ) ).
% case_prod_eta
thf(fact_120_case__prodE2,axiom,
! [Q: $o > $o,P: a > a > $o,Z2: product_prod_a_a] :
( ( Q @ ( produc6436628058953941356_a_a_o @ P @ Z2 ) )
=> ~ ! [X3: a,Y4: a] :
( ( Z2
= ( product_Pair_a_a @ X3 @ Y4 ) )
=> ~ ( Q @ ( P @ X3 @ Y4 ) ) ) ) ).
% case_prodE2
thf(fact_121_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_122_mem__Collect__eq,axiom,
! [A2: product_prod_a_a,P: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ A2 @ ( collec3336397797384452498od_a_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_123_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_124_mem__Collect__eq,axiom,
! [A2: int,P: int > $o] :
( ( member_int @ A2 @ ( collect_int @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_125_mem__Collect__eq,axiom,
! [A2: complex,P: complex > $o] :
( ( member_complex @ A2 @ ( collect_complex @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_126_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_127_Collect__mem__eq,axiom,
! [A: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_128_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_129_Collect__mem__eq,axiom,
! [A: set_int] :
( ( collect_int
@ ^ [X2: int] : ( member_int @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_130_Collect__mem__eq,axiom,
! [A: set_complex] :
( ( collect_complex
@ ^ [X2: complex] : ( member_complex @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_131_Collect__cong,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ! [X3: product_prod_a_a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec3336397797384452498od_a_a @ P )
= ( collec3336397797384452498od_a_a @ Q ) ) ) ).
% Collect_cong
thf(fact_132_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_133_Collect__cong,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_int @ P )
= ( collect_int @ Q ) ) ) ).
% Collect_cong
thf(fact_134_Collect__cong,axiom,
! [P: complex > $o,Q: complex > $o] :
( ! [X3: complex] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_complex @ P )
= ( collect_complex @ Q ) ) ) ).
% Collect_cong
thf(fact_135_Pair__inject,axiom,
! [A2: a,B: a,A3: a,B2: a] :
( ( ( product_Pair_a_a @ A2 @ B )
= ( product_Pair_a_a @ A3 @ B2 ) )
=> ~ ( ( A2 = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_136_Pair__inject,axiom,
! [A2: num,B: num,A3: num,B2: num] :
( ( ( product_Pair_num_num @ A2 @ B )
= ( product_Pair_num_num @ A3 @ B2 ) )
=> ~ ( ( A2 = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_137_prod__cases,axiom,
! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
( ! [A5: a,B5: a] : ( P @ ( product_Pair_a_a @ A5 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_138_prod__cases,axiom,
! [P: product_prod_num_num > $o,P2: product_prod_num_num] :
( ! [A5: num,B5: num] : ( P @ ( product_Pair_num_num @ A5 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_139_surj__pair,axiom,
! [P2: product_prod_a_a] :
? [X3: a,Y4: a] :
( P2
= ( product_Pair_a_a @ X3 @ Y4 ) ) ).
% surj_pair
thf(fact_140_surj__pair,axiom,
! [P2: product_prod_num_num] :
? [X3: num,Y4: num] :
( P2
= ( product_Pair_num_num @ X3 @ Y4 ) ) ).
% surj_pair
thf(fact_141_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_a_a] :
~ ! [A5: a,B5: a] :
( Y
!= ( product_Pair_a_a @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_142_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_num_num] :
~ ! [A5: num,B5: num] :
( Y
!= ( product_Pair_num_num @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_143_rev__image__eqI,axiom,
! [X: product_prod_a_a,A: set_Product_prod_a_a,B: product_prod_a_a,F: product_prod_a_a > product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member1426531477525435216od_a_a @ B @ ( image_4636654165204879301od_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_144_rev__image__eqI,axiom,
! [X: product_prod_a_a,A: set_Product_prod_a_a,B: nat,F: product_prod_a_a > nat] :
( ( member1426531477525435216od_a_a @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_9053670898913107890_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_145_rev__image__eqI,axiom,
! [X: product_prod_a_a,A: set_Product_prod_a_a,B: a,F: product_prod_a_a > a] :
( ( member1426531477525435216od_a_a @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_3437945252899457948_a_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_146_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: product_prod_a_a,F: nat > product_prod_a_a] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member1426531477525435216od_a_a @ B @ ( image_372941888232738320od_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_147_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_148_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: a,F: nat > a] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_nat_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_149_rev__image__eqI,axiom,
! [X: a,A: set_a,B: product_prod_a_a,F: a > product_prod_a_a] :
( ( member_a @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member1426531477525435216od_a_a @ B @ ( image_7400625782589995694od_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_150_rev__image__eqI,axiom,
! [X: a,A: set_a,B: nat,F: a > nat] :
( ( member_a @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_151_rev__image__eqI,axiom,
! [X: a,A: set_a,B: a,F: a > a] :
( ( member_a @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_152_ball__imageD,axiom,
! [F: a > product_prod_a_a,A: set_a,P: product_prod_a_a > $o] :
( ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( image_7400625782589995694od_a_a @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_153_ball__imageD,axiom,
! [F: product_prod_a_a > product_prod_a_a,A: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( image_4636654165204879301od_a_a @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_154_ball__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_155_ball__imageD,axiom,
! [F: nat > product_prod_a_a,A: set_nat,P: product_prod_a_a > $o] :
( ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( image_372941888232738320od_a_a @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_156_ball__imageD,axiom,
! [F: product_prod_a_a > nat,A: set_Product_prod_a_a,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_9053670898913107890_a_nat @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_157_image__cong,axiom,
! [M2: set_Product_prod_a_a,N2: set_Product_prod_a_a,F: product_prod_a_a > product_prod_a_a,G: product_prod_a_a > product_prod_a_a] :
( ( M2 = N2 )
=> ( ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_4636654165204879301od_a_a @ F @ M2 )
= ( image_4636654165204879301od_a_a @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_158_image__cong,axiom,
! [M2: set_Product_prod_a_a,N2: set_Product_prod_a_a,F: product_prod_a_a > nat,G: product_prod_a_a > nat] :
( ( M2 = N2 )
=> ( ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_9053670898913107890_a_nat @ F @ M2 )
= ( image_9053670898913107890_a_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_159_image__cong,axiom,
! [M2: set_nat,N2: set_nat,F: nat > nat,G: nat > nat] :
( ( M2 = N2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_nat @ F @ M2 )
= ( image_nat_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_160_image__cong,axiom,
! [M2: set_nat,N2: set_nat,F: nat > product_prod_a_a,G: nat > product_prod_a_a] :
( ( M2 = N2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_372941888232738320od_a_a @ F @ M2 )
= ( image_372941888232738320od_a_a @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_161_image__cong,axiom,
! [M2: set_a,N2: set_a,F: a > product_prod_a_a,G: a > product_prod_a_a] :
( ( M2 = N2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_7400625782589995694od_a_a @ F @ M2 )
= ( image_7400625782589995694od_a_a @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_162_bex__imageD,axiom,
! [F: a > product_prod_a_a,A: set_a,P: product_prod_a_a > $o] :
( ? [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ ( image_7400625782589995694od_a_a @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_163_bex__imageD,axiom,
! [F: product_prod_a_a > product_prod_a_a,A: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ? [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ ( image_4636654165204879301od_a_a @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_164_bex__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_165_bex__imageD,axiom,
! [F: nat > product_prod_a_a,A: set_nat,P: product_prod_a_a > $o] :
( ? [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ ( image_372941888232738320od_a_a @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_166_bex__imageD,axiom,
! [F: product_prod_a_a > nat,A: set_Product_prod_a_a,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_9053670898913107890_a_nat @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_167_image__iff,axiom,
! [Z2: product_prod_a_a,F: a > product_prod_a_a,A: set_a] :
( ( member1426531477525435216od_a_a @ Z2 @ ( image_7400625782589995694od_a_a @ F @ A ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_168_image__iff,axiom,
! [Z2: product_prod_a_a,F: product_prod_a_a > product_prod_a_a,A: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z2 @ ( image_4636654165204879301od_a_a @ F @ A ) )
= ( ? [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_169_image__iff,axiom,
! [Z2: product_prod_a_a,F: nat > product_prod_a_a,A: set_nat] :
( ( member1426531477525435216od_a_a @ Z2 @ ( image_372941888232738320od_a_a @ F @ A ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_170_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_171_image__iff,axiom,
! [Z2: nat,F: product_prod_a_a > nat,A: set_Product_prod_a_a] :
( ( member_nat @ Z2 @ ( image_9053670898913107890_a_nat @ F @ A ) )
= ( ? [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_172_imageI,axiom,
! [X: product_prod_a_a,A: set_Product_prod_a_a,F: product_prod_a_a > product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A )
=> ( member1426531477525435216od_a_a @ ( F @ X ) @ ( image_4636654165204879301od_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_173_imageI,axiom,
! [X: product_prod_a_a,A: set_Product_prod_a_a,F: product_prod_a_a > nat] :
( ( member1426531477525435216od_a_a @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_9053670898913107890_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_174_imageI,axiom,
! [X: product_prod_a_a,A: set_Product_prod_a_a,F: product_prod_a_a > a] :
( ( member1426531477525435216od_a_a @ X @ A )
=> ( member_a @ ( F @ X ) @ ( image_3437945252899457948_a_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_175_imageI,axiom,
! [X: nat,A: set_nat,F: nat > product_prod_a_a] :
( ( member_nat @ X @ A )
=> ( member1426531477525435216od_a_a @ ( F @ X ) @ ( image_372941888232738320od_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_176_imageI,axiom,
! [X: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_177_imageI,axiom,
! [X: nat,A: set_nat,F: nat > a] :
( ( member_nat @ X @ A )
=> ( member_a @ ( F @ X ) @ ( image_nat_a @ F @ A ) ) ) ).
% imageI
thf(fact_178_imageI,axiom,
! [X: a,A: set_a,F: a > product_prod_a_a] :
( ( member_a @ X @ A )
=> ( member1426531477525435216od_a_a @ ( F @ X ) @ ( image_7400625782589995694od_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_179_imageI,axiom,
! [X: a,A: set_a,F: a > nat] :
( ( member_a @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_180_imageI,axiom,
! [X: a,A: set_a,F: a > a] :
( ( member_a @ X @ A )
=> ( member_a @ ( F @ X ) @ ( image_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_181_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_182_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_183_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_184_Compr__image__eq,axiom,
! [F: complex > a,A: set_complex,P: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ ( image_complex_a @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_complex_a @ F
@ ( collect_complex
@ ^ [X2: complex] :
( ( member_complex @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_185_Compr__image__eq,axiom,
! [F: a > nat,A: set_a,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_a_nat @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_a_nat @ F
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_186_Compr__image__eq,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_187_Compr__image__eq,axiom,
! [F: int > nat,A: set_int,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_int_nat @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_int_nat @ F
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_188_Compr__image__eq,axiom,
! [F: complex > nat,A: set_complex,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_complex_nat @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_complex_nat @ F
@ ( collect_complex
@ ^ [X2: complex] :
( ( member_complex @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_189_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_190_Compr__image__eq,axiom,
! [F: nat > int,A: set_nat,P: int > $o] :
( ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ ( image_nat_int @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_nat_int @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_191_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_192_image__image,axiom,
! [F: a > product_prod_a_a,G: a > a,A: set_a] :
( ( image_7400625782589995694od_a_a @ F @ ( image_a_a @ G @ A ) )
= ( image_7400625782589995694od_a_a
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_193_image__image,axiom,
! [F: a > product_prod_a_a,G: nat > a,A: set_nat] :
( ( image_7400625782589995694od_a_a @ F @ ( image_nat_a @ G @ A ) )
= ( image_372941888232738320od_a_a
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_194_image__image,axiom,
! [F: nat > nat,G: product_prod_a_a > nat,A: set_Product_prod_a_a] :
( ( image_nat_nat @ F @ ( image_9053670898913107890_a_nat @ G @ A ) )
= ( image_9053670898913107890_a_nat
@ ^ [X2: product_prod_a_a] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_195_image__image,axiom,
! [F: nat > product_prod_a_a,G: a > nat,A: set_a] :
( ( image_372941888232738320od_a_a @ F @ ( image_a_nat @ G @ A ) )
= ( image_7400625782589995694od_a_a
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_196_image__image,axiom,
! [F: nat > product_prod_a_a,G: nat > nat,A: set_nat] :
( ( image_372941888232738320od_a_a @ F @ ( image_nat_nat @ G @ A ) )
= ( image_372941888232738320od_a_a
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_197_image__image,axiom,
! [F: product_prod_a_a > nat,G: a > product_prod_a_a,A: set_a] :
( ( image_9053670898913107890_a_nat @ F @ ( image_7400625782589995694od_a_a @ G @ A ) )
= ( image_a_nat
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_198_image__image,axiom,
! [F: product_prod_a_a > nat,G: nat > product_prod_a_a,A: set_nat] :
( ( image_9053670898913107890_a_nat @ F @ ( image_372941888232738320od_a_a @ G @ A ) )
= ( image_nat_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_199_image__image,axiom,
! [F: a > product_prod_a_a,G: product_prod_a_a > a,A: set_Product_prod_a_a] :
( ( image_7400625782589995694od_a_a @ F @ ( image_3437945252899457948_a_a_a @ G @ A ) )
= ( image_4636654165204879301od_a_a
@ ^ [X2: product_prod_a_a] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_200_image__image,axiom,
! [F: product_prod_a_a > product_prod_a_a,G: a > product_prod_a_a,A: set_a] :
( ( image_4636654165204879301od_a_a @ F @ ( image_7400625782589995694od_a_a @ G @ A ) )
= ( image_7400625782589995694od_a_a
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_201_imageE,axiom,
! [B: product_prod_a_a,F: product_prod_a_a > product_prod_a_a,A: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B @ ( image_4636654165204879301od_a_a @ F @ A ) )
=> ~ ! [X3: product_prod_a_a] :
( ( B
= ( F @ X3 ) )
=> ~ ( member1426531477525435216od_a_a @ X3 @ A ) ) ) ).
% imageE
thf(fact_202_imageE,axiom,
! [B: product_prod_a_a,F: nat > product_prod_a_a,A: set_nat] :
( ( member1426531477525435216od_a_a @ B @ ( image_372941888232738320od_a_a @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_203_imageE,axiom,
! [B: product_prod_a_a,F: a > product_prod_a_a,A: set_a] :
( ( member1426531477525435216od_a_a @ B @ ( image_7400625782589995694od_a_a @ F @ A ) )
=> ~ ! [X3: a] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_a @ X3 @ A ) ) ) ).
% imageE
thf(fact_204_imageE,axiom,
! [B: nat,F: product_prod_a_a > nat,A: set_Product_prod_a_a] :
( ( member_nat @ B @ ( image_9053670898913107890_a_nat @ F @ A ) )
=> ~ ! [X3: product_prod_a_a] :
( ( B
= ( F @ X3 ) )
=> ~ ( member1426531477525435216od_a_a @ X3 @ A ) ) ) ).
% imageE
thf(fact_205_imageE,axiom,
! [B: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_206_imageE,axiom,
! [B: nat,F: a > nat,A: set_a] :
( ( member_nat @ B @ ( image_a_nat @ F @ A ) )
=> ~ ! [X3: a] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_a @ X3 @ A ) ) ) ).
% imageE
thf(fact_207_imageE,axiom,
! [B: a,F: product_prod_a_a > a,A: set_Product_prod_a_a] :
( ( member_a @ B @ ( image_3437945252899457948_a_a_a @ F @ A ) )
=> ~ ! [X3: product_prod_a_a] :
( ( B
= ( F @ X3 ) )
=> ~ ( member1426531477525435216od_a_a @ X3 @ A ) ) ) ).
% imageE
thf(fact_208_imageE,axiom,
! [B: a,F: nat > a,A: set_nat] :
( ( member_a @ B @ ( image_nat_a @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_209_imageE,axiom,
! [B: a,F: a > a,A: set_a] :
( ( member_a @ B @ ( image_a_a @ F @ A ) )
=> ~ ! [X3: a] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_a @ X3 @ A ) ) ) ).
% imageE
thf(fact_210_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_211_finite__SigmaI,axiom,
! [A: set_a,B4: a > set_nat] :
( ( finite_finite_a @ A )
=> ( ! [A5: a] :
( ( member_a @ A5 @ A )
=> ( finite_finite_nat @ ( B4 @ A5 ) ) )
=> ( finite6644898363146130708_a_nat @ ( product_Sigma_a_nat @ A @ B4 ) ) ) ) ).
% finite_SigmaI
thf(fact_212_finite__SigmaI,axiom,
! [A: set_a,B4: a > set_int] :
( ( finite_finite_a @ A )
=> ( ! [A5: a] :
( ( member_a @ A5 @ A )
=> ( finite_finite_int @ ( B4 @ A5 ) ) )
=> ( finite2467047343636934000_a_int @ ( product_Sigma_a_int @ A @ B4 ) ) ) ) ).
% finite_SigmaI
thf(fact_213_finite__SigmaI,axiom,
! [A: set_a,B4: a > set_complex] :
( ( finite_finite_a @ A )
=> ( ! [A5: a] :
( ( member_a @ A5 @ A )
=> ( finite3207457112153483333omplex @ ( B4 @ A5 ) ) )
=> ( finite228341146693599474omplex @ ( produc73771002080876644omplex @ A @ B4 ) ) ) ) ).
% finite_SigmaI
thf(fact_214_finite__SigmaI,axiom,
! [A: set_nat,B4: nat > set_a] :
( ( finite_finite_nat @ A )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ( finite_finite_a @ ( B4 @ A5 ) ) )
=> ( finite659689790015031866_nat_a @ ( product_Sigma_nat_a @ A @ B4 ) ) ) ) ).
% finite_SigmaI
thf(fact_215_finite__SigmaI,axiom,
! [A: set_nat,B4: nat > set_nat] :
( ( finite_finite_nat @ A )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ( finite_finite_nat @ ( B4 @ A5 ) ) )
=> ( finite6177210948735845034at_nat @ ( produc457027306803732586at_nat @ A @ B4 ) ) ) ) ).
% finite_SigmaI
thf(fact_216_finite__SigmaI,axiom,
! [A: set_nat,B4: nat > set_int] :
( ( finite_finite_nat @ A )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ( finite_finite_int @ ( B4 @ A5 ) ) )
=> ( finite1999359929226648326at_int @ ( produc454536836294682310at_int @ A @ B4 ) ) ) ) ).
% finite_SigmaI
thf(fact_217_finite__SigmaI,axiom,
! [A: set_nat,B4: nat > set_complex] :
( ( finite_finite_nat @ A )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ( finite3207457112153483333omplex @ ( B4 @ A5 ) ) )
=> ( finite1477110953402811016omplex @ ( produc7489356884331829576omplex @ A @ B4 ) ) ) ) ).
% finite_SigmaI
thf(fact_218_finite__SigmaI,axiom,
! [A: set_int,B4: int > set_a] :
( ( finite_finite_int @ A )
=> ( ! [A5: int] :
( ( member_int @ A5 @ A )
=> ( finite_finite_a @ ( B4 @ A5 ) ) )
=> ( finite7637916859199361758_int_a @ ( product_Sigma_int_a @ A @ B4 ) ) ) ) ).
% finite_SigmaI
thf(fact_219_finite__SigmaI,axiom,
! [A: set_int,B4: int > set_nat] :
( ( finite_finite_int @ A )
=> ( ! [A5: int] :
( ( member_int @ A5 @ A )
=> ( finite_finite_nat @ ( B4 @ A5 ) ) )
=> ( finite7176564660636899590nt_nat @ ( produc1456381018704787142nt_nat @ A @ B4 ) ) ) ) ).
% finite_SigmaI
thf(fact_220_finite__SigmaI,axiom,
! [A: set_int,B4: int > set_int] :
( ( finite_finite_int @ A )
=> ( ! [A5: int] :
( ( member_int @ A5 @ A )
=> ( finite_finite_int @ ( B4 @ A5 ) ) )
=> ( finite2998713641127702882nt_int @ ( produc1453890548195736866nt_int @ A @ B4 ) ) ) ) ).
% finite_SigmaI
thf(fact_221_finite__imageI,axiom,
! [F2: set_a,H: a > a] :
( ( finite_finite_a @ F2 )
=> ( finite_finite_a @ ( image_a_a @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_222_finite__imageI,axiom,
! [F2: set_a,H: a > nat] :
( ( finite_finite_a @ F2 )
=> ( finite_finite_nat @ ( image_a_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_223_finite__imageI,axiom,
! [F2: set_a,H: a > int] :
( ( finite_finite_a @ F2 )
=> ( finite_finite_int @ ( image_a_int @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_224_finite__imageI,axiom,
! [F2: set_a,H: a > complex] :
( ( finite_finite_a @ F2 )
=> ( finite3207457112153483333omplex @ ( image_a_complex @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_225_finite__imageI,axiom,
! [F2: set_nat,H: nat > a] :
( ( finite_finite_nat @ F2 )
=> ( finite_finite_a @ ( image_nat_a @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_226_finite__imageI,axiom,
! [F2: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F2 )
=> ( finite_finite_nat @ ( image_nat_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_227_finite__imageI,axiom,
! [F2: set_nat,H: nat > int] :
( ( finite_finite_nat @ F2 )
=> ( finite_finite_int @ ( image_nat_int @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_228_finite__imageI,axiom,
! [F2: set_nat,H: nat > complex] :
( ( finite_finite_nat @ F2 )
=> ( finite3207457112153483333omplex @ ( image_nat_complex @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_229_finite__imageI,axiom,
! [F2: set_int,H: int > a] :
( ( finite_finite_int @ F2 )
=> ( finite_finite_a @ ( image_int_a @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_230_finite__imageI,axiom,
! [F2: set_int,H: int > nat] :
( ( finite_finite_int @ F2 )
=> ( finite_finite_nat @ ( image_int_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_231_calculation,axiom,
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
@ ( finite4795055649997197647od_a_a
@ ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X2: a,Y2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 )
@ ( product_Sigma_a_a @ m
@ ^ [Uu: a] : m ) )
& ( ord_less_a @ X2 @ Y2 ) ) ) ) ) )
= ( minus_minus_nat @ ( power_power_nat @ ( finite_card_a @ m ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
@ ( finite4795055649997197647od_a_a
@ ( image_7400625782589995694od_a_a
@ ^ [X2: a] : ( product_Pair_a_a @ X2 @ X2 )
@ m ) ) ) ) ).
% calculation
thf(fact_232_finite__Collect__conjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P ) )
| ( finite_finite_a @ ( collect_a @ Q ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X2: a] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_233_finite__Collect__conjI,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( ( finite6544458595007987280od_a_a @ ( collec3336397797384452498od_a_a @ P ) )
| ( finite6544458595007987280od_a_a @ ( collec3336397797384452498od_a_a @ Q ) ) )
=> ( finite6544458595007987280od_a_a
@ ( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_234_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_235_finite__Collect__conjI,axiom,
! [P: int > $o,Q: int > $o] :
( ( ( finite_finite_int @ ( collect_int @ P ) )
| ( finite_finite_int @ ( collect_int @ Q ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X2: int] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_236_finite__Collect__conjI,axiom,
! [P: complex > $o,Q: complex > $o] :
( ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
| ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X2: complex] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_237_finite__Collect__disjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X2: a] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P ) )
& ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_238_finite__Collect__disjI,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a
@ ( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite6544458595007987280od_a_a @ ( collec3336397797384452498od_a_a @ P ) )
& ( finite6544458595007987280od_a_a @ ( collec3336397797384452498od_a_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_239_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_240_finite__Collect__disjI,axiom,
! [P: int > $o,Q: int > $o] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X2: int] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite_int @ ( collect_int @ P ) )
& ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_241_finite__Collect__disjI,axiom,
! [P: complex > $o,Q: complex > $o] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X2: complex] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
& ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_242_a,axiom,
( finite6544458595007987280od_a_a
@ ( product_Sigma_a_a @ m
@ ^ [Uu: a] : m ) ) ).
% a
thf(fact_243_power__numeral,axiom,
! [K: num,L: num] :
( ( power_power_complex @ ( numera6690914467698888265omplex @ K ) @ ( numeral_numeral_nat @ L ) )
= ( numera6690914467698888265omplex @ ( pow @ K @ L ) ) ) ).
% power_numeral
thf(fact_244_power__numeral,axiom,
! [K: num,L: num] :
( ( power_power_nat @ ( numeral_numeral_nat @ K ) @ ( numeral_numeral_nat @ L ) )
= ( numeral_numeral_nat @ ( pow @ K @ L ) ) ) ).
% power_numeral
thf(fact_245_power__numeral,axiom,
! [K: num,L: num] :
( ( power_power_real @ ( numeral_numeral_real @ K ) @ ( numeral_numeral_nat @ L ) )
= ( numeral_numeral_real @ ( pow @ K @ L ) ) ) ).
% power_numeral
thf(fact_246_power__numeral,axiom,
! [K: num,L: num] :
( ( power_8040749407984259932d_enat @ ( numera1916890842035813515d_enat @ K ) @ ( numeral_numeral_nat @ L ) )
= ( numera1916890842035813515d_enat @ ( pow @ K @ L ) ) ) ).
% power_numeral
thf(fact_247_power__numeral,axiom,
! [K: num,L: num] :
( ( power_power_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_nat @ L ) )
= ( numeral_numeral_int @ ( pow @ K @ L ) ) ) ).
% power_numeral
thf(fact_248_finite__Diff,axiom,
! [A: set_a,B4: set_a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( minus_minus_set_a @ A @ B4 ) ) ) ).
% finite_Diff
thf(fact_249_finite__Diff,axiom,
! [A: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B4 ) ) ) ).
% finite_Diff
thf(fact_250_finite__Diff,axiom,
! [A: set_int,B4: set_int] :
( ( finite_finite_int @ A )
=> ( finite_finite_int @ ( minus_minus_set_int @ A @ B4 ) ) ) ).
% finite_Diff
thf(fact_251_finite__Diff,axiom,
! [A: set_complex,B4: set_complex] :
( ( finite3207457112153483333omplex @ A )
=> ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A @ B4 ) ) ) ).
% finite_Diff
thf(fact_252_finite__Diff,axiom,
! [A: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A )
=> ( finite6544458595007987280od_a_a @ ( minus_6817036919807184750od_a_a @ A @ B4 ) ) ) ).
% finite_Diff
thf(fact_253_finite__Diff2,axiom,
! [B4: set_a,A: set_a] :
( ( finite_finite_a @ B4 )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A @ B4 ) )
= ( finite_finite_a @ A ) ) ) ).
% finite_Diff2
thf(fact_254_finite__Diff2,axiom,
! [B4: set_nat,A: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B4 ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_255_finite__Diff2,axiom,
! [B4: set_int,A: set_int] :
( ( finite_finite_int @ B4 )
=> ( ( finite_finite_int @ ( minus_minus_set_int @ A @ B4 ) )
= ( finite_finite_int @ A ) ) ) ).
% finite_Diff2
thf(fact_256_finite__Diff2,axiom,
! [B4: set_complex,A: set_complex] :
( ( finite3207457112153483333omplex @ B4 )
=> ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A @ B4 ) )
= ( finite3207457112153483333omplex @ A ) ) ) ).
% finite_Diff2
thf(fact_257_finite__Diff2,axiom,
! [B4: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ B4 )
=> ( ( finite6544458595007987280od_a_a @ ( minus_6817036919807184750od_a_a @ A @ B4 ) )
= ( finite6544458595007987280od_a_a @ A ) ) ) ).
