TPTP Problem File: SLH0730^1.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 : Eval_FO/0005_Ailamazyan/prob_05021_217162__16354752_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1863 ( 583 unt; 583 typ; 0 def)
% Number of atoms : 3647 (1460 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 11947 ( 380 ~; 35 |; 257 &;9866 @)
% ( 0 <=>;1409 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 75 ( 74 usr)
% Number of type conns : 2585 (2585 >; 0 *; 0 +; 0 <<)
% Number of symbols : 512 ( 509 usr; 37 con; 0-5 aty)
% Number of variables : 4228 ( 602 ^;3553 !; 73 ?;4228 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:13:22.143
%------------------------------------------------------------------------------
% Could-be-implicit typings (74)
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thf(sy_v__092_060phi_062,type,
phi: fo_fmla_a_b ).
thf(sy_v_i,type,
i2: nat ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_t,type,
t: produc8208987855206359447_a_nat ).
% Relevant facts (1276)
thf(fact_0_t__def_I2_J,axiom,
( ad
= ( act_edom_a_b @ phi @ i ) ) ).
% t_def(2)
thf(fact_1_t__def_I3_J,axiom,
( ad
= ( act_edom_a_b @ ( fo_Exists_a_b @ i2 @ phi ) @ i ) ) ).
% t_def(3)
thf(fact_2_X__def,axiom,
( x
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ ad )
@ ( proj_f6037546931041448027_a_nat @ phi
@ ( collec5629555741568564177_a_nat
@ ^ [Sigma: nat > sum_sum_a_nat] : ( esat_a_b @ phi @ i @ Sigma @ top_to795618464972521135_a_nat ) ) ) ) ) ).
% X_def
thf(fact_3_calculation,axiom,
( x
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ ad )
@ ( proj_f6037546931041448027_a_nat @ ( fo_Exists_a_b @ i2 @ phi )
@ ( collec5629555741568564177_a_nat
@ ^ [Sigma: nat > sum_sum_a_nat] : ( esat_a_b @ phi @ i @ Sigma @ top_to795618464972521135_a_nat ) ) ) ) ) ).
% calculation
thf(fact_4_local_Owf,axiom,
fo_wf_a_b @ phi @ i @ t ).
% local.wf
thf(fact_5_fv__eq_I1_J,axiom,
( ( fv_fo_fmla_list_a_b @ ( fo_Exists_a_b @ i2 @ phi ) )
= ( fv_fo_fmla_list_a_b @ phi ) ) ).
% fv_eq(1)
thf(fact_6_image__ident,axiom,
! [Y: set_o] :
( ( image_o_o2
@ ^ [X: $o] : X
@ Y )
= Y ) ).
% image_ident
thf(fact_7_image__ident,axiom,
! [Y: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat
@ ^ [X: list_Sum_sum_a_nat] : X
@ Y )
= Y ) ).
% image_ident
thf(fact_8_fo__fmla_Oinject_I7_J,axiom,
! [X71: nat,X72: fo_fmla_a_b,Y71: nat,Y72: fo_fmla_a_b] :
( ( ( fo_Exists_a_b @ X71 @ X72 )
= ( fo_Exists_a_b @ Y71 @ Y72 ) )
= ( ( X71 = Y71 )
& ( X72 = Y72 ) ) ) ).
% fo_fmla.inject(7)
thf(fact_9_image__eqI,axiom,
! [B: list_Sum_sum_a_nat,F: $o > list_Sum_sum_a_nat,X2: $o,A: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A )
=> ( member408289922725080238_a_nat @ B @ ( image_733135354807394034_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_10_image__eqI,axiom,
! [B: sum_sum_a_nat,F: $o > sum_sum_a_nat,X2: $o,A: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A )
=> ( member_Sum_sum_a_nat @ B @ ( image_4139480514073730540_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_11_image__eqI,axiom,
! [B: $o,F: list_Sum_sum_a_nat > $o,X2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member408289922725080238_a_nat @ X2 @ A )
=> ( member_o @ B @ ( image_3984946558445957976_nat_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_12_image__eqI,axiom,
! [B: sum_sum_a_nat,F: list_Sum_sum_a_nat > sum_sum_a_nat,X2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member408289922725080238_a_nat @ X2 @ A )
=> ( member_Sum_sum_a_nat @ B @ ( image_6705138127771971519_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_13_image__eqI,axiom,
! [B: $o,F: sum_sum_a_nat > $o,X2: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_Sum_sum_a_nat @ X2 @ A )
=> ( member_o @ B @ ( image_6095136190293192542_nat_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_14_image__eqI,axiom,
! [B: list_Sum_sum_a_nat,F: sum_sum_a_nat > list_Sum_sum_a_nat,X2: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_Sum_sum_a_nat @ X2 @ A )
=> ( member408289922725080238_a_nat @ B @ ( image_7077900747146408011_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_15_image__eqI,axiom,
! [B: sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,X2: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_Sum_sum_a_nat @ X2 @ A )
=> ( member_Sum_sum_a_nat @ B @ ( image_7142520692256960453_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_16_image__eqI,axiom,
! [B: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,X2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member408289922725080238_a_nat @ X2 @ A )
=> ( member408289922725080238_a_nat @ B @ ( image_5081948215111134021_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_17_image__eqI,axiom,
! [B: $o,F: $o > $o,X2: $o,A: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A )
=> ( member_o @ B @ ( image_o_o2 @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_18_UNIV__I,axiom,
! [X2: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X2 @ top_to6433055325616222389_a_nat ) ).
% UNIV_I
thf(fact_19_UNIV__I,axiom,
! [X2: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X2 @ top_to795618464972521135_a_nat ) ).
% UNIV_I
thf(fact_20_UNIV__I,axiom,
! [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ).
% UNIV_I
thf(fact_21_iso__tuple__UNIV__I,axiom,
! [X2: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X2 @ top_to6433055325616222389_a_nat ) ).
% iso_tuple_UNIV_I
thf(fact_22_iso__tuple__UNIV__I,axiom,
! [X2: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X2 @ top_to795618464972521135_a_nat ) ).
% iso_tuple_UNIV_I
thf(fact_23_iso__tuple__UNIV__I,axiom,
! [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ).
% iso_tuple_UNIV_I
thf(fact_24_rangeE,axiom,
! [B: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ B @ ( image_5081948215111134021_a_nat @ F @ top_to6433055325616222389_a_nat ) )
=> ~ ! [X3: list_Sum_sum_a_nat] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_25_rangeE,axiom,
! [B: $o,F: sum_sum_a_nat > $o] :
( ( member_o @ B @ ( image_6095136190293192542_nat_o @ F @ top_to795618464972521135_a_nat ) )
=> ~ ! [X3: sum_sum_a_nat] :
( B
= ( ~ ( F @ X3 ) ) ) ) ).
% rangeE
thf(fact_26_rangeE,axiom,
! [B: list_Sum_sum_a_nat,F: sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ B @ ( image_7077900747146408011_a_nat @ F @ top_to795618464972521135_a_nat ) )
=> ~ ! [X3: sum_sum_a_nat] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_27_rangeE,axiom,
! [B: sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ B @ ( image_7142520692256960453_a_nat @ F @ top_to795618464972521135_a_nat ) )
=> ~ ! [X3: sum_sum_a_nat] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_28_rangeE,axiom,
! [B: $o,F: $o > $o] :
( ( member_o @ B @ ( image_o_o2 @ F @ top_top_set_o ) )
=> ~ ! [X3: $o] :
( B
= ( ~ ( F @ X3 ) ) ) ) ).
% rangeE
thf(fact_29_rangeE,axiom,
! [B: list_Sum_sum_a_nat,F: $o > list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ B @ ( image_733135354807394034_a_nat @ F @ top_top_set_o ) )
=> ~ ! [X3: $o] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_30_rangeE,axiom,
! [B: sum_sum_a_nat,F: $o > sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ B @ ( image_4139480514073730540_a_nat @ F @ top_top_set_o ) )
=> ~ ! [X3: $o] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_31_range__composition,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( image_5081948215111134021_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( F @ ( G @ X ) )
@ top_to6433055325616222389_a_nat )
= ( image_5081948215111134021_a_nat @ F @ ( image_5081948215111134021_a_nat @ G @ top_to6433055325616222389_a_nat ) ) ) ).
% range_composition
thf(fact_32_range__composition,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,G: sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( image_7077900747146408011_a_nat
@ ^ [X: sum_sum_a_nat] : ( F @ ( G @ X ) )
@ top_to795618464972521135_a_nat )
= ( image_5081948215111134021_a_nat @ F @ ( image_7077900747146408011_a_nat @ G @ top_to795618464972521135_a_nat ) ) ) ).
% range_composition
thf(fact_33_range__composition,axiom,
! [F: $o > $o,G: sum_sum_a_nat > $o] :
( ( image_6095136190293192542_nat_o
@ ^ [X: sum_sum_a_nat] : ( F @ ( G @ X ) )
@ top_to795618464972521135_a_nat )
= ( image_o_o2 @ F @ ( image_6095136190293192542_nat_o @ G @ top_to795618464972521135_a_nat ) ) ) ).
% range_composition
thf(fact_34_range__composition,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,G: $o > list_Sum_sum_a_nat] :
( ( image_733135354807394034_a_nat
@ ^ [X: $o] : ( F @ ( G @ X ) )
@ top_top_set_o )
= ( image_5081948215111134021_a_nat @ F @ ( image_733135354807394034_a_nat @ G @ top_top_set_o ) ) ) ).
% range_composition
thf(fact_35_range__composition,axiom,
! [F: $o > $o,G: $o > $o] :
( ( image_o_o2
@ ^ [X: $o] : ( F @ ( G @ X ) )
@ top_top_set_o )
= ( image_o_o2 @ F @ ( image_o_o2 @ G @ top_top_set_o ) ) ) ).
% range_composition
thf(fact_36_surjD,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,Y2: list_Sum_sum_a_nat] :
( ( ( image_5081948215111134021_a_nat @ F @ top_to6433055325616222389_a_nat )
= top_to6433055325616222389_a_nat )
=> ? [X3: list_Sum_sum_a_nat] :
( Y2
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_37_surjD,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Y2: sum_sum_a_nat] :
( ( ( image_7142520692256960453_a_nat @ F @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat )
=> ? [X3: sum_sum_a_nat] :
( Y2
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_38_surjD,axiom,
! [F: sum_sum_a_nat > $o,Y2: $o] :
( ( ( image_6095136190293192542_nat_o @ F @ top_to795618464972521135_a_nat )
= top_top_set_o )
=> ? [X3: sum_sum_a_nat] :
( Y2
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_39_surjD,axiom,
! [F: $o > sum_sum_a_nat,Y2: sum_sum_a_nat] :
( ( ( image_4139480514073730540_a_nat @ F @ top_top_set_o )
= top_to795618464972521135_a_nat )
=> ? [X3: $o] :
( Y2
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_40_surjD,axiom,
! [F: $o > $o,Y2: $o] :
( ( ( image_o_o2 @ F @ top_top_set_o )
= top_top_set_o )
=> ? [X3: $o] :
( Y2
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_41_surjE,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,Y2: list_Sum_sum_a_nat] :
( ( ( image_5081948215111134021_a_nat @ F @ top_to6433055325616222389_a_nat )
= top_to6433055325616222389_a_nat )
=> ~ ! [X3: list_Sum_sum_a_nat] :
( Y2
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_42_surjE,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Y2: sum_sum_a_nat] :
( ( ( image_7142520692256960453_a_nat @ F @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat )
=> ~ ! [X3: sum_sum_a_nat] :
( Y2
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_43_surjE,axiom,
! [F: sum_sum_a_nat > $o,Y2: $o] :
( ( ( image_6095136190293192542_nat_o @ F @ top_to795618464972521135_a_nat )
= top_top_set_o )
=> ~ ! [X3: sum_sum_a_nat] :
( Y2
= ( ~ ( F @ X3 ) ) ) ) ).
% surjE
thf(fact_44_surjE,axiom,
! [F: $o > sum_sum_a_nat,Y2: sum_sum_a_nat] :
( ( ( image_4139480514073730540_a_nat @ F @ top_top_set_o )
= top_to795618464972521135_a_nat )
=> ~ ! [X3: $o] :
( Y2
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_45_surjE,axiom,
! [F: $o > $o,Y2: $o] :
( ( ( image_o_o2 @ F @ top_top_set_o )
= top_top_set_o )
=> ~ ! [X3: $o] :
( Y2
= ( ~ ( F @ X3 ) ) ) ) ).
% surjE
thf(fact_46_top__set__def,axiom,
( top_to9106040778512017686_a_nat
= ( collec5629555741568564177_a_nat @ top_to660882137183040839_nat_o ) ) ).
% top_set_def
thf(fact_47_top__set__def,axiom,
( top_to795618464972521135_a_nat
= ( collec7073057861543223018_a_nat @ top_to1565196397637005550_nat_o ) ) ).
% top_set_def
thf(fact_48_top__set__def,axiom,
( top_top_set_o
= ( collect_o @ top_top_o_o ) ) ).
% top_set_def
thf(fact_49_act__edom_Osimps_I7_J,axiom,
! [N: nat,Phi: fo_fmla_a_b,I: product_prod_b_nat > set_list_a] :
( ( act_edom_a_b @ ( fo_Exists_a_b @ N @ Phi ) @ I )
= ( act_edom_a_b @ Phi @ I ) ) ).
% act_edom.simps(7)
thf(fact_50_UNIV__witness,axiom,
? [X3: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X3 @ top_to6433055325616222389_a_nat ) ).
% UNIV_witness
thf(fact_51_UNIV__witness,axiom,
? [X3: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X3 @ top_to795618464972521135_a_nat ) ).
% UNIV_witness
thf(fact_52_UNIV__witness,axiom,
? [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).
% UNIV_witness
thf(fact_53_UNIV__eq__I,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ! [X3: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X3 @ A )
=> ( top_to6433055325616222389_a_nat = A ) ) ).
% UNIV_eq_I
thf(fact_54_UNIV__eq__I,axiom,
! [A: set_Sum_sum_a_nat] :
( ! [X3: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X3 @ A )
=> ( top_to795618464972521135_a_nat = A ) ) ).
% UNIV_eq_I
thf(fact_55_UNIV__eq__I,axiom,
! [A: set_o] :
( ! [X3: $o] : ( member_o @ X3 @ A )
=> ( top_top_set_o = A ) ) ).
% UNIV_eq_I
thf(fact_56_rev__image__eqI,axiom,
! [X2: $o,A: set_o,B: $o,F: $o > $o] :
( ( member_o @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_o @ B @ ( image_o_o2 @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_57_rev__image__eqI,axiom,
! [X2: $o,A: set_o,B: list_Sum_sum_a_nat,F: $o > list_Sum_sum_a_nat] :
( ( member_o @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member408289922725080238_a_nat @ B @ ( image_733135354807394034_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_58_rev__image__eqI,axiom,
! [X2: $o,A: set_o,B: sum_sum_a_nat,F: $o > sum_sum_a_nat] :
( ( member_o @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_Sum_sum_a_nat @ B @ ( image_4139480514073730540_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_59_rev__image__eqI,axiom,
! [X2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: $o,F: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_o @ B @ ( image_3984946558445957976_nat_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_60_rev__image__eqI,axiom,
! [X2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member408289922725080238_a_nat @ B @ ( image_5081948215111134021_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_61_rev__image__eqI,axiom,
! [X2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: sum_sum_a_nat,F: list_Sum_sum_a_nat > sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_Sum_sum_a_nat @ B @ ( image_6705138127771971519_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_62_rev__image__eqI,axiom,
! [X2: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: $o,F: sum_sum_a_nat > $o] :
( ( member_Sum_sum_a_nat @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_o @ B @ ( image_6095136190293192542_nat_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_63_rev__image__eqI,axiom,
! [X2: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: list_Sum_sum_a_nat,F: sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member408289922725080238_a_nat @ B @ ( image_7077900747146408011_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_64_rev__image__eqI,axiom,
! [X2: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_Sum_sum_a_nat @ B @ ( image_7142520692256960453_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_65_ball__imageD,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ ( image_5081948215111134021_a_nat @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_66_ball__imageD,axiom,
! [F: $o > $o,A: set_o,P: $o > $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ ( image_o_o2 @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_67_image__cong,axiom,
! [M: set_o,N2: set_o,F: $o > $o,G: $o > $o] :
( ( M = N2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_o_o2 @ F @ M )
= ( image_o_o2 @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_68_image__cong,axiom,
! [M: set_li6526943997496501093_a_nat,N2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( M = N2 )
=> ( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_5081948215111134021_a_nat @ F @ M )
= ( image_5081948215111134021_a_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_69_bex__imageD,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ? [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ ( image_5081948215111134021_a_nat @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_70_bex__imageD,axiom,
! [F: $o > $o,A: set_o,P: $o > $o] :
( ? [X4: $o] :
( ( member_o @ X4 @ ( image_o_o2 @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: $o] :
( ( member_o @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_71_image__iff,axiom,
! [Z: $o,F: $o > $o,A: set_o] :
( ( member_o @ Z @ ( image_o_o2 @ F @ A ) )
= ( ? [X: $o] :
( ( member_o @ X @ A )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_72_image__iff,axiom,
! [Z: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ Z @ ( image_5081948215111134021_a_nat @ F @ A ) )
= ( ? [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_73_imageI,axiom,
! [X2: $o,A: set_o,F: $o > $o] :
( ( member_o @ X2 @ A )
=> ( member_o @ ( F @ X2 ) @ ( image_o_o2 @ F @ A ) ) ) ).
% imageI
thf(fact_74_imageI,axiom,
! [X2: $o,A: set_o,F: $o > list_Sum_sum_a_nat] :
( ( member_o @ X2 @ A )
=> ( member408289922725080238_a_nat @ ( F @ X2 ) @ ( image_733135354807394034_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_75_imageI,axiom,
! [X2: $o,A: set_o,F: $o > sum_sum_a_nat] :
( ( member_o @ X2 @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X2 ) @ ( image_4139480514073730540_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_76_imageI,axiom,
! [X2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ X2 @ A )
=> ( member_o @ ( F @ X2 ) @ ( image_3984946558445957976_nat_o @ F @ A ) ) ) ).
% imageI
thf(fact_77_imageI,axiom,
! [X2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A )
=> ( member408289922725080238_a_nat @ ( F @ X2 ) @ ( image_5081948215111134021_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_78_imageI,axiom,
! [X2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X2 ) @ ( image_6705138127771971519_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_79_imageI,axiom,
! [X2: sum_sum_a_nat,A: set_Sum_sum_a_nat,F: sum_sum_a_nat > $o] :
( ( member_Sum_sum_a_nat @ X2 @ A )
=> ( member_o @ ( F @ X2 ) @ ( image_6095136190293192542_nat_o @ F @ A ) ) ) ).
% imageI
thf(fact_80_imageI,axiom,
! [X2: sum_sum_a_nat,A: set_Sum_sum_a_nat,F: sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ A )
=> ( member408289922725080238_a_nat @ ( F @ X2 ) @ ( image_7077900747146408011_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_81_imageI,axiom,
! [X2: sum_sum_a_nat,A: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X2 ) @ ( image_7142520692256960453_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_82_UNIV__def,axiom,
( top_to9106040778512017686_a_nat
= ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] : $true ) ) ).
% UNIV_def
thf(fact_83_UNIV__def,axiom,
( top_to795618464972521135_a_nat
= ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] : $true ) ) ).
% UNIV_def
thf(fact_84_UNIV__def,axiom,
( top_top_set_o
= ( collect_o
@ ^ [X: $o] : $true ) ) ).
% UNIV_def
thf(fact_85_Compr__image__eq,axiom,
! [F: $o > $o,A: set_o,P: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( image_o_o2 @ F @ A ) )
& ( P @ X ) ) )
= ( image_o_o2 @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_86_Compr__image__eq,axiom,
! [F: sum_sum_a_nat > $o,A: set_Sum_sum_a_nat,P: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( image_6095136190293192542_nat_o @ F @ A ) )
& ( P @ X ) ) )
= ( image_6095136190293192542_nat_o @ F
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_87_Compr__image__eq,axiom,
! [F: $o > sum_sum_a_nat,A: set_o,P: sum_sum_a_nat > $o] :
( ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( image_4139480514073730540_a_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_4139480514073730540_a_nat @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_88_Compr__image__eq,axiom,
! [F: list_Sum_sum_a_nat > $o,A: set_li6526943997496501093_a_nat,P: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( image_3984946558445957976_nat_o @ F @ A ) )
& ( P @ X ) ) )
= ( image_3984946558445957976_nat_o @ F
@ ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_89_Compr__image__eq,axiom,
! [F: $o > list_Sum_sum_a_nat,A: set_o,P: list_Sum_sum_a_nat > $o] :
( ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ ( image_733135354807394034_a_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_733135354807394034_a_nat @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_90_Compr__image__eq,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( image_7142520692256960453_a_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_7142520692256960453_a_nat @ F
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_91_Compr__image__eq,axiom,
! [F: ( nat > sum_sum_a_nat ) > $o,A: set_na3699693778330250182_a_nat,P: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( image_2376713081370839351_nat_o @ F @ A ) )
& ( P @ X ) ) )
= ( image_2376713081370839351_nat_o @ F
@ ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_92_Compr__image__eq,axiom,
! [F: $o > nat > sum_sum_a_nat,A: set_o,P: ( nat > sum_sum_a_nat ) > $o] :
( ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ ( image_3967413701311303635_a_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_3967413701311303635_a_nat @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_93_Compr__image__eq,axiom,
! [F: sum_sum_a_nat > list_Sum_sum_a_nat,A: set_Sum_sum_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ ( image_7077900747146408011_a_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_7077900747146408011_a_nat @ F
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_94_Compr__image__eq,axiom,
! [F: list_Sum_sum_a_nat > sum_sum_a_nat,A: set_li6526943997496501093_a_nat,P: sum_sum_a_nat > $o] :
( ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( image_6705138127771971519_a_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_6705138127771971519_a_nat @ F
@ ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_95_image__image,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat @ F @ ( image_5081948215111134021_a_nat @ G @ A ) )
= ( image_5081948215111134021_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_96_image__image,axiom,
! [F: $o > $o,G: $o > $o,A: set_o] :
( ( image_o_o2 @ F @ ( image_o_o2 @ G @ A ) )
= ( image_o_o2
@ ^ [X: $o] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_97_imageE,axiom,
! [B: $o,F: $o > $o,A: set_o] :
( ( member_o @ B @ ( image_o_o2 @ F @ A ) )
=> ~ ! [X3: $o] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A ) ) ) ).
% imageE
thf(fact_98_imageE,axiom,
! [B: $o,F: list_Sum_sum_a_nat > $o,A: set_li6526943997496501093_a_nat] :
( ( member_o @ B @ ( image_3984946558445957976_nat_o @ F @ A ) )
=> ~ ! [X3: list_Sum_sum_a_nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member408289922725080238_a_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_99_imageE,axiom,
! [B: $o,F: sum_sum_a_nat > $o,A: set_Sum_sum_a_nat] :
( ( member_o @ B @ ( image_6095136190293192542_nat_o @ F @ A ) )
=> ~ ! [X3: sum_sum_a_nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_Sum_sum_a_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_100_imageE,axiom,
! [B: list_Sum_sum_a_nat,F: $o > list_Sum_sum_a_nat,A: set_o] :
( ( member408289922725080238_a_nat @ B @ ( image_733135354807394034_a_nat @ F @ A ) )
=> ~ ! [X3: $o] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A ) ) ) ).
% imageE
thf(fact_101_imageE,axiom,
! [B: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ B @ ( image_5081948215111134021_a_nat @ F @ A ) )
=> ~ ! [X3: list_Sum_sum_a_nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member408289922725080238_a_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_102_imageE,axiom,
! [B: list_Sum_sum_a_nat,F: sum_sum_a_nat > list_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ B @ ( image_7077900747146408011_a_nat @ F @ A ) )
=> ~ ! [X3: sum_sum_a_nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_Sum_sum_a_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_103_imageE,axiom,
! [B: sum_sum_a_nat,F: $o > sum_sum_a_nat,A: set_o] :
( ( member_Sum_sum_a_nat @ B @ ( image_4139480514073730540_a_nat @ F @ A ) )
=> ~ ! [X3: $o] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A ) ) ) ).