% finite_Diff2
thf(fact_258_finite__cartesian__product,axiom,
! [A: set_a,B4: set_nat] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ B4 )
=> ( finite6644898363146130708_a_nat
@ ( product_Sigma_a_nat @ A
@ ^ [Uu: a] : B4 ) ) ) ) ).
% finite_cartesian_product
thf(fact_259_finite__cartesian__product,axiom,
! [A: set_a,B4: set_int] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_int @ B4 )
=> ( finite2467047343636934000_a_int
@ ( product_Sigma_a_int @ A
@ ^ [Uu: a] : B4 ) ) ) ) ).
% finite_cartesian_product
thf(fact_260_finite__cartesian__product,axiom,
! [A: set_a,B4: set_complex] :
( ( finite_finite_a @ A )
=> ( ( finite3207457112153483333omplex @ B4 )
=> ( finite228341146693599474omplex
@ ( produc73771002080876644omplex @ A
@ ^ [Uu: a] : B4 ) ) ) ) ).
% finite_cartesian_product
thf(fact_261_finite__cartesian__product,axiom,
! [A: set_nat,B4: set_a] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ B4 )
=> ( finite659689790015031866_nat_a
@ ( product_Sigma_nat_a @ A
@ ^ [Uu: nat] : B4 ) ) ) ) ).
% finite_cartesian_product
thf(fact_262_finite__cartesian__product,axiom,
! [A: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B4 )
=> ( finite6177210948735845034at_nat
@ ( produc457027306803732586at_nat @ A
@ ^ [Uu: nat] : B4 ) ) ) ) ).
% finite_cartesian_product
thf(fact_263_finite__cartesian__product,axiom,
! [A: set_nat,B4: set_int] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_int @ B4 )
=> ( finite1999359929226648326at_int
@ ( produc454536836294682310at_int @ A
@ ^ [Uu: nat] : B4 ) ) ) ) ).
% finite_cartesian_product
thf(fact_264_finite__cartesian__product,axiom,
! [A: set_nat,B4: set_complex] :
( ( finite_finite_nat @ A )
=> ( ( finite3207457112153483333omplex @ B4 )
=> ( finite1477110953402811016omplex
@ ( produc7489356884331829576omplex @ A
@ ^ [Uu: nat] : B4 ) ) ) ) ).
% finite_cartesian_product
thf(fact_265_finite__cartesian__product,axiom,
! [A: set_int,B4: set_a] :
( ( finite_finite_int @ A )
=> ( ( finite_finite_a @ B4 )
=> ( finite7637916859199361758_int_a
@ ( product_Sigma_int_a @ A
@ ^ [Uu: int] : B4 ) ) ) ) ).
% finite_cartesian_product
thf(fact_266_finite__cartesian__product,axiom,
! [A: set_int,B4: set_nat] :
( ( finite_finite_int @ A )
=> ( ( finite_finite_nat @ B4 )
=> ( finite7176564660636899590nt_nat
@ ( produc1456381018704787142nt_nat @ A
@ ^ [Uu: int] : B4 ) ) ) ) ).
% finite_cartesian_product
thf(fact_267_finite__cartesian__product,axiom,
! [A: set_int,B4: set_int] :
( ( finite_finite_int @ A )
=> ( ( finite_finite_int @ B4 )
=> ( finite2998713641127702882nt_int
@ ( produc1453890548195736866nt_int @ A
@ ^ [Uu: int] : B4 ) ) ) ) ).
% finite_cartesian_product
thf(fact_268_DiffI,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ A )
=> ( ~ ( member_nat @ C @ B4 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A @ B4 ) ) ) ) ).
% DiffI
thf(fact_269_DiffI,axiom,
! [C: a,A: set_a,B4: set_a] :
( ( member_a @ C @ A )
=> ( ~ ( member_a @ C @ B4 )
=> ( member_a @ C @ ( minus_minus_set_a @ A @ B4 ) ) ) ) ).
% DiffI
thf(fact_270_DiffI,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ A )
=> ( ~ ( member1426531477525435216od_a_a @ C @ B4 )
=> ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A @ B4 ) ) ) ) ).
% DiffI
thf(fact_271_Diff__iff,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B4 ) )
= ( ( member_nat @ C @ A )
& ~ ( member_nat @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_272_Diff__iff,axiom,
! [C: a,A: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B4 ) )
= ( ( member_a @ C @ A )
& ~ ( member_a @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_273_Diff__iff,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A @ B4 ) )
= ( ( member1426531477525435216od_a_a @ C @ A )
& ~ ( member1426531477525435216od_a_a @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_274_Diff__idemp,axiom,
! [A: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( minus_6817036919807184750od_a_a @ ( minus_6817036919807184750od_a_a @ A @ B4 ) @ B4 )
= ( minus_6817036919807184750od_a_a @ A @ B4 ) ) ).
% Diff_idemp
thf(fact_275_mem__case__prodI,axiom,
! [Z2: product_prod_a_a,C: a > a > set_Product_prod_a_a,A2: a,B: a] :
( ( member1426531477525435216od_a_a @ Z2 @ ( C @ A2 @ B ) )
=> ( member1426531477525435216od_a_a @ Z2 @ ( produc5766521956407364827od_a_a @ C @ ( product_Pair_a_a @ A2 @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_276_mem__case__prodI,axiom,
! [Z2: nat,C: a > a > set_nat,A2: a,B: a] :
( ( member_nat @ Z2 @ ( C @ A2 @ B ) )
=> ( member_nat @ Z2 @ ( produc153843693180602034et_nat @ C @ ( product_Pair_a_a @ A2 @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_277_mem__case__prodI,axiom,
! [Z2: a,C: a > a > set_a,A2: a,B: a] :
( ( member_a @ Z2 @ ( C @ A2 @ B ) )
=> ( member_a @ Z2 @ ( produc9217457822752978994_set_a @ C @ ( product_Pair_a_a @ A2 @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_278_mem__case__prodI,axiom,
! [Z2: product_prod_a_a,C: num > num > set_Product_prod_a_a,A2: num,B: num] :
( ( member1426531477525435216od_a_a @ Z2 @ ( C @ A2 @ B ) )
=> ( member1426531477525435216od_a_a @ Z2 @ ( produc1804361614519313061od_a_a @ C @ ( product_Pair_num_num @ A2 @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_279_mem__case__prodI,axiom,
! [Z2: nat,C: num > num > set_nat,A2: num,B: num] :
( ( member_nat @ Z2 @ ( C @ A2 @ B ) )
=> ( member_nat @ Z2 @ ( produc1361121860356118632et_nat @ C @ ( product_Pair_num_num @ A2 @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_280_mem__case__prodI,axiom,
! [Z2: a,C: num > num > set_a,A2: num,B: num] :
( ( member_a @ Z2 @ ( C @ A2 @ B ) )
=> ( member_a @ Z2 @ ( produc2987288209340884988_set_a @ C @ ( product_Pair_num_num @ A2 @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_281_mem__case__prodI2,axiom,
! [P2: product_prod_a_a,Z2: product_prod_a_a,C: a > a > set_Product_prod_a_a] :
( ! [A5: a,B5: a] :
( ( P2
= ( product_Pair_a_a @ A5 @ B5 ) )
=> ( member1426531477525435216od_a_a @ Z2 @ ( C @ A5 @ B5 ) ) )
=> ( member1426531477525435216od_a_a @ Z2 @ ( produc5766521956407364827od_a_a @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_282_mem__case__prodI2,axiom,
! [P2: product_prod_a_a,Z2: nat,C: a > a > set_nat] :
( ! [A5: a,B5: a] :
( ( P2
= ( product_Pair_a_a @ A5 @ B5 ) )
=> ( member_nat @ Z2 @ ( C @ A5 @ B5 ) ) )
=> ( member_nat @ Z2 @ ( produc153843693180602034et_nat @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_283_mem__case__prodI2,axiom,
! [P2: product_prod_a_a,Z2: a,C: a > a > set_a] :
( ! [A5: a,B5: a] :
( ( P2
= ( product_Pair_a_a @ A5 @ B5 ) )
=> ( member_a @ Z2 @ ( C @ A5 @ B5 ) ) )
=> ( member_a @ Z2 @ ( produc9217457822752978994_set_a @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_284_mem__case__prodI2,axiom,
! [P2: product_prod_num_num,Z2: product_prod_a_a,C: num > num > set_Product_prod_a_a] :
( ! [A5: num,B5: num] :
( ( P2
= ( product_Pair_num_num @ A5 @ B5 ) )
=> ( member1426531477525435216od_a_a @ Z2 @ ( C @ A5 @ B5 ) ) )
=> ( member1426531477525435216od_a_a @ Z2 @ ( produc1804361614519313061od_a_a @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_285_mem__case__prodI2,axiom,
! [P2: product_prod_num_num,Z2: nat,C: num > num > set_nat] :
( ! [A5: num,B5: num] :
( ( P2
= ( product_Pair_num_num @ A5 @ B5 ) )
=> ( member_nat @ Z2 @ ( C @ A5 @ B5 ) ) )
=> ( member_nat @ Z2 @ ( produc1361121860356118632et_nat @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_286_mem__case__prodI2,axiom,
! [P2: product_prod_num_num,Z2: a,C: num > num > set_a] :
( ! [A5: num,B5: num] :
( ( P2
= ( product_Pair_num_num @ A5 @ B5 ) )
=> ( member_a @ Z2 @ ( C @ A5 @ B5 ) ) )
=> ( member_a @ Z2 @ ( produc2987288209340884988_set_a @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_287_verit__comp__simplify1_I1_J,axiom,
! [A2: real] :
~ ( ord_less_real @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_288_verit__comp__simplify1_I1_J,axiom,
! [A2: num] :
~ ( ord_less_num @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_289_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_290_verit__comp__simplify1_I1_J,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_291_verit__comp__simplify1_I1_J,axiom,
! [A2: a] :
~ ( ord_less_a @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_292_DiffE,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B4 ) )
=> ~ ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B4 ) ) ) ).
% DiffE
thf(fact_293_DiffE,axiom,
! [C: a,A: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B4 ) )
=> ~ ( ( member_a @ C @ A )
=> ( member_a @ C @ B4 ) ) ) ).
% DiffE
thf(fact_294_DiffE,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A @ B4 ) )
=> ~ ( ( member1426531477525435216od_a_a @ C @ A )
=> ( member1426531477525435216od_a_a @ C @ B4 ) ) ) ).
% DiffE
thf(fact_295_DiffD1,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B4 ) )
=> ( member_nat @ C @ A ) ) ).
% DiffD1
thf(fact_296_DiffD1,axiom,
! [C: a,A: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B4 ) )
=> ( member_a @ C @ A ) ) ).
% DiffD1
thf(fact_297_DiffD1,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A @ B4 ) )
=> ( member1426531477525435216od_a_a @ C @ A ) ) ).
% DiffD1
thf(fact_298_DiffD2,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B4 ) )
=> ~ ( member_nat @ C @ B4 ) ) ).
% DiffD2
thf(fact_299_DiffD2,axiom,
! [C: a,A: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B4 ) )
=> ~ ( member_a @ C @ B4 ) ) ).
% DiffD2
thf(fact_300_DiffD2,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A @ B4 ) )
=> ~ ( member1426531477525435216od_a_a @ C @ B4 ) ) ).
% DiffD2
thf(fact_301_set__diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A6 )
& ~ ( member_a @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_302_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A6 )
& ~ ( member_nat @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_303_set__diff__eq,axiom,
( minus_minus_set_int
= ( ^ [A6: set_int,B6: set_int] :
( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A6 )
& ~ ( member_int @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_304_set__diff__eq,axiom,
( minus_811609699411566653omplex
= ( ^ [A6: set_complex,B6: set_complex] :
( collect_complex
@ ^ [X2: complex] :
( ( member_complex @ X2 @ A6 )
& ~ ( member_complex @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_305_set__diff__eq,axiom,
( minus_6817036919807184750od_a_a
= ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A6 )
& ~ ( member1426531477525435216od_a_a @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_306_minus__set__def,axiom,
( minus_minus_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ( minus_minus_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A6 )
@ ^ [X2: a] : ( member_a @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_307_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_308_minus__set__def,axiom,
( minus_minus_set_int
= ( ^ [A6: set_int,B6: set_int] :
( collect_int
@ ( minus_minus_int_o
@ ^ [X2: int] : ( member_int @ X2 @ A6 )
@ ^ [X2: int] : ( member_int @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_309_minus__set__def,axiom,
( minus_811609699411566653omplex
= ( ^ [A6: set_complex,B6: set_complex] :
( collect_complex
@ ( minus_8727706125548526216plex_o
@ ^ [X2: complex] : ( member_complex @ X2 @ A6 )
@ ^ [X2: complex] : ( member_complex @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_310_minus__set__def,axiom,
( minus_6817036919807184750od_a_a
= ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( collec3336397797384452498od_a_a
@ ( minus_4793868396798367471_a_a_o
@ ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ A6 )
@ ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_311_mem__case__prodE,axiom,
! [Z2: product_prod_a_a,C: a > a > set_Product_prod_a_a,P2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z2 @ ( produc5766521956407364827od_a_a @ C @ P2 ) )
=> ~ ! [X3: a,Y4: a] :
( ( P2
= ( product_Pair_a_a @ X3 @ Y4 ) )
=> ~ ( member1426531477525435216od_a_a @ Z2 @ ( C @ X3 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_312_mem__case__prodE,axiom,
! [Z2: nat,C: a > a > set_nat,P2: product_prod_a_a] :
( ( member_nat @ Z2 @ ( produc153843693180602034et_nat @ C @ P2 ) )
=> ~ ! [X3: a,Y4: a] :
( ( P2
= ( product_Pair_a_a @ X3 @ Y4 ) )
=> ~ ( member_nat @ Z2 @ ( C @ X3 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_313_mem__case__prodE,axiom,
! [Z2: a,C: a > a > set_a,P2: product_prod_a_a] :
( ( member_a @ Z2 @ ( produc9217457822752978994_set_a @ C @ P2 ) )
=> ~ ! [X3: a,Y4: a] :
( ( P2
= ( product_Pair_a_a @ X3 @ Y4 ) )
=> ~ ( member_a @ Z2 @ ( C @ X3 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_314_mem__case__prodE,axiom,
! [Z2: product_prod_a_a,C: num > num > set_Product_prod_a_a,P2: product_prod_num_num] :
( ( member1426531477525435216od_a_a @ Z2 @ ( produc1804361614519313061od_a_a @ C @ P2 ) )
=> ~ ! [X3: num,Y4: num] :
( ( P2
= ( product_Pair_num_num @ X3 @ Y4 ) )
=> ~ ( member1426531477525435216od_a_a @ Z2 @ ( C @ X3 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_315_mem__case__prodE,axiom,
! [Z2: nat,C: num > num > set_nat,P2: product_prod_num_num] :
( ( member_nat @ Z2 @ ( produc1361121860356118632et_nat @ C @ P2 ) )
=> ~ ! [X3: num,Y4: num] :
( ( P2
= ( product_Pair_num_num @ X3 @ Y4 ) )
=> ~ ( member_nat @ Z2 @ ( C @ X3 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_316_mem__case__prodE,axiom,
! [Z2: a,C: num > num > set_a,P2: product_prod_num_num] :
( ( member_a @ Z2 @ ( produc2987288209340884988_set_a @ C @ P2 ) )
=> ~ ! [X3: num,Y4: num] :
( ( P2
= ( product_Pair_num_num @ X3 @ Y4 ) )
=> ~ ( member_a @ Z2 @ ( C @ X3 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_317_power__commutes,axiom,
! [A2: complex,N: nat] :
( ( times_times_complex @ ( power_power_complex @ A2 @ N ) @ A2 )
= ( times_times_complex @ A2 @ ( power_power_complex @ A2 @ N ) ) ) ).
% power_commutes
thf(fact_318_power__commutes,axiom,
! [A2: nat,N: nat] :
( ( times_times_nat @ ( power_power_nat @ A2 @ N ) @ A2 )
= ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ).
% power_commutes
thf(fact_319_power__commutes,axiom,
! [A2: real,N: nat] :
( ( times_times_real @ ( power_power_real @ A2 @ N ) @ A2 )
= ( times_times_real @ A2 @ ( power_power_real @ A2 @ N ) ) ) ).
% power_commutes
thf(fact_320_power__commutes,axiom,
! [A2: int,N: nat] :
( ( times_times_int @ ( power_power_int @ A2 @ N ) @ A2 )
= ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ).
% power_commutes
thf(fact_321_power__commutes,axiom,
! [A2: extended_enat,N: nat] :
( ( times_7803423173614009249d_enat @ ( power_8040749407984259932d_enat @ A2 @ N ) @ A2 )
= ( times_7803423173614009249d_enat @ A2 @ ( power_8040749407984259932d_enat @ A2 @ N ) ) ) ).
% power_commutes
thf(fact_322_power__mult__distrib,axiom,
! [A2: complex,B: complex,N: nat] :
( ( power_power_complex @ ( times_times_complex @ A2 @ B ) @ N )
= ( times_times_complex @ ( power_power_complex @ A2 @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_323_power__mult__distrib,axiom,
! [A2: nat,B: nat,N: nat] :
( ( power_power_nat @ ( times_times_nat @ A2 @ B ) @ N )
= ( times_times_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_324_power__mult__distrib,axiom,
! [A2: real,B: real,N: nat] :
( ( power_power_real @ ( times_times_real @ A2 @ B ) @ N )
= ( times_times_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_325_power__mult__distrib,axiom,
! [A2: int,B: int,N: nat] :
( ( power_power_int @ ( times_times_int @ A2 @ B ) @ N )
= ( times_times_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_326_power__mult__distrib,axiom,
! [A2: extended_enat,B: extended_enat,N: nat] :
( ( power_8040749407984259932d_enat @ ( times_7803423173614009249d_enat @ A2 @ B ) @ N )
= ( times_7803423173614009249d_enat @ ( power_8040749407984259932d_enat @ A2 @ N ) @ ( power_8040749407984259932d_enat @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_327_power__commuting__commutes,axiom,
! [X: complex,Y: complex,N: nat] :
( ( ( times_times_complex @ X @ Y )
= ( times_times_complex @ Y @ X ) )
=> ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ Y )
= ( times_times_complex @ Y @ ( power_power_complex @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_328_power__commuting__commutes,axiom,
! [X: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X @ Y )
= ( times_times_nat @ Y @ X ) )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y )
= ( times_times_nat @ Y @ ( power_power_nat @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_329_power__commuting__commutes,axiom,
! [X: real,Y: real,N: nat] :
( ( ( times_times_real @ X @ Y )
= ( times_times_real @ Y @ X ) )
=> ( ( times_times_real @ ( power_power_real @ X @ N ) @ Y )
= ( times_times_real @ Y @ ( power_power_real @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_330_power__commuting__commutes,axiom,
! [X: int,Y: int,N: nat] :
( ( ( times_times_int @ X @ Y )
= ( times_times_int @ Y @ X ) )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y )
= ( times_times_int @ Y @ ( power_power_int @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_331_power__commuting__commutes,axiom,
! [X: extended_enat,Y: extended_enat,N: nat] :
( ( ( times_7803423173614009249d_enat @ X @ Y )
= ( times_7803423173614009249d_enat @ Y @ X ) )
=> ( ( times_7803423173614009249d_enat @ ( power_8040749407984259932d_enat @ X @ N ) @ Y )
= ( times_7803423173614009249d_enat @ Y @ ( power_8040749407984259932d_enat @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_332_power__mult,axiom,
! [A2: nat,M: nat,N: nat] :
( ( power_power_nat @ A2 @ ( times_times_nat @ M @ N ) )
= ( power_power_nat @ ( power_power_nat @ A2 @ M ) @ N ) ) ).
% power_mult
thf(fact_333_power__mult,axiom,
! [A2: real,M: nat,N: nat] :
( ( power_power_real @ A2 @ ( times_times_nat @ M @ N ) )
= ( power_power_real @ ( power_power_real @ A2 @ M ) @ N ) ) ).
% power_mult
thf(fact_334_power__mult,axiom,
! [A2: complex,M: nat,N: nat] :
( ( power_power_complex @ A2 @ ( times_times_nat @ M @ N ) )
= ( power_power_complex @ ( power_power_complex @ A2 @ M ) @ N ) ) ).
% power_mult
thf(fact_335_power__mult,axiom,
! [A2: int,M: nat,N: nat] :
( ( power_power_int @ A2 @ ( times_times_nat @ M @ N ) )
= ( power_power_int @ ( power_power_int @ A2 @ M ) @ N ) ) ).
% power_mult
thf(fact_336_swap__product,axiom,
! [A: set_num,B4: set_num] :
( ( image_4941118702469476421um_num
@ ( produc64540874165560627um_num
@ ^ [I2: num,J2: num] : ( product_Pair_num_num @ J2 @ I2 ) )
@ ( produc4368061533121756414um_num @ A
@ ^ [Uu: num] : B4 ) )
= ( produc4368061533121756414um_num @ B4
@ ^ [Uu: num] : A ) ) ).
% swap_product
thf(fact_337_swap__product,axiom,
! [A: set_a,B4: set_a] :
( ( image_4636654165204879301od_a_a
@ ( produc408267641121961211od_a_a
@ ^ [I2: a,J2: a] : ( product_Pair_a_a @ J2 @ I2 ) )
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : B4 ) )
= ( product_Sigma_a_a @ B4
@ ^ [Uu: a] : A ) ) ).
% swap_product
thf(fact_338_image__paired__Times,axiom,
! [F: nat > nat,G: nat > nat,A: set_nat,B4: set_nat] :
( ( image_5168914502847457605at_nat
@ ( produc2626176000494625587at_nat
@ ^ [X2: nat,Y2: nat] : ( product_Pair_nat_nat @ ( F @ X2 ) @ ( G @ Y2 ) ) )
@ ( produc457027306803732586at_nat @ A
@ ^ [Uu: nat] : B4 ) )
= ( produc457027306803732586at_nat @ ( image_nat_nat @ F @ A )
@ ^ [Uu: nat] : ( image_nat_nat @ G @ B4 ) ) ) ).
% image_paired_Times
thf(fact_339_image__paired__Times,axiom,
! [F: a > num,G: a > num,A: set_a,B4: set_a] :
( ( image_8319527084546292275um_num
@ ( produc7593592049749559913um_num
@ ^ [X2: a,Y2: a] : ( product_Pair_num_num @ ( F @ X2 ) @ ( G @ Y2 ) ) )
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : B4 ) )
= ( produc4368061533121756414um_num @ ( image_a_num @ F @ A )
@ ^ [Uu: num] : ( image_a_num @ G @ B4 ) ) ) ).
% image_paired_Times
thf(fact_340_image__paired__Times,axiom,
! [F: a > a,G: a > a,A: set_a,B4: set_a] :
( ( image_4636654165204879301od_a_a
@ ( produc408267641121961211od_a_a
@ ^ [X2: a,Y2: a] : ( product_Pair_a_a @ ( F @ X2 ) @ ( G @ Y2 ) ) )
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : B4 ) )
= ( product_Sigma_a_a @ ( image_a_a @ F @ A )
@ ^ [Uu: a] : ( image_a_a @ G @ B4 ) ) ) ).
% image_paired_Times
thf(fact_341_image__paired__Times,axiom,
! [F: a > product_prod_a_a,G: nat > nat,A: set_a,B4: set_nat] :
( ( image_5821456874911110186_a_nat
@ ( produc5937631023085385034_a_nat
@ ^ [X2: a,Y2: nat] : ( produc6483765539767234976_a_nat @ ( F @ X2 ) @ ( G @ Y2 ) ) )
@ ( product_Sigma_a_nat @ A
@ ^ [Uu: a] : B4 ) )
= ( produc1049071135499013807_a_nat @ ( image_7400625782589995694od_a_a @ F @ A )
@ ^ [Uu: product_prod_a_a] : ( image_nat_nat @ G @ B4 ) ) ) ).
% image_paired_Times
thf(fact_342_image__paired__Times,axiom,
! [F: nat > nat,G: a > product_prod_a_a,A: set_nat,B4: set_a] :
( ( image_7019153534300696310od_a_a
@ ( produc132582535617199182od_a_a
@ ^ [X2: nat,Y2: a] : ( produc7026408565941641214od_a_a @ ( F @ X2 ) @ ( G @ Y2 ) ) )
@ ( product_Sigma_nat_a @ A
@ ^ [Uu: nat] : B4 ) )
= ( produc1591714161673420045od_a_a @ ( image_nat_nat @ F @ A )
@ ^ [Uu: nat] : ( image_7400625782589995694od_a_a @ G @ B4 ) ) ) ).
% image_paired_Times
thf(fact_343_image__paired__Times,axiom,
! [F: nat > nat,G: nat > product_prod_a_a,A: set_nat,B4: set_nat] :
( ( image_8420896010980471816od_a_a
@ ( produc7616861372559947674od_a_a
@ ^ [X2: nat,Y2: nat] : ( produc7026408565941641214od_a_a @ ( F @ X2 ) @ ( G @ Y2 ) ) )
@ ( produc457027306803732586at_nat @ A
@ ^ [Uu: nat] : B4 ) )
= ( produc1591714161673420045od_a_a @ ( image_nat_nat @ F @ A )
@ ^ [Uu: nat] : ( image_372941888232738320od_a_a @ G @ B4 ) ) ) ).
% image_paired_Times
thf(fact_344_image__paired__Times,axiom,
! [F: nat > nat,G: product_prod_a_a > nat,A: set_nat,B4: set_Product_prod_a_a] :
( ( image_2440655731066075118at_nat
@ ( produc2309353493998781206at_nat
@ ^ [X2: nat,Y2: product_prod_a_a] : ( product_Pair_nat_nat @ ( F @ X2 ) @ ( G @ Y2 ) ) )
@ ( produc1591714161673420045od_a_a @ A
@ ^ [Uu: nat] : B4 ) )
= ( produc457027306803732586at_nat @ ( image_nat_nat @ F @ A )
@ ^ [Uu: nat] : ( image_9053670898913107890_a_nat @ G @ B4 ) ) ) ).
% image_paired_Times
thf(fact_345_image__paired__Times,axiom,
! [F: nat > product_prod_a_a,G: nat > nat,A: set_nat,B4: set_nat] :
( ( image_3170123979049421410_a_nat
@ ( produc2366089340628897268_a_nat
@ ^ [X2: nat,Y2: nat] : ( produc6483765539767234976_a_nat @ ( F @ X2 ) @ ( G @ Y2 ) ) )
@ ( produc457027306803732586at_nat @ A
@ ^ [Uu: nat] : B4 ) )
= ( produc1049071135499013807_a_nat @ ( image_372941888232738320od_a_a @ F @ A )
@ ^ [Uu: product_prod_a_a] : ( image_nat_nat @ G @ B4 ) ) ) ).
% image_paired_Times
thf(fact_346_image__paired__Times,axiom,
! [F: product_prod_a_a > nat,G: nat > nat,A: set_Product_prod_a_a,B4: set_nat] :
( ( image_2251685427393900820at_nat
@ ( produc413714314474088180at_nat
@ ^ [X2: product_prod_a_a,Y2: nat] : ( product_Pair_nat_nat @ ( F @ X2 ) @ ( G @ Y2 ) ) )
@ ( produc1049071135499013807_a_nat @ A
@ ^ [Uu: product_prod_a_a] : B4 ) )
= ( produc457027306803732586at_nat @ ( image_9053670898913107890_a_nat @ F @ A )
@ ^ [Uu: nat] : ( image_nat_nat @ G @ B4 ) ) ) ).