% imageE
thf(fact_104_imageE,axiom,
! [B: sum_sum_a_nat,F: list_Sum_sum_a_nat > sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member_Sum_sum_a_nat @ B @ ( image_6705138127771971519_a_nat @ F @ A ) )
=> ~ ! [X3: list_Sum_sum_a_nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member408289922725080238_a_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_105_imageE,axiom,
! [B: sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ B @ ( image_7142520692256960453_a_nat @ F @ A ) )
=> ~ ! [X3: sum_sum_a_nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_Sum_sum_a_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_106_range__eqI,axiom,
! [B: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,X2: list_Sum_sum_a_nat] :
( ( B
= ( F @ X2 ) )
=> ( member408289922725080238_a_nat @ B @ ( image_5081948215111134021_a_nat @ F @ top_to6433055325616222389_a_nat ) ) ) ).
% range_eqI
thf(fact_107_range__eqI,axiom,
! [B: $o,F: sum_sum_a_nat > $o,X2: sum_sum_a_nat] :
( ( B
= ( F @ X2 ) )
=> ( member_o @ B @ ( image_6095136190293192542_nat_o @ F @ top_to795618464972521135_a_nat ) ) ) ).
% range_eqI
thf(fact_108_range__eqI,axiom,
! [B: list_Sum_sum_a_nat,F: sum_sum_a_nat > list_Sum_sum_a_nat,X2: sum_sum_a_nat] :
( ( B
= ( F @ X2 ) )
=> ( member408289922725080238_a_nat @ B @ ( image_7077900747146408011_a_nat @ F @ top_to795618464972521135_a_nat ) ) ) ).
% range_eqI
thf(fact_109_range__eqI,axiom,
! [B: sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,X2: sum_sum_a_nat] :
( ( B
= ( F @ X2 ) )
=> ( member_Sum_sum_a_nat @ B @ ( image_7142520692256960453_a_nat @ F @ top_to795618464972521135_a_nat ) ) ) ).
% range_eqI
thf(fact_110_range__eqI,axiom,
! [B: $o,F: $o > $o,X2: $o] :
( ( B
= ( F @ X2 ) )
=> ( member_o @ B @ ( image_o_o2 @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_111_range__eqI,axiom,
! [B: list_Sum_sum_a_nat,F: $o > list_Sum_sum_a_nat,X2: $o] :
( ( B
= ( F @ X2 ) )
=> ( member408289922725080238_a_nat @ B @ ( image_733135354807394034_a_nat @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_112_range__eqI,axiom,
! [B: sum_sum_a_nat,F: $o > sum_sum_a_nat,X2: $o] :
( ( B
= ( F @ X2 ) )
=> ( member_Sum_sum_a_nat @ B @ ( image_4139480514073730540_a_nat @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_113_surj__def,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( ( image_5081948215111134021_a_nat @ F @ top_to6433055325616222389_a_nat )
= top_to6433055325616222389_a_nat )
= ( ! [Y3: list_Sum_sum_a_nat] :
? [X: list_Sum_sum_a_nat] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_114_surj__def,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat] :
( ( ( image_7142520692256960453_a_nat @ F @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat )
= ( ! [Y3: sum_sum_a_nat] :
? [X: sum_sum_a_nat] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_115_surj__def,axiom,
! [F: sum_sum_a_nat > $o] :
( ( ( image_6095136190293192542_nat_o @ F @ top_to795618464972521135_a_nat )
= top_top_set_o )
= ( ! [Y3: $o] :
? [X: sum_sum_a_nat] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_116_surj__def,axiom,
! [F: $o > sum_sum_a_nat] :
( ( ( image_4139480514073730540_a_nat @ F @ top_top_set_o )
= top_to795618464972521135_a_nat )
= ( ! [Y3: sum_sum_a_nat] :
? [X: $o] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_117_surj__def,axiom,
! [F: $o > $o] :
( ( ( image_o_o2 @ F @ top_top_set_o )
= top_top_set_o )
= ( ! [Y3: $o] :
? [X: $o] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_118_rangeI,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,X2: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ ( F @ X2 ) @ ( image_5081948215111134021_a_nat @ F @ top_to6433055325616222389_a_nat ) ) ).
% rangeI
thf(fact_119_rangeI,axiom,
! [F: sum_sum_a_nat > $o,X2: sum_sum_a_nat] : ( member_o @ ( F @ X2 ) @ ( image_6095136190293192542_nat_o @ F @ top_to795618464972521135_a_nat ) ) ).
% rangeI
thf(fact_120_rangeI,axiom,
! [F: sum_sum_a_nat > list_Sum_sum_a_nat,X2: sum_sum_a_nat] : ( member408289922725080238_a_nat @ ( F @ X2 ) @ ( image_7077900747146408011_a_nat @ F @ top_to795618464972521135_a_nat ) ) ).
% rangeI
thf(fact_121_rangeI,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,X2: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ ( F @ X2 ) @ ( image_7142520692256960453_a_nat @ F @ top_to795618464972521135_a_nat ) ) ).
% rangeI
thf(fact_122_rangeI,axiom,
! [F: $o > $o,X2: $o] : ( member_o @ ( F @ X2 ) @ ( image_o_o2 @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_123_rangeI,axiom,
! [F: $o > list_Sum_sum_a_nat,X2: $o] : ( member408289922725080238_a_nat @ ( F @ X2 ) @ ( image_733135354807394034_a_nat @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_124_rangeI,axiom,
! [F: $o > sum_sum_a_nat,X2: $o] : ( member_Sum_sum_a_nat @ ( F @ X2 ) @ ( image_4139480514073730540_a_nat @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_125_surjI,axiom,
! [G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ! [X3: list_Sum_sum_a_nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_5081948215111134021_a_nat @ G @ top_to6433055325616222389_a_nat )
= top_to6433055325616222389_a_nat ) ) ).
% surjI
thf(fact_126_surjI,axiom,
! [G: sum_sum_a_nat > sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ! [X3: sum_sum_a_nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_7142520692256960453_a_nat @ G @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat ) ) ).
% surjI
thf(fact_127_surjI,axiom,
! [G: sum_sum_a_nat > $o,F: $o > sum_sum_a_nat] :
( ! [X3: $o] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_6095136190293192542_nat_o @ G @ top_to795618464972521135_a_nat )
= top_top_set_o ) ) ).
% surjI
thf(fact_128_surjI,axiom,
! [G: $o > sum_sum_a_nat,F: sum_sum_a_nat > $o] :
( ! [X3: sum_sum_a_nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_4139480514073730540_a_nat @ G @ top_top_set_o )
= top_to795618464972521135_a_nat ) ) ).
% surjI
thf(fact_129_surjI,axiom,
! [G: $o > $o,F: $o > $o] :
( ! [X3: $o] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_o_o2 @ G @ top_top_set_o )
= top_top_set_o ) ) ).
% surjI
thf(fact_130_fv__eq_I2_J,axiom,
( ( fv_fo_fmla_list_a_b @ ( fo_Forall_a_b @ i2 @ phi ) )
= ( fv_fo_fmla_list_a_b @ phi ) ) ).
% fv_eq(2)
thf(fact_131_None,axiom,
( ( pos_nat @ i2 @ ( fv_fo_fmla_list_a_b @ phi ) )
= none_nat ) ).
% None
thf(fact_132__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062AD_An_AX_O_A_092_060lbrakk_062t_A_061_A_IAD_M_An_M_AX_J_059_AAD_A_061_Aact__edom_A_092_060phi_062_AI_059_AAD_A_061_Aact__edom_A_IExists_Ai_A_092_060phi_062_J_AI_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [AD: set_a] :
( ? [N3: nat,X5: set_li6526943997496501093_a_nat] :
( t
= ( produc1626021218680649873_a_nat @ AD @ ( produc4622690550020908860_a_nat @ N3 @ X5 ) ) )
=> ( ( AD
= ( act_edom_a_b @ phi @ i ) )
=> ( AD
!= ( act_edom_a_b @ ( fo_Exists_a_b @ i2 @ phi ) @ i ) ) ) ) ).
% \<open>\<And>thesis. (\<And>AD n X. \<lbrakk>t = (AD, n, X); AD = act_edom \<phi> I; AD = act_edom (Exists i \<phi>) I\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_133_t__def_I1_J,axiom,
( t
= ( produc1626021218680649873_a_nat @ ad @ ( produc4622690550020908860_a_nat @ n @ x ) ) ) ).
% t_def(1)
thf(fact_134_proj__fmla__def,axiom,
( proj_f6037546931041448027_a_nat
= ( ^ [Phi2: fo_fmla_a_b,R: set_na3699693778330250182_a_nat] : ( proj_v3643391342904276326_a_nat @ R @ ( fv_fo_fmla_list_a_b @ Phi2 ) ) ) ) ).
% proj_fmla_def
thf(fact_135_Sup_OSUP__identity__eq,axiom,
! [Sup: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( Sup
@ ( image_5081948215111134021_a_nat
@ ^ [X: list_Sum_sum_a_nat] : X
@ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_identity_eq
thf(fact_136_Sup_OSUP__identity__eq,axiom,
! [Sup: set_o > $o,A: set_o] :
( ( Sup
@ ( image_o_o2
@ ^ [X: $o] : X
@ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_identity_eq
thf(fact_137_Inf_OINF__identity__eq,axiom,
! [Inf: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( Inf
@ ( image_5081948215111134021_a_nat
@ ^ [X: list_Sum_sum_a_nat] : X
@ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_identity_eq
thf(fact_138_Inf_OINF__identity__eq,axiom,
! [Inf: set_o > $o,A: set_o] :
( ( Inf
@ ( image_o_o2
@ ^ [X: $o] : X
@ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_identity_eq
thf(fact_139_fo__wf__X,axiom,
! [Phi: fo_fmla_a_b,I: product_prod_b_nat > set_list_a,AD2: set_a,N: nat,X6: set_li6526943997496501093_a_nat] :
( ( fo_wf_a_b @ Phi @ I @ ( produc1626021218680649873_a_nat @ AD2 @ ( produc4622690550020908860_a_nat @ N @ X6 ) ) )
=> ( X6
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ AD2 )
@ ( proj_f6037546931041448027_a_nat @ Phi
@ ( collec5629555741568564177_a_nat
@ ^ [Sigma: nat > sum_sum_a_nat] : ( esat_a_b @ Phi @ I @ Sigma @ top_to795618464972521135_a_nat ) ) ) ) ) ) ).
% fo_wf_X
thf(fact_140_Sup_OSUP__cong,axiom,
! [A: set_o,B2: set_o,C: $o > $o,D: $o > $o,Sup: set_o > $o] :
( ( A = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_o_o2 @ C @ A ) )
= ( Sup @ ( image_o_o2 @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_141_Sup_OSUP__cong,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C: list_Sum_sum_a_nat > list_Sum_sum_a_nat,D: list_Sum_sum_a_nat > list_Sum_sum_a_nat,Sup: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat] :
( ( A = B2 )
=> ( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_5081948215111134021_a_nat @ C @ A ) )
= ( Sup @ ( image_5081948215111134021_a_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_142_Inf_OINF__cong,axiom,
! [A: set_o,B2: set_o,C: $o > $o,D: $o > $o,Inf: set_o > $o] :
( ( A = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_o_o2 @ C @ A ) )
= ( Inf @ ( image_o_o2 @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_143_Inf_OINF__cong,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C: list_Sum_sum_a_nat > list_Sum_sum_a_nat,D: list_Sum_sum_a_nat > list_Sum_sum_a_nat,Inf: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat] :
( ( A = B2 )
=> ( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_5081948215111134021_a_nat @ C @ A ) )
= ( Inf @ ( image_5081948215111134021_a_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_144_top__empty__eq,axiom,
( top_to7646693901462700136_nat_o
= ( ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ top_to6433055325616222389_a_nat ) ) ) ).
% top_empty_eq
thf(fact_145_top__empty__eq,axiom,
( top_to1565196397637005550_nat_o
= ( ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ top_to795618464972521135_a_nat ) ) ) ).
% top_empty_eq
thf(fact_146_top__empty__eq,axiom,
( top_top_o_o
= ( ^ [X: $o] : ( member_o @ X @ top_top_set_o ) ) ) ).
% top_empty_eq
thf(fact_147_mem__Collect__eq,axiom,
! [A2: $o,P: $o > $o] :
( ( member_o @ A2 @ ( collect_o @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_148_mem__Collect__eq,axiom,
! [A2: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ A2 @ ( collec7555443234367654128_a_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_149_mem__Collect__eq,axiom,
! [A2: sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( member_Sum_sum_a_nat @ A2 @ ( collec7073057861543223018_a_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_150_mem__Collect__eq,axiom,
! [A2: nat > sum_sum_a_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ( member8690443509505302927_a_nat @ A2 @ ( collec5629555741568564177_a_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_151_Collect__mem__eq,axiom,
! [A: set_o] :
( ( collect_o
@ ^ [X: $o] : ( member_o @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_152_Collect__mem__eq,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_153_Collect__mem__eq,axiom,
! [A: set_Sum_sum_a_nat] :
( ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_154_Collect__mem__eq,axiom,
! [A: set_na3699693778330250182_a_nat] :
( ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_155_Collect__cong,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,Q: ( nat > sum_sum_a_nat ) > $o] :
( ! [X3: nat > sum_sum_a_nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec5629555741568564177_a_nat @ P )
= ( collec5629555741568564177_a_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_156_fo__fmla_Oinject_I8_J,axiom,
! [X81: nat,X82: fo_fmla_a_b,Y81: nat,Y82: fo_fmla_a_b] :
( ( ( fo_Forall_a_b @ X81 @ X82 )
= ( fo_Forall_a_b @ Y81 @ Y82 ) )
= ( ( X81 = Y81 )
& ( X82 = Y82 ) ) ) ).
% fo_fmla.inject(8)
thf(fact_157_fv__ex__all_I2_J,axiom,
! [I2: nat,Phi: fo_fmla_a_b] :
( ( ( pos_nat @ I2 @ ( fv_fo_fmla_list_a_b @ Phi ) )
= none_nat )
=> ( ( fv_fo_fmla_list_a_b @ ( fo_Forall_a_b @ I2 @ Phi ) )
= ( fv_fo_fmla_list_a_b @ Phi ) ) ) ).
% fv_ex_all(2)
thf(fact_158_top__empty__eq2,axiom,
( top_to2708995584971643772_nat_o
= ( ^ [X: set_a,Y3: produc5986856040925105290_a_nat] : ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ X @ Y3 ) @ top_to6603017186293284989_a_nat ) ) ) ).
% top_empty_eq2
thf(fact_159_top__empty__eq2,axiom,
( top_to7174986363109593079_nat_o
= ( ^ [X: nat,Y3: set_li6526943997496501093_a_nat] : ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ X @ Y3 ) @ top_to6009278897662198000_a_nat ) ) ) ).
% top_empty_eq2
thf(fact_160_pred__equals__eq2,axiom,
! [R2: set_Pr8537526423911375309_a_nat,S: set_Pr8537526423911375309_a_nat] :
( ( ( ^ [X: set_a,Y3: produc5986856040925105290_a_nat] : ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ X @ Y3 ) @ R2 ) )
= ( ^ [X: set_a,Y3: produc5986856040925105290_a_nat] : ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ X @ Y3 ) @ S ) ) )
= ( R2 = S ) ) ).
% pred_equals_eq2
thf(fact_161_pred__equals__eq2,axiom,
! [R2: set_Pr1465208939327205952_a_nat,S: set_Pr1465208939327205952_a_nat] :
( ( ( ^ [X: nat,Y3: set_li6526943997496501093_a_nat] : ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ X @ Y3 ) @ R2 ) )
= ( ^ [X: nat,Y3: set_li6526943997496501093_a_nat] : ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ X @ Y3 ) @ S ) ) )
= ( R2 = S ) ) ).
% pred_equals_eq2
thf(fact_162_sp__equiv__pair_Ocases,axiom,
! [X2: produc6346555946419678479_a_nat] :
~ ! [A3: set_a,B3: produc5986856040925105290_a_nat,A4: set_a,B4: produc5986856040925105290_a_nat] :
( X2
!= ( produc5405793250563226055_a_nat @ ( produc1626021218680649873_a_nat @ A3 @ B3 ) @ ( produc1626021218680649873_a_nat @ A4 @ B4 ) ) ) ).
% sp_equiv_pair.cases
thf(fact_163_sp__equiv__pair_Ocases,axiom,
! [X2: produc5621125494677191285_a_nat] :
~ ! [A3: nat,B3: set_li6526943997496501093_a_nat,A4: nat,B4: set_li6526943997496501093_a_nat] :
( X2
!= ( produc5595853527594814125_a_nat @ ( produc4622690550020908860_a_nat @ A3 @ B3 ) @ ( produc4622690550020908860_a_nat @ A4 @ B4 ) ) ) ).
% sp_equiv_pair.cases
thf(fact_164_fo__adom_Ocases,axiom,
! [X2: produc8208987855206359447_a_nat] :
~ ! [AD: set_a,N3: nat,X5: set_li6526943997496501093_a_nat] :
( X2
!= ( produc1626021218680649873_a_nat @ AD @ ( produc4622690550020908860_a_nat @ N3 @ X5 ) ) ) ).
% fo_adom.cases
thf(fact_165_fo__fin_Ocases,axiom,
! [X2: produc8208987855206359447_a_nat] :
~ ! [AD: set_a,N3: nat,X5: set_li6526943997496501093_a_nat] :
( X2
!= ( produc1626021218680649873_a_nat @ AD @ ( produc4622690550020908860_a_nat @ N3 @ X5 ) ) ) ).
% fo_fin.cases
thf(fact_166_fo__fmla_Odistinct_I55_J,axiom,
! [X71: nat,X72: fo_fmla_a_b,X81: nat,X82: fo_fmla_a_b] :
( ( fo_Exists_a_b @ X71 @ X72 )
!= ( fo_Forall_a_b @ X81 @ X82 ) ) ).
% fo_fmla.distinct(55)
thf(fact_167_act__edom_Osimps_I8_J,axiom,
! [N: nat,Phi: fo_fmla_a_b,I: product_prod_b_nat > set_list_a] :
( ( act_edom_a_b @ ( fo_Forall_a_b @ N @ Phi ) @ I )
= ( act_edom_a_b @ Phi @ I ) ) ).
% act_edom.simps(8)
thf(fact_168_fv__ex__all_I1_J,axiom,
! [I2: nat,Phi: fo_fmla_a_b] :
( ( ( pos_nat @ I2 @ ( fv_fo_fmla_list_a_b @ Phi ) )
= none_nat )
=> ( ( fv_fo_fmla_list_a_b @ ( fo_Exists_a_b @ I2 @ Phi ) )
= ( fv_fo_fmla_list_a_b @ Phi ) ) ) ).
% fv_ex_all(1)
thf(fact_169_eval__abs__def,axiom,
( eval_abs_a_b
= ( ^ [Phi2: fo_fmla_a_b,I3: product_prod_b_nat > set_list_a] :
( produc1626021218680649873_a_nat @ ( act_edom_a_b @ Phi2 @ I3 )
@ ( produc4622690550020908860_a_nat @ ( nfv_a_b @ Phi2 )
@ ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ ( act_edom_a_b @ Phi2 @ I3 ) )
@ ( proj_f6037546931041448027_a_nat @ Phi2
@ ( collec5629555741568564177_a_nat
@ ^ [Sigma: nat > sum_sum_a_nat] : ( esat_a_b @ Phi2 @ I3 @ Sigma @ top_to795618464972521135_a_nat ) ) ) ) ) ) ) ) ).
% eval_abs_def
thf(fact_170_old_Oprod_Oinject,axiom,
! [A2: set_a,B: produc5986856040925105290_a_nat,A5: set_a,B5: produc5986856040925105290_a_nat] :
( ( ( produc1626021218680649873_a_nat @ A2 @ B )
= ( produc1626021218680649873_a_nat @ A5 @ B5 ) )
= ( ( A2 = A5 )
& ( B = B5 ) ) ) ).
% old.prod.inject
thf(fact_171_old_Oprod_Oinject,axiom,
! [A2: nat,B: set_li6526943997496501093_a_nat,A5: nat,B5: set_li6526943997496501093_a_nat] :
( ( ( produc4622690550020908860_a_nat @ A2 @ B )
= ( produc4622690550020908860_a_nat @ A5 @ B5 ) )
= ( ( A2 = A5 )
& ( B = B5 ) ) ) ).
% old.prod.inject
thf(fact_172_prod_Oinject,axiom,
! [X1: set_a,X22: produc5986856040925105290_a_nat,Y1: set_a,Y22: produc5986856040925105290_a_nat] :
( ( ( produc1626021218680649873_a_nat @ X1 @ X22 )
= ( produc1626021218680649873_a_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_173_prod_Oinject,axiom,
! [X1: nat,X22: set_li6526943997496501093_a_nat,Y1: nat,Y22: set_li6526943997496501093_a_nat] :
( ( ( produc4622690550020908860_a_nat @ X1 @ X22 )
= ( produc4622690550020908860_a_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_174_fo__adom_Osimps,axiom,
! [AD2: set_a,N: nat,X6: set_li6526943997496501093_a_nat] :
( ( fo_adom_a_nat @ ( produc1626021218680649873_a_nat @ AD2 @ ( produc4622690550020908860_a_nat @ N @ X6 ) ) )
= AD2 ) ).
% fo_adom.simps
thf(fact_175_fo__adom_Oelims,axiom,
! [X2: produc8208987855206359447_a_nat,Y2: set_a] :
( ( ( fo_adom_a_nat @ X2 )
= Y2 )
=> ~ ! [AD: set_a] :
( ? [N3: nat,X5: set_li6526943997496501093_a_nat] :
( X2
= ( produc1626021218680649873_a_nat @ AD @ ( produc4622690550020908860_a_nat @ N3 @ X5 ) ) )
=> ( Y2 != AD ) ) ) ).
% fo_adom.elims
thf(fact_176_eval__ajoin_Ocases,axiom,
! [X2: produc8481584770077217079_a_nat] :
~ ! [Ns_phi: list_nat,AD_phi: set_a,Uu: nat,X_phi: set_li6526943997496501093_a_nat,Ns_psi: list_nat,AD_psi: set_a,Uv: nat,X_psi: set_li6526943997496501093_a_nat] :
( X2
!= ( produc5266839609353528303_a_nat @ Ns_phi @ ( produc6936860417230850472_a_nat @ ( produc1626021218680649873_a_nat @ AD_phi @ ( produc4622690550020908860_a_nat @ Uu @ X_phi ) ) @ ( produc5478379421572013840_a_nat @ Ns_psi @ ( produc1626021218680649873_a_nat @ AD_psi @ ( produc4622690550020908860_a_nat @ Uv @ X_psi ) ) ) ) ) ) ).
% eval_ajoin.cases
thf(fact_177_eval__forall_Ocases,axiom,
! [X2: produc3169862784723210109_a_nat] :
~ ! [I4: nat,Ns: list_nat,AD: set_a,Uu: nat,X5: set_li6526943997496501093_a_nat] :
( X2
!= ( produc1383156333761652911_a_nat @ I4 @ ( produc5478379421572013840_a_nat @ Ns @ ( produc1626021218680649873_a_nat @ AD @ ( produc4622690550020908860_a_nat @ Uu @ X5 ) ) ) ) ) ).
% eval_forall.cases
thf(fact_178_eval__neg_Ocases,axiom,
! [X2: produc7015024766196696536_a_nat] :
~ ! [Ns: list_nat,AD: set_a,Uu: nat,X5: set_li6526943997496501093_a_nat] :
( X2
!= ( produc5478379421572013840_a_nat @ Ns @ ( produc1626021218680649873_a_nat @ AD @ ( produc4622690550020908860_a_nat @ Uu @ X5 ) ) ) ) ).
% eval_neg.cases
thf(fact_179_old_Oprod_Oexhaust,axiom,
! [Y2: produc8208987855206359447_a_nat] :
~ ! [A3: set_a,B3: produc5986856040925105290_a_nat] :
( Y2
!= ( produc1626021218680649873_a_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_180_old_Oprod_Oexhaust,axiom,
! [Y2: produc5986856040925105290_a_nat] :
~ ! [A3: nat,B3: set_li6526943997496501093_a_nat] :
( Y2
!= ( produc4622690550020908860_a_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_181_surj__pair,axiom,
! [P2: produc8208987855206359447_a_nat] :
? [X3: set_a,Y4: produc5986856040925105290_a_nat] :
( P2
= ( produc1626021218680649873_a_nat @ X3 @ Y4 ) ) ).
% surj_pair
thf(fact_182_surj__pair,axiom,
! [P2: produc5986856040925105290_a_nat] :
? [X3: nat,Y4: set_li6526943997496501093_a_nat] :
( P2
= ( produc4622690550020908860_a_nat @ X3 @ Y4 ) ) ).