% image_paired_Times
thf(fact_347_image__paired__Times,axiom,
! [F: a > product_prod_a_a,G: nat > product_prod_a_a,A: set_a,B4: set_nat] :
( ( image_2366706423754990947od_a_a
@ ( produc231606628065897539od_a_a
@ ^ [X2: a,Y2: nat] : ( produc7886510207707329367od_a_a @ ( F @ X2 ) @ ( G @ Y2 ) ) )
@ ( product_Sigma_a_nat @ A
@ ^ [Uu: a] : B4 ) )
= ( produc5899993699339346696od_a_a @ ( image_7400625782589995694od_a_a @ F @ A )
@ ^ [Uu: product_prod_a_a] : ( image_372941888232738320od_a_a @ G @ B4 ) ) ) ).
% image_paired_Times
thf(fact_348_Diff__infinite__finite,axiom,
! [T: set_a,S: set_a] :
( ( finite_finite_a @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_349_Diff__infinite__finite,axiom,
! [T: set_nat,S: set_nat] :
( ( finite_finite_nat @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_350_Diff__infinite__finite,axiom,
! [T: set_int,S: set_int] :
( ( finite_finite_int @ T )
=> ( ~ ( finite_finite_int @ S )
=> ~ ( finite_finite_int @ ( minus_minus_set_int @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_351_Diff__infinite__finite,axiom,
! [T: set_complex,S: set_complex] :
( ( finite3207457112153483333omplex @ T )
=> ( ~ ( finite3207457112153483333omplex @ S )
=> ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_352_Diff__infinite__finite,axiom,
! [T: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ T )
=> ( ~ ( finite6544458595007987280od_a_a @ S )
=> ~ ( finite6544458595007987280od_a_a @ ( minus_6817036919807184750od_a_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_353_power__numeral__even,axiom,
! [Z2: complex,W: num] :
( ( power_power_complex @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_complex @ ( power_power_complex @ Z2 @ ( numeral_numeral_nat @ W ) ) @ ( power_power_complex @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_354_power__numeral__even,axiom,
! [Z2: nat,W: num] :
( ( power_power_nat @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_nat @ ( power_power_nat @ Z2 @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_355_power__numeral__even,axiom,
! [Z2: real,W: num] :
( ( power_power_real @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_real @ ( power_power_real @ Z2 @ ( numeral_numeral_nat @ W ) ) @ ( power_power_real @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_356_power__numeral__even,axiom,
! [Z2: int,W: num] :
( ( power_power_int @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_int @ ( power_power_int @ Z2 @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_357_power__numeral__even,axiom,
! [Z2: extended_enat,W: num] :
( ( power_8040749407984259932d_enat @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_7803423173614009249d_enat @ ( power_8040749407984259932d_enat @ Z2 @ ( numeral_numeral_nat @ W ) ) @ ( power_8040749407984259932d_enat @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_358_power4__eq__xxxx,axiom,
! [X: complex] :
( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_359_power4__eq__xxxx,axiom,
! [X: nat] :
( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_360_power4__eq__xxxx,axiom,
! [X: real] :
( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_real @ ( times_times_real @ ( times_times_real @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_361_power4__eq__xxxx,axiom,
! [X: int] :
( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_362_power4__eq__xxxx,axiom,
! [X: extended_enat] :
( ( power_8040749407984259932d_enat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_7803423173614009249d_enat @ ( times_7803423173614009249d_enat @ ( times_7803423173614009249d_enat @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_363_power2__eq__square,axiom,
! [A2: complex] :
( ( power_power_complex @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_complex @ A2 @ A2 ) ) ).
% power2_eq_square
thf(fact_364_power2__eq__square,axiom,
! [A2: nat] :
( ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_nat @ A2 @ A2 ) ) ).
% power2_eq_square
thf(fact_365_power2__eq__square,axiom,
! [A2: real] :
( ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_real @ A2 @ A2 ) ) ).
% power2_eq_square
thf(fact_366_power2__eq__square,axiom,
! [A2: int] :
( ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_int @ A2 @ A2 ) ) ).
% power2_eq_square
thf(fact_367_power2__eq__square,axiom,
! [A2: extended_enat] :
( ( power_8040749407984259932d_enat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_7803423173614009249d_enat @ A2 @ A2 ) ) ).
% power2_eq_square
thf(fact_368_power__even__eq,axiom,
! [A2: nat,N: nat] :
( ( power_power_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_nat @ ( power_power_nat @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_369_power__even__eq,axiom,
! [A2: real,N: nat] :
( ( power_power_real @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_real @ ( power_power_real @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_370_power__even__eq,axiom,
! [A2: complex,N: nat] :
( ( power_power_complex @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_complex @ ( power_power_complex @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_371_power__even__eq,axiom,
! [A2: int,N: nat] :
( ( power_power_int @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_int @ ( power_power_int @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_372_pigeonhole__infinite__rel,axiom,
! [A: set_a,B4: set_a,R: a > a > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B4 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B4 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B4 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_373_pigeonhole__infinite__rel,axiom,
! [A: set_a,B4: set_nat,R: a > nat > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ B4 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B4 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B4 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_374_pigeonhole__infinite__rel,axiom,
! [A: set_a,B4: set_int,R: a > int > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_int @ B4 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ? [Xa: int] :
( ( member_int @ Xa @ B4 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: int] :
( ( member_int @ X3 @ B4 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_375_pigeonhole__infinite__rel,axiom,
! [A: set_a,B4: set_complex,R: a > complex > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite3207457112153483333omplex @ B4 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ B4 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: complex] :
( ( member_complex @ X3 @ B4 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_376_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B4: set_a,R: nat > a > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B4 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B4 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_377_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B4: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B4 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B4 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_378_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B4: set_int,R: nat > int > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_int @ B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ? [Xa: int] :
( ( member_int @ Xa @ B4 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: int] :
( ( member_int @ X3 @ B4 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_379_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B4: set_complex,R: nat > complex > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite3207457112153483333omplex @ B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ B4 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: complex] :
( ( member_complex @ X3 @ B4 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_380_pigeonhole__infinite__rel,axiom,
! [A: set_int,B4: set_a,R: int > a > $o] :
( ~ ( finite_finite_int @ A )
=> ( ( finite_finite_a @ B4 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B4 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B4 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A4: int] :
( ( member_int @ A4 @ A )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_381_pigeonhole__infinite__rel,axiom,
! [A: set_int,B4: set_nat,R: int > nat > $o] :
( ~ ( finite_finite_int @ A )
=> ( ( finite_finite_nat @ B4 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B4 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B4 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A4: int] :
( ( member_int @ A4 @ A )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_382_not__finite__existsD,axiom,
! [P: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P ) )
=> ? [X_1: a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_383_not__finite__existsD,axiom,
! [P: product_prod_a_a > $o] :
( ~ ( finite6544458595007987280od_a_a @ ( collec3336397797384452498od_a_a @ P ) )
=> ? [X_1: product_prod_a_a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_384_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_385_not__finite__existsD,axiom,
! [P: int > $o] :
( ~ ( finite_finite_int @ ( collect_int @ P ) )
=> ? [X_1: int] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_386_not__finite__existsD,axiom,
! [P: complex > $o] :
( ~ ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
=> ? [X_1: complex] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_387_pigeonhole__infinite,axiom,
! [A: set_a,F: a > a] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_a @ ( image_a_a @ F @ A ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_388_pigeonhole__infinite,axiom,
! [A: set_a,F: a > nat] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ ( image_a_nat @ F @ A ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_389_pigeonhole__infinite,axiom,
! [A: set_a,F: a > int] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_int @ ( image_a_int @ F @ A ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_390_pigeonhole__infinite,axiom,
! [A: set_a,F: a > complex] :
( ~ ( finite_finite_a @ A )
=> ( ( finite3207457112153483333omplex @ ( image_a_complex @ F @ A ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_391_pigeonhole__infinite,axiom,
! [A: set_nat,F: nat > a] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ ( image_nat_a @ F @ A ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_392_pigeonhole__infinite,axiom,
! [A: set_nat,F: nat > nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F @ A ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_393_pigeonhole__infinite,axiom,
! [A: set_nat,F: nat > int] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_int @ ( image_nat_int @ F @ A ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_394_pigeonhole__infinite,axiom,
! [A: set_nat,F: nat > complex] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite3207457112153483333omplex @ ( image_nat_complex @ F @ A ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_395_pigeonhole__infinite,axiom,
! [A: set_int,F: int > a] :
( ~ ( finite_finite_int @ A )
=> ( ( finite_finite_a @ ( image_int_a @ F @ A ) )
=> ? [X3: int] :
( ( member_int @ X3 @ A )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A4: int] :
( ( member_int @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_396_pigeonhole__infinite,axiom,
! [A: set_int,F: int > nat] :
( ~ ( finite_finite_int @ A )
=> ( ( finite_finite_nat @ ( image_int_nat @ F @ A ) )
=> ? [X3: int] :
( ( member_int @ X3 @ A )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A4: int] :
( ( member_int @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_397__092_060open_062card_A_I_123_Ix_M_Ay_J_O_A_Ix_M_Ay_J_A_092_060in_062_AM_A_092_060times_062_AM_A_092_060and_062_Ax_A_060_Ay_125_A_092_060union_062_A_123_Ix_M_Ay_J_O_A_Ix_M_Ay_J_A_092_060in_062_AM_A_092_060times_062_AM_A_092_060and_062_Ay_A_060_Ax_125_J_A_061_Acard_A_IM_A_092_060times_062_AM_A_N_A_123_Ix_M_Ay_J_O_A_Ix_M_Ay_J_A_092_060in_062_AM_A_092_060times_062_AM_A_092_060and_062_Ax_A_061_Ay_125_J_092_060close_062,axiom,
( ( finite4795055649997197647od_a_a
@ ( sup_su3048258781599657691od_a_a
@ ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X2: a,Y2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 )
@ ( product_Sigma_a_a @ m
@ ^ [Uu: a] : m ) )
& ( ord_less_a @ X2 @ Y2 ) ) ) )
@ ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X2: a,Y2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 )
@ ( product_Sigma_a_a @ m
@ ^ [Uu: a] : m ) )
& ( ord_less_a @ Y2 @ X2 ) ) ) ) ) )
= ( finite4795055649997197647od_a_a
@ ( minus_6817036919807184750od_a_a
@ ( product_Sigma_a_a @ m
@ ^ [Uu: a] : m )
@ ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X2: a,Y2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 )
@ ( product_Sigma_a_a @ m
@ ^ [Uu: a] : m ) )
& ( X2 = Y2 ) ) ) ) ) ) ) ).
% \<open>card ({(x, y). (x, y) \<in> M \<times> M \<and> x < y} \<union> {(x, y). (x, y) \<in> M \<times> M \<and> y < x}) = card (M \<times> M - {(x, y). (x, y) \<in> M \<times> M \<and> x = y})\<close>
thf(fact_398_left__diff__distrib__numeral,axiom,
! [A2: complex,B: complex,V: num] :
( ( times_times_complex @ ( minus_minus_complex @ A2 @ B ) @ ( numera6690914467698888265omplex @ V ) )
= ( minus_minus_complex @ ( times_times_complex @ A2 @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_399_left__diff__distrib__numeral,axiom,
! [A2: real,B: real,V: num] :
( ( times_times_real @ ( minus_minus_real @ A2 @ B ) @ ( numeral_numeral_real @ V ) )
= ( minus_minus_real @ ( times_times_real @ A2 @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_400_left__diff__distrib__numeral,axiom,
! [A2: int,B: int,V: num] :
( ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ ( numeral_numeral_int @ V ) )
= ( minus_minus_int @ ( times_times_int @ A2 @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_401_right__diff__distrib__numeral,axiom,
! [V: num,B: complex,C: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( minus_minus_complex @ B @ C ) )
= ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_402_right__diff__distrib__numeral,axiom,
! [V: num,B: real,C: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_403_right__diff__distrib__numeral,axiom,
! [V: num,B: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_404_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z2 ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_405_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z2: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z2 ) )
= ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_406_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z2: extended_enat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ W ) @ Z2 ) )
= ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_407_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z2: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z2 ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_408_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_409_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_410_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( numera1916890842035813515d_enat @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_411_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_412_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_413_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_414_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_415_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_416_card__cartesian__product,axiom,
! [A: set_a,B4: set_nat] :
( ( finite5799836783309606997_a_nat
@ ( product_Sigma_a_nat @ A
@ ^ [Uu: a] : B4 ) )
= ( times_times_nat @ ( finite_card_a @ A ) @ ( finite_card_nat @ B4 ) ) ) ).
% card_cartesian_product
thf(fact_417_card__cartesian__product,axiom,
! [A: set_a,B4: set_complex] :
( ( finite7492710888111979059omplex
@ ( produc73771002080876644omplex @ A
@ ^ [Uu: a] : B4 ) )
= ( times_times_nat @ ( finite_card_a @ A ) @ ( finite_card_complex @ B4 ) ) ) ).
% card_cartesian_product
thf(fact_418_card__cartesian__product,axiom,
! [A: set_nat,B4: set_a] :
( ( finite9038000247033283963_nat_a
@ ( product_Sigma_nat_a @ A
@ ^ [Uu: nat] : B4 ) )
= ( times_times_nat @ ( finite_card_nat @ A ) @ ( finite_card_a @ B4 ) ) ) ).
% card_cartesian_product
thf(fact_419_card__cartesian__product,axiom,
! [A: set_nat,B4: set_nat] :
( ( finite711546835091564841at_nat
@ ( produc457027306803732586at_nat @ A
@ ^ [Uu: nat] : B4 ) )
= ( times_times_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B4 ) ) ) ).
% card_cartesian_product
thf(fact_420_card__cartesian__product,axiom,
! [A: set_nat,B4: set_complex] :
( ( finite3383162986784156423omplex
@ ( produc7489356884331829576omplex @ A
@ ^ [Uu: nat] : B4 ) )
= ( times_times_nat @ ( finite_card_nat @ A ) @ ( finite_card_complex @ B4 ) ) ) ).
% card_cartesian_product
thf(fact_421_card__cartesian__product,axiom,
! [A: set_complex,B4: set_a] :
( ( finite5862851395709996573plex_a
@ ( produc5920925292428845446plex_a @ A
@ ^ [Uu: complex] : B4 ) )
= ( times_times_nat @ ( finite_card_complex @ A ) @ ( finite_card_a @ B4 ) ) ) ).
% card_cartesian_product
thf(fact_422_card__cartesian__product,axiom,
! [A: set_complex,B4: set_nat] :
( ( finite319700565923552263ex_nat
@ ( produc1885768171911791432ex_nat @ A
@ ^ [Uu: complex] : B4 ) )
= ( times_times_nat @ ( finite_card_complex @ A ) @ ( finite_card_nat @ B4 ) ) ) ).
% card_cartesian_product
thf(fact_423_card__cartesian__product,axiom,
! [A: set_complex,B4: set_complex] :
( ( finite7505797800800641509omplex
@ ( produc4778231967749636134omplex @ A
@ ^ [Uu: complex] : B4 ) )
= ( times_times_nat @ ( finite_card_complex @ A ) @ ( finite_card_complex @ B4 ) ) ) ).
% card_cartesian_product
thf(fact_424_card__cartesian__product,axiom,
! [A: set_a,B4: set_a] :
( ( finite4795055649997197647od_a_a
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : B4 ) )
= ( times_times_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B4 ) ) ) ).
% card_cartesian_product
thf(fact_425_card__cartesian__product,axiom,
! [A: set_a,B4: set_Product_prod_a_a] :
( ( finite6893194910719049976od_a_a
@ ( produc6342321021181284593od_a_a @ A
@ ^ [Uu: a] : B4 ) )
= ( times_times_nat @ ( finite_card_a @ A ) @ ( finite4795055649997197647od_a_a @ B4 ) ) ) ).
% card_cartesian_product
thf(fact_426_set__times__image,axiom,
( times_times_set_nat
= ( ^ [S2: set_nat,T2: set_nat] :
( image_2486076414777270412at_nat @ ( produc6842872674320459806at_nat @ times_times_nat )
@ ( produc457027306803732586at_nat @ S2
@ ^ [Uu: nat] : T2 ) ) ) ) ).
% set_times_image
thf(fact_427_set__times__image,axiom,
( times_times_set_num
= ( ^ [S2: set_num,T2: set_num] :
( image_8225983159724657258um_num @ ( produc8694714728609142524um_num @ times_times_num )
@ ( produc4368061533121756414um_num @ S2
@ ^ [Uu: num] : T2 ) ) ) ) ).
% set_times_image
thf(fact_428_set__times__image,axiom,
( times_times_set_real
= ( ^ [S2: set_real,T2: set_real] :
( image_4246084775291713952l_real @ ( produc313441363659479858l_real @ times_times_real )
@ ( produc1020309166918817442l_real @ S2
@ ^ [Uu: real] : T2 ) ) ) ) ).
% set_times_image
thf(fact_429_set__times__image,axiom,
( times_times_set_int
= ( ^ [S2: set_int,T2: set_int] :
( image_5042161079198086560nt_int @ ( produc8211389475949308722nt_int @ times_times_int )
@ ( produc1453890548195736866nt_int @ S2
@ ^ [Uu: int] : T2 ) ) ) ) ).
% set_times_image
thf(fact_430_set__times__image,axiom,
( times_2438108612031896577d_enat
= ( ^ [S2: set_Extended_enat,T2: set_Extended_enat] :
( image_3712855581125276676d_enat @ ( produc797783881074051898d_enat @ times_7803423173614009249d_enat )
@ ( produc5797200089183630280d_enat @ S2
@ ^ [Uu: extended_enat] : T2 ) ) ) ) ).
% set_times_image
thf(fact_431_infinite__cartesian__product,axiom,
! [A: set_a,B4: set_nat] :
( ~ ( finite_finite_a @ A )
=> ( ~ ( finite_finite_nat @ B4 )
=> ~ ( finite6644898363146130708_a_nat
@ ( product_Sigma_a_nat @ A
@ ^ [Uu: a] : B4 ) ) ) ) ).
% infinite_cartesian_product
thf(fact_432_infinite__cartesian__product,axiom,
! [A: set_a,B4: set_int] :
( ~ ( finite_finite_a @ A )
=> ( ~ ( finite_finite_int @ B4 )
=> ~ ( finite2467047343636934000_a_int
@ ( product_Sigma_a_int @ A
@ ^ [Uu: a] : B4 ) ) ) ) ).
% infinite_cartesian_product
thf(fact_433_infinite__cartesian__product,axiom,
! [A: set_a,B4: set_complex] :
( ~ ( finite_finite_a @ A )
=> ( ~ ( finite3207457112153483333omplex @ B4 )
=> ~ ( finite228341146693599474omplex
@ ( produc73771002080876644omplex @ A
@ ^ [Uu: a] : B4 ) ) ) ) ).
% infinite_cartesian_product
thf(fact_434_infinite__cartesian__product,axiom,
! [A: set_nat,B4: set_a] :
( ~ ( finite_finite_nat @ A )
=> ( ~ ( finite_finite_a @ B4 )
=> ~ ( finite659689790015031866_nat_a
@ ( product_Sigma_nat_a @ A
@ ^ [Uu: nat] : B4 ) ) ) ) ).
% infinite_cartesian_product
thf(fact_435_infinite__cartesian__product,axiom,
! [A: set_nat,B4: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ~ ( finite_finite_nat @ B4 )
=> ~ ( finite6177210948735845034at_nat
@ ( produc457027306803732586at_nat @ A
@ ^ [Uu: nat] : B4 ) ) ) ) ).
% infinite_cartesian_product
thf(fact_436_infinite__cartesian__product,axiom,
! [A: set_nat,B4: set_int] :
( ~ ( finite_finite_nat @ A )
=> ( ~ ( finite_finite_int @ B4 )
=> ~ ( finite1999359929226648326at_int
@ ( produc454536836294682310at_int @ A
@ ^ [Uu: nat] : B4 ) ) ) ) ).
% infinite_cartesian_product
thf(fact_437_infinite__cartesian__product,axiom,
! [A: set_nat,B4: set_complex] :
( ~ ( finite_finite_nat @ A )
=> ( ~ ( finite3207457112153483333omplex @ B4 )
=> ~ ( finite1477110953402811016omplex
@ ( produc7489356884331829576omplex @ A
@ ^ [Uu: nat] : B4 ) ) ) ) ).
% infinite_cartesian_product
thf(fact_438_infinite__cartesian__product,axiom,
! [A: set_int,B4: set_a] :
( ~ ( finite_finite_int @ A )
=> ( ~ ( finite_finite_a @ B4 )
=> ~ ( finite7637916859199361758_int_a
@ ( product_Sigma_int_a @ A
@ ^ [Uu: int] : B4 ) ) ) ) ).
% infinite_cartesian_product
thf(fact_439_infinite__cartesian__product,axiom,
! [A: set_int,B4: set_nat] :
( ~ ( finite_finite_int @ A )
=> ( ~ ( finite_finite_nat @ B4 )
=> ~ ( finite7176564660636899590nt_nat
@ ( produc1456381018704787142nt_nat @ A
@ ^ [Uu: int] : B4 ) ) ) ) ).
% infinite_cartesian_product
thf(fact_440_infinite__cartesian__product,axiom,
! [A: set_int,B4: set_int] :
( ~ ( finite_finite_int @ A )
=> ( ~ ( finite_finite_int @ B4 )
=> ~ ( finite2998713641127702882nt_int
@ ( produc1453890548195736866nt_int @ A
@ ^ [Uu: int] : B4 ) ) ) ) ).
% infinite_cartesian_product
thf(fact_441_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_442_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_real @ M )
= ( numeral_numeral_real @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_443_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numera1916890842035813515d_enat @ M )
= ( numera1916890842035813515d_enat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_444_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_445_sup_Oright__idem,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) @ B )
= ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ).
% sup.right_idem
thf(fact_446_sup__left__idem,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ X @ Y ) )
= ( sup_su3048258781599657691od_a_a @ X @ Y ) ) ).
% sup_left_idem
thf(fact_447_sup_Oleft__idem,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A2 @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) )
= ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ).
% sup.left_idem
thf(fact_448_sup__idem,axiom,
! [X: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_449_sup_Oidem,axiom,
! [A2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_450_Un__iff,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) )
= ( ( member_nat @ C @ A )
| ( member_nat @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_451_Un__iff,axiom,
! [C: a,A: set_a,B4: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B4 ) )
= ( ( member_a @ C @ A )
| ( member_a @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_452_Un__iff,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A @ B4 ) )
= ( ( member1426531477525435216od_a_a @ C @ A )
| ( member1426531477525435216od_a_a @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_453_UnCI,axiom,
! [C: nat,B4: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C @ B4 )
=> ( member_nat @ C @ A ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% UnCI
thf(fact_454_UnCI,axiom,
! [C: a,B4: set_a,A: set_a] :
( ( ~ ( member_a @ C @ B4 )
=> ( member_a @ C @ A ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% UnCI
thf(fact_455_UnCI,axiom,
! [C: product_prod_a_a,B4: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ~ ( member1426531477525435216od_a_a @ C @ B4 )
=> ( member1426531477525435216od_a_a @ C @ A ) )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A @ B4 ) ) ) ).
% UnCI
thf(fact_456_finite__Un,axiom,
! [F2: set_a,G2: set_a] :
( ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G2 ) )
= ( ( finite_finite_a @ F2 )
& ( finite_finite_a @ G2 ) ) ) ).
% finite_Un
thf(fact_457_finite__Un,axiom,
! [F2: set_nat,G2: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) )
= ( ( finite_finite_nat @ F2 )
& ( finite_finite_nat @ G2 ) ) ) ).
% finite_Un
thf(fact_458_finite__Un,axiom,
! [F2: set_int,G2: set_int] :
( ( finite_finite_int @ ( sup_sup_set_int @ F2 @ G2 ) )
= ( ( finite_finite_int @ F2 )
& ( finite_finite_int @ G2 ) ) ) ).
% finite_Un
thf(fact_459_finite__Un,axiom,
! [F2: set_complex,G2: set_complex] :
( ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ F2 @ G2 ) )
= ( ( finite3207457112153483333omplex @ F2 )
& ( finite3207457112153483333omplex @ G2 ) ) ) ).
% finite_Un
thf(fact_460_finite__Un,axiom,
! [F2: set_Product_prod_a_a,G2: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ ( sup_su3048258781599657691od_a_a @ F2 @ G2 ) )
= ( ( finite6544458595007987280od_a_a @ F2 )
& ( finite6544458595007987280od_a_a @ G2 ) ) ) ).
% finite_Un
thf(fact_461_Un__Diff__cancel2,axiom,
! [B4: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( minus_6817036919807184750od_a_a @ B4 @ A ) @ A )
= ( sup_su3048258781599657691od_a_a @ B4 @ A ) ) ).
% Un_Diff_cancel2
thf(fact_462_Un__Diff__cancel,axiom,
! [A: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A @ ( minus_6817036919807184750od_a_a @ B4 @ A ) )
= ( sup_su3048258781599657691od_a_a @ A @ B4 ) ) ).
% Un_Diff_cancel
thf(fact_463_semiring__norm_I13_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% semiring_norm(13)
thf(fact_464_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times_num @ M @ one )
= M ) ).
% semiring_norm(11)
thf(fact_465_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times_num @ one @ N )
= N ) ).
% semiring_norm(12)
thf(fact_466_set__times__intro,axiom,
! [A2: nat,C2: set_nat,B: nat,D: set_nat] :
( ( member_nat @ A2 @ C2 )
=> ( ( member_nat @ B @ D )
=> ( member_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_set_nat @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_467_set__times__intro,axiom,
! [A2: num,C2: set_num,B: num,D: set_num] :
( ( member_num @ A2 @ C2 )
=> ( ( member_num @ B @ D )
=> ( member_num @ ( times_times_num @ A2 @ B ) @ ( times_times_set_num @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_468_set__times__intro,axiom,
! [A2: real,C2: set_real,B: real,D: set_real] :
( ( member_real @ A2 @ C2 )
=> ( ( member_real @ B @ D )
=> ( member_real @ ( times_times_real @ A2 @ B ) @ ( times_times_set_real @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_469_set__times__intro,axiom,
! [A2: int,C2: set_int,B: int,D: set_int] :
( ( member_int @ A2 @ C2 )
=> ( ( member_int @ B @ D )
=> ( member_int @ ( times_times_int @ A2 @ B ) @ ( times_times_set_int @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_470_set__times__intro,axiom,
! [A2: extended_enat,C2: set_Extended_enat,B: extended_enat,D: set_Extended_enat] :
( ( member_Extended_enat @ A2 @ C2 )
=> ( ( member_Extended_enat @ B @ D )
=> ( member_Extended_enat @ ( times_7803423173614009249d_enat @ A2 @ B ) @ ( times_2438108612031896577d_enat @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_471_semiring__norm_I78_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(78)
thf(fact_472_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% semiring_norm(75)
thf(fact_473_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_474_power__mult__numeral,axiom,
! [A2: nat,M: num,N: num] :
( ( power_power_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_475_power__mult__numeral,axiom,
! [A2: real,M: num,N: num] :
( ( power_power_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_476_power__mult__numeral,axiom,
! [A2: complex,M: num,N: num] :
( ( power_power_complex @ ( power_power_complex @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_complex @ A2 @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_477_power__mult__numeral,axiom,
! [A2: int,M: num,N: num] :
( ( power_power_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_478_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_479_Un__left__commute,axiom,
! [A: set_Product_prod_a_a,B4: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A @ ( sup_su3048258781599657691od_a_a @ B4 @ C2 ) )
= ( sup_su3048258781599657691od_a_a @ B4 @ ( sup_su3048258781599657691od_a_a @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_480_Un__left__absorb,axiom,
! [A: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A @ ( sup_su3048258781599657691od_a_a @ A @ B4 ) )
= ( sup_su3048258781599657691od_a_a @ A @ B4 ) ) ).