% surj_pair
thf(fact_183_prod__cases,axiom,
! [P: produc8208987855206359447_a_nat > $o,P2: produc8208987855206359447_a_nat] :
( ! [A3: set_a,B3: produc5986856040925105290_a_nat] : ( P @ ( produc1626021218680649873_a_nat @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_184_prod__cases,axiom,
! [P: produc5986856040925105290_a_nat > $o,P2: produc5986856040925105290_a_nat] :
( ! [A3: nat,B3: set_li6526943997496501093_a_nat] : ( P @ ( produc4622690550020908860_a_nat @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_185_Pair__inject,axiom,
! [A2: set_a,B: produc5986856040925105290_a_nat,A5: set_a,B5: produc5986856040925105290_a_nat] :
( ( ( produc1626021218680649873_a_nat @ A2 @ B )
= ( produc1626021218680649873_a_nat @ A5 @ B5 ) )
=> ~ ( ( A2 = A5 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_186_Pair__inject,axiom,
! [A2: nat,B: set_li6526943997496501093_a_nat,A5: nat,B5: set_li6526943997496501093_a_nat] :
( ( ( produc4622690550020908860_a_nat @ A2 @ B )
= ( produc4622690550020908860_a_nat @ A5 @ B5 ) )
=> ~ ( ( A2 = A5 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_187_prod__cases3,axiom,
! [Y2: produc8208987855206359447_a_nat] :
~ ! [A3: set_a,B3: nat,C2: set_li6526943997496501093_a_nat] :
( Y2
!= ( produc1626021218680649873_a_nat @ A3 @ ( produc4622690550020908860_a_nat @ B3 @ C2 ) ) ) ).
% prod_cases3
thf(fact_188_prod__induct3,axiom,
! [P: produc8208987855206359447_a_nat > $o,X2: produc8208987855206359447_a_nat] :
( ! [A3: set_a,B3: nat,C2: set_li6526943997496501093_a_nat] : ( P @ ( produc1626021218680649873_a_nat @ A3 @ ( produc4622690550020908860_a_nat @ B3 @ C2 ) ) )
=> ( P @ X2 ) ) ).
% prod_induct3
thf(fact_189_same__fstI,axiom,
! [P: set_a > $o,X2: set_a,Y5: produc5986856040925105290_a_nat,Y2: produc5986856040925105290_a_nat,R2: set_a > set_Pr3551440518166979541_a_nat] :
( ( P @ X2 )
=> ( ( member8591614759246846750_a_nat @ ( produc5595853527594814125_a_nat @ Y5 @ Y2 ) @ ( R2 @ X2 ) )
=> ( member8751153172265411512_a_nat @ ( produc5405793250563226055_a_nat @ ( produc1626021218680649873_a_nat @ X2 @ Y5 ) @ ( produc1626021218680649873_a_nat @ X2 @ Y2 ) ) @ ( same_f7968148086661921148_a_nat @ P @ R2 ) ) ) ) ).
% same_fstI
thf(fact_190_same__fstI,axiom,
! [P: nat > $o,X2: nat,Y5: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat,R2: nat > set_Pr8343503725489915207_a_nat] :
( ( P @ X2 )
=> ( ( member5874747487503750032_a_nat @ ( produc7939031333681902295_a_nat @ Y5 @ Y2 ) @ ( R2 @ X2 ) )
=> ( member8591614759246846750_a_nat @ ( produc5595853527594814125_a_nat @ ( produc4622690550020908860_a_nat @ X2 @ Y5 ) @ ( produc4622690550020908860_a_nat @ X2 @ Y2 ) ) @ ( same_f6449513938715440935_a_nat @ P @ R2 ) ) ) ) ).
% same_fstI
thf(fact_191_fold__atLeastAtMost__nat_Ocases,axiom,
! [X2: produc6237156593867757212_a_nat] :
~ ! [F2: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,A3: nat,B3: nat,Acc: set_li6526943997496501093_a_nat] :
( X2
!= ( produc6069846434347434318_a_nat @ F2 @ ( produc4479492211097124851_a_nat @ A3 @ ( produc4622690550020908860_a_nat @ B3 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_192_fo__adom_Opelims,axiom,
! [X2: produc8208987855206359447_a_nat,Y2: set_a] :
( ( ( fo_adom_a_nat @ X2 )
= Y2 )
=> ( ( accp_P3058278626556831054_a_nat @ fo_adom_rel_a_nat @ X2 )
=> ~ ! [AD: set_a,N3: nat,X5: set_li6526943997496501093_a_nat] :
( ( X2
= ( produc1626021218680649873_a_nat @ AD @ ( produc4622690550020908860_a_nat @ N3 @ X5 ) ) )
=> ( ( Y2 = AD )
=> ~ ( accp_P3058278626556831054_a_nat @ fo_adom_rel_a_nat @ ( produc1626021218680649873_a_nat @ AD @ ( produc4622690550020908860_a_nat @ N3 @ X5 ) ) ) ) ) ) ) ).
% fo_adom.pelims
thf(fact_193_asymI,axiom,
! [R3: set_Pr7343886759072863943_a_nat] :
( ! [X3: sum_sum_a_nat,Y4: sum_sum_a_nat] :
( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X3 @ Y4 ) @ R3 )
=> ~ ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ Y4 @ X3 ) @ R3 ) )
=> ( asym_o9163613717866838065_a_nat @ top_to795618464972521135_a_nat @ R3 ) ) ).
% asymI
thf(fact_194_asymI,axiom,
! [R3: set_Product_prod_o_o] :
( ! [X3: $o,Y4: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y4 ) @ R3 )
=> ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y4 @ X3 ) @ R3 ) )
=> ( asym_on_o @ top_top_set_o @ R3 ) ) ).
% asymI
thf(fact_195_asym__onI,axiom,
! [A: set_o,R3: set_Product_prod_o_o] :
( ! [X3: $o,Y4: $o] :
( ( member_o @ X3 @ A )
=> ( ( member_o @ Y4 @ A )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y4 ) @ R3 )
=> ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y4 @ X3 ) @ R3 ) ) ) )
=> ( asym_on_o @ A @ R3 ) ) ).
% asym_onI
thf(fact_196_asym__onI,axiom,
! [A: set_li6526943997496501093_a_nat,R3: set_Pr4870381170404451655_a_nat] :
( ! [X3: list_Sum_sum_a_nat,Y4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ( ( member408289922725080238_a_nat @ Y4 @ A )
=> ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ X3 @ Y4 ) @ R3 )
=> ~ ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Y4 @ X3 ) @ R3 ) ) ) )
=> ( asym_o6235823786712371895_a_nat @ A @ R3 ) ) ).
% asym_onI
thf(fact_197_asym__onI,axiom,
! [A: set_Sum_sum_a_nat,R3: set_Pr7343886759072863943_a_nat] :
( ! [X3: sum_sum_a_nat,Y4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ A )
=> ( ( member_Sum_sum_a_nat @ Y4 @ A )
=> ( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X3 @ Y4 ) @ R3 )
=> ~ ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ Y4 @ X3 ) @ R3 ) ) ) )
=> ( asym_o9163613717866838065_a_nat @ A @ R3 ) ) ).
% asym_onI
thf(fact_198_asym__onD,axiom,
! [A: set_o,R3: set_Product_prod_o_o,X2: $o,Y2: $o] :
( ( asym_on_o @ A @ R3 )
=> ( ( member_o @ X2 @ A )
=> ( ( member_o @ Y2 @ A )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y2 ) @ R3 )
=> ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y2 @ X2 ) @ R3 ) ) ) ) ) ).
% asym_onD
thf(fact_199_asym__onD,axiom,
! [A: set_li6526943997496501093_a_nat,R3: set_Pr4870381170404451655_a_nat,X2: list_Sum_sum_a_nat,Y2: list_Sum_sum_a_nat] :
( ( asym_o6235823786712371895_a_nat @ A @ R3 )
=> ( ( member408289922725080238_a_nat @ X2 @ A )
=> ( ( member408289922725080238_a_nat @ Y2 @ A )
=> ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ X2 @ Y2 ) @ R3 )
=> ~ ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Y2 @ X2 ) @ R3 ) ) ) ) ) ).
% asym_onD
thf(fact_200_asym__onD,axiom,
! [A: set_Sum_sum_a_nat,R3: set_Pr7343886759072863943_a_nat,X2: sum_sum_a_nat,Y2: sum_sum_a_nat] :
( ( asym_o9163613717866838065_a_nat @ A @ R3 )
=> ( ( member_Sum_sum_a_nat @ X2 @ A )
=> ( ( member_Sum_sum_a_nat @ Y2 @ A )
=> ( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X2 @ Y2 ) @ R3 )
=> ~ ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ Y2 @ X2 ) @ R3 ) ) ) ) ) ).
% asym_onD
thf(fact_201_asymD,axiom,
! [R3: set_Pr7343886759072863943_a_nat,X2: sum_sum_a_nat,Y2: sum_sum_a_nat] :
( ( asym_o9163613717866838065_a_nat @ top_to795618464972521135_a_nat @ R3 )
=> ( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X2 @ Y2 ) @ R3 )
=> ~ ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ Y2 @ X2 ) @ R3 ) ) ) ).
% asymD
thf(fact_202_asymD,axiom,
! [R3: set_Product_prod_o_o,X2: $o,Y2: $o] :
( ( asym_on_o @ top_top_set_o @ R3 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y2 ) @ R3 )
=> ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y2 @ X2 ) @ R3 ) ) ) ).
% asymD
thf(fact_203_asym__iff,axiom,
! [R3: set_Pr7343886759072863943_a_nat] :
( ( asym_o9163613717866838065_a_nat @ top_to795618464972521135_a_nat @ R3 )
= ( ! [X: sum_sum_a_nat,Y3: sum_sum_a_nat] :
( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X @ Y3 ) @ R3 )
=> ~ ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ Y3 @ X ) @ R3 ) ) ) ) ).
% asym_iff
thf(fact_204_asym__iff,axiom,
! [R3: set_Product_prod_o_o] :
( ( asym_on_o @ top_top_set_o @ R3 )
= ( ! [X: $o,Y3: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y3 ) @ R3 )
=> ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y3 @ X ) @ R3 ) ) ) ) ).
% asym_iff
thf(fact_205_asymp__asym__eq,axiom,
! [R3: set_Pr7343886759072863943_a_nat] :
( ( asymp_7353567122270816157_a_nat @ top_to795618464972521135_a_nat
@ ^ [X: sum_sum_a_nat,Y3: sum_sum_a_nat] : ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X @ Y3 ) @ R3 ) )
= ( asym_o9163613717866838065_a_nat @ top_to795618464972521135_a_nat @ R3 ) ) ).
% asymp_asym_eq
thf(fact_206_asymp__asym__eq,axiom,
! [R3: set_Product_prod_o_o] :
( ( asymp_on_o @ top_top_set_o
@ ^ [X: $o,Y3: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y3 ) @ R3 ) )
= ( asym_on_o @ top_top_set_o @ R3 ) ) ).
% asymp_asym_eq
thf(fact_207_asym__inv__image,axiom,
! [R2: set_Pr7343886759072863943_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ( asym_o9163613717866838065_a_nat @ top_to795618464972521135_a_nat @ R2 )
=> ( asym_o9163613717866838065_a_nat @ top_to795618464972521135_a_nat @ ( inv_im4626614748035979026_a_nat @ R2 @ F ) ) ) ).
% asym_inv_image
thf(fact_208_asym__inv__image,axiom,
! [R2: set_Pr7343886759072863943_a_nat,F: $o > sum_sum_a_nat] :
( ( asym_o9163613717866838065_a_nat @ top_to795618464972521135_a_nat @ R2 )
=> ( asym_on_o @ top_top_set_o @ ( inv_im6968873752965879979_nat_o @ R2 @ F ) ) ) ).
% asym_inv_image
thf(fact_209_asym__inv__image,axiom,
! [R2: set_Product_prod_o_o,F: sum_sum_a_nat > $o] :
( ( asym_on_o @ top_top_set_o @ R2 )
=> ( asym_o9163613717866838065_a_nat @ top_to795618464972521135_a_nat @ ( inv_im5013218076746417977_a_nat @ R2 @ F ) ) ) ).
% asym_inv_image
thf(fact_210_asym__inv__image,axiom,
! [R2: set_Product_prod_o_o,F: $o > $o] :
( ( asym_on_o @ top_top_set_o @ R2 )
=> ( asym_on_o @ top_top_set_o @ ( inv_image_o_o @ R2 @ F ) ) ) ).