% Un_left_absorb
thf(fact_481_Un__commute,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_482_Un__absorb,axiom,
! [A: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A @ A )
= A ) ).
% Un_absorb
thf(fact_483_Un__assoc,axiom,
! [A: set_Product_prod_a_a,B4: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ A @ B4 ) @ C2 )
= ( sup_su3048258781599657691od_a_a @ A @ ( sup_su3048258781599657691od_a_a @ B4 @ C2 ) ) ) ).
% Un_assoc
thf(fact_484_ball__Un,axiom,
! [A: set_Product_prod_a_a,B4: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( sup_su3048258781599657691od_a_a @ A @ B4 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A )
=> ( P @ X2 ) )
& ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ B4 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_485_bex__Un,axiom,
! [A: set_Product_prod_a_a,B4: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ( ? [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( sup_su3048258781599657691od_a_a @ A @ B4 ) )
& ( P @ X2 ) ) )
= ( ? [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A )
& ( P @ X2 ) )
| ? [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ B4 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_486_UnI2,axiom,
! [C: nat,B4: set_nat,A: set_nat] :
( ( member_nat @ C @ B4 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% UnI2
thf(fact_487_UnI2,axiom,
! [C: a,B4: set_a,A: set_a] :
( ( member_a @ C @ B4 )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% UnI2
thf(fact_488_UnI2,axiom,
! [C: product_prod_a_a,B4: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ B4 )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A @ B4 ) ) ) ).
% UnI2
thf(fact_489_UnI1,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ A )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% UnI1
thf(fact_490_UnI1,axiom,
! [C: a,A: set_a,B4: set_a] :
( ( member_a @ C @ A )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% UnI1
thf(fact_491_UnI1,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ A )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A @ B4 ) ) ) ).
% UnI1
thf(fact_492_UnE,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) )
=> ( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ B4 ) ) ) ).
% UnE
thf(fact_493_UnE,axiom,
! [C: a,A: set_a,B4: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B4 ) )
=> ( ~ ( member_a @ C @ A )
=> ( member_a @ C @ B4 ) ) ) ).
% UnE
thf(fact_494_UnE,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A @ B4 ) )
=> ( ~ ( member1426531477525435216od_a_a @ C @ A )
=> ( member1426531477525435216od_a_a @ C @ B4 ) ) ) ).
% UnE
thf(fact_495_sup__left__commute,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ Y @ Z2 ) )
= ( sup_su3048258781599657691od_a_a @ Y @ ( sup_su3048258781599657691od_a_a @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_496_sup_Oleft__commute,axiom,
! [B: set_Product_prod_a_a,A2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ B @ ( sup_su3048258781599657691od_a_a @ A2 @ C ) )
= ( sup_su3048258781599657691od_a_a @ A2 @ ( sup_su3048258781599657691od_a_a @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_497_sup__commute,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [X2: set_Product_prod_a_a,Y2: set_Product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ Y2 @ X2 ) ) ) ).
% sup_commute
thf(fact_498_sup_Ocommute,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [A4: set_Product_prod_a_a,B3: set_Product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ B3 @ A4 ) ) ) ).
% sup.commute
thf(fact_499_sup__assoc,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ X @ Y ) @ Z2 )
= ( sup_su3048258781599657691od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_500_sup_Oassoc,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) @ C )
= ( sup_su3048258781599657691od_a_a @ A2 @ ( sup_su3048258781599657691od_a_a @ B @ C ) ) ) ).
% sup.assoc
thf(fact_501_inf__sup__aci_I5_J,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [X2: set_Product_prod_a_a,Y2: set_Product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ Y2 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_502_inf__sup__aci_I6_J,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ X @ Y ) @ Z2 )
= ( sup_su3048258781599657691od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_503_inf__sup__aci_I7_J,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ Y @ Z2 ) )
= ( sup_su3048258781599657691od_a_a @ Y @ ( sup_su3048258781599657691od_a_a @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_504_inf__sup__aci_I8_J,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ X @ Y ) )
= ( sup_su3048258781599657691od_a_a @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_505_Sigma__Un__distrib1,axiom,
! [I: set_a,J: set_a,C2: a > set_a] :
( ( product_Sigma_a_a @ ( sup_sup_set_a @ I @ J ) @ C2 )
= ( sup_su3048258781599657691od_a_a @ ( product_Sigma_a_a @ I @ C2 ) @ ( product_Sigma_a_a @ J @ C2 ) ) ) ).
% Sigma_Un_distrib1
thf(fact_506_Un__def,axiom,
( sup_sup_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A6 )
| ( member_a @ X2 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_507_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A6 )
| ( member_nat @ X2 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_508_Un__def,axiom,
( sup_sup_set_int
= ( ^ [A6: set_int,B6: set_int] :
( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A6 )
| ( member_int @ X2 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_509_Un__def,axiom,
( sup_sup_set_complex
= ( ^ [A6: set_complex,B6: set_complex] :
( collect_complex
@ ^ [X2: complex] :
( ( member_complex @ X2 @ A6 )
| ( member_complex @ X2 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_510_Un__def,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A6 )
| ( member1426531477525435216od_a_a @ X2 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_511_Collect__disj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_512_Collect__disj__eq,axiom,
! [P: int > $o,Q: int > $o] :
( ( collect_int
@ ^ [X2: int] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_513_Collect__disj__eq,axiom,
! [P: complex > $o,Q: complex > $o] :
( ( collect_complex
@ ^ [X2: complex] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup_set_complex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_514_Collect__disj__eq,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_su3048258781599657691od_a_a @ ( collec3336397797384452498od_a_a @ P ) @ ( collec3336397797384452498od_a_a @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_515_finite__set__times,axiom,
! [S3: set_nat,T3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ( finite_finite_nat @ T3 )
=> ( finite_finite_nat @ ( times_times_set_nat @ S3 @ T3 ) ) ) ) ).
% finite_set_times
thf(fact_516_finite__set__times,axiom,
! [S3: set_int,T3: set_int] :
( ( finite_finite_int @ S3 )
=> ( ( finite_finite_int @ T3 )
=> ( finite_finite_int @ ( times_times_set_int @ S3 @ T3 ) ) ) ) ).
% finite_set_times
thf(fact_517_finite__set__times,axiom,
! [S3: set_complex,T3: set_complex] :
( ( finite3207457112153483333omplex @ S3 )
=> ( ( finite3207457112153483333omplex @ T3 )
=> ( finite3207457112153483333omplex @ ( times_6048082448287401577omplex @ S3 @ T3 ) ) ) ) ).
% finite_set_times
thf(fact_518_Sigma__Un__distrib2,axiom,
! [I: set_a,A: a > set_a,B4: a > set_a] :
( ( product_Sigma_a_a @ I
@ ^ [I2: a] : ( sup_sup_set_a @ ( A @ I2 ) @ ( B4 @ I2 ) ) )
= ( sup_su3048258781599657691od_a_a @ ( product_Sigma_a_a @ I @ A ) @ ( product_Sigma_a_a @ I @ B4 ) ) ) ).
% Sigma_Un_distrib2
thf(fact_519_Times__Un__distrib1,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( product_Sigma_a_a @ ( sup_sup_set_a @ A @ B4 )
@ ^ [Uu: a] : C2 )
= ( sup_su3048258781599657691od_a_a
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : C2 )
@ ( product_Sigma_a_a @ B4
@ ^ [Uu: a] : C2 ) ) ) ).
% Times_Un_distrib1
thf(fact_520_psubset__card__mono,axiom,
! [B4: set_a,A: set_a] :
( ( finite_finite_a @ B4 )
=> ( ( ord_less_set_a @ A @ B4 )
=> ( ord_less_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B4 ) ) ) ) ).
% psubset_card_mono
thf(fact_521_psubset__card__mono,axiom,
! [B4: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ B4 )
=> ( ( ord_le6819997720685908915od_a_a @ A @ B4 )
=> ( ord_less_nat @ ( finite4795055649997197647od_a_a @ A ) @ ( finite4795055649997197647od_a_a @ B4 ) ) ) ) ).
% psubset_card_mono
thf(fact_522_psubset__card__mono,axiom,
! [B4: set_nat,A: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_set_nat @ A @ B4 )
=> ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B4 ) ) ) ) ).
% psubset_card_mono
thf(fact_523_psubset__card__mono,axiom,
! [B4: set_int,A: set_int] :
( ( finite_finite_int @ B4 )
=> ( ( ord_less_set_int @ A @ B4 )
=> ( ord_less_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B4 ) ) ) ) ).
% psubset_card_mono
thf(fact_524_psubset__card__mono,axiom,
! [B4: set_complex,A: set_complex] :
( ( finite3207457112153483333omplex @ B4 )
=> ( ( ord_less_set_complex @ A @ B4 )
=> ( ord_less_nat @ ( finite_card_complex @ A ) @ ( finite_card_complex @ B4 ) ) ) ) ).
% psubset_card_mono
thf(fact_525_sup_Ostrict__coboundedI2,axiom,
! [C: set_Product_prod_a_a,B: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ C @ B )
=> ( ord_le6819997720685908915od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_526_sup_Ostrict__coboundedI2,axiom,
! [C: real,B: real,A2: real] :
( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ ( sup_sup_real @ A2 @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_527_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_528_sup_Ostrict__coboundedI2,axiom,
! [C: int,B: int,A2: int] :
( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ ( sup_sup_int @ A2 @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_529_sup_Ostrict__coboundedI1,axiom,
! [C: set_Product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ C @ A2 )
=> ( ord_le6819997720685908915od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_530_sup_Ostrict__coboundedI1,axiom,
! [C: real,A2: real,B: real] :
( ( ord_less_real @ C @ A2 )
=> ( ord_less_real @ C @ ( sup_sup_real @ A2 @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_531_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ C @ A2 )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_532_sup_Ostrict__coboundedI1,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ C @ A2 )
=> ( ord_less_int @ C @ ( sup_sup_int @ A2 @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_533_sup_Ostrict__order__iff,axiom,
( ord_le6819997720685908915od_a_a
= ( ^ [B3: set_Product_prod_a_a,A4: set_Product_prod_a_a] :
( ( A4
= ( sup_su3048258781599657691od_a_a @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_534_sup_Ostrict__order__iff,axiom,
( ord_less_real
= ( ^ [B3: real,A4: real] :
( ( A4
= ( sup_sup_real @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_535_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( A4
= ( sup_sup_nat @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_536_sup_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [B3: int,A4: int] :
( ( A4
= ( sup_sup_int @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_537_sup_Ostrict__boundedE,axiom,
! [B: set_Product_prod_a_a,C: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ ( sup_su3048258781599657691od_a_a @ B @ C ) @ A2 )
=> ~ ( ( ord_le6819997720685908915od_a_a @ B @ A2 )
=> ~ ( ord_le6819997720685908915od_a_a @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_538_sup_Ostrict__boundedE,axiom,
! [B: real,C: real,A2: real] :
( ( ord_less_real @ ( sup_sup_real @ B @ C ) @ A2 )
=> ~ ( ( ord_less_real @ B @ A2 )
=> ~ ( ord_less_real @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_539_sup_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A2 )
=> ~ ( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_540_sup_Ostrict__boundedE,axiom,
! [B: int,C: int,A2: int] :
( ( ord_less_int @ ( sup_sup_int @ B @ C ) @ A2 )
=> ~ ( ( ord_less_int @ B @ A2 )
=> ~ ( ord_less_int @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_541_sup_Oabsorb4,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ A2 @ B )
=> ( ( sup_su3048258781599657691od_a_a @ A2 @ B )
= B ) ) ).
% sup.absorb4
thf(fact_542_sup_Oabsorb4,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( sup_sup_real @ A2 @ B )
= B ) ) ).
% sup.absorb4
thf(fact_543_sup_Oabsorb4,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( sup_sup_nat @ A2 @ B )
= B ) ) ).
% sup.absorb4
thf(fact_544_sup_Oabsorb4,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( sup_sup_int @ A2 @ B )
= B ) ) ).
% sup.absorb4
thf(fact_545_sup_Oabsorb3,axiom,
! [B: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ B @ A2 )
=> ( ( sup_su3048258781599657691od_a_a @ A2 @ B )
= A2 ) ) ).
% sup.absorb3
thf(fact_546_sup_Oabsorb3,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ B @ A2 )
=> ( ( sup_sup_real @ A2 @ B )
= A2 ) ) ).
% sup.absorb3
thf(fact_547_sup_Oabsorb3,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( sup_sup_nat @ A2 @ B )
= A2 ) ) ).
% sup.absorb3
thf(fact_548_sup_Oabsorb3,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( sup_sup_int @ A2 @ B )
= A2 ) ) ).
% sup.absorb3
thf(fact_549_less__supI2,axiom,
! [X: set_Product_prod_a_a,B: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ X @ B )
=> ( ord_le6819997720685908915od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ) ).
% less_supI2
thf(fact_550_less__supI2,axiom,
! [X: real,B: real,A2: real] :
( ( ord_less_real @ X @ B )
=> ( ord_less_real @ X @ ( sup_sup_real @ A2 @ B ) ) ) ).
% less_supI2
thf(fact_551_less__supI2,axiom,
! [X: nat,B: nat,A2: nat] :
( ( ord_less_nat @ X @ B )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% less_supI2
thf(fact_552_less__supI2,axiom,
! [X: int,B: int,A2: int] :
( ( ord_less_int @ X @ B )
=> ( ord_less_int @ X @ ( sup_sup_int @ A2 @ B ) ) ) ).
% less_supI2
thf(fact_553_less__supI1,axiom,
! [X: set_Product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ X @ A2 )
=> ( ord_le6819997720685908915od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ) ).
% less_supI1
thf(fact_554_less__supI1,axiom,
! [X: real,A2: real,B: real] :
( ( ord_less_real @ X @ A2 )
=> ( ord_less_real @ X @ ( sup_sup_real @ A2 @ B ) ) ) ).
% less_supI1
thf(fact_555_less__supI1,axiom,
! [X: nat,A2: nat,B: nat] :
( ( ord_less_nat @ X @ A2 )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% less_supI1
thf(fact_556_less__supI1,axiom,
! [X: int,A2: int,B: int] :
( ( ord_less_int @ X @ A2 )
=> ( ord_less_int @ X @ ( sup_sup_int @ A2 @ B ) ) ) ).
% less_supI1
thf(fact_557_image__Un,axiom,
! [F: nat > nat,A: set_nat,B4: set_nat] :
( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A @ B4 ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B4 ) ) ) ).
% image_Un
thf(fact_558_image__Un,axiom,
! [F: a > product_prod_a_a,A: set_a,B4: set_a] :
( ( image_7400625782589995694od_a_a @ F @ ( sup_sup_set_a @ A @ B4 ) )
= ( sup_su3048258781599657691od_a_a @ ( image_7400625782589995694od_a_a @ F @ A ) @ ( image_7400625782589995694od_a_a @ F @ B4 ) ) ) ).
% image_Un
thf(fact_559_image__Un,axiom,
! [F: nat > product_prod_a_a,A: set_nat,B4: set_nat] :
( ( image_372941888232738320od_a_a @ F @ ( sup_sup_set_nat @ A @ B4 ) )
= ( sup_su3048258781599657691od_a_a @ ( image_372941888232738320od_a_a @ F @ A ) @ ( image_372941888232738320od_a_a @ F @ B4 ) ) ) ).
% image_Un
thf(fact_560_image__Un,axiom,
! [F: product_prod_a_a > nat,A: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( image_9053670898913107890_a_nat @ F @ ( sup_su3048258781599657691od_a_a @ A @ B4 ) )
= ( sup_sup_set_nat @ ( image_9053670898913107890_a_nat @ F @ A ) @ ( image_9053670898913107890_a_nat @ F @ B4 ) ) ) ).
% image_Un
thf(fact_561_image__Un,axiom,
! [F: product_prod_a_a > product_prod_a_a,A: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( image_4636654165204879301od_a_a @ F @ ( sup_su3048258781599657691od_a_a @ A @ B4 ) )
= ( sup_su3048258781599657691od_a_a @ ( image_4636654165204879301od_a_a @ F @ A ) @ ( image_4636654165204879301od_a_a @ F @ B4 ) ) ) ).
% image_Un
thf(fact_562_finite__UnI,axiom,
! [F2: set_a,G2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( finite_finite_a @ G2 )
=> ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_563_finite__UnI,axiom,
! [F2: set_nat,G2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( finite_finite_nat @ G2 )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_564_finite__UnI,axiom,
! [F2: set_int,G2: set_int] :
( ( finite_finite_int @ F2 )
=> ( ( finite_finite_int @ G2 )
=> ( finite_finite_int @ ( sup_sup_set_int @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_565_finite__UnI,axiom,
! [F2: set_complex,G2: set_complex] :
( ( finite3207457112153483333omplex @ F2 )
=> ( ( finite3207457112153483333omplex @ G2 )
=> ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_566_finite__UnI,axiom,
! [F2: set_Product_prod_a_a,G2: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F2 )
=> ( ( finite6544458595007987280od_a_a @ G2 )
=> ( finite6544458595007987280od_a_a @ ( sup_su3048258781599657691od_a_a @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_567_Un__infinite,axiom,
! [S: set_a,T: set_a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_568_Un__infinite,axiom,
! [S: set_nat,T: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) ) ).
% Un_infinite
thf(fact_569_Un__infinite,axiom,
! [S: set_int,T: set_int] :
( ~ ( finite_finite_int @ S )
=> ~ ( finite_finite_int @ ( sup_sup_set_int @ S @ T ) ) ) ).
% Un_infinite
thf(fact_570_Un__infinite,axiom,
! [S: set_complex,T: set_complex] :
( ~ ( finite3207457112153483333omplex @ S )
=> ~ ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ S @ T ) ) ) ).
% Un_infinite
thf(fact_571_Un__infinite,axiom,
! [S: set_Product_prod_a_a,T: set_Product_prod_a_a] :
( ~ ( finite6544458595007987280od_a_a @ S )
=> ~ ( finite6544458595007987280od_a_a @ ( sup_su3048258781599657691od_a_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_572_infinite__Un,axiom,
! [S: set_a,T: set_a] :
( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) )
= ( ~ ( finite_finite_a @ S )
| ~ ( finite_finite_a @ T ) ) ) ).
% infinite_Un
thf(fact_573_infinite__Un,axiom,
! [S: set_nat,T: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_Un
thf(fact_574_infinite__Un,axiom,
! [S: set_int,T: set_int] :
( ( ~ ( finite_finite_int @ ( sup_sup_set_int @ S @ T ) ) )
= ( ~ ( finite_finite_int @ S )
| ~ ( finite_finite_int @ T ) ) ) ).
% infinite_Un
thf(fact_575_infinite__Un,axiom,
! [S: set_complex,T: set_complex] :
( ( ~ ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ S @ T ) ) )
= ( ~ ( finite3207457112153483333omplex @ S )
| ~ ( finite3207457112153483333omplex @ T ) ) ) ).
% infinite_Un
thf(fact_576_infinite__Un,axiom,
! [S: set_Product_prod_a_a,T: set_Product_prod_a_a] :
( ( ~ ( finite6544458595007987280od_a_a @ ( sup_su3048258781599657691od_a_a @ S @ T ) ) )
= ( ~ ( finite6544458595007987280od_a_a @ S )
| ~ ( finite6544458595007987280od_a_a @ T ) ) ) ).
% infinite_Un
thf(fact_577_Un__Diff,axiom,
! [A: set_Product_prod_a_a,B4: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( minus_6817036919807184750od_a_a @ ( sup_su3048258781599657691od_a_a @ A @ B4 ) @ C2 )
= ( sup_su3048258781599657691od_a_a @ ( minus_6817036919807184750od_a_a @ A @ C2 ) @ ( minus_6817036919807184750od_a_a @ B4 @ C2 ) ) ) ).
% Un_Diff
thf(fact_578_finite__psubset__induct,axiom,
! [A: set_a,P: set_a > $o] :
( ( finite_finite_a @ A )
=> ( ! [A7: set_a] :
( ( finite_finite_a @ A7 )
=> ( ! [B7: set_a] :
( ( ord_less_set_a @ B7 @ A7 )
=> ( P @ B7 ) )
=> ( P @ A7 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_579_finite__psubset__induct,axiom,
! [A: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ A )
=> ( ! [A7: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A7 )
=> ( ! [B7: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ B7 @ A7 )
=> ( P @ B7 ) )
=> ( P @ A7 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_580_finite__psubset__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ! [B7: set_nat] :
( ( ord_less_set_nat @ B7 @ A7 )
=> ( P @ B7 ) )
=> ( P @ A7 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_581_finite__psubset__induct,axiom,
! [A: set_int,P: set_int > $o] :
( ( finite_finite_int @ A )
=> ( ! [A7: set_int] :
( ( finite_finite_int @ A7 )
=> ( ! [B7: set_int] :
( ( ord_less_set_int @ B7 @ A7 )
=> ( P @ B7 ) )
=> ( P @ A7 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_582_finite__psubset__induct,axiom,
! [A: set_complex,P: set_complex > $o] :
( ( finite3207457112153483333omplex @ A )
=> ( ! [A7: set_complex] :
( ( finite3207457112153483333omplex @ A7 )
=> ( ! [B7: set_complex] :
( ( ord_less_set_complex @ B7 @ A7 )
=> ( P @ B7 ) )
=> ( P @ A7 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_583_psubset__imp__ex__mem,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A @ B4 )
=> ? [B5: nat] : ( member_nat @ B5 @ ( minus_minus_set_nat @ B4 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_584_psubset__imp__ex__mem,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_set_a @ A @ B4 )
=> ? [B5: a] : ( member_a @ B5 @ ( minus_minus_set_a @ B4 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_585_psubset__imp__ex__mem,axiom,
! [A: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ A @ B4 )
=> ? [B5: product_prod_a_a] : ( member1426531477525435216od_a_a @ B5 @ ( minus_6817036919807184750od_a_a @ B4 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_586_nat__seg__image__imp__finite,axiom,
! [A: set_a,F: nat > a,N: nat] :
( ( A
= ( image_nat_a @ F
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N ) ) ) )
=> ( finite_finite_a @ A ) ) ).
% nat_seg_image_imp_finite
thf(fact_587_nat__seg__image__imp__finite,axiom,
! [A: set_Product_prod_a_a,F: nat > product_prod_a_a,N: nat] :
( ( A
= ( image_372941888232738320od_a_a @ F
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N ) ) ) )
=> ( finite6544458595007987280od_a_a @ A ) ) ).
% nat_seg_image_imp_finite
thf(fact_588_nat__seg__image__imp__finite,axiom,
! [A: set_nat,F: nat > nat,N: nat] :
( ( A
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N ) ) ) )
=> ( finite_finite_nat @ A ) ) ).
% nat_seg_image_imp_finite
thf(fact_589_nat__seg__image__imp__finite,axiom,
! [A: set_int,F: nat > int,N: nat] :
( ( A
= ( image_nat_int @ F
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N ) ) ) )
=> ( finite_finite_int @ A ) ) ).
% nat_seg_image_imp_finite
thf(fact_590_nat__seg__image__imp__finite,axiom,
! [A: set_complex,F: nat > complex,N: nat] :
( ( A
= ( image_nat_complex @ F
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N ) ) ) )
=> ( finite3207457112153483333omplex @ A ) ) ).
% nat_seg_image_imp_finite
thf(fact_591_finite__conv__nat__seg__image,axiom,
( finite_finite_a
= ( ^ [A6: set_a] :
? [N3: nat,F3: nat > a] :
( A6
= ( image_nat_a @ F3
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N3 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_592_finite__conv__nat__seg__image,axiom,
( finite6544458595007987280od_a_a
= ( ^ [A6: set_Product_prod_a_a] :
? [N3: nat,F3: nat > product_prod_a_a] :
( A6
= ( image_372941888232738320od_a_a @ F3
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N3 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_593_finite__conv__nat__seg__image,axiom,
( finite_finite_nat
= ( ^ [A6: set_nat] :
? [N3: nat,F3: nat > nat] :
( A6
= ( image_nat_nat @ F3
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N3 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_594_finite__conv__nat__seg__image,axiom,
( finite_finite_int
= ( ^ [A6: set_int] :
? [N3: nat,F3: nat > int] :
( A6
= ( image_nat_int @ F3
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N3 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_595_finite__conv__nat__seg__image,axiom,
( finite3207457112153483333omplex
= ( ^ [A6: set_complex] :
? [N3: nat,F3: nat > complex] :
( A6
= ( image_nat_complex @ F3
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N3 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_596_set__times__elim,axiom,
! [X: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ X @ ( times_times_set_nat @ A @ B4 ) )
=> ~ ! [A5: nat,B5: nat] :
( ( X
= ( times_times_nat @ A5 @ B5 ) )
=> ( ( member_nat @ A5 @ A )
=> ~ ( member_nat @ B5 @ B4 ) ) ) ) ).
% set_times_elim
thf(fact_597_set__times__elim,axiom,
! [X: num,A: set_num,B4: set_num] :
( ( member_num @ X @ ( times_times_set_num @ A @ B4 ) )
=> ~ ! [A5: num,B5: num] :
( ( X
= ( times_times_num @ A5 @ B5 ) )
=> ( ( member_num @ A5 @ A )
=> ~ ( member_num @ B5 @ B4 ) ) ) ) ).
% set_times_elim
thf(fact_598_set__times__elim,axiom,
! [X: real,A: set_real,B4: set_real] :
( ( member_real @ X @ ( times_times_set_real @ A @ B4 ) )
=> ~ ! [A5: real,B5: real] :
( ( X
= ( times_times_real @ A5 @ B5 ) )
=> ( ( member_real @ A5 @ A )
=> ~ ( member_real @ B5 @ B4 ) ) ) ) ).
% set_times_elim
thf(fact_599_set__times__elim,axiom,
! [X: int,A: set_int,B4: set_int] :
( ( member_int @ X @ ( times_times_set_int @ A @ B4 ) )
=> ~ ! [A5: int,B5: int] :
( ( X
= ( times_times_int @ A5 @ B5 ) )
=> ( ( member_int @ A5 @ A )
=> ~ ( member_int @ B5 @ B4 ) ) ) ) ).
% set_times_elim
thf(fact_600_set__times__elim,axiom,
! [X: extended_enat,A: set_Extended_enat,B4: set_Extended_enat] :
( ( member_Extended_enat @ X @ ( times_2438108612031896577d_enat @ A @ B4 ) )
=> ~ ! [A5: extended_enat,B5: extended_enat] :
( ( X
= ( times_7803423173614009249d_enat @ A5 @ B5 ) )
=> ( ( member_Extended_enat @ A5 @ A )
=> ~ ( member_Extended_enat @ B5 @ B4 ) ) ) ) ).