% asym_inv_image
thf(fact_211_asym__lex__prod,axiom,
! [R_A: set_Pr7343886759072863943_a_nat,R_B: set_Pr7343886759072863943_a_nat] :
( ( asym_o9163613717866838065_a_nat @ top_to795618464972521135_a_nat @ R_A )
=> ( ( asym_o9163613717866838065_a_nat @ top_to795618464972521135_a_nat @ R_B )
=> ( asym_o8224450772295581977_a_nat @ top_to4556246370645296151_a_nat @ ( lex_pr6689677931783198063_a_nat @ R_A @ R_B ) ) ) ) ).
% asym_lex_prod
thf(fact_212_asym__lex__prod,axiom,
! [R_A: set_Pr7343886759072863943_a_nat,R_B: set_Product_prod_o_o] :
( ( asym_o9163613717866838065_a_nat @ top_to795618464972521135_a_nat @ R_A )
=> ( ( asym_on_o @ top_top_set_o @ R_B )
=> ( asym_o8502058357588188722_nat_o @ top_to6060290738076295856_nat_o @ ( lex_pr687214353484183816_nat_o @ R_A @ R_B ) ) ) ) ).
% asym_lex_prod
thf(fact_213_asym__lex__prod,axiom,
! [R_A: set_Product_prod_o_o,R_B: set_Pr7343886759072863943_a_nat] :
( ( asym_on_o @ top_top_set_o @ R_A )
=> ( ( asym_o9163613717866838065_a_nat @ top_to795618464972521135_a_nat @ R_B )
=> ( asym_o8740793336417624384_a_nat @ top_to4859158508675448510_a_nat @ ( lex_pr7954930714119497622_a_nat @ R_A @ R_B ) ) ) ) ).
% asym_lex_prod
thf(fact_214_asym__lex__prod,axiom,
! [R_A: set_Product_prod_o_o,R_B: set_Product_prod_o_o] :
( ( asym_on_o @ top_top_set_o @ R_A )
=> ( ( asym_on_o @ top_top_set_o @ R_B )
=> ( asym_o744349338269926489od_o_o @ top_to7721136755696657239od_o_o @ ( lex_prod_o_o @ R_A @ R_B ) ) ) ) ).
% asym_lex_prod
thf(fact_215_asymp__onI,axiom,
! [A: set_o,R2: $o > $o > $o] :
( ! [X3: $o,Y4: $o] :
( ( member_o @ X3 @ A )
=> ( ( member_o @ Y4 @ A )
=> ( ( R2 @ X3 @ Y4 )
=> ~ ( R2 @ Y4 @ X3 ) ) ) )
=> ( asymp_on_o @ A @ R2 ) ) ).
% asymp_onI
thf(fact_216_asymp__onI,axiom,
! [A: set_li6526943997496501093_a_nat,R2: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
( ! [X3: list_Sum_sum_a_nat,Y4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ( ( member408289922725080238_a_nat @ Y4 @ A )
=> ( ( R2 @ X3 @ Y4 )
=> ~ ( R2 @ Y4 @ X3 ) ) ) )
=> ( asymp_8879505906235046947_a_nat @ A @ R2 ) ) ).
% asymp_onI
thf(fact_217_asymp__onI,axiom,
! [A: set_Sum_sum_a_nat,R2: sum_sum_a_nat > sum_sum_a_nat > $o] :
( ! [X3: sum_sum_a_nat,Y4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ A )
=> ( ( member_Sum_sum_a_nat @ Y4 @ A )
=> ( ( R2 @ X3 @ Y4 )
=> ~ ( R2 @ Y4 @ X3 ) ) ) )
=> ( asymp_7353567122270816157_a_nat @ A @ R2 ) ) ).
% asymp_onI
thf(fact_218_asympI,axiom,
! [R2: sum_sum_a_nat > sum_sum_a_nat > $o] :
( ! [X3: sum_sum_a_nat,Y4: sum_sum_a_nat] :
( ( R2 @ X3 @ Y4 )
=> ~ ( R2 @ Y4 @ X3 ) )
=> ( asymp_7353567122270816157_a_nat @ top_to795618464972521135_a_nat @ R2 ) ) ).
% asympI
thf(fact_219_asympI,axiom,
! [R2: $o > $o > $o] :
( ! [X3: $o,Y4: $o] :
( ( R2 @ X3 @ Y4 )
=> ~ ( R2 @ Y4 @ X3 ) )
=> ( asymp_on_o @ top_top_set_o @ R2 ) ) ).
% asympI
thf(fact_220_in__lex__prod,axiom,
! [A2: set_a,B: produc5986856040925105290_a_nat,A5: set_a,B5: produc5986856040925105290_a_nat,R3: set_Pr5845495582615845127_set_a,S2: set_Pr3551440518166979541_a_nat] :
( ( member8751153172265411512_a_nat @ ( produc5405793250563226055_a_nat @ ( produc1626021218680649873_a_nat @ A2 @ B ) @ ( produc1626021218680649873_a_nat @ A5 @ B5 ) ) @ ( lex_pr3097501987248043641_a_nat @ R3 @ S2 ) )
= ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ A2 @ A5 ) @ R3 )
| ( ( A2 = A5 )
& ( member8591614759246846750_a_nat @ ( produc5595853527594814125_a_nat @ B @ B5 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_221_in__lex__prod,axiom,
! [A2: nat,B: set_li6526943997496501093_a_nat,A5: nat,B5: set_li6526943997496501093_a_nat,R3: set_Pr1261947904930325089at_nat,S2: set_Pr8343503725489915207_a_nat] :
( ( member8591614759246846750_a_nat @ ( produc5595853527594814125_a_nat @ ( produc4622690550020908860_a_nat @ A2 @ B ) @ ( produc4622690550020908860_a_nat @ A5 @ B5 ) ) @ ( lex_pr5038142128417811748_a_nat @ R3 @ S2 ) )
= ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A5 ) @ R3 )
| ( ( A2 = A5 )
& ( member5874747487503750032_a_nat @ ( produc7939031333681902295_a_nat @ B @ B5 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_222_asymp__onD,axiom,
! [A: set_o,R2: $o > $o > $o,X2: $o,Y2: $o] :
( ( asymp_on_o @ A @ R2 )
=> ( ( member_o @ X2 @ A )
=> ( ( member_o @ Y2 @ A )
=> ( ( R2 @ X2 @ Y2 )
=> ~ ( R2 @ Y2 @ X2 ) ) ) ) ) ).
% asymp_onD
thf(fact_223_asymp__onD,axiom,
! [A: set_li6526943997496501093_a_nat,R2: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o,X2: list_Sum_sum_a_nat,Y2: list_Sum_sum_a_nat] :
( ( asymp_8879505906235046947_a_nat @ A @ R2 )
=> ( ( member408289922725080238_a_nat @ X2 @ A )
=> ( ( member408289922725080238_a_nat @ Y2 @ A )
=> ( ( R2 @ X2 @ Y2 )
=> ~ ( R2 @ Y2 @ X2 ) ) ) ) ) ).
% asymp_onD
thf(fact_224_asymp__onD,axiom,
! [A: set_Sum_sum_a_nat,R2: sum_sum_a_nat > sum_sum_a_nat > $o,X2: sum_sum_a_nat,Y2: sum_sum_a_nat] :
( ( asymp_7353567122270816157_a_nat @ A @ R2 )
=> ( ( member_Sum_sum_a_nat @ X2 @ A )
=> ( ( member_Sum_sum_a_nat @ Y2 @ A )
=> ( ( R2 @ X2 @ Y2 )
=> ~ ( R2 @ Y2 @ X2 ) ) ) ) ) ).
% asymp_onD
thf(fact_225_asympD,axiom,
! [R2: sum_sum_a_nat > sum_sum_a_nat > $o,X2: sum_sum_a_nat,Y2: sum_sum_a_nat] :
( ( asymp_7353567122270816157_a_nat @ top_to795618464972521135_a_nat @ R2 )
=> ( ( R2 @ X2 @ Y2 )
=> ~ ( R2 @ Y2 @ X2 ) ) ) ).
% asympD
thf(fact_226_asympD,axiom,
! [R2: $o > $o > $o,X2: $o,Y2: $o] :
( ( asymp_on_o @ top_top_set_o @ R2 )
=> ( ( R2 @ X2 @ Y2 )
=> ~ ( R2 @ Y2 @ X2 ) ) ) ).
% asympD
thf(fact_227_fo__nmlz__proj__sub,axiom,
! [AD2: set_a,Phi: fo_fmla_a_b,R2: set_na3699693778330250182_a_nat] : ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ AD2 ) @ ( proj_f6037546931041448027_a_nat @ Phi @ R2 ) ) @ ( nall_tuples_a @ AD2 @ ( nfv_a_b @ Phi ) ) ) ).
% fo_nmlz_proj_sub
thf(fact_228_map__prod__surj,axiom,
! [F: $o > $o,G: $o > $o] :
( ( ( image_o_o2 @ F @ top_top_set_o )
= top_top_set_o )
=> ( ( ( image_o_o2 @ G @ top_top_set_o )
= top_top_set_o )
=> ( ( image_9131363867636255685od_o_o @ ( produc1928436952130685430_o_o_o @ F @ G ) @ top_to7721136755696657239od_o_o )
= top_to7721136755696657239od_o_o ) ) ) ).
% map_prod_surj
thf(fact_229_map__prod__surj,axiom,
! [F: sum_sum_a_nat > $o,G: $o > $o] :
( ( ( image_6095136190293192542_nat_o @ F @ top_to795618464972521135_a_nat )
= top_top_set_o )
=> ( ( ( image_o_o2 @ G @ top_top_set_o )
= top_top_set_o )
=> ( ( image_3134416316614865452od_o_o @ ( produc1182733323763000399_o_o_o @ F @ G ) @ top_to6060290738076295856_nat_o )
= top_to7721136755696657239od_o_o ) ) ) ).
% map_prod_surj
thf(fact_230_map__prod__surj,axiom,
! [F: $o > sum_sum_a_nat,G: $o > $o] :
( ( ( image_4139480514073730540_a_nat @ F @ top_top_set_o )
= top_to795618464972521135_a_nat )
=> ( ( ( image_o_o2 @ G @ top_top_set_o )
= top_top_set_o )
=> ( ( image_4150549323899317662_nat_o @ ( produc6140416349899342301at_o_o @ F @ G ) @ top_to7721136755696657239od_o_o )
= top_to6060290738076295856_nat_o ) ) ) ).
% map_prod_surj
thf(fact_231_map__prod__surj,axiom,
! [F: $o > $o,G: sum_sum_a_nat > $o] :
( ( ( image_o_o2 @ F @ top_top_set_o )
= top_top_set_o )
=> ( ( ( image_6095136190293192542_nat_o @ G @ top_to795618464972521135_a_nat )
= top_top_set_o )
=> ( ( image_133203050957209246od_o_o @ ( produc3003466691699731407_nat_o @ F @ G ) @ top_to4859158508675448510_a_nat )
= top_to7721136755696657239od_o_o ) ) ) ).
% map_prod_surj
thf(fact_232_map__prod__surj,axiom,
! [F: $o > $o,G: $o > sum_sum_a_nat] :
( ( ( image_o_o2 @ F @ top_top_set_o )
= top_top_set_o )
=> ( ( ( image_4139480514073730540_a_nat @ G @ top_top_set_o )
= top_to795618464972521135_a_nat )
=> ( ( image_4389284302728753324_a_nat @ ( produc1047811015480269405_a_nat @ F @ G ) @ top_to7721136755696657239od_o_o )
= top_to4859158508675448510_a_nat ) ) ) ).
% map_prod_surj
thf(fact_233_map__prod__surj,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: $o > $o] :
( ( ( image_7142520692256960453_a_nat @ F @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat )
=> ( ( ( image_o_o2 @ G @ top_top_set_o )
= top_top_set_o )
=> ( ( image_2781668254213443077_nat_o @ ( produc5618155173899674166at_o_o @ F @ G ) @ top_to6060290738076295856_nat_o )
= top_to6060290738076295856_nat_o ) ) ) ).
% map_prod_surj
thf(fact_234_map__prod__surj,axiom,
! [F: sum_sum_a_nat > $o,G: sum_sum_a_nat > $o] :
( ( ( image_6095136190293192542_nat_o @ F @ top_to795618464972521135_a_nat )
= top_top_set_o )
=> ( ( ( image_6095136190293192542_nat_o @ G @ top_to795618464972521135_a_nat )
= top_top_set_o )
=> ( ( image_7987694018126110469od_o_o @ ( produc2481205515700063272_nat_o @ F @ G ) @ top_to4556246370645296151_a_nat )
= top_to7721136755696657239od_o_o ) ) ) ).