% set_times_elim
thf(fact_601_card__less__sym__Diff,axiom,
! [A: set_a,B4: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B4 )
=> ( ( ord_less_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B4 ) )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A @ B4 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B4 @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_602_card__less__sym__Diff,axiom,
! [A: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B4 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B4 ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B4 @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_603_card__less__sym__Diff,axiom,
! [A: set_int,B4: set_int] :
( ( finite_finite_int @ A )
=> ( ( finite_finite_int @ B4 )
=> ( ( ord_less_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B4 ) )
=> ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A @ B4 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ B4 @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_604_card__less__sym__Diff,axiom,
! [A: set_complex,B4: set_complex] :
( ( finite3207457112153483333omplex @ A )
=> ( ( finite3207457112153483333omplex @ B4 )
=> ( ( ord_less_nat @ ( finite_card_complex @ A ) @ ( finite_card_complex @ B4 ) )
=> ( ord_less_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A @ B4 ) ) @ ( finite_card_complex @ ( minus_811609699411566653omplex @ B4 @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_605_card__less__sym__Diff,axiom,
! [A: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A )
=> ( ( finite6544458595007987280od_a_a @ B4 )
=> ( ( ord_less_nat @ ( finite4795055649997197647od_a_a @ A ) @ ( finite4795055649997197647od_a_a @ B4 ) )
=> ( ord_less_nat @ ( finite4795055649997197647od_a_a @ ( minus_6817036919807184750od_a_a @ A @ B4 ) ) @ ( finite4795055649997197647od_a_a @ ( minus_6817036919807184750od_a_a @ B4 @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_606_less__exp,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% less_exp
thf(fact_607_pow_Osimps_I1_J,axiom,
! [X: num] :
( ( pow @ X @ one )
= X ) ).
% pow.simps(1)
thf(fact_608_mult__numeral__1__right,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ ( numeral_numeral_nat @ one ) )
= A2 ) ).
% mult_numeral_1_right
thf(fact_609_mult__numeral__1__right,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ ( numeral_numeral_real @ one ) )
= A2 ) ).
% mult_numeral_1_right
thf(fact_610_mult__numeral__1__right,axiom,
! [A2: extended_enat] :
( ( times_7803423173614009249d_enat @ A2 @ ( numera1916890842035813515d_enat @ one ) )
= A2 ) ).
% mult_numeral_1_right
thf(fact_611_mult__numeral__1__right,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ ( numeral_numeral_int @ one ) )
= A2 ) ).
% mult_numeral_1_right
thf(fact_612_mult__numeral__1,axiom,
! [A2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A2 )
= A2 ) ).
% mult_numeral_1
thf(fact_613_mult__numeral__1,axiom,
! [A2: real] :
( ( times_times_real @ ( numeral_numeral_real @ one ) @ A2 )
= A2 ) ).
% mult_numeral_1
thf(fact_614_mult__numeral__1,axiom,
! [A2: extended_enat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ one ) @ A2 )
= A2 ) ).
% mult_numeral_1
thf(fact_615_mult__numeral__1,axiom,
! [A2: int] :
( ( times_times_int @ ( numeral_numeral_int @ one ) @ A2 )
= A2 ) ).
% mult_numeral_1
thf(fact_616__092_060open_062card_A_123_Ix_M_Ay_J_O_A_Ix_M_Ay_J_A_092_060in_062_AM_A_092_060times_062_AM_A_092_060and_062_Ax_A_060_Ay_125_A_L_Acard_A_123_Ix_M_Ay_J_O_A_Ix_M_Ay_J_A_092_060in_062_AM_A_092_060times_062_AM_A_092_060and_062_Ay_A_060_Ax_125_A_061_Acard_A_I_123_Ix_M_Ay_J_O_A_Ix_M_Ay_J_A_092_060in_062_AM_A_092_060times_062_AM_A_092_060and_062_Ax_A_060_Ay_125_A_092_060union_062_A_123_Ix_M_Ay_J_O_A_Ix_M_Ay_J_A_092_060in_062_AM_A_092_060times_062_AM_A_092_060and_062_Ay_A_060_Ax_125_J_092_060close_062,axiom,
( ( plus_plus_nat
@ ( finite4795055649997197647od_a_a
@ ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X2: a,Y2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 )
@ ( product_Sigma_a_a @ m
@ ^ [Uu: a] : m ) )
& ( ord_less_a @ X2 @ Y2 ) ) ) ) )
@ ( finite4795055649997197647od_a_a
@ ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X2: a,Y2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 )
@ ( product_Sigma_a_a @ m
@ ^ [Uu: a] : m ) )
& ( ord_less_a @ Y2 @ X2 ) ) ) ) ) )
= ( finite4795055649997197647od_a_a
@ ( sup_su3048258781599657691od_a_a
@ ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X2: a,Y2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 )
@ ( product_Sigma_a_a @ m
@ ^ [Uu: a] : m ) )
& ( ord_less_a @ X2 @ Y2 ) ) ) )
@ ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X2: a,Y2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 )
@ ( product_Sigma_a_a @ m
@ ^ [Uu: a] : m ) )
& ( ord_less_a @ Y2 @ X2 ) ) ) ) ) ) ) ).
% \<open>card {(x, y). (x, y) \<in> M \<times> M \<and> x < y} + card {(x, y). (x, y) \<in> M \<times> M \<and> y < x} = card ({(x, y). (x, y) \<in> M \<times> M \<and> x < y} \<union> {(x, y). (x, y) \<in> M \<times> M \<and> y < x})\<close>
thf(fact_617_translation__subtract__diff,axiom,
! [A2: int,S3: set_int,T3: set_int] :
( ( image_int_int
@ ^ [X2: int] : ( minus_minus_int @ X2 @ A2 )
@ ( minus_minus_set_int @ S3 @ T3 ) )
= ( minus_minus_set_int
@ ( image_int_int
@ ^ [X2: int] : ( minus_minus_int @ X2 @ A2 )
@ S3 )
@ ( image_int_int
@ ^ [X2: int] : ( minus_minus_int @ X2 @ A2 )
@ T3 ) ) ) ).
% translation_subtract_diff
thf(fact_618_translation__subtract__diff,axiom,
! [A2: complex,S3: set_complex,T3: set_complex] :
( ( image_1468599708987790691omplex
@ ^ [X2: complex] : ( minus_minus_complex @ X2 @ A2 )
@ ( minus_811609699411566653omplex @ S3 @ T3 ) )
= ( minus_811609699411566653omplex
@ ( image_1468599708987790691omplex
@ ^ [X2: complex] : ( minus_minus_complex @ X2 @ A2 )
@ S3 )
@ ( image_1468599708987790691omplex
@ ^ [X2: complex] : ( minus_minus_complex @ X2 @ A2 )
@ T3 ) ) ) ).
% translation_subtract_diff
thf(fact_619_translation__subtract__diff,axiom,
! [A2: real,S3: set_real,T3: set_real] :
( ( image_real_real
@ ^ [X2: real] : ( minus_minus_real @ X2 @ A2 )
@ ( minus_minus_set_real @ S3 @ T3 ) )
= ( minus_minus_set_real
@ ( image_real_real
@ ^ [X2: real] : ( minus_minus_real @ X2 @ A2 )
@ S3 )
@ ( image_real_real
@ ^ [X2: real] : ( minus_minus_real @ X2 @ A2 )
@ T3 ) ) ) ).
% translation_subtract_diff
thf(fact_620_four__x__squared,axiom,
! [X: real] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% four_x_squared
thf(fact_621_enat__ord__number_I2_J,axiom,
! [M: num,N: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(2)
thf(fact_622_case__prod__Pair__iden,axiom,
! [P2: product_prod_a_a] :
( ( produc408267641121961211od_a_a @ product_Pair_a_a @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_623_case__prod__Pair__iden,axiom,
! [P2: product_prod_num_num] :
( ( produc64540874165560627um_num @ product_Pair_num_num @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_624_divmod__algorithm__code_I4_J,axiom,
! [M: num,N: num] :
( ( unique5405566460079783412od_nat @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( produc2626176000494625587at_nat
@ ^ [Q2: nat,R2: nat] : ( product_Pair_nat_nat @ Q2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R2 ) )
@ ( unique5405566460079783412od_nat @ M @ N ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_625_divmod__algorithm__code_I4_J,axiom,
! [M: num,N: num] :
( ( unique5403075989570733136od_int @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( produc4245557441103728435nt_int
@ ^ [Q2: int,R2: int] : ( product_Pair_int_int @ Q2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) )
@ ( unique5403075989570733136od_int @ M @ N ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_626_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_627_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_628_set__plus__intro,axiom,
! [A2: nat,C2: set_nat,B: nat,D: set_nat] :
( ( member_nat @ A2 @ C2 )
=> ( ( member_nat @ B @ D )
=> ( member_nat @ ( plus_plus_nat @ A2 @ B ) @ ( plus_plus_set_nat @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_629_set__plus__intro,axiom,
! [A2: num,C2: set_num,B: num,D: set_num] :
( ( member_num @ A2 @ C2 )
=> ( ( member_num @ B @ D )
=> ( member_num @ ( plus_plus_num @ A2 @ B ) @ ( plus_plus_set_num @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_630_set__plus__intro,axiom,
! [A2: int,C2: set_int,B: int,D: set_int] :
( ( member_int @ A2 @ C2 )
=> ( ( member_int @ B @ D )
=> ( member_int @ ( plus_plus_int @ A2 @ B ) @ ( plus_plus_set_int @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_631_set__plus__intro,axiom,
! [A2: extended_enat,C2: set_Extended_enat,B: extended_enat,D: set_Extended_enat] :
( ( member_Extended_enat @ A2 @ C2 )
=> ( ( member_Extended_enat @ B @ D )
=> ( member_Extended_enat @ ( plus_p3455044024723400733d_enat @ A2 @ B ) @ ( plus_p3482335003337316477d_enat @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_632_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_nat @ N3 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_633_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_634_add__numeral__left,axiom,
! [V: num,W: num,Z2: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_635_add__numeral__left,axiom,
! [V: num,W: num,Z2: real] :
( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z2 ) )
= ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_636_add__numeral__left,axiom,
! [V: num,W: num,Z2: extended_enat] :
( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ W ) @ Z2 ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_637_add__numeral__left,axiom,
! [V: num,W: num,Z2: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z2 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_638_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_639_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_640_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( numera1916890842035813515d_enat @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_641_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_642_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_643_diff__diff__left,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ K )
= ( minus_minus_nat @ I3 @ ( plus_plus_nat @ J3 @ K ) ) ) ).
% diff_diff_left
thf(fact_644_distrib__right__numeral,axiom,
! [A2: nat,B: nat,V: num] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ ( numeral_numeral_nat @ V ) )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_645_distrib__right__numeral,axiom,
! [A2: real,B: real,V: num] :
( ( times_times_real @ ( plus_plus_real @ A2 @ B ) @ ( numeral_numeral_real @ V ) )
= ( plus_plus_real @ ( times_times_real @ A2 @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_646_distrib__right__numeral,axiom,
! [A2: extended_enat,B: extended_enat,V: num] :
( ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A2 @ B ) @ ( numera1916890842035813515d_enat @ V ) )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A2 @ ( numera1916890842035813515d_enat @ V ) ) @ ( times_7803423173614009249d_enat @ B @ ( numera1916890842035813515d_enat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_647_distrib__right__numeral,axiom,
! [A2: int,B: int,V: num] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ ( numeral_numeral_int @ V ) )
= ( plus_plus_int @ ( times_times_int @ A2 @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_648_distrib__left__numeral,axiom,
! [V: num,B: nat,C: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_649_distrib__left__numeral,axiom,
! [V: num,B: real,C: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_650_distrib__left__numeral,axiom,
! [V: num,B: extended_enat,C: extended_enat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ B @ C ) )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ B ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_651_distrib__left__numeral,axiom,
! [V: num,B: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_652_psubsetD,axiom,
! [A: set_Product_prod_a_a,B4: set_Product_prod_a_a,C: product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ A @ B4 )
=> ( ( member1426531477525435216od_a_a @ C @ A )
=> ( member1426531477525435216od_a_a @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_653_psubsetD,axiom,
! [A: set_nat,B4: set_nat,C: nat] :
( ( ord_less_set_nat @ A @ B4 )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_654_psubsetD,axiom,
! [A: set_a,B4: set_a,C: a] :
( ( ord_less_set_a @ A @ B4 )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_655_less__set__def,axiom,
( ord_le6819997720685908915od_a_a
= ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ord_le3684186632072642282_a_a_o
@ ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ A6 )
@ ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_656_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ord_less_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_657_less__set__def,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ord_less_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A6 )
@ ^ [X2: a] : ( member_a @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_658_bounded__nat__set__is__finite,axiom,
! [N2: set_nat,N: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ N2 )
=> ( ord_less_nat @ X3 @ N ) )
=> ( finite_finite_nat @ N2 ) ) ).
% bounded_nat_set_is_finite
thf(fact_659_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M3: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N4 )
=> ( ord_less_nat @ X2 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_660_enat__less__induct,axiom,
! [P: extended_enat > $o,N: extended_enat] :
( ! [N5: extended_enat] :
( ! [M4: extended_enat] :
( ( ord_le72135733267957522d_enat @ M4 @ N5 )
=> ( P @ M4 ) )
=> ( P @ N5 ) )
=> ( P @ N ) ) ).
% enat_less_induct
thf(fact_661_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_662_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_663_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N5: nat] :
( ~ ( P @ N5 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N5 )
& ~ ( P @ M4 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_664_trans__less__add2,axiom,
! [I3: nat,J3: nat,M: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ M @ J3 ) ) ) ).
% trans_less_add2
thf(fact_665_trans__less__add1,axiom,
! [I3: nat,J3: nat,M: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ J3 @ M ) ) ) ).
% trans_less_add1
thf(fact_666_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N5: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N5 )
=> ( P @ M4 ) )
=> ( P @ N5 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_667_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_668_less__not__refl3,axiom,
! [S3: nat,T3: nat] :
( ( ord_less_nat @ S3 @ T3 )
=> ( S3 != T3 ) ) ).
% less_not_refl3
thf(fact_669_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_670_add__less__mono1,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J3 @ K ) ) ) ).
% add_less_mono1
thf(fact_671_not__add__less2,axiom,
! [J3: nat,I3: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J3 @ I3 ) @ I3 ) ).
% not_add_less2
thf(fact_672_not__add__less1,axiom,
! [I3: nat,J3: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I3 @ J3 ) @ I3 ) ).
% not_add_less1
thf(fact_673_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_674_add__less__mono,axiom,
! [I3: nat,J3: nat,K: nat,L: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ) ).
% add_less_mono
thf(fact_675_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_676_add__lessD1,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I3 @ J3 ) @ K )
=> ( ord_less_nat @ I3 @ K ) ) ).
% add_lessD1
thf(fact_677_is__num__normalize_I1_J,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_678_set__plus__elim,axiom,
! [X: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ X @ ( plus_plus_set_nat @ A @ B4 ) )
=> ~ ! [A5: nat,B5: nat] :
( ( X
= ( plus_plus_nat @ A5 @ B5 ) )
=> ( ( member_nat @ A5 @ A )
=> ~ ( member_nat @ B5 @ B4 ) ) ) ) ).
% set_plus_elim
thf(fact_679_set__plus__elim,axiom,
! [X: num,A: set_num,B4: set_num] :
( ( member_num @ X @ ( plus_plus_set_num @ A @ B4 ) )
=> ~ ! [A5: num,B5: num] :
( ( X
= ( plus_plus_num @ A5 @ B5 ) )
=> ( ( member_num @ A5 @ A )
=> ~ ( member_num @ B5 @ B4 ) ) ) ) ).
% set_plus_elim
thf(fact_680_set__plus__elim,axiom,
! [X: int,A: set_int,B4: set_int] :
( ( member_int @ X @ ( plus_plus_set_int @ A @ B4 ) )
=> ~ ! [A5: int,B5: int] :
( ( X
= ( plus_plus_int @ A5 @ B5 ) )
=> ( ( member_int @ A5 @ A )
=> ~ ( member_int @ B5 @ B4 ) ) ) ) ).
% set_plus_elim
thf(fact_681_set__plus__elim,axiom,
! [X: extended_enat,A: set_Extended_enat,B4: set_Extended_enat] :
( ( member_Extended_enat @ X @ ( plus_p3482335003337316477d_enat @ A @ B4 ) )
=> ~ ! [A5: extended_enat,B5: extended_enat] :
( ( X
= ( plus_p3455044024723400733d_enat @ A5 @ B5 ) )
=> ( ( member_Extended_enat @ A5 @ A )
=> ~ ( member_Extended_enat @ B5 @ B4 ) ) ) ) ).
% set_plus_elim
thf(fact_682_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I3: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( P @ K2 )
& ( ord_less_nat @ K2 @ I3 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_683_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_684_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_685_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_686_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_687_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_688_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_689_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_690_less__diff__conv,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_nat @ I3 @ ( minus_minus_nat @ J3 @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ J3 ) ) ).
% less_diff_conv
thf(fact_691_left__add__mult__distrib,axiom,
! [I3: nat,U: nat,J3: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J3 @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I3 @ J3 ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_692_translation__diff,axiom,
! [A2: int,S3: set_int,T3: set_int] :
( ( image_int_int @ ( plus_plus_int @ A2 ) @ ( minus_minus_set_int @ S3 @ T3 ) )
= ( minus_minus_set_int @ ( image_int_int @ ( plus_plus_int @ A2 ) @ S3 ) @ ( image_int_int @ ( plus_plus_int @ A2 ) @ T3 ) ) ) ).
% translation_diff
thf(fact_693_sup__set__def,axiom,
( sup_sup_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ( sup_sup_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A6 )
@ ^ [X2: a] : ( member_a @ X2 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_694_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_695_sup__set__def,axiom,
( sup_sup_set_int
= ( ^ [A6: set_int,B6: set_int] :
( collect_int
@ ( sup_sup_int_o
@ ^ [X2: int] : ( member_int @ X2 @ A6 )
@ ^ [X2: int] : ( member_int @ X2 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_696_sup__set__def,axiom,
( sup_sup_set_complex
= ( ^ [A6: set_complex,B6: set_complex] :
( collect_complex
@ ( sup_sup_complex_o
@ ^ [X2: complex] : ( member_complex @ X2 @ A6 )
@ ^ [X2: complex] : ( member_complex @ X2 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_697_sup__set__def,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( collec3336397797384452498od_a_a
@ ( sup_su1640154476453168578_a_a_o
@ ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ A6 )
@ ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_698_set__plus__image,axiom,
( plus_plus_set_nat
= ( ^ [S2: set_nat,T2: set_nat] :
( image_2486076414777270412at_nat @ ( produc6842872674320459806at_nat @ plus_plus_nat )
@ ( produc457027306803732586at_nat @ S2
@ ^ [Uu: nat] : T2 ) ) ) ) ).
% set_plus_image
thf(fact_699_set__plus__image,axiom,
( plus_plus_set_num
= ( ^ [S2: set_num,T2: set_num] :
( image_8225983159724657258um_num @ ( produc8694714728609142524um_num @ plus_plus_num )
@ ( produc4368061533121756414um_num @ S2
@ ^ [Uu: num] : T2 ) ) ) ) ).
% set_plus_image
thf(fact_700_set__plus__image,axiom,
( plus_plus_set_int
= ( ^ [S2: set_int,T2: set_int] :
( image_5042161079198086560nt_int @ ( produc8211389475949308722nt_int @ plus_plus_int )
@ ( produc1453890548195736866nt_int @ S2
@ ^ [Uu: int] : T2 ) ) ) ) ).
% set_plus_image
thf(fact_701_set__plus__image,axiom,
( plus_p3482335003337316477d_enat
= ( ^ [S2: set_Extended_enat,T2: set_Extended_enat] :
( image_3712855581125276676d_enat @ ( produc797783881074051898d_enat @ plus_p3455044024723400733d_enat )
@ ( produc5797200089183630280d_enat @ S2
@ ^ [Uu: extended_enat] : T2 ) ) ) ) ).
% set_plus_image
thf(fact_702_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_Bit0
thf(fact_703_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit0 @ N ) )
= ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% numeral_Bit0
thf(fact_704_numeral__Bit0,axiom,
! [N: num] :
( ( numera1916890842035813515d_enat @ ( bit0 @ N ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ ( numera1916890842035813515d_enat @ N ) ) ) ).
% numeral_Bit0
thf(fact_705_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_Bit0
thf(fact_706_power__add,axiom,
! [A2: complex,M: nat,N: nat] :
( ( power_power_complex @ A2 @ ( plus_plus_nat @ M @ N ) )
= ( times_times_complex @ ( power_power_complex @ A2 @ M ) @ ( power_power_complex @ A2 @ N ) ) ) ).
% power_add
thf(fact_707_power__add,axiom,
! [A2: nat,M: nat,N: nat] :
( ( power_power_nat @ A2 @ ( plus_plus_nat @ M @ N ) )
= ( times_times_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) ) ) ).
% power_add
thf(fact_708_power__add,axiom,
! [A2: real,M: nat,N: nat] :
( ( power_power_real @ A2 @ ( plus_plus_nat @ M @ N ) )
= ( times_times_real @ ( power_power_real @ A2 @ M ) @ ( power_power_real @ A2 @ N ) ) ) ).
% power_add
thf(fact_709_power__add,axiom,
! [A2: int,M: nat,N: nat] :
( ( power_power_int @ A2 @ ( plus_plus_nat @ M @ N ) )
= ( times_times_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) ) ) ).
% power_add
thf(fact_710_power__add,axiom,
! [A2: extended_enat,M: nat,N: nat] :
( ( power_8040749407984259932d_enat @ A2 @ ( plus_plus_nat @ M @ N ) )
= ( times_7803423173614009249d_enat @ ( power_8040749407984259932d_enat @ A2 @ M ) @ ( power_8040749407984259932d_enat @ A2 @ N ) ) ) ).
% power_add
thf(fact_711_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_code(2)
thf(fact_712_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit0 @ N ) )
= ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% numeral_code(2)
thf(fact_713_numeral__code_I2_J,axiom,
! [N: num] :
( ( numera1916890842035813515d_enat @ ( bit0 @ N ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ ( numera1916890842035813515d_enat @ N ) ) ) ).
% numeral_code(2)
thf(fact_714_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_code(2)
thf(fact_715_diff__left__imp__eq,axiom,
! [A2: int,B: int,C: int] :
( ( ( minus_minus_int @ A2 @ B )
= ( minus_minus_int @ A2 @ C ) )
=> ( B = C ) ) ).
% diff_left_imp_eq
thf(fact_716_diff__left__imp__eq,axiom,
! [A2: complex,B: complex,C: complex] :
( ( ( minus_minus_complex @ A2 @ B )
= ( minus_minus_complex @ A2 @ C ) )
=> ( B = C ) ) ).
% diff_left_imp_eq
thf(fact_717_diff__left__imp__eq,axiom,
! [A2: real,B: real,C: real] :
( ( ( minus_minus_real @ A2 @ B )
= ( minus_minus_real @ A2 @ C ) )
=> ( B = C ) ) ).
% diff_left_imp_eq
thf(fact_718_diff__commute,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I3 @ K ) @ J3 ) ) ).
% diff_commute
thf(fact_719_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_720_less__imp__diff__less,axiom,
! [J3: nat,K: nat,N: nat] :
( ( ord_less_nat @ J3 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J3 @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_721_mult__2,axiom,
! [Z2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z2 )
= ( plus_plus_nat @ Z2 @ Z2 ) ) ).
% mult_2
thf(fact_722_mult__2,axiom,
! [Z2: real] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z2 )
= ( plus_plus_real @ Z2 @ Z2 ) ) ).
% mult_2
thf(fact_723_mult__2,axiom,
! [Z2: extended_enat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ Z2 )
= ( plus_p3455044024723400733d_enat @ Z2 @ Z2 ) ) ).
% mult_2
thf(fact_724_mult__2,axiom,
! [Z2: int] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z2 )
= ( plus_plus_int @ Z2 @ Z2 ) ) ).
% mult_2
thf(fact_725_mult__2__right,axiom,
! [Z2: nat] :
( ( times_times_nat @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ Z2 @ Z2 ) ) ).
% mult_2_right
thf(fact_726_mult__2__right,axiom,
! [Z2: real] :
( ( times_times_real @ Z2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
= ( plus_plus_real @ Z2 @ Z2 ) ) ).
% mult_2_right
thf(fact_727_mult__2__right,axiom,
! [Z2: extended_enat] :
( ( times_7803423173614009249d_enat @ Z2 @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) )
= ( plus_p3455044024723400733d_enat @ Z2 @ Z2 ) ) ).
% mult_2_right
thf(fact_728_mult__2__right,axiom,
! [Z2: int] :
( ( times_times_int @ Z2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ Z2 @ Z2 ) ) ).
% mult_2_right
thf(fact_729_left__add__twice,axiom,
! [A2: nat,B: nat] :
( ( plus_plus_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) @ B ) ) ).
% left_add_twice
thf(fact_730_left__add__twice,axiom,
! [A2: real,B: real] :
( ( plus_plus_real @ A2 @ ( plus_plus_real @ A2 @ B ) )
= ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A2 ) @ B ) ) ).
% left_add_twice
thf(fact_731_left__add__twice,axiom,
! [A2: extended_enat,B: extended_enat] :
( ( plus_p3455044024723400733d_enat @ A2 @ ( plus_p3455044024723400733d_enat @ A2 @ B ) )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ A2 ) @ B ) ) ).
% left_add_twice
thf(fact_732_left__add__twice,axiom,
! [A2: int,B: int] :
( ( plus_plus_int @ A2 @ ( plus_plus_int @ A2 @ B ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) @ B ) ) ).
% left_add_twice
thf(fact_733_power2__sum,axiom,
! [X: complex,Y: complex] :
( ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_734_power2__sum,axiom,
! [X: nat,Y: nat] :
( ( power_power_nat @ ( plus_plus_nat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_735_power2__sum,axiom,
! [X: real,Y: real] :
( ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_736_power2__sum,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( power_8040749407984259932d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( power_8040749407984259932d_enat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8040749407984259932d_enat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_7803423173614009249d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_737_power2__sum,axiom,
! [X: int,Y: int] :
( ( power_power_int @ ( plus_plus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_738_power2__diff,axiom,
! [X: complex,Y: complex] :
( ( power_power_complex @ ( minus_minus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_diff
thf(fact_739_power2__diff,axiom,
! [X: real,Y: real] :
( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_diff
thf(fact_740_power2__diff,axiom,
! [X: int,Y: int] :
( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_diff
thf(fact_741_add__diff__cancel,axiom,
! [A2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel
thf(fact_742_add__diff__cancel,axiom,
! [A2: complex,B: complex] :
( ( minus_minus_complex @ ( plus_plus_complex @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel
thf(fact_743_add__diff__cancel,axiom,
! [A2: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel
thf(fact_744_diff__add__cancel,axiom,
! [A2: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A2 @ B ) @ B )
= A2 ) ).
% diff_add_cancel
thf(fact_745_diff__add__cancel,axiom,
! [A2: complex,B: complex] :
( ( plus_plus_complex @ ( minus_minus_complex @ A2 @ B ) @ B )
= A2 ) ).
% diff_add_cancel
thf(fact_746_diff__add__cancel,axiom,
! [A2: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A2 @ B ) @ B )
= A2 ) ).