% map_prod_surj
thf(fact_235_map__prod__surj,axiom,
! [F: sum_sum_a_nat > $o,G: $o > sum_sum_a_nat] :
( ( ( image_6095136190293192542_nat_o @ F @ top_to795618464972521135_a_nat )
= top_top_set_o )
=> ( ( ( image_4139480514073730540_a_nat @ G @ top_top_set_o )
= top_to795618464972521135_a_nat )
=> ( ( image_3020403233042878739_a_nat @ ( produc525549839480601270_a_nat @ F @ G ) @ top_to6060290738076295856_nat_o )
= top_to4859158508675448510_a_nat ) ) ) ).
% map_prod_surj
thf(fact_236_map__prod__surj,axiom,
! [F: $o > sum_sum_a_nat,G: sum_sum_a_nat > $o] :
( ( ( image_4139480514073730540_a_nat @ F @ top_top_set_o )
= top_to795618464972521135_a_nat )
=> ( ( ( image_6095136190293192542_nat_o @ G @ top_to795618464972521135_a_nat )
= top_top_set_o )
=> ( ( image_4079455202402477687_nat_o @ ( produc2892572387668450742_nat_o @ F @ G ) @ top_to4859158508675448510_a_nat )
= top_to6060290738076295856_nat_o ) ) ) ).
% map_prod_surj
thf(fact_237_map__prod__surj,axiom,
! [F: $o > sum_sum_a_nat,G: $o > sum_sum_a_nat] :
( ( ( image_4139480514073730540_a_nat @ F @ top_top_set_o )
= top_to795618464972521135_a_nat )
=> ( ( ( image_4139480514073730540_a_nat @ G @ top_top_set_o )
= top_to795618464972521135_a_nat )
=> ( ( image_7984595726292654213_a_nat @ ( produc936916711448988740_a_nat @ F @ G ) @ top_to7721136755696657239od_o_o )
= top_to4556246370645296151_a_nat ) ) ) ).
% map_prod_surj
thf(fact_238_ssubst__Pair__rhs,axiom,
! [R3: set_a,S2: produc5986856040925105290_a_nat,R2: set_Pr8537526423911375309_a_nat,S3: produc5986856040925105290_a_nat] :
( ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ R3 @ S2 ) @ R2 )
=> ( ( S3 = S2 )
=> ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ R3 @ S3 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_239_ssubst__Pair__rhs,axiom,
! [R3: nat,S2: set_li6526943997496501093_a_nat,R2: set_Pr1465208939327205952_a_nat,S3: set_li6526943997496501093_a_nat] :
( ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ R3 @ S2 ) @ R2 )
=> ( ( S3 = S2 )
=> ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ R3 @ S3 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_240_wfP__imp__asymp,axiom,
! [R3: sum_sum_a_nat > sum_sum_a_nat > $o] :
( ( wfP_Sum_sum_a_nat @ R3 )
=> ( asymp_7353567122270816157_a_nat @ top_to795618464972521135_a_nat @ R3 ) ) ).
% wfP_imp_asymp
thf(fact_241_wfP__imp__asymp,axiom,
! [R3: $o > $o > $o] :
( ( wfP_o @ R3 )
=> ( asymp_on_o @ top_top_set_o @ R3 ) ) ).
% wfP_imp_asymp
thf(fact_242_curryI,axiom,
! [F: produc8208987855206359447_a_nat > $o,A2: set_a,B: produc5986856040925105290_a_nat] :
( ( F @ ( produc1626021218680649873_a_nat @ A2 @ B ) )
=> ( produc6259199808005150713_nat_o @ F @ A2 @ B ) ) ).
% curryI
thf(fact_243_curryI,axiom,
! [F: produc5986856040925105290_a_nat > $o,A2: nat,B: set_li6526943997496501093_a_nat] :
( ( F @ ( produc4622690550020908860_a_nat @ A2 @ B ) )
=> ( produc4568951772394601358_nat_o @ F @ A2 @ B ) ) ).
% curryI
thf(fact_244_subsetI,axiom,
! [A: set_o,B2: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member_o @ X3 @ B2 ) )
=> ( ord_less_eq_set_o @ A @ B2 ) ) ).
% subsetI
thf(fact_245_subsetI,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ( member408289922725080238_a_nat @ X3 @ B2 ) )
=> ( ord_le1147066620699065093_a_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_246_subsetI,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ A )
=> ( member_Sum_sum_a_nat @ X3 @ B2 ) )
=> ( ord_le1325389633284124927_a_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_247_mem__Sigma__iff,axiom,
! [A2: $o,B: $o,A: set_o,B2: $o > set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ ( product_Sigma_o_o @ A @ B2 ) )
= ( ( member_o @ A2 @ A )
& ( member_o @ B @ ( B2 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_248_mem__Sigma__iff,axiom,
! [A2: $o,B: sum_sum_a_nat,A: set_o,B2: $o > set_Sum_sum_a_nat] :
( ( member6925473634427674423_a_nat @ ( produc6394072109761784702_a_nat @ A2 @ B ) @ ( produc3081175752665019439_a_nat @ A @ B2 ) )
= ( ( member_o @ A2 @ A )
& ( member_Sum_sum_a_nat @ B @ ( B2 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_249_mem__Sigma__iff,axiom,
! [A2: sum_sum_a_nat,B: $o,A: set_Sum_sum_a_nat,B2: sum_sum_a_nat > set_o] :
( ( member6686738655598238761_nat_o @ ( produc8349727785981246704_nat_o @ A2 @ B ) @ ( produc5036831428884481441_nat_o @ A @ B2 ) )
= ( ( member_Sum_sum_a_nat @ A2 @ A )
& ( member_o @ B @ ( B2 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_250_mem__Sigma__iff,axiom,
! [A2: $o,B: list_Sum_sum_a_nat,A: set_o,B2: $o > set_li6526943997496501093_a_nat] :
( ( member2772425197143436605_a_nat @ ( produc7378284081089837956_a_nat @ A2 @ B ) @ ( produc5640064037308512181_a_nat @ A @ B2 ) )
= ( ( member_o @ A2 @ A )
& ( member408289922725080238_a_nat @ B @ ( B2 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_251_mem__Sigma__iff,axiom,
! [A2: list_Sum_sum_a_nat,B: $o,A: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat > set_o] :
( ( member1161018691129610787_nat_o @ ( produc1406723247873626090_nat_o @ A2 @ B ) @ ( produc8891875240947076123_nat_o @ A @ B2 ) )
= ( ( member408289922725080238_a_nat @ A2 @ A )
& ( member_o @ B @ ( B2 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_252_mem__Sigma__iff,axiom,
! [A2: sum_sum_a_nat,B: sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: sum_sum_a_nat > set_Sum_sum_a_nat] :
( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ A2 @ B ) @ ( produc3014407214216980488_a_nat @ A @ B2 ) )
= ( ( member_Sum_sum_a_nat @ A2 @ A )
& ( member_Sum_sum_a_nat @ B @ ( B2 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_253_mem__Sigma__iff,axiom,
! [A2: nat,B: set_li6526943997496501093_a_nat,A: set_nat,B2: nat > set_se4330304633200676677_a_nat] :
( ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ A2 @ B ) @ ( produc3542699778996723019_a_nat @ A @ B2 ) )
= ( ( member_nat @ A2 @ A )
& ( member5553968465346197646_a_nat @ B @ ( B2 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_254_mem__Sigma__iff,axiom,
! [A2: list_Sum_sum_a_nat,B: sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat > set_Sum_sum_a_nat] :
( ( member6493994536747010826_a_nat @ ( produc5977302043283085393_a_nat @ A2 @ B ) @ ( produc1079312233239997058_a_nat @ A @ B2 ) )
= ( ( member408289922725080238_a_nat @ A2 @ A )
& ( member_Sum_sum_a_nat @ B @ ( B2 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_255_mem__Sigma__iff,axiom,
! [A2: sum_sum_a_nat,B: list_Sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( member312212173870163606_a_nat @ ( produc6350064662657521885_a_nat @ A2 @ B ) @ ( produc1452074852614433550_a_nat @ A @ B2 ) )
= ( ( member_Sum_sum_a_nat @ A2 @ A )
& ( member408289922725080238_a_nat @ B @ ( B2 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_256_mem__Sigma__iff,axiom,
! [A2: list_Sum_sum_a_nat,B: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ A2 @ B ) @ ( produc5624668685630548104_a_nat @ A @ B2 ) )
= ( ( member408289922725080238_a_nat @ A2 @ A )
& ( member408289922725080238_a_nat @ B @ ( B2 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_257_SigmaI,axiom,
! [A2: $o,A: set_o,B: $o,B2: $o > set_o] :
( ( member_o @ A2 @ A )
=> ( ( member_o @ B @ ( B2 @ A2 ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ ( product_Sigma_o_o @ A @ B2 ) ) ) ) ).
% SigmaI
thf(fact_258_SigmaI,axiom,
! [A2: $o,A: set_o,B: sum_sum_a_nat,B2: $o > set_Sum_sum_a_nat] :
( ( member_o @ A2 @ A )
=> ( ( member_Sum_sum_a_nat @ B @ ( B2 @ A2 ) )
=> ( member6925473634427674423_a_nat @ ( produc6394072109761784702_a_nat @ A2 @ B ) @ ( produc3081175752665019439_a_nat @ A @ B2 ) ) ) ) ).
% SigmaI
thf(fact_259_SigmaI,axiom,
! [A2: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: $o,B2: sum_sum_a_nat > set_o] :
( ( member_Sum_sum_a_nat @ A2 @ A )
=> ( ( member_o @ B @ ( B2 @ A2 ) )
=> ( member6686738655598238761_nat_o @ ( produc8349727785981246704_nat_o @ A2 @ B ) @ ( produc5036831428884481441_nat_o @ A @ B2 ) ) ) ) ).
% SigmaI
thf(fact_260_SigmaI,axiom,
! [A2: $o,A: set_o,B: list_Sum_sum_a_nat,B2: $o > set_li6526943997496501093_a_nat] :
( ( member_o @ A2 @ A )
=> ( ( member408289922725080238_a_nat @ B @ ( B2 @ A2 ) )
=> ( member2772425197143436605_a_nat @ ( produc7378284081089837956_a_nat @ A2 @ B ) @ ( produc5640064037308512181_a_nat @ A @ B2 ) ) ) ) ).
% SigmaI
thf(fact_261_SigmaI,axiom,
! [A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: $o,B2: list_Sum_sum_a_nat > set_o] :
( ( member408289922725080238_a_nat @ A2 @ A )
=> ( ( member_o @ B @ ( B2 @ A2 ) )
=> ( member1161018691129610787_nat_o @ ( produc1406723247873626090_nat_o @ A2 @ B ) @ ( produc8891875240947076123_nat_o @ A @ B2 ) ) ) ) ).
% SigmaI
thf(fact_262_SigmaI,axiom,
! [A2: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: sum_sum_a_nat,B2: sum_sum_a_nat > set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A2 @ A )
=> ( ( member_Sum_sum_a_nat @ B @ ( B2 @ A2 ) )
=> ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ A2 @ B ) @ ( produc3014407214216980488_a_nat @ A @ B2 ) ) ) ) ).
% SigmaI
thf(fact_263_SigmaI,axiom,
! [A2: nat,A: set_nat,B: set_li6526943997496501093_a_nat,B2: nat > set_se4330304633200676677_a_nat] :
( ( member_nat @ A2 @ A )
=> ( ( member5553968465346197646_a_nat @ B @ ( B2 @ A2 ) )
=> ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ A2 @ B ) @ ( produc3542699778996723019_a_nat @ A @ B2 ) ) ) ) ).
% SigmaI
thf(fact_264_SigmaI,axiom,
! [A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: sum_sum_a_nat,B2: list_Sum_sum_a_nat > set_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ A2 @ A )
=> ( ( member_Sum_sum_a_nat @ B @ ( B2 @ A2 ) )
=> ( member6493994536747010826_a_nat @ ( produc5977302043283085393_a_nat @ A2 @ B ) @ ( produc1079312233239997058_a_nat @ A @ B2 ) ) ) ) ).
% SigmaI
thf(fact_265_SigmaI,axiom,
! [A2: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: list_Sum_sum_a_nat,B2: sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( member_Sum_sum_a_nat @ A2 @ A )
=> ( ( member408289922725080238_a_nat @ B @ ( B2 @ A2 ) )
=> ( member312212173870163606_a_nat @ ( produc6350064662657521885_a_nat @ A2 @ B ) @ ( produc1452074852614433550_a_nat @ A @ B2 ) ) ) ) ).