% diff_add_cancel
thf(fact_747_add__diff__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_748_add__diff__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_749_add__diff__cancel__left,axiom,
! [C: complex,A2: complex,B: complex] :
( ( minus_minus_complex @ ( plus_plus_complex @ C @ A2 ) @ ( plus_plus_complex @ C @ B ) )
= ( minus_minus_complex @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_750_add__diff__cancel__left,axiom,
! [C: real,A2: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B ) )
= ( minus_minus_real @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_751_add__diff__cancel__left_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_752_add__diff__cancel__left_H,axiom,
! [A2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_753_add__diff__cancel__left_H,axiom,
! [A2: complex,B: complex] :
( ( minus_minus_complex @ ( plus_plus_complex @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_754_add__diff__cancel__left_H,axiom,
! [A2: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_755_add__diff__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_756_add__diff__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_757_add__diff__cancel__right,axiom,
! [A2: complex,C: complex,B: complex] :
( ( minus_minus_complex @ ( plus_plus_complex @ A2 @ C ) @ ( plus_plus_complex @ B @ C ) )
= ( minus_minus_complex @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_758_add__diff__cancel__right,axiom,
! [A2: real,C: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_759_add__diff__cancel__right_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_760_add__diff__cancel__right_H,axiom,
! [A2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_761_add__diff__cancel__right_H,axiom,
! [A2: complex,B: complex] :
( ( minus_minus_complex @ ( plus_plus_complex @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_762_add__diff__cancel__right_H,axiom,
! [A2: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_763_add__less__cancel__right,axiom,
! [A2: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_764_add__less__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_765_add__less__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_766_add__less__cancel__left,axiom,
! [C: real,A2: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_767_add__less__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_768_add__less__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_769_add__left__cancel,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_770_add__left__cancel,axiom,
! [A2: int,B: int,C: int] :
( ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_771_add__right__cancel,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_772_add__right__cancel,axiom,
! [B: int,A2: int,C: int] :
( ( ( plus_plus_int @ B @ A2 )
= ( plus_plus_int @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_773_semiring__norm_I6_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(6)
thf(fact_774_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_775_power__add__numeral2,axiom,
! [A2: complex,M: num,N: num,B: complex] :
( ( times_times_complex @ ( power_power_complex @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A2 @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_complex @ ( power_power_complex @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_776_power__add__numeral2,axiom,
! [A2: nat,M: num,N: num,B: nat] :
( ( times_times_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_777_power__add__numeral2,axiom,
! [A2: real,M: num,N: num,B: real] :
( ( times_times_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_778_power__add__numeral2,axiom,
! [A2: int,M: num,N: num,B: int] :
( ( times_times_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_779_power__add__numeral2,axiom,
! [A2: extended_enat,M: num,N: num,B: extended_enat] :
( ( times_7803423173614009249d_enat @ ( power_8040749407984259932d_enat @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( times_7803423173614009249d_enat @ ( power_8040749407984259932d_enat @ A2 @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_7803423173614009249d_enat @ ( power_8040749407984259932d_enat @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_780_power__add__numeral,axiom,
! [A2: complex,M: num,N: num] :
( ( times_times_complex @ ( power_power_complex @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A2 @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_complex @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_781_power__add__numeral,axiom,
! [A2: nat,M: num,N: num] :
( ( times_times_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_782_power__add__numeral,axiom,
! [A2: real,M: num,N: num] :
( ( times_times_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_783_power__add__numeral,axiom,
! [A2: int,M: num,N: num] :
( ( times_times_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_784_power__add__numeral,axiom,
! [A2: extended_enat,M: num,N: num] :
( ( times_7803423173614009249d_enat @ ( power_8040749407984259932d_enat @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( power_8040749407984259932d_enat @ A2 @ ( numeral_numeral_nat @ N ) ) )
= ( power_8040749407984259932d_enat @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_785_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_786_finite__set__plus,axiom,
! [S3: set_nat,T3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ( finite_finite_nat @ T3 )
=> ( finite_finite_nat @ ( plus_plus_set_nat @ S3 @ T3 ) ) ) ) ).
% finite_set_plus
thf(fact_787_finite__set__plus,axiom,
! [S3: set_int,T3: set_int] :
( ( finite_finite_int @ S3 )
=> ( ( finite_finite_int @ T3 )
=> ( finite_finite_int @ ( plus_plus_set_int @ S3 @ T3 ) ) ) ) ).
% finite_set_plus
thf(fact_788_finite__set__plus,axiom,
! [S3: set_complex,T3: set_complex] :
( ( finite3207457112153483333omplex @ S3 )
=> ( ( finite3207457112153483333omplex @ T3 )
=> ( finite3207457112153483333omplex @ ( plus_p7052360327008956141omplex @ S3 @ T3 ) ) ) ) ).
% finite_set_plus
thf(fact_789_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_790_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A2 @ B ) @ C )
= ( times_times_real @ A2 @ ( times_times_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_791_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A2 @ B ) @ C )
= ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_792_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: extended_enat,B: extended_enat,C: extended_enat] :
( ( times_7803423173614009249d_enat @ ( times_7803423173614009249d_enat @ A2 @ B ) @ C )
= ( times_7803423173614009249d_enat @ A2 @ ( times_7803423173614009249d_enat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_793_mult_Oassoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_794_mult_Oassoc,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A2 @ B ) @ C )
= ( times_times_real @ A2 @ ( times_times_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_795_mult_Oassoc,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A2 @ B ) @ C )
= ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_796_mult_Oassoc,axiom,
! [A2: extended_enat,B: extended_enat,C: extended_enat] :
( ( times_7803423173614009249d_enat @ ( times_7803423173614009249d_enat @ A2 @ B ) @ C )
= ( times_7803423173614009249d_enat @ A2 @ ( times_7803423173614009249d_enat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_797_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B3: nat] : ( times_times_nat @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_798_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A4: real,B3: real] : ( times_times_real @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_799_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A4: int,B3: int] : ( times_times_int @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_800_mult_Ocommute,axiom,
( times_7803423173614009249d_enat
= ( ^ [A4: extended_enat,B3: extended_enat] : ( times_7803423173614009249d_enat @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_801_mult_Oleft__commute,axiom,
! [B: nat,A2: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A2 @ C ) )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_802_mult_Oleft__commute,axiom,
! [B: real,A2: real,C: real] :
( ( times_times_real @ B @ ( times_times_real @ A2 @ C ) )
= ( times_times_real @ A2 @ ( times_times_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_803_mult_Oleft__commute,axiom,
! [B: int,A2: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A2 @ C ) )
= ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_804_mult_Oleft__commute,axiom,
! [B: extended_enat,A2: extended_enat,C: extended_enat] :
( ( times_7803423173614009249d_enat @ B @ ( times_7803423173614009249d_enat @ A2 @ C ) )
= ( times_7803423173614009249d_enat @ A2 @ ( times_7803423173614009249d_enat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_805_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_806_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_807_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: extended_enat,B: extended_enat,C: extended_enat] :
( ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ A2 @ B ) @ C )
= ( plus_p3455044024723400733d_enat @ A2 @ ( plus_p3455044024723400733d_enat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_808_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I3: nat,J3: nat,K: nat,L: nat] :
( ( ( I3 = J3 )
& ( K = L ) )
=> ( ( plus_plus_nat @ I3 @ K )
= ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_809_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I3: int,J3: int,K: int,L: int] :
( ( ( I3 = J3 )
& ( K = L ) )
=> ( ( plus_plus_int @ I3 @ K )
= ( plus_plus_int @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_810_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I3: extended_enat,J3: extended_enat,K: extended_enat,L: extended_enat] :
( ( ( I3 = J3 )
& ( K = L ) )
=> ( ( plus_p3455044024723400733d_enat @ I3 @ K )
= ( plus_p3455044024723400733d_enat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_811_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A2: nat,B: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_812_group__cancel_Oadd1,axiom,
! [A: int,K: int,A2: int,B: int] :
( ( A
= ( plus_plus_int @ K @ A2 ) )
=> ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_813_group__cancel_Oadd1,axiom,
! [A: extended_enat,K: extended_enat,A2: extended_enat,B: extended_enat] :
( ( A
= ( plus_p3455044024723400733d_enat @ K @ A2 ) )
=> ( ( plus_p3455044024723400733d_enat @ A @ B )
= ( plus_p3455044024723400733d_enat @ K @ ( plus_p3455044024723400733d_enat @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_814_group__cancel_Oadd2,axiom,
! [B4: nat,K: nat,B: nat,A2: nat] :
( ( B4
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A2 @ B4 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_815_group__cancel_Oadd2,axiom,
! [B4: int,K: int,B: int,A2: int] :
( ( B4
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A2 @ B4 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_816_group__cancel_Oadd2,axiom,
! [B4: extended_enat,K: extended_enat,B: extended_enat,A2: extended_enat] :
( ( B4
= ( plus_p3455044024723400733d_enat @ K @ B ) )
=> ( ( plus_p3455044024723400733d_enat @ A2 @ B4 )
= ( plus_p3455044024723400733d_enat @ K @ ( plus_p3455044024723400733d_enat @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_817_add_Oassoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_818_add_Oassoc,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_819_add_Oassoc,axiom,
! [A2: extended_enat,B: extended_enat,C: extended_enat] :
( ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ A2 @ B ) @ C )
= ( plus_p3455044024723400733d_enat @ A2 @ ( plus_p3455044024723400733d_enat @ B @ C ) ) ) ).
% add.assoc
thf(fact_820_add_Oleft__cancel,axiom,
! [A2: int,B: int,C: int] :
( ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ A2 @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_821_add_Oright__cancel,axiom,
! [B: int,A2: int,C: int] :
( ( ( plus_plus_int @ B @ A2 )
= ( plus_plus_int @ C @ A2 ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_822_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B3: nat] : ( plus_plus_nat @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_823_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B3: int] : ( plus_plus_int @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_824_add_Ocommute,axiom,
( plus_p3455044024723400733d_enat
= ( ^ [A4: extended_enat,B3: extended_enat] : ( plus_p3455044024723400733d_enat @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_825_add_Oleft__commute,axiom,
! [B: nat,A2: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A2 @ C ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_826_add_Oleft__commute,axiom,
! [B: int,A2: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A2 @ C ) )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_827_add_Oleft__commute,axiom,
! [B: extended_enat,A2: extended_enat,C: extended_enat] :
( ( plus_p3455044024723400733d_enat @ B @ ( plus_p3455044024723400733d_enat @ A2 @ C ) )
= ( plus_p3455044024723400733d_enat @ A2 @ ( plus_p3455044024723400733d_enat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_828_add__left__imp__eq,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_829_add__left__imp__eq,axiom,
! [A2: int,B: int,C: int] :
( ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ A2 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_830_add__right__imp__eq,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_831_add__right__imp__eq,axiom,
! [B: int,A2: int,C: int] :
( ( ( plus_plus_int @ B @ A2 )
= ( plus_plus_int @ C @ A2 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_832_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_833_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A2 @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_834_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: complex,C: complex,B: complex] :
( ( minus_minus_complex @ ( minus_minus_complex @ A2 @ C ) @ B )
= ( minus_minus_complex @ ( minus_minus_complex @ A2 @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_835_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A2 @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A2 @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_836_diff__eq__diff__eq,axiom,
! [A2: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A2 @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( A2 = B )
= ( C = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_837_diff__eq__diff__eq,axiom,
! [A2: complex,B: complex,C: complex,D2: complex] :
( ( ( minus_minus_complex @ A2 @ B )
= ( minus_minus_complex @ C @ D2 ) )
=> ( ( A2 = B )
= ( C = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_838_diff__eq__diff__eq,axiom,
! [A2: real,B: real,C: real,D2: real] :
( ( ( minus_minus_real @ A2 @ B )
= ( minus_minus_real @ C @ D2 ) )
=> ( ( A2 = B )
= ( C = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_839_add__less__imp__less__right,axiom,
! [A2: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_real @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_840_add__less__imp__less__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_841_add__less__imp__less__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_842_add__less__imp__less__left,axiom,
! [C: real,A2: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_real @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_843_add__less__imp__less__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_844_add__less__imp__less__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_845_add__strict__right__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_846_add__strict__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_847_add__strict__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_848_add__strict__left__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_849_add__strict__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_850_add__strict__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_851_add__strict__mono,axiom,
! [A2: extended_enat,B: extended_enat,C: extended_enat,D2: extended_enat] :
( ( ord_le72135733267957522d_enat @ A2 @ B )
=> ( ( ord_le72135733267957522d_enat @ C @ D2 )
=> ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A2 @ C ) @ ( plus_p3455044024723400733d_enat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_852_add__strict__mono,axiom,
! [A2: real,B: real,C: real,D2: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ C @ D2 )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_853_add__strict__mono,axiom,
! [A2: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_854_add__strict__mono,axiom,
! [A2: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_855_add__mono__thms__linordered__field_I1_J,axiom,
! [I3: real,J3: real,K: real,L: real] :
( ( ( ord_less_real @ I3 @ J3 )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_856_add__mono__thms__linordered__field_I1_J,axiom,
! [I3: nat,J3: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J3 )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_857_add__mono__thms__linordered__field_I1_J,axiom,
! [I3: int,J3: int,K: int,L: int] :
( ( ( ord_less_int @ I3 @ J3 )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_858_add__mono__thms__linordered__field_I2_J,axiom,
! [I3: real,J3: real,K: real,L: real] :
( ( ( I3 = J3 )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_859_add__mono__thms__linordered__field_I2_J,axiom,
! [I3: nat,J3: nat,K: nat,L: nat] :
( ( ( I3 = J3 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_860_add__mono__thms__linordered__field_I2_J,axiom,
! [I3: int,J3: int,K: int,L: int] :
( ( ( I3 = J3 )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_861_add__mono__thms__linordered__field_I5_J,axiom,
! [I3: real,J3: real,K: real,L: real] :
( ( ( ord_less_real @ I3 @ J3 )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_862_add__mono__thms__linordered__field_I5_J,axiom,
! [I3: nat,J3: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J3 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_863_add__mono__thms__linordered__field_I5_J,axiom,
! [I3: int,J3: int,K: int,L: int] :
( ( ( ord_less_int @ I3 @ J3 )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_864_diff__strict__right__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_865_diff__strict__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_866_diff__strict__left__mono,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_real @ B @ A2 )
=> ( ord_less_real @ ( minus_minus_real @ C @ A2 ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_867_diff__strict__left__mono,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ord_less_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_868_diff__eq__diff__less,axiom,
! [A2: real,B: real,C: real,D2: real] :
( ( ( minus_minus_real @ A2 @ B )
= ( minus_minus_real @ C @ D2 ) )
=> ( ( ord_less_real @ A2 @ B )
= ( ord_less_real @ C @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_869_diff__eq__diff__less,axiom,
! [A2: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A2 @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( ord_less_int @ A2 @ B )
= ( ord_less_int @ C @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_870_diff__strict__mono,axiom,
! [A2: real,B: real,D2: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ D2 @ C )
=> ( ord_less_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_871_diff__strict__mono,axiom,
! [A2: int,B: int,D2: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ D2 @ C )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_872_group__cancel_Osub1,axiom,
! [A: int,K: int,A2: int,B: int] :
( ( A
= ( plus_plus_int @ K @ A2 ) )
=> ( ( minus_minus_int @ A @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A2 @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_873_group__cancel_Osub1,axiom,
! [A: complex,K: complex,A2: complex,B: complex] :
( ( A
= ( plus_plus_complex @ K @ A2 ) )
=> ( ( minus_minus_complex @ A @ B )
= ( plus_plus_complex @ K @ ( minus_minus_complex @ A2 @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_874_group__cancel_Osub1,axiom,
! [A: real,K: real,A2: real,B: real] :
( ( A
= ( plus_plus_real @ K @ A2 ) )
=> ( ( minus_minus_real @ A @ B )
= ( plus_plus_real @ K @ ( minus_minus_real @ A2 @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_875_diff__eq__eq,axiom,
! [A2: int,B: int,C: int] :
( ( ( minus_minus_int @ A2 @ B )
= C )
= ( A2
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_876_diff__eq__eq,axiom,
! [A2: complex,B: complex,C: complex] :
( ( ( minus_minus_complex @ A2 @ B )
= C )
= ( A2
= ( plus_plus_complex @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_877_diff__eq__eq,axiom,
! [A2: real,B: real,C: real] :
( ( ( minus_minus_real @ A2 @ B )
= C )
= ( A2
= ( plus_plus_real @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_878_eq__diff__eq,axiom,
! [A2: int,C: int,B: int] :
( ( A2
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A2 @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_879_eq__diff__eq,axiom,
! [A2: complex,C: complex,B: complex] :
( ( A2
= ( minus_minus_complex @ C @ B ) )
= ( ( plus_plus_complex @ A2 @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_880_eq__diff__eq,axiom,
! [A2: real,C: real,B: real] :
( ( A2
= ( minus_minus_real @ C @ B ) )
= ( ( plus_plus_real @ A2 @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_881_add__diff__eq,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ A2 @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_882_add__diff__eq,axiom,
! [A2: complex,B: complex,C: complex] :
( ( plus_plus_complex @ A2 @ ( minus_minus_complex @ B @ C ) )
= ( minus_minus_complex @ ( plus_plus_complex @ A2 @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_883_add__diff__eq,axiom,
! [A2: real,B: real,C: real] :
( ( plus_plus_real @ A2 @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A2 @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_884_diff__diff__eq2,axiom,
! [A2: int,B: int,C: int] :
( ( minus_minus_int @ A2 @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_885_diff__diff__eq2,axiom,
! [A2: complex,B: complex,C: complex] :
( ( minus_minus_complex @ A2 @ ( minus_minus_complex @ B @ C ) )
= ( minus_minus_complex @ ( plus_plus_complex @ A2 @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_886_diff__diff__eq2,axiom,
! [A2: real,B: real,C: real] :
( ( minus_minus_real @ A2 @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A2 @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_887_diff__add__eq,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A2 @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_888_diff__add__eq,axiom,
! [A2: complex,B: complex,C: complex] :
( ( plus_plus_complex @ ( minus_minus_complex @ A2 @ B ) @ C )
= ( minus_minus_complex @ ( plus_plus_complex @ A2 @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_889_diff__add__eq,axiom,
! [A2: real,B: real,C: real] :
( ( plus_plus_real @ ( minus_minus_real @ A2 @ B ) @ C )
= ( minus_minus_real @ ( plus_plus_real @ A2 @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_890_diff__add__eq__diff__diff__swap,axiom,
! [A2: int,B: int,C: int] :
( ( minus_minus_int @ A2 @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_891_diff__add__eq__diff__diff__swap,axiom,
! [A2: complex,B: complex,C: complex] :
( ( minus_minus_complex @ A2 @ ( plus_plus_complex @ B @ C ) )
= ( minus_minus_complex @ ( minus_minus_complex @ A2 @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_892_diff__add__eq__diff__diff__swap,axiom,
! [A2: real,B: real,C: real] :
( ( minus_minus_real @ A2 @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ ( minus_minus_real @ A2 @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_893_add__implies__diff,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ( plus_plus_nat @ C @ B )
= A2 )
=> ( C
= ( minus_minus_nat @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_894_add__implies__diff,axiom,
! [C: int,B: int,A2: int] :
( ( ( plus_plus_int @ C @ B )
= A2 )
=> ( C
= ( minus_minus_int @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_895_add__implies__diff,axiom,
! [C: complex,B: complex,A2: complex] :
( ( ( plus_plus_complex @ C @ B )
= A2 )
=> ( C
= ( minus_minus_complex @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_896_add__implies__diff,axiom,
! [C: real,B: real,A2: real] :
( ( ( plus_plus_real @ C @ B )
= A2 )
=> ( C
= ( minus_minus_real @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_897_diff__diff__eq,axiom,
! [A2: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C )
= ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_898_diff__diff__eq,axiom,
! [A2: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A2 @ B ) @ C )
= ( minus_minus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_899_diff__diff__eq,axiom,
! [A2: complex,B: complex,C: complex] :
( ( minus_minus_complex @ ( minus_minus_complex @ A2 @ B ) @ C )
= ( minus_minus_complex @ A2 @ ( plus_plus_complex @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_900_diff__diff__eq,axiom,
! [A2: real,B: real,C: real] :
( ( minus_minus_real @ ( minus_minus_real @ A2 @ B ) @ C )
= ( minus_minus_real @ A2 @ ( plus_plus_real @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_901_less__diff__eq,axiom,
! [A2: real,C: real,B: real] :
( ( ord_less_real @ A2 @ ( minus_minus_real @ C @ B ) )
= ( ord_less_real @ ( plus_plus_real @ A2 @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_902_less__diff__eq,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_int @ A2 @ ( minus_minus_int @ C @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_903_diff__less__eq,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ ( minus_minus_real @ A2 @ B ) @ C )
= ( ord_less_real @ A2 @ ( plus_plus_real @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_904_diff__less__eq,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A2 @ B ) @ C )
= ( ord_less_int @ A2 @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_905_less__add__iff2,axiom,
! [A2: real,E: real,C: real,B: real,D2: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A2 @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D2 ) )
= ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A2 ) @ E ) @ D2 ) ) ) ).
% less_add_iff2
thf(fact_906_less__add__iff2,axiom,
! [A2: int,E: int,C: int,B: int,D2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
= ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A2 ) @ E ) @ D2 ) ) ) ).
% less_add_iff2
thf(fact_907_less__add__iff1,axiom,
! [A2: real,E: real,C: real,B: real,D2: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A2 @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D2 ) )
= ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A2 @ B ) @ E ) @ C ) @ D2 ) ) ).
% less_add_iff1
thf(fact_908_less__add__iff1,axiom,
! [A2: int,E: int,C: int,B: int,D2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
= ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ E ) @ C ) @ D2 ) ) ).
% less_add_iff1
thf(fact_909_mult__diff__mult,axiom,
! [X: complex,Y: complex,A2: complex,B: complex] :
( ( minus_minus_complex @ ( times_times_complex @ X @ Y ) @ ( times_times_complex @ A2 @ B ) )
= ( plus_plus_complex @ ( times_times_complex @ X @ ( minus_minus_complex @ Y @ B ) ) @ ( times_times_complex @ ( minus_minus_complex @ X @ A2 ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_910_mult__diff__mult,axiom,
! [X: real,Y: real,A2: real,B: real] :
( ( minus_minus_real @ ( times_times_real @ X @ Y ) @ ( times_times_real @ A2 @ B ) )
= ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A2 ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_911_mult__diff__mult,axiom,
! [X: int,Y: int,A2: int,B: int] :
( ( minus_minus_int @ ( times_times_int @ X @ Y ) @ ( times_times_int @ A2 @ B ) )
= ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A2 ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_912_square__diff__square__factored,axiom,
! [X: complex,Y: complex] :
( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ ( times_times_complex @ Y @ Y ) )
= ( times_times_complex @ ( plus_plus_complex @ X @ Y ) @ ( minus_minus_complex @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_913_square__diff__square__factored,axiom,
! [X: real,Y: real] :
( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= ( times_times_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_914_square__diff__square__factored,axiom,
! [X: int,Y: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_915_eq__add__iff2,axiom,
! [A2: complex,E: complex,C: complex,B: complex,D2: complex] :
( ( ( plus_plus_complex @ ( times_times_complex @ A2 @ E ) @ C )
= ( plus_plus_complex @ ( times_times_complex @ B @ E ) @ D2 ) )
= ( C
= ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ B @ A2 ) @ E ) @ D2 ) ) ) ).
% eq_add_iff2
thf(fact_916_eq__add__iff2,axiom,
! [A2: real,E: real,C: real,B: real,D2: real] :
( ( ( plus_plus_real @ ( times_times_real @ A2 @ E ) @ C )
= ( plus_plus_real @ ( times_times_real @ B @ E ) @ D2 ) )
= ( C
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A2 ) @ E ) @ D2 ) ) ) ).
% eq_add_iff2
thf(fact_917_eq__add__iff2,axiom,
! [A2: int,E: int,C: int,B: int,D2: int] :
( ( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A2 ) @ E ) @ D2 ) ) ) ).
% eq_add_iff2
thf(fact_918_eq__add__iff1,axiom,
! [A2: complex,E: complex,C: complex,B: complex,D2: complex] :
( ( ( plus_plus_complex @ ( times_times_complex @ A2 @ E ) @ C )
= ( plus_plus_complex @ ( times_times_complex @ B @ E ) @ D2 ) )
= ( ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ A2 @ B ) @ E ) @ C )
= D2 ) ) ).
% eq_add_iff1
thf(fact_919_eq__add__iff1,axiom,
! [A2: real,E: real,C: real,B: real,D2: real] :
( ( ( plus_plus_real @ ( times_times_real @ A2 @ E ) @ C )
= ( plus_plus_real @ ( times_times_real @ B @ E ) @ D2 ) )
= ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A2 @ B ) @ E ) @ C )
= D2 ) ) ).
% eq_add_iff1
thf(fact_920_eq__add__iff1,axiom,
! [A2: int,E: int,C: int,B: int,D2: int] :
( ( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ E ) @ C )
= D2 ) ) ).
% eq_add_iff1
thf(fact_921_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_922_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_923_boolean__algebra__cancel_Osup2,axiom,
! [B4: set_Product_prod_a_a,K: set_Product_prod_a_a,B: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( B4
= ( sup_su3048258781599657691od_a_a @ K @ B ) )
=> ( ( sup_su3048258781599657691od_a_a @ A2 @ B4 )
= ( sup_su3048258781599657691od_a_a @ K @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_924_boolean__algebra__cancel_Osup1,axiom,
! [A: set_Product_prod_a_a,K: set_Product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( A
= ( sup_su3048258781599657691od_a_a @ K @ A2 ) )
=> ( ( sup_su3048258781599657691od_a_a @ A @ B )
= ( sup_su3048258781599657691od_a_a @ K @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_925_ring__class_Oring__distribs_I2_J,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A2 @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_926_ring__class_Oring__distribs_I2_J,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_927_ring__class_Oring__distribs_I1_J,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ A2 @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A2 @ B ) @ ( times_times_real @ A2 @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_928_ring__class_Oring__distribs_I1_J,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ A2 @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_929_comm__semiring__class_Odistrib,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_930_comm__semiring__class_Odistrib,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A2 @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_931_comm__semiring__class_Odistrib,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_932_comm__semiring__class_Odistrib,axiom,
! [A2: extended_enat,B: extended_enat,C: extended_enat] :
( ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A2 @ B ) @ C )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A2 @ C ) @ ( times_7803423173614009249d_enat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_933_distrib__left,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ A2 @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_nat @ A2 @ C ) ) ) ).
% distrib_left
thf(fact_934_distrib__left,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ A2 @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A2 @ B ) @ ( times_times_real @ A2 @ C ) ) ) ).
% distrib_left
thf(fact_935_distrib__left,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ A2 @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% distrib_left
thf(fact_936_distrib__left,axiom,
! [A2: extended_enat,B: extended_enat,C: extended_enat] :
( ( times_7803423173614009249d_enat @ A2 @ ( plus_p3455044024723400733d_enat @ B @ C ) )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A2 @ B ) @ ( times_7803423173614009249d_enat @ A2 @ C ) ) ) ).
% distrib_left
thf(fact_937_distrib__right,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_938_distrib__right,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A2 @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% distrib_right
thf(fact_939_distrib__right,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_940_distrib__right,axiom,
! [A2: extended_enat,B: extended_enat,C: extended_enat] :
( ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A2 @ B ) @ C )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A2 @ C ) @ ( times_7803423173614009249d_enat @ B @ C ) ) ) ).
% distrib_right
thf(fact_941_combine__common__factor,axiom,
! [A2: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A2 @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_942_combine__common__factor,axiom,
! [A2: real,E: real,B: real,C: real] :
( ( plus_plus_real @ ( times_times_real @ A2 @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A2 @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_943_combine__common__factor,axiom,
! [A2: int,E: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_944_combine__common__factor,axiom,
! [A2: extended_enat,E: extended_enat,B: extended_enat,C: extended_enat] :
( ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A2 @ E ) @ ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ B @ E ) @ C ) )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A2 @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_945_left__diff__distrib,axiom,
! [A2: complex,B: complex,C: complex] :
( ( times_times_complex @ ( minus_minus_complex @ A2 @ B ) @ C )
= ( minus_minus_complex @ ( times_times_complex @ A2 @ C ) @ ( times_times_complex @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_946_left__diff__distrib,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ ( minus_minus_real @ A2 @ B ) @ C )
= ( minus_minus_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_947_left__diff__distrib,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_948_right__diff__distrib,axiom,
! [A2: complex,B: complex,C: complex] :
( ( times_times_complex @ A2 @ ( minus_minus_complex @ B @ C ) )
= ( minus_minus_complex @ ( times_times_complex @ A2 @ B ) @ ( times_times_complex @ A2 @ C ) ) ) ).
% right_diff_distrib
thf(fact_949_right__diff__distrib,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ A2 @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A2 @ B ) @ ( times_times_real @ A2 @ C ) ) ) ).
% right_diff_distrib
thf(fact_950_right__diff__distrib,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ A2 @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% right_diff_distrib
thf(fact_951_left__diff__distrib_H,axiom,
! [B: complex,C: complex,A2: complex] :
( ( times_times_complex @ ( minus_minus_complex @ B @ C ) @ A2 )
= ( minus_minus_complex @ ( times_times_complex @ B @ A2 ) @ ( times_times_complex @ C @ A2 ) ) ) ).
% left_diff_distrib'
thf(fact_952_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A2 )
= ( minus_minus_nat @ ( times_times_nat @ B @ A2 ) @ ( times_times_nat @ C @ A2 ) ) ) ).
% left_diff_distrib'
thf(fact_953_left__diff__distrib_H,axiom,
! [B: real,C: real,A2: real] :
( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A2 )
= ( minus_minus_real @ ( times_times_real @ B @ A2 ) @ ( times_times_real @ C @ A2 ) ) ) ).
% left_diff_distrib'
thf(fact_954_left__diff__distrib_H,axiom,
! [B: int,C: int,A2: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A2 )
= ( minus_minus_int @ ( times_times_int @ B @ A2 ) @ ( times_times_int @ C @ A2 ) ) ) ).
% left_diff_distrib'
thf(fact_955_right__diff__distrib_H,axiom,
! [A2: complex,B: complex,C: complex] :
( ( times_times_complex @ A2 @ ( minus_minus_complex @ B @ C ) )
= ( minus_minus_complex @ ( times_times_complex @ A2 @ B ) @ ( times_times_complex @ A2 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_956_right__diff__distrib_H,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ A2 @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_nat @ A2 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_957_right__diff__distrib_H,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ A2 @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A2 @ B ) @ ( times_times_real @ A2 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_958_right__diff__distrib_H,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ A2 @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_959_add__diff__add,axiom,
! [A2: int,C: int,B: int,D2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D2 ) )
= ( plus_plus_int @ ( minus_minus_int @ A2 @ B ) @ ( minus_minus_int @ C @ D2 ) ) ) ).
% add_diff_add
thf(fact_960_add__diff__add,axiom,
! [A2: complex,C: complex,B: complex,D2: complex] :
( ( minus_minus_complex @ ( plus_plus_complex @ A2 @ C ) @ ( plus_plus_complex @ B @ D2 ) )
= ( plus_plus_complex @ ( minus_minus_complex @ A2 @ B ) @ ( minus_minus_complex @ C @ D2 ) ) ) ).
% add_diff_add
thf(fact_961_add__diff__add,axiom,
! [A2: real,C: real,B: real,D2: real] :
( ( minus_minus_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B @ D2 ) )
= ( plus_plus_real @ ( minus_minus_real @ A2 @ B ) @ ( minus_minus_real @ C @ D2 ) ) ) ).
% add_diff_add
thf(fact_962_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A2: real,B: real] :
( ~ ( ord_less_real @ A2 @ B )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A2 @ B ) )
= A2 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_963_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A2: nat,B: nat] :
( ~ ( ord_less_nat @ A2 @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A2 @ B ) )
= A2 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_964_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A2: int,B: int] :
( ~ ( ord_less_int @ A2 @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A2 @ B ) )
= A2 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_965_sup__Un__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( sup_sup_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ R )
@ ^ [X2: nat] : ( member_nat @ X2 @ S ) )
= ( ^ [X2: nat] : ( member_nat @ X2 @ ( sup_sup_set_nat @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_966_sup__Un__eq,axiom,
! [R: set_a,S: set_a] :
( ( sup_sup_a_o
@ ^ [X2: a] : ( member_a @ X2 @ R )
@ ^ [X2: a] : ( member_a @ X2 @ S ) )
= ( ^ [X2: a] : ( member_a @ X2 @ ( sup_sup_set_a @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_967_sup__Un__eq,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( sup_su1640154476453168578_a_a_o
@ ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ R )
@ ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ S ) )
= ( ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ ( sup_su3048258781599657691od_a_a @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_968__092_060open_0622_A_K_Acard_A_123_Ix_M_Ay_J_O_A_Ix_M_Ay_J_A_092_060in_062_AM_A_092_060times_062_AM_A_092_060and_062_Ax_A_060_Ay_125_A_061_Acard_A_123_Ix_M_Ay_J_O_A_Ix_M_Ay_J_A_092_060in_062_AM_A_092_060times_062_AM_A_092_060and_062_Ax_A_060_Ay_125_A_L_Acard_A_I_I_092_060lambda_062x_O_A_Isnd_Ax_M_Afst_Ax_J_J_A_096_A_123_Ix_M_Ay_J_O_A_Ix_M_Ay_J_A_092_060in_062_AM_A_092_060times_062_AM_A_092_060and_062_Ax_A_060_Ay_125_J_092_060close_062,axiom,
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
@ ( finite4795055649997197647od_a_a
@ ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X2: a,Y2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 )
@ ( product_Sigma_a_a @ m
@ ^ [Uu: a] : m ) )
& ( ord_less_a @ X2 @ Y2 ) ) ) ) ) )
= ( plus_plus_nat
@ ( finite4795055649997197647od_a_a
@ ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X2: a,Y2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 )
@ ( product_Sigma_a_a @ m
@ ^ [Uu: a] : m ) )
& ( ord_less_a @ X2 @ Y2 ) ) ) ) )
@ ( finite4795055649997197647od_a_a
@ ( image_4636654165204879301od_a_a
@ ^ [X2: product_prod_a_a] : ( product_Pair_a_a @ ( product_snd_a_a @ X2 ) @ ( product_fst_a_a @ X2 ) )
@ ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X2: a,Y2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 )
@ ( product_Sigma_a_a @ m
@ ^ [Uu: a] : m ) )
& ( ord_less_a @ X2 @ Y2 ) ) ) ) ) ) ) ) ).
% \<open>2 * card {(x, y). (x, y) \<in> M \<times> M \<and> x < y} = card {(x, y). (x, y) \<in> M \<times> M \<and> x < y} + card ((\<lambda>x. (snd x, fst x)) ` {(x, y). (x, y) \<in> M \<times> M \<and> x < y})\<close>
thf(fact_969_sup__Un__eq2,axiom,
! [R: set_Pr8218934625190621173um_num,S: set_Pr8218934625190621173um_num] :
( ( sup_sup_num_num_o
@ ^ [X2: num,Y2: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X2 @ Y2 ) @ R )
@ ^ [X2: num,Y2: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X2 @ Y2 ) @ S ) )
= ( ^ [X2: num,Y2: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X2 @ Y2 ) @ ( sup_su4061117120043295689um_num @ R @ S ) ) ) ) ).
% sup_Un_eq2
thf(fact_970_sup__Un__eq2,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( sup_sup_a_a_o
@ ^ [X2: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ R )
@ ^ [X2: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ S ) )
= ( ^ [X2: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ ( sup_su3048258781599657691od_a_a @ R @ S ) ) ) ) ).
% sup_Un_eq2
thf(fact_971_even__diff__nat,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% even_diff_nat
thf(fact_972_unbounded__k__infinite,axiom,
! [K: nat,S: set_nat] :
( ! [M5: nat] :
( ( ord_less_nat @ K @ M5 )
=> ? [N6: nat] :
( ( ord_less_nat @ M5 @ N6 )
& ( member_nat @ N6 @ S ) ) )
=> ~ ( finite_finite_nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_973_infinite__nat__iff__unbounded,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M3: nat] :
? [N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ( member_nat @ N3 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_974_dvd__add__triv__left__iff,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( dvd_dvd_nat @ A2 @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_975_dvd__add__triv__left__iff,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ A2 @ B ) )
= ( dvd_dvd_int @ A2 @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_976_dvd__add__triv__right__iff,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( dvd_dvd_nat @ A2 @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_977_dvd__add__triv__right__iff,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B @ A2 ) )
= ( dvd_dvd_int @ A2 @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_978_dvd__add__times__triv__left__iff,axiom,
! [A2: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ ( times_times_nat @ C @ A2 ) @ B ) )
= ( dvd_dvd_nat @ A2 @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_979_dvd__add__times__triv__left__iff,axiom,
! [A2: real,C: real,B: real] :
( ( dvd_dvd_real @ A2 @ ( plus_plus_real @ ( times_times_real @ C @ A2 ) @ B ) )
= ( dvd_dvd_real @ A2 @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_980_dvd__add__times__triv__left__iff,axiom,
! [A2: int,C: int,B: int] :
( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ ( times_times_int @ C @ A2 ) @ B ) )
= ( dvd_dvd_int @ A2 @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_981_dvd__add__times__triv__right__iff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A2 ) ) )
= ( dvd_dvd_nat @ A2 @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_982_dvd__add__times__triv__right__iff,axiom,
! [A2: real,B: real,C: real] :
( ( dvd_dvd_real @ A2 @ ( plus_plus_real @ B @ ( times_times_real @ C @ A2 ) ) )
= ( dvd_dvd_real @ A2 @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_983_dvd__add__times__triv__right__iff,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B @ ( times_times_int @ C @ A2 ) ) )
= ( dvd_dvd_int @ A2 @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_984_prod_Ocollapse,axiom,
! [Prod: product_prod_num_num] :
( ( product_Pair_num_num @ ( product_fst_num_num @ Prod ) @ ( product_snd_num_num @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_985_prod_Ocollapse,axiom,
! [Prod: product_prod_a_a] :
( ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_986__092_060open_062card_A_123_Ix_M_Ay_J_O_A_Ix_M_Ay_J_A_092_060in_062_AM_A_092_060times_062_AM_A_092_060and_062_Ax_A_060_Ay_125_A_L_Acard_A_I_I_092_060lambda_062x_O_A_Isnd_Ax_M_Afst_Ax_J_J_A_096_A_123_Ix_M_Ay_J_O_A_Ix_M_Ay_J_A_092_060in_062_AM_A_092_060times_062_AM_A_092_060and_062_Ax_A_060_Ay_125_J_A_061_Acard_A_123_Ix_M_Ay_J_O_A_Ix_M_Ay_J_A_092_060in_062_AM_A_092_060times_062_AM_A_092_060and_062_Ax_A_060_Ay_125_A_L_Acard_A_123_Ix_M_Ay_J_O_A_Ix_M_Ay_J_A_092_060in_062_AM_A_092_060times_062_AM_A_092_060and_062_Ay_A_060_Ax_125_092_060close_062,axiom,
( ( plus_plus_nat
@ ( finite4795055649997197647od_a_a
@ ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X2: a,Y2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 )
@ ( product_Sigma_a_a @ m
@ ^ [Uu: a] : m ) )
& ( ord_less_a @ X2 @ Y2 ) ) ) ) )
@ ( finite4795055649997197647od_a_a
@ ( image_4636654165204879301od_a_a
@ ^ [X2: product_prod_a_a] : ( product_Pair_a_a @ ( product_snd_a_a @ X2 ) @ ( product_fst_a_a @ X2 ) )
@ ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X2: a,Y2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 )
@ ( product_Sigma_a_a @ m
@ ^ [Uu: a] : m ) )
& ( ord_less_a @ X2 @ Y2 ) ) ) ) ) ) )
= ( plus_plus_nat
@ ( finite4795055649997197647od_a_a
@ ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X2: a,Y2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 )
@ ( product_Sigma_a_a @ m
@ ^ [Uu: a] : m ) )
& ( ord_less_a @ X2 @ Y2 ) ) ) ) )
@ ( finite4795055649997197647od_a_a
@ ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X2: a,Y2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 )
@ ( product_Sigma_a_a @ m
@ ^ [Uu: a] : m ) )
& ( ord_less_a @ Y2 @ X2 ) ) ) ) ) ) ) ).
% \<open>card {(x, y). (x, y) \<in> M \<times> M \<and> x < y} + card ((\<lambda>x. (snd x, fst x)) ` {(x, y). (x, y) \<in> M \<times> M \<and> x < y}) = card {(x, y). (x, y) \<in> M \<times> M \<and> x < y} + card {(x, y). (x, y) \<in> M \<times> M \<and> y < x}\<close>
thf(fact_987_even__mult__iff,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A2 @ B ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_mult_iff
thf(fact_988_even__mult__iff,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A2 @ B ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
| ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_mult_iff
thf(fact_989_even__add,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A2 @ B ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_add
thf(fact_990_even__add,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A2 @ B ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_add
thf(fact_991_odd__add,axiom,
! [A2: nat,B: nat] :
( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A2 @ B ) ) )
= ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) )
!= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% odd_add
thf(fact_992_odd__add,axiom,
! [A2: int,B: int] :
( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A2 @ B ) ) )
= ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
!= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% odd_add
thf(fact_993_even__diff,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A2 @ B ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A2 @ B ) ) ) ).
% even_diff
thf(fact_994_prod_Osplit__sel__asm,axiom,
! [P: $o > $o,F: a > a > $o,Prod: product_prod_a_a] :
( ( P @ ( produc6436628058953941356_a_a_o @ F @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) ) )
& ~ ( P @ ( F @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_995_prod_Osplit__sel,axiom,
! [P: $o > $o,F: a > a > $o,Prod: product_prod_a_a] :
( ( P @ ( produc6436628058953941356_a_a_o @ F @ Prod ) )
= ( ( Prod
= ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) ) )
=> ( P @ ( F @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_996_case__prod__beta,axiom,
( produc6436628058953941356_a_a_o
= ( ^ [F3: a > a > $o,P3: product_prod_a_a] : ( F3 @ ( product_fst_a_a @ P3 ) @ ( product_snd_a_a @ P3 ) ) ) ) ).
% case_prod_beta
thf(fact_997_split__beta,axiom,
( produc6436628058953941356_a_a_o
= ( ^ [F3: a > a > $o,Prod2: product_prod_a_a] : ( F3 @ ( product_fst_a_a @ Prod2 ) @ ( product_snd_a_a @ Prod2 ) ) ) ) ).
% split_beta
thf(fact_998_Product__Type_OCollect__case__prodD,axiom,
! [X: product_prod_a_a,A: a > a > $o] :
( ( member1426531477525435216od_a_a @ X @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ A ) ) )
=> ( A @ ( product_fst_a_a @ X ) @ ( product_snd_a_a @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_999_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_1000_case__prod__unfold,axiom,
( produc6436628058953941356_a_a_o
= ( ^ [C3: a > a > $o,P3: product_prod_a_a] : ( C3 @ ( product_fst_a_a @ P3 ) @ ( product_snd_a_a @ P3 ) ) ) ) ).
% case_prod_unfold
thf(fact_1001_case__prod__beta_H,axiom,
( produc6436628058953941356_a_a_o
= ( ^ [F3: a > a > $o,X2: product_prod_a_a] : ( F3 @ ( product_fst_a_a @ X2 ) @ ( product_snd_a_a @ X2 ) ) ) ) ).
% case_prod_beta'
thf(fact_1002_mem__Times__iff,axiom,
! [X: produc3498347346309940967od_a_a,A: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( member6330455413206600464od_a_a @ X
@ ( produc5899993699339346696od_a_a @ A
@ ^ [Uu: product_prod_a_a] : B4 ) )
= ( ( member1426531477525435216od_a_a @ ( produc6349652284990284099od_a_a @ X ) @ A )
& ( member1426531477525435216od_a_a @ ( produc988467003200404613od_a_a @ X ) @ B4 ) ) ) ).
% mem_Times_iff
thf(fact_1003_mem__Times__iff,axiom,
! [X: produc6073246360284067238_a_nat,A: set_Product_prod_a_a,B4: set_nat] :
( ( member1268613965720311869_a_nat @ X
@ ( produc1049071135499013807_a_nat @ A
@ ^ [Uu: product_prod_a_a] : B4 ) )
= ( ( member1426531477525435216od_a_a @ ( produc5384542433974473908_a_nat @ X ) @ A )
& ( member_nat @ ( produc5937524592981195506_a_nat @ X ) @ B4 ) ) ) ).
% mem_Times_iff
thf(fact_1004_mem__Times__iff,axiom,
! [X: produc3802892049952890430_a_a_a,A: set_Product_prod_a_a,B4: set_a] :
( ( member2829916517802586983_a_a_a @ X
@ ( produc2379640491490746847_a_a_a @ A
@ ^ [Uu: product_prod_a_a] : B4 ) )
= ( ( member1426531477525435216od_a_a @ ( produc2553620009333216026_a_a_a @ X ) @ A )
& ( member_a @ ( produc7473117258747215964_a_a_a @ X ) @ B4 ) ) ) ).
% mem_Times_iff
thf(fact_1005_mem__Times__iff,axiom,
! [X: produc2100646355360341836od_a_a,A: set_nat,B4: set_Product_prod_a_a] :
( ( member6519385997651362275od_a_a @ X
@ ( produc1591714161673420045od_a_a @ A
@ ^ [Uu: nat] : B4 ) )
= ( ( member_nat @ ( produc5927185460148880146od_a_a @ X ) @ A )
& ( member1426531477525435216od_a_a @ ( produc6480167619155601744od_a_a @ X ) @ B4 ) ) ) ).
% mem_Times_iff
thf(fact_1006_mem__Times__iff,axiom,
! [X: product_prod_nat_nat,A: set_nat,B4: set_nat] :
( ( member8440522571783428010at_nat @ X
@ ( produc457027306803732586at_nat @ A
@ ^ [Uu: nat] : B4 ) )
= ( ( member_nat @ ( product_fst_nat_nat @ X ) @ A )
& ( member_nat @ ( product_snd_nat_nat @ X ) @ B4 ) ) ) ).
% mem_Times_iff
thf(fact_1007_mem__Times__iff,axiom,
! [X: product_prod_nat_a,A: set_nat,B4: set_a] :
( ( member8962352052110095674_nat_a @ X
@ ( product_Sigma_nat_a @ A
@ ^ [Uu: nat] : B4 ) )
= ( ( member_nat @ ( product_fst_nat_a @ X ) @ A )
& ( member_a @ ( product_snd_nat_a @ X ) @ B4 ) ) ) ).
% mem_Times_iff
thf(fact_1008_mem__Times__iff,axiom,
! [X: produc4044097585999906000od_a_a,A: set_a,B4: set_Product_prod_a_a] :
( ( member3071122053849602553od_a_a @ X
@ ( produc6342321021181284593od_a_a @ A
@ ^ [Uu: a] : B4 ) )
= ( ( member_a @ ( produc6516300539023753772od_a_a @ X ) @ A )
& ( member1426531477525435216od_a_a @ ( produc2212425751582977902od_a_a @ X ) @ B4 ) ) ) ).
% mem_Times_iff
thf(fact_1009_mem__Times__iff,axiom,
! [X: product_prod_a_nat,A: set_a,B4: set_nat] :
( ( member5724188588386418708_a_nat @ X
@ ( product_Sigma_a_nat @ A
@ ^ [Uu: a] : B4 ) )
= ( ( member_a @ ( product_fst_a_nat @ X ) @ A )
& ( member_nat @ ( product_snd_a_nat @ X ) @ B4 ) ) ) ).
% mem_Times_iff
thf(fact_1010_mem__Times__iff,axiom,
! [X: product_prod_a_a,A: set_a,B4: set_a] :
( ( member1426531477525435216od_a_a @ X
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : B4 ) )
= ( ( member_a @ ( product_fst_a_a @ X ) @ A )
& ( member_a @ ( product_snd_a_a @ X ) @ B4 ) ) ) ).
% mem_Times_iff
thf(fact_1011_surjective__pairing,axiom,
! [T3: product_prod_num_num] :
( T3
= ( product_Pair_num_num @ ( product_fst_num_num @ T3 ) @ ( product_snd_num_num @ T3 ) ) ) ).
% surjective_pairing
thf(fact_1012_surjective__pairing,axiom,
! [T3: product_prod_a_a] :
( T3
= ( product_Pair_a_a @ ( product_fst_a_a @ T3 ) @ ( product_snd_a_a @ T3 ) ) ) ).
% surjective_pairing
thf(fact_1013_prod_Oexhaust__sel,axiom,
! [Prod: product_prod_num_num] :
( Prod
= ( product_Pair_num_num @ ( product_fst_num_num @ Prod ) @ ( product_snd_num_num @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_1014_prod_Oexhaust__sel,axiom,
! [Prod: product_prod_a_a] :
( Prod
= ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_1015_prod__eq__iff,axiom,
( ( ^ [Y5: product_prod_a_a,Z3: product_prod_a_a] : ( Y5 = Z3 ) )
= ( ^ [S4: product_prod_a_a,T4: product_prod_a_a] :
( ( ( product_fst_a_a @ S4 )
= ( product_fst_a_a @ T4 ) )
& ( ( product_snd_a_a @ S4 )
= ( product_snd_a_a @ T4 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_1016_prod__eqI,axiom,
! [P2: product_prod_a_a,Q3: product_prod_a_a] :
( ( ( product_fst_a_a @ P2 )
= ( product_fst_a_a @ Q3 ) )
=> ( ( ( product_snd_a_a @ P2 )
= ( product_snd_a_a @ Q3 ) )
=> ( P2 = Q3 ) ) ) ).
% prod_eqI
thf(fact_1017_prod_Oexpand,axiom,
! [Prod: product_prod_a_a,Prod3: product_prod_a_a] :
( ( ( ( product_fst_a_a @ Prod )
= ( product_fst_a_a @ Prod3 ) )
& ( ( product_snd_a_a @ Prod )
= ( product_snd_a_a @ Prod3 ) ) )
=> ( Prod = Prod3 ) ) ).
% prod.expand
thf(fact_1018_snd__conv,axiom,
! [X1: num,X22: num] :
( ( product_snd_num_num @ ( product_Pair_num_num @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_1019_snd__conv,axiom,
! [X1: a,X22: a] :
( ( product_snd_a_a @ ( product_Pair_a_a @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_1020_snd__eqD,axiom,
! [X: num,Y: num,A2: num] :
( ( ( product_snd_num_num @ ( product_Pair_num_num @ X @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_1021_snd__eqD,axiom,
! [X: a,Y: a,A2: a] :
( ( ( product_snd_a_a @ ( product_Pair_a_a @ X @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_1022_fst__conv,axiom,
! [X1: num,X22: num] :
( ( product_fst_num_num @ ( product_Pair_num_num @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_1023_fst__conv,axiom,
! [X1: a,X22: a] :
( ( product_fst_a_a @ ( product_Pair_a_a @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_1024_fst__eqD,axiom,
! [X: num,Y: num,A2: num] :
( ( ( product_fst_num_num @ ( product_Pair_num_num @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_1025_fst__eqD,axiom,
! [X: a,Y: a,A2: a] :
( ( ( product_fst_a_a @ ( product_Pair_a_a @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_1026_dvdE,axiom,
! [B: nat,A2: nat] :
( ( dvd_dvd_nat @ B @ A2 )
=> ~ ! [K3: nat] :
( A2
!= ( times_times_nat @ B @ K3 ) ) ) ).
% dvdE
thf(fact_1027_dvdE,axiom,
! [B: real,A2: real] :
( ( dvd_dvd_real @ B @ A2 )
=> ~ ! [K3: real] :
( A2
!= ( times_times_real @ B @ K3 ) ) ) ).
% dvdE
thf(fact_1028_dvdE,axiom,
! [B: int,A2: int] :
( ( dvd_dvd_int @ B @ A2 )
=> ~ ! [K3: int] :
( A2
!= ( times_times_int @ B @ K3 ) ) ) ).
% dvdE
thf(fact_1029_dvdE,axiom,
! [B: extended_enat,A2: extended_enat] :
( ( dvd_dv3785147216227455552d_enat @ B @ A2 )
=> ~ ! [K3: extended_enat] :
( A2
!= ( times_7803423173614009249d_enat @ B @ K3 ) ) ) ).
% dvdE
thf(fact_1030_dvdI,axiom,
! [A2: nat,B: nat,K: nat] :
( ( A2
= ( times_times_nat @ B @ K ) )
=> ( dvd_dvd_nat @ B @ A2 ) ) ).
% dvdI
thf(fact_1031_dvdI,axiom,
! [A2: real,B: real,K: real] :
( ( A2
= ( times_times_real @ B @ K ) )
=> ( dvd_dvd_real @ B @ A2 ) ) ).
% dvdI
thf(fact_1032_dvdI,axiom,
! [A2: int,B: int,K: int] :
( ( A2
= ( times_times_int @ B @ K ) )
=> ( dvd_dvd_int @ B @ A2 ) ) ).
% dvdI
thf(fact_1033_dvdI,axiom,
! [A2: extended_enat,B: extended_enat,K: extended_enat] :
( ( A2
= ( times_7803423173614009249d_enat @ B @ K ) )
=> ( dvd_dv3785147216227455552d_enat @ B @ A2 ) ) ).
% dvdI
thf(fact_1034_dvd__def,axiom,
( dvd_dvd_nat
= ( ^ [B3: nat,A4: nat] :
? [K2: nat] :
( A4
= ( times_times_nat @ B3 @ K2 ) ) ) ) ).
% dvd_def
thf(fact_1035_dvd__def,axiom,
( dvd_dvd_real
= ( ^ [B3: real,A4: real] :
? [K2: real] :
( A4
= ( times_times_real @ B3 @ K2 ) ) ) ) ).
% dvd_def
thf(fact_1036_dvd__def,axiom,
( dvd_dvd_int
= ( ^ [B3: int,A4: int] :
? [K2: int] :
( A4
= ( times_times_int @ B3 @ K2 ) ) ) ) ).
% dvd_def
thf(fact_1037_dvd__def,axiom,
( dvd_dv3785147216227455552d_enat
= ( ^ [B3: extended_enat,A4: extended_enat] :
? [K2: extended_enat] :
( A4
= ( times_7803423173614009249d_enat @ B3 @ K2 ) ) ) ) ).
% dvd_def
thf(fact_1038_dvd__mult,axiom,
! [A2: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ C )
=> ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% dvd_mult
thf(fact_1039_dvd__mult,axiom,
! [A2: real,C: real,B: real] :
( ( dvd_dvd_real @ A2 @ C )
=> ( dvd_dvd_real @ A2 @ ( times_times_real @ B @ C ) ) ) ).
% dvd_mult
thf(fact_1040_dvd__mult,axiom,
! [A2: int,C: int,B: int] :
( ( dvd_dvd_int @ A2 @ C )
=> ( dvd_dvd_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% dvd_mult
thf(fact_1041_dvd__mult,axiom,
! [A2: extended_enat,C: extended_enat,B: extended_enat] :
( ( dvd_dv3785147216227455552d_enat @ A2 @ C )
=> ( dvd_dv3785147216227455552d_enat @ A2 @ ( times_7803423173614009249d_enat @ B @ C ) ) ) ).
% dvd_mult
thf(fact_1042_dvd__mult2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B )
=> ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_1043_dvd__mult2,axiom,
! [A2: real,B: real,C: real] :
( ( dvd_dvd_real @ A2 @ B )
=> ( dvd_dvd_real @ A2 @ ( times_times_real @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_1044_dvd__mult2,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ A2 @ B )
=> ( dvd_dvd_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_1045_dvd__mult2,axiom,
! [A2: extended_enat,B: extended_enat,C: extended_enat] :
( ( dvd_dv3785147216227455552d_enat @ A2 @ B )
=> ( dvd_dv3785147216227455552d_enat @ A2 @ ( times_7803423173614009249d_enat @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_1046_dvd__mult__left,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ C )
=> ( dvd_dvd_nat @ A2 @ C ) ) ).
% dvd_mult_left
thf(fact_1047_dvd__mult__left,axiom,
! [A2: real,B: real,C: real] :
( ( dvd_dvd_real @ ( times_times_real @ A2 @ B ) @ C )
=> ( dvd_dvd_real @ A2 @ C ) ) ).
% dvd_mult_left
thf(fact_1048_dvd__mult__left,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ C )
=> ( dvd_dvd_int @ A2 @ C ) ) ).
% dvd_mult_left
thf(fact_1049_dvd__mult__left,axiom,
! [A2: extended_enat,B: extended_enat,C: extended_enat] :
( ( dvd_dv3785147216227455552d_enat @ ( times_7803423173614009249d_enat @ A2 @ B ) @ C )
=> ( dvd_dv3785147216227455552d_enat @ A2 @ C ) ) ).
% dvd_mult_left
thf(fact_1050_dvd__triv__left,axiom,
! [A2: nat,B: nat] : ( dvd_dvd_nat @ A2 @ ( times_times_nat @ A2 @ B ) ) ).
% dvd_triv_left
thf(fact_1051_dvd__triv__left,axiom,
! [A2: real,B: real] : ( dvd_dvd_real @ A2 @ ( times_times_real @ A2 @ B ) ) ).
% dvd_triv_left
thf(fact_1052_dvd__triv__left,axiom,
! [A2: int,B: int] : ( dvd_dvd_int @ A2 @ ( times_times_int @ A2 @ B ) ) ).
% dvd_triv_left
thf(fact_1053_dvd__triv__left,axiom,
! [A2: extended_enat,B: extended_enat] : ( dvd_dv3785147216227455552d_enat @ A2 @ ( times_7803423173614009249d_enat @ A2 @ B ) ) ).
% dvd_triv_left
thf(fact_1054_mult__dvd__mono,axiom,
! [A2: nat,B: nat,C: nat,D2: nat] :
( ( dvd_dvd_nat @ A2 @ B )
=> ( ( dvd_dvd_nat @ C @ D2 )
=> ( dvd_dvd_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ).
% mult_dvd_mono
thf(fact_1055_mult__dvd__mono,axiom,
! [A2: real,B: real,C: real,D2: real] :
( ( dvd_dvd_real @ A2 @ B )
=> ( ( dvd_dvd_real @ C @ D2 )
=> ( dvd_dvd_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ D2 ) ) ) ) ).
% mult_dvd_mono
thf(fact_1056_mult__dvd__mono,axiom,
! [A2: int,B: int,C: int,D2: int] :
( ( dvd_dvd_int @ A2 @ B )
=> ( ( dvd_dvd_int @ C @ D2 )
=> ( dvd_dvd_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ).
% mult_dvd_mono
thf(fact_1057_mult__dvd__mono,axiom,
! [A2: extended_enat,B: extended_enat,C: extended_enat,D2: extended_enat] :
( ( dvd_dv3785147216227455552d_enat @ A2 @ B )
=> ( ( dvd_dv3785147216227455552d_enat @ C @ D2 )
=> ( dvd_dv3785147216227455552d_enat @ ( times_7803423173614009249d_enat @ A2 @ C ) @ ( times_7803423173614009249d_enat @ B @ D2 ) ) ) ) ).
% mult_dvd_mono
thf(fact_1058_dvd__mult__right,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ C )
=> ( dvd_dvd_nat @ B @ C ) ) ).
% dvd_mult_right
thf(fact_1059_dvd__mult__right,axiom,
! [A2: real,B: real,C: real] :
( ( dvd_dvd_real @ ( times_times_real @ A2 @ B ) @ C )
=> ( dvd_dvd_real @ B @ C ) ) ).
% dvd_mult_right
thf(fact_1060_dvd__mult__right,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ C )
=> ( dvd_dvd_int @ B @ C ) ) ).
% dvd_mult_right
thf(fact_1061_dvd__mult__right,axiom,
! [A2: extended_enat,B: extended_enat,C: extended_enat] :
( ( dvd_dv3785147216227455552d_enat @ ( times_7803423173614009249d_enat @ A2 @ B ) @ C )
=> ( dvd_dv3785147216227455552d_enat @ B @ C ) ) ).
% dvd_mult_right
thf(fact_1062_dvd__triv__right,axiom,
! [A2: nat,B: nat] : ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B @ A2 ) ) ).
% dvd_triv_right
thf(fact_1063_dvd__triv__right,axiom,
! [A2: real,B: real] : ( dvd_dvd_real @ A2 @ ( times_times_real @ B @ A2 ) ) ).
% dvd_triv_right
thf(fact_1064_dvd__triv__right,axiom,
! [A2: int,B: int] : ( dvd_dvd_int @ A2 @ ( times_times_int @ B @ A2 ) ) ).
% dvd_triv_right
thf(fact_1065_dvd__triv__right,axiom,
! [A2: extended_enat,B: extended_enat] : ( dvd_dv3785147216227455552d_enat @ A2 @ ( times_7803423173614009249d_enat @ B @ A2 ) ) ).
% dvd_triv_right
thf(fact_1066_dvd__add,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B )
=> ( ( dvd_dvd_nat @ A2 @ C )
=> ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_1067_dvd__add,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ A2 @ B )
=> ( ( dvd_dvd_int @ A2 @ C )
=> ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_1068_dvd__add,axiom,
! [A2: extended_enat,B: extended_enat,C: extended_enat] :
( ( dvd_dv3785147216227455552d_enat @ A2 @ B )
=> ( ( dvd_dv3785147216227455552d_enat @ A2 @ C )
=> ( dvd_dv3785147216227455552d_enat @ A2 @ ( plus_p3455044024723400733d_enat @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_1069_dvd__add__left__iff,axiom,
! [A2: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ C )
=> ( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B @ C ) )
= ( dvd_dvd_nat @ A2 @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_1070_dvd__add__left__iff,axiom,
! [A2: int,C: int,B: int] :
( ( dvd_dvd_int @ A2 @ C )
=> ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B @ C ) )
= ( dvd_dvd_int @ A2 @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_1071_dvd__add__right__iff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B )
=> ( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B @ C ) )
= ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_1072_dvd__add__right__iff,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ A2 @ B )
=> ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B @ C ) )
= ( dvd_dvd_int @ A2 @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_1073_dvd__diff,axiom,
! [X: int,Y: int,Z2: int] :
( ( dvd_dvd_int @ X @ Y )
=> ( ( dvd_dvd_int @ X @ Z2 )
=> ( dvd_dvd_int @ X @ ( minus_minus_int @ Y @ Z2 ) ) ) ) ).
% dvd_diff
thf(fact_1074_dvd__diff,axiom,
! [X: complex,Y: complex,Z2: complex] :
( ( dvd_dvd_complex @ X @ Y )
=> ( ( dvd_dvd_complex @ X @ Z2 )
=> ( dvd_dvd_complex @ X @ ( minus_minus_complex @ Y @ Z2 ) ) ) ) ).
% dvd_diff
thf(fact_1075_dvd__diff,axiom,
! [X: real,Y: real,Z2: real] :
( ( dvd_dvd_real @ X @ Y )
=> ( ( dvd_dvd_real @ X @ Z2 )
=> ( dvd_dvd_real @ X @ ( minus_minus_real @ Y @ Z2 ) ) ) ) ).
% dvd_diff
thf(fact_1076_dvd__power__same,axiom,
! [X: nat,Y: nat,N: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) ) ) ).
% dvd_power_same
thf(fact_1077_dvd__power__same,axiom,
! [X: real,Y: real,N: nat] :
( ( dvd_dvd_real @ X @ Y )
=> ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) ) ) ).
% dvd_power_same
thf(fact_1078_dvd__power__same,axiom,
! [X: complex,Y: complex,N: nat] :
( ( dvd_dvd_complex @ X @ Y )
=> ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) ) ) ).
% dvd_power_same
thf(fact_1079_dvd__power__same,axiom,
! [X: int,Y: int,N: nat] :
( ( dvd_dvd_int @ X @ Y )
=> ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) ) ) ).
% dvd_power_same
thf(fact_1080_dvd__diff__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ M )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_1081_snd__def,axiom,
( product_snd_a_a
= ( produc8815886927560695506_a_a_a
@ ^ [X12: a,X23: a] : X23 ) ) ).
% snd_def
thf(fact_1082_fst__def,axiom,
( product_fst_a_a
= ( produc8815886927560695506_a_a_a
@ ^ [X12: a,X23: a] : X12 ) ) ).
% fst_def
thf(fact_1083_strict__subset__divisors__dvd,axiom,
! [A2: complex,B: complex] :
( ( ord_less_set_complex
@ ( collect_complex
@ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ A2 ) )
@ ( collect_complex
@ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ B ) ) )
= ( ( dvd_dvd_complex @ A2 @ B )
& ~ ( dvd_dvd_complex @ B @ A2 ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_1084_strict__subset__divisors__dvd,axiom,
! [A2: nat,B: nat] :
( ( ord_less_set_nat
@ ( collect_nat
@ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A2 ) )
@ ( collect_nat
@ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B ) ) )
= ( ( dvd_dvd_nat @ A2 @ B )
& ~ ( dvd_dvd_nat @ B @ A2 ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_1085_strict__subset__divisors__dvd,axiom,
! [A2: int,B: int] :
( ( ord_less_set_int
@ ( collect_int
@ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A2 ) )
@ ( collect_int
@ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B ) ) )
= ( ( dvd_dvd_int @ A2 @ B )
& ~ ( dvd_dvd_int @ B @ A2 ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_1086_dvd__minus__self,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
= ( ( ord_less_nat @ N @ M )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_minus_self
thf(fact_1087_even__diff__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% even_diff_iff
thf(fact_1088_bezout1__nat,axiom,
! [A2: nat,B: nat] :
? [D3: nat,X3: nat,Y4: nat] :
( ( dvd_dvd_nat @ D3 @ A2 )
& ( dvd_dvd_nat @ D3 @ B )
& ( ( ( minus_minus_nat @ ( times_times_nat @ A2 @ X3 ) @ ( times_times_nat @ B @ Y4 ) )
= D3 )
| ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A2 @ Y4 ) )
= D3 ) ) ) ).
% bezout1_nat
thf(fact_1089_bezout__lemma__nat,axiom,
! [D2: nat,A2: nat,B: nat,X: nat,Y: nat] :
( ( dvd_dvd_nat @ D2 @ A2 )
=> ( ( dvd_dvd_nat @ D2 @ B )
=> ( ( ( ( times_times_nat @ A2 @ X )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D2 ) )
| ( ( times_times_nat @ B @ X )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ Y ) @ D2 ) ) )
=> ? [X3: nat,Y4: nat] :
( ( dvd_dvd_nat @ D2 @ A2 )
& ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ A2 @ B ) )
& ( ( ( times_times_nat @ A2 @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ Y4 ) @ D2 ) )
| ( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ Y4 ) @ D2 ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_1090_bezout__add__nat,axiom,
! [A2: nat,B: nat] :
? [D3: nat,X3: nat,Y4: nat] :
( ( dvd_dvd_nat @ D3 @ A2 )
& ( dvd_dvd_nat @ D3 @ B )
& ( ( ( times_times_nat @ A2 @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D3 ) )
| ( ( times_times_nat @ B @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ Y4 ) @ D3 ) ) ) ) ).
% bezout_add_nat
thf(fact_1091_zdvd__period,axiom,
! [A2: int,D2: int,X: int,T3: int,C: int] :
( ( dvd_dvd_int @ A2 @ D2 )
=> ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ X @ T3 ) )
= ( dvd_dvd_int @ A2 @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D2 ) ) @ T3 ) ) ) ) ).
% zdvd_period
thf(fact_1092_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1093_i0__less,axiom,
! [N: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
= ( N != zero_z5237406670263579293d_enat ) ) ).
% i0_less
thf(fact_1094_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1095_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1096_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_1097_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1098_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_1099_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1100_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1101_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1102_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1103_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1104_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1105_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_1106_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1107_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1108_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1109_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_1110_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1111_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_1112_zdvd__not__zless,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ord_less_int @ M @ N )
=> ~ ( dvd_dvd_int @ N @ M ) ) ) ).
% zdvd_not_zless
thf(fact_1113_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1114_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1115_zmult__zless__mono2,axiom,
! [I3: int,J3: int,K: int] :
( ( ord_less_int @ I3 @ J3 )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I3 ) @ ( times_times_int @ K @ J3 ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1116_realpow__pos__nth__unique,axiom,
! [N: nat,A2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ? [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
& ( ( power_power_real @ X3 @ N )
= A2 )
& ! [Y6: real] :
( ( ( ord_less_real @ zero_zero_real @ Y6 )
& ( ( power_power_real @ Y6 @ N )
= A2 ) )
=> ( Y6 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1117_realpow__pos__nth,axiom,
! [N: nat,A2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ( ( power_power_real @ R3 @ N )
= A2 ) ) ) ) ).
% realpow_pos_nth
thf(fact_1118_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
=> ( ~ ( P @ N5 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N5 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1119_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1120_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1121_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1122_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1123_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1124_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1125_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_1126_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1127_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( P @ A5 @ B5 )
= ( P @ B5 @ A5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
=> ( ! [A5: nat,B5: nat] :
( ( P @ A5 @ B5 )
=> ( P @ A5 @ ( plus_plus_nat @ A5 @ B5 ) ) )
=> ( P @ A2 @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1128_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1129_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1130_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1131_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_1132_minusinfinity,axiom,
! [D2: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X3: int,K3: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D2 ) ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_1133_plusinfinity,axiom,
! [D2: int,P4: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X3: int,K3: int] :
( ( P4 @ X3 )
= ( P4 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D2 ) ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [X_12: int] : ( P4 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_1134_enat__0__less__mult__iff,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
= ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
& ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).
% enat_0_less_mult_iff
thf(fact_1135_not__iless0,axiom,
! [N: extended_enat] :
~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% not_iless0
thf(fact_1136_finite__divisors__int,axiom,
! [I3: int] :
( ( I3 != zero_zero_int )
=> ( finite_finite_int
@ ( collect_int
@ ^ [D4: int] : ( dvd_dvd_int @ D4 @ I3 ) ) ) ) ).
% finite_divisors_int
thf(fact_1137_less__imp__add__positive,axiom,
! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I3 @ K3 )
= J3 ) ) ) ).
% less_imp_add_positive
thf(fact_1138_mult__less__mono1,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J3 @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1139_mult__less__mono2,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I3 ) @ ( times_times_nat @ K @ J3 ) ) ) ) ).
% mult_less_mono2
thf(fact_1140_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1141_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1142_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_1143_nat__dvd__not__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_1144_dvd__pos__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ M @ N )
=> ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% dvd_pos_nat
thf(fact_1145_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_1146_nat__power__less__imp__less,axiom,
! [I3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I3 )
=> ( ( ord_less_nat @ ( power_power_nat @ I3 @ M ) @ ( power_power_nat @ I3 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_1147_nat__diff__split,axiom,
! [P: nat > $o,A2: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B ) )
= ( ( ( ord_less_nat @ A2 @ B )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A2
= ( plus_plus_nat @ B @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1148_nat__diff__split__asm,axiom,
! [P: nat > $o,A2: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B ) )
= ( ~ ( ( ( ord_less_nat @ A2 @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A2
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1149_nat__mult__dvd__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_1150_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_1151_bezout__add__strong__nat,axiom,
! [A2: nat,B: nat] :
( ( A2 != zero_zero_nat )
=> ? [D3: nat,X3: nat,Y4: nat] :
( ( dvd_dvd_nat @ D3 @ A2 )
& ( dvd_dvd_nat @ D3 @ B )
& ( ( times_times_nat @ A2 @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D3 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_1152_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_1153_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1154_finite__divisors__nat,axiom,
! [M: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ M ) ) ) ) ).
% finite_divisors_nat
thf(fact_1155_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% pos2
thf(fact_1156_odd__pos,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% odd_pos
thf(fact_1157_zdvd__reduce,axiom,
! [K: int,N: int,M: int] :
( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
= ( dvd_dvd_int @ K @ N ) ) ).
% zdvd_reduce
thf(fact_1158_finite__nth__roots,axiom,
! [N: nat,C: complex] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Z: complex] :
( ( power_power_complex @ Z @ N )
= C ) ) ) ) ).
% finite_nth_roots
thf(fact_1159_idiff__0,axiom,
! [N: extended_enat] :
( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
= zero_z5237406670263579293d_enat ) ).
% idiff_0
thf(fact_1160_idiff__0__right,axiom,
! [N: extended_enat] :
( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
= N ) ).
% idiff_0_right
thf(fact_1161_finite__interval__int4,axiom,
! [A2: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( ord_less_int @ A2 @ I2 )
& ( ord_less_int @ I2 @ B ) ) ) ) ).
% finite_interval_int4
thf(fact_1162_iadd__is__0,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( plus_p3455044024723400733d_enat @ M @ N )
= zero_z5237406670263579293d_enat )
= ( ( M = zero_z5237406670263579293d_enat )
& ( N = zero_z5237406670263579293d_enat ) ) ) ).
% iadd_is_0
thf(fact_1163_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1164_imult__is__0,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( times_7803423173614009249d_enat @ M @ N )
= zero_z5237406670263579293d_enat )
= ( ( M = zero_z5237406670263579293d_enat )
| ( N = zero_z5237406670263579293d_enat ) ) ) ).
% imult_is_0
thf(fact_1165_card__nth__roots,axiom,
! [C: complex,N: nat] :
( ( C != zero_zero_complex )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [Z: complex] :
( ( power_power_complex @ Z @ N )
= C ) ) )
= N ) ) ) ).
% card_nth_roots
thf(fact_1166_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_1167_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_1168_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1169_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1170_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1171_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y32: num] :
( ( ( bit1 @ X32 )
= ( bit1 @ Y32 ) )
= ( X32 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_1172_semiring__norm_I90_J,axiom,
! [M: num,N: num] :
( ( ( bit1 @ M )
= ( bit1 @ N ) )
= ( M = N ) ) ).
% semiring_norm(90)
thf(fact_1173_semiring__norm_I73_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(73)
thf(fact_1174_diff__diff__cancel,axiom,
! [I3: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I3 ) )
= I3 ) ) ).
% diff_diff_cancel
thf(fact_1175_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_1176_finite__interval__int1,axiom,
! [A2: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( ord_less_eq_int @ A2 @ I2 )
& ( ord_less_eq_int @ I2 @ B ) ) ) ) ).
% finite_interval_int1
thf(fact_1177_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_1178_semiring__norm_I88_J,axiom,
! [M: num,N: num] :
( ( bit0 @ M )
!= ( bit1 @ N ) ) ).
% semiring_norm(88)
thf(fact_1179_semiring__norm_I89_J,axiom,
! [M: num,N: num] :
( ( bit1 @ M )
!= ( bit0 @ N ) ) ).
% semiring_norm(89)
thf(fact_1180_semiring__norm_I72_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(72)
thf(fact_1181_semiring__norm_I84_J,axiom,
! [N: num] :
( one
!= ( bit1 @ N ) ) ).
% semiring_norm(84)
thf(fact_1182_semiring__norm_I86_J,axiom,
! [M: num] :
( ( bit1 @ M )
!= one ) ).
% semiring_norm(86)
thf(fact_1183_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% semiring_norm(70)
thf(fact_1184_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_1185_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_1186_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J3: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J3 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J3 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1187_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J3: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J3 @ K ) @ I3 )
= ( minus_minus_nat @ ( plus_plus_nat @ J3 @ I3 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1188_Nat_Odiff__diff__right,axiom,
! [K: nat,J3: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( minus_minus_nat @ I3 @ ( minus_minus_nat @ J3 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K ) @ J3 ) ) ) ).
% Nat.diff_diff_right
thf(fact_1189_semiring__norm_I80_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(80)
thf(fact_1190_enat__ord__number_I1_J,axiom,
! [M: num,N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(1)
thf(fact_1191_finite__interval__int3,axiom,
! [A2: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( ord_less_int @ A2 @ I2 )
& ( ord_less_eq_int @ I2 @ B ) ) ) ) ).
% finite_interval_int3
thf(fact_1192_finite__interval__int2,axiom,
! [A2: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( ord_less_eq_int @ A2 @ I2 )
& ( ord_less_int @ I2 @ B ) ) ) ) ).
% finite_interval_int2
thf(fact_1193_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1194_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1195_semiring__norm_I7_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(7)
thf(fact_1196_semiring__norm_I9_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(9)
thf(fact_1197_semiring__norm_I14_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).
% semiring_norm(14)
thf(fact_1198_semiring__norm_I15_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).
% semiring_norm(15)
thf(fact_1199_semiring__norm_I74_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(74)
thf(fact_1200_semiring__norm_I79_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(79)
thf(fact_1201_semiring__norm_I81_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(81)
thf(fact_1202_semiring__norm_I77_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% semiring_norm(77)
thf(fact_1203_semiring__norm_I10_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).
% semiring_norm(10)
thf(fact_1204_semiring__norm_I8_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ one )
= ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% semiring_norm(8)
thf(fact_1205_semiring__norm_I5_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ one )
= ( bit1 @ M ) ) ).
% semiring_norm(5)
thf(fact_1206_semiring__norm_I4_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% semiring_norm(4)
thf(fact_1207_semiring__norm_I3_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit0 @ N ) )
= ( bit1 @ N ) ) ).
% semiring_norm(3)
thf(fact_1208_semiring__norm_I16_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).
% semiring_norm(16)
thf(fact_1209_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1210_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_1211_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_1212_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1213_i0__lb,axiom,
! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).
% i0_lb
thf(fact_1214_ile0__eq,axiom,
! [N: extended_enat] :
( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
= ( N = zero_z5237406670263579293d_enat ) ) ).
% ile0_eq
thf(fact_1215_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1216_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1217_mult__le__mono,axiom,
! [I3: nat,J3: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J3 @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1218_mult__le__mono1,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J3 @ K ) ) ) ).
% mult_le_mono1
thf(fact_1219_mult__le__mono2,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I3 ) @ ( times_times_nat @ K @ J3 ) ) ) ).
% mult_le_mono2
thf(fact_1220_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J3: nat] :
( ! [I4: nat,J4: nat] :
( ( ord_less_nat @ I4 @ J4 )
=> ( ord_less_nat @ ( F @ I4 ) @ ( F @ J4 ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J3 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1221_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_1222_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_1223_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_1224_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_1225_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_1226_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% less_eq_real_def
thf(fact_1227_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M3: nat] :
? [N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( member_nat @ N3 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_1228_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M3: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N4 )
=> ( ord_less_eq_nat @ X2 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1229_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N5: nat] : ( ord_less_eq_nat @ N5 @ ( F @ N5 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_1230_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_1231_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_1232_verit__eq__simplify_I14_J,axiom,
! [X22: num,X32: num] :
( ( bit0 @ X22 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_1233_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1234_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1235_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1236_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_1237_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1238_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1239_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_1240_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
? [K2: nat] :
( N3
= ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1241_trans__le__add2,axiom,
! [I3: nat,J3: nat,M: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ M @ J3 ) ) ) ).
% trans_le_add2
thf(fact_1242_trans__le__add1,axiom,
! [I3: nat,J3: nat,M: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ J3 @ M ) ) ) ).
% trans_le_add1
thf(fact_1243_add__le__mono1,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J3 @ K ) ) ) ).
% add_le_mono1
thf(fact_1244_add__le__mono,axiom,
! [I3: nat,J3: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ) ).
% add_le_mono
thf(fact_1245_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N5: nat] :
( L
= ( plus_plus_nat @ K @ N5 ) ) ) ).
% le_Suc_ex
thf(fact_1246_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1247_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1248_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1249_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1250_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1251_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one )
=> ( ! [X24: num] :
( Y
!= ( bit0 @ X24 ) )
=> ~ ! [X33: num] :
( Y
!= ( bit1 @ X33 ) ) ) ) ).
% num.exhaust
thf(fact_1252_xor__num_Ocases,axiom,
! [X: product_prod_num_num] :
( ( X
!= ( product_Pair_num_num @ one @ one ) )
=> ( ! [N5: num] :
( X
!= ( product_Pair_num_num @ one @ ( bit0 @ N5 ) ) )
=> ( ! [N5: num] :
( X
!= ( product_Pair_num_num @ one @ ( bit1 @ N5 ) ) )
=> ( ! [M5: num] :
( X
!= ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) )
=> ( ! [M5: num,N5: num] :
( X
!= ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N5 ) ) )
=> ( ! [M5: num,N5: num] :
( X
!= ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N5 ) ) )
=> ( ! [M5: num] :
( X
!= ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) )
=> ( ! [M5: num,N5: num] :
( X
!= ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N5 ) ) )
=> ~ ! [M5: num,N5: num] :
( X
!= ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N5 ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.cases
thf(fact_1253_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K3 )
=> ~ ( P @ I5 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1254_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M5: nat,N5: nat] :
( ( ord_less_nat @ M5 @ N5 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N5 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1255_diff__less__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1256_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1257_le__diff__conv,axiom,
! [J3: nat,K: nat,I3: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J3 @ K ) @ I3 )
= ( ord_less_eq_nat @ J3 @ ( plus_plus_nat @ I3 @ K ) ) ) ).
% le_diff_conv
thf(fact_1258_Nat_Ole__diff__conv2,axiom,
! [K: nat,J3: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ J3 @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ J3 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1259_Nat_Odiff__add__assoc,axiom,
! [K: nat,J3: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J3 ) @ K )
= ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J3 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1260_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J3: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J3 @ I3 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J3 @ K ) @ I3 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1261_Nat_Ole__imp__diff__is__add,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ( minus_minus_nat @ J3 @ I3 )
= K )
= ( J3
= ( plus_plus_nat @ K @ I3 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1262_less__eq__dvd__minus,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( dvd_dvd_nat @ M @ N )
= ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_1263_dvd__diffD1,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ M )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% dvd_diffD1
thf(fact_1264_dvd__diffD,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% dvd_diffD
thf(fact_1265_add__diff__assoc__enat,axiom,
! [Z2: extended_enat,Y: extended_enat,X: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Z2 @ Y )
=> ( ( plus_p3455044024723400733d_enat @ X @ ( minus_3235023915231533773d_enat @ Y @ Z2 ) )
= ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ Z2 ) ) ) ).
% add_diff_assoc_enat
% Conjectures (1)
thf(conj_0,conjecture,
( ( minus_minus_nat @ ( power_power_nat @ ( finite_card_a @ m ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
@ ( finite4795055649997197647od_a_a
@ ( image_7400625782589995694od_a_a
@ ^ [X2: a] : ( product_Pair_a_a @ X2 @ X2 )
@ m ) ) )
= ( minus_minus_nat @ ( power_power_nat @ ( finite_card_a @ m ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( finite_card_a @ m ) ) ) ).
%------------------------------------------------------------------------------