% SigmaI
thf(fact_266_SigmaI,axiom,
! [A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: list_Sum_sum_a_nat,B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A2 @ A )
=> ( ( member408289922725080238_a_nat @ B @ ( B2 @ A2 ) )
=> ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ A2 @ B ) @ ( produc5624668685630548104_a_nat @ A @ B2 ) ) ) ) ).
% SigmaI
thf(fact_267_map__prod__simp,axiom,
! [F: set_a > set_a,G: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,A2: set_a,B: produc5986856040925105290_a_nat] :
( ( produc5812298523581686028_a_nat @ F @ G @ ( produc1626021218680649873_a_nat @ A2 @ B ) )
= ( produc1626021218680649873_a_nat @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_268_map__prod__simp,axiom,
! [F: set_a > nat,G: produc5986856040925105290_a_nat > set_li6526943997496501093_a_nat,A2: set_a,B: produc5986856040925105290_a_nat] :
( ( produc5294700245526393121_a_nat @ F @ G @ ( produc1626021218680649873_a_nat @ A2 @ B ) )
= ( produc4622690550020908860_a_nat @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_269_map__prod__simp,axiom,
! [F: nat > set_a,G: set_li6526943997496501093_a_nat > produc5986856040925105290_a_nat,A2: nat,B: set_li6526943997496501093_a_nat] :
( ( produc3382755674669967065_a_nat @ F @ G @ ( produc4622690550020908860_a_nat @ A2 @ B ) )
= ( produc1626021218680649873_a_nat @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_270_map__prod__simp,axiom,
! [F: nat > nat,G: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,A2: nat,B: set_li6526943997496501093_a_nat] :
( ( produc6714918141915149656_a_nat @ F @ G @ ( produc4622690550020908860_a_nat @ A2 @ B ) )
= ( produc4622690550020908860_a_nat @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_271_map__prod__imageI,axiom,
! [A2: set_a,B: produc5986856040925105290_a_nat,R2: set_Pr8537526423911375309_a_nat,F: set_a > set_a,G: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat] :
( ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ A2 @ B ) @ R2 )
=> ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_6916668542591666485_a_nat @ ( produc5812298523581686028_a_nat @ F @ G ) @ R2 ) ) ) ).
% map_prod_imageI
thf(fact_272_map__prod__imageI,axiom,
! [A2: set_a,B: produc5986856040925105290_a_nat,R2: set_Pr8537526423911375309_a_nat,F: set_a > nat,G: produc5986856040925105290_a_nat > set_li6526943997496501093_a_nat] :
( ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ A2 @ B ) @ R2 )
=> ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_3110382862205598248_a_nat @ ( produc5294700245526393121_a_nat @ F @ G ) @ R2 ) ) ) ).
% map_prod_imageI
thf(fact_273_map__prod__imageI,axiom,
! [A2: nat,B: set_li6526943997496501093_a_nat,R2: set_Pr1465208939327205952_a_nat,F: nat > set_a,G: set_li6526943997496501093_a_nat > produc5986856040925105290_a_nat] :
( ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ A2 @ B ) @ R2 )
=> ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_9052215851415248424_a_nat @ ( produc3382755674669967065_a_nat @ F @ G ) @ R2 ) ) ) ).
% map_prod_imageI
thf(fact_274_map__prod__imageI,axiom,
! [A2: nat,B: set_li6526943997496501093_a_nat,R2: set_Pr1465208939327205952_a_nat,F: nat > nat,G: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat] :
( ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ A2 @ B ) @ R2 )
=> ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_933174759805388315_a_nat @ ( produc6714918141915149656_a_nat @ F @ G ) @ R2 ) ) ) ).
% map_prod_imageI
thf(fact_275_UNIV__Times__UNIV,axiom,
( ( produc3014407214216980488_a_nat @ top_to795618464972521135_a_nat
@ ^ [Uu2: sum_sum_a_nat] : top_to795618464972521135_a_nat )
= top_to4556246370645296151_a_nat ) ).
% UNIV_Times_UNIV
thf(fact_276_UNIV__Times__UNIV,axiom,
( ( produc5036831428884481441_nat_o @ top_to795618464972521135_a_nat
@ ^ [Uu2: sum_sum_a_nat] : top_top_set_o )
= top_to6060290738076295856_nat_o ) ).
% UNIV_Times_UNIV
thf(fact_277_UNIV__Times__UNIV,axiom,
( ( produc3081175752665019439_a_nat @ top_top_set_o
@ ^ [Uu2: $o] : top_to795618464972521135_a_nat )
= top_to4859158508675448510_a_nat ) ).
% UNIV_Times_UNIV
thf(fact_278_UNIV__Times__UNIV,axiom,
( ( product_Sigma_o_o @ top_top_set_o
@ ^ [Uu2: $o] : top_top_set_o )
= top_to7721136755696657239od_o_o ) ).
% UNIV_Times_UNIV
thf(fact_279_wfP__eq__minimal,axiom,
( wfP_o
= ( ^ [R4: $o > $o > $o] :
! [Q2: set_o] :
( ? [X: $o] : ( member_o @ X @ Q2 )
=> ? [X: $o] :
( ( member_o @ X @ Q2 )
& ! [Y3: $o] :
( ( R4 @ Y3 @ X )
=> ~ ( member_o @ Y3 @ Q2 ) ) ) ) ) ) ).
% wfP_eq_minimal
thf(fact_280_wfP__eq__minimal,axiom,
( wfP_li5146069326679374858_a_nat
= ( ^ [R4: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
! [Q2: set_li6526943997496501093_a_nat] :
( ? [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ Q2 )
=> ? [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ Q2 )
& ! [Y3: list_Sum_sum_a_nat] :
( ( R4 @ Y3 @ X )
=> ~ ( member408289922725080238_a_nat @ Y3 @ Q2 ) ) ) ) ) ) ).
% wfP_eq_minimal
thf(fact_281_wfP__eq__minimal,axiom,
( wfP_Sum_sum_a_nat
= ( ^ [R4: sum_sum_a_nat > sum_sum_a_nat > $o] :
! [Q2: set_Sum_sum_a_nat] :
( ? [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ Q2 )
=> ? [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ Q2 )
& ! [Y3: sum_sum_a_nat] :
( ( R4 @ Y3 @ X )
=> ~ ( member_Sum_sum_a_nat @ Y3 @ Q2 ) ) ) ) ) ) ).
% wfP_eq_minimal
thf(fact_282_in__mono,axiom,
! [A: set_o,B2: set_o,X2: $o] :
( ( ord_less_eq_set_o @ A @ B2 )
=> ( ( member_o @ X2 @ A )
=> ( member_o @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_283_in__mono,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,X2: list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B2 )
=> ( ( member408289922725080238_a_nat @ X2 @ A )
=> ( member408289922725080238_a_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_284_in__mono,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,X2: sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( ( member_Sum_sum_a_nat @ X2 @ A )
=> ( member_Sum_sum_a_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_285_subsetD,axiom,
! [A: set_o,B2: set_o,C3: $o] :
( ( ord_less_eq_set_o @ A @ B2 )
=> ( ( member_o @ C3 @ A )
=> ( member_o @ C3 @ B2 ) ) ) ).
% subsetD
thf(fact_286_subsetD,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C3: list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B2 )
=> ( ( member408289922725080238_a_nat @ C3 @ A )
=> ( member408289922725080238_a_nat @ C3 @ B2 ) ) ) ).
% subsetD
thf(fact_287_subsetD,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C3: sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( ( member_Sum_sum_a_nat @ C3 @ A )
=> ( member_Sum_sum_a_nat @ C3 @ B2 ) ) ) ).
% subsetD
thf(fact_288_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A6: set_o,B6: set_o] :
! [X: $o] :
( ( member_o @ X @ A6 )
=> ( member_o @ X @ B6 ) ) ) ) ).
% subset_eq
thf(fact_289_subset__eq,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A6: set_li6526943997496501093_a_nat,B6: set_li6526943997496501093_a_nat] :
! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A6 )
=> ( member408289922725080238_a_nat @ X @ B6 ) ) ) ) ).
% subset_eq
thf(fact_290_subset__eq,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [A6: set_Sum_sum_a_nat,B6: set_Sum_sum_a_nat] :
! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A6 )
=> ( member_Sum_sum_a_nat @ X @ B6 ) ) ) ) ).
% subset_eq
thf(fact_291_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A6: set_o,B6: set_o] :
! [T: $o] :
( ( member_o @ T @ A6 )
=> ( member_o @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_292_subset__iff,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A6: set_li6526943997496501093_a_nat,B6: set_li6526943997496501093_a_nat] :
! [T: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ T @ A6 )
=> ( member408289922725080238_a_nat @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_293_subset__iff,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [A6: set_Sum_sum_a_nat,B6: set_Sum_sum_a_nat] :
! [T: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ T @ A6 )
=> ( member_Sum_sum_a_nat @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_294_Collect__mono,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,Q: ( nat > sum_sum_a_nat ) > $o] :
( ! [X3: nat > sum_sum_a_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le8108555184339247974_a_nat @ ( collec5629555741568564177_a_nat @ P ) @ ( collec5629555741568564177_a_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_295_Collect__mono__iff,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,Q: ( nat > sum_sum_a_nat ) > $o] :
( ( ord_le8108555184339247974_a_nat @ ( collec5629555741568564177_a_nat @ P ) @ ( collec5629555741568564177_a_nat @ Q ) )
= ( ! [X: nat > sum_sum_a_nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_296_prop__restrict,axiom,
! [X2: $o,Z2: set_o,X6: set_o,P: $o > $o] :
( ( member_o @ X2 @ Z2 )
=> ( ( ord_less_eq_set_o @ Z2
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ X6 )
& ( P @ X ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_297_prop__restrict,axiom,
! [X2: list_Sum_sum_a_nat,Z2: set_li6526943997496501093_a_nat,X6: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ X2 @ Z2 )
=> ( ( ord_le1147066620699065093_a_nat @ Z2
@ ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ X6 )
& ( P @ X ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_298_prop__restrict,axiom,
! [X2: sum_sum_a_nat,Z2: set_Sum_sum_a_nat,X6: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( member_Sum_sum_a_nat @ X2 @ Z2 )
=> ( ( ord_le1325389633284124927_a_nat @ Z2
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ X6 )
& ( P @ X ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_299_prop__restrict,axiom,
! [X2: nat > sum_sum_a_nat,Z2: set_na3699693778330250182_a_nat,X6: set_na3699693778330250182_a_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ( member8690443509505302927_a_nat @ X2 @ Z2 )
=> ( ( ord_le8108555184339247974_a_nat @ Z2
@ ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ X6 )
& ( P @ X ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_300_Collect__restrict,axiom,
! [X6: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ X6 )
& ( P @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_301_Collect__restrict,axiom,
! [X6: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ord_le1147066620699065093_a_nat
@ ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ X6 )
& ( P @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_302_Collect__restrict,axiom,
! [X6: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ord_le1325389633284124927_a_nat
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ X6 )
& ( P @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_303_Collect__restrict,axiom,
! [X6: set_na3699693778330250182_a_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ord_le8108555184339247974_a_nat
@ ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ X6 )
& ( P @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_304_image__diff__subset,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( minus_7395159227704179404_a_nat @ ( image_5081948215111134021_a_nat @ F @ A ) @ ( image_5081948215111134021_a_nat @ F @ B2 ) ) @ ( image_5081948215111134021_a_nat @ F @ ( minus_7395159227704179404_a_nat @ A @ B2 ) ) ) ).
% image_diff_subset
thf(fact_305_image__diff__subset,axiom,
! [F: $o > $o,A: set_o,B2: set_o] : ( ord_less_eq_set_o @ ( minus_minus_set_o @ ( image_o_o2 @ F @ A ) @ ( image_o_o2 @ F @ B2 ) ) @ ( image_o_o2 @ F @ ( minus_minus_set_o @ A @ B2 ) ) ) ).
% image_diff_subset
thf(fact_306_Collect__subset,axiom,
! [A: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_307_Collect__subset,axiom,
! [A: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ord_le1147066620699065093_a_nat
@ ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_308_Collect__subset,axiom,
! [A: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ord_le1325389633284124927_a_nat
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_309_Collect__subset,axiom,
! [A: set_na3699693778330250182_a_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ord_le8108555184339247974_a_nat
@ ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_310_map__prod__surj__on,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,A7: set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat,B7: set_li6526943997496501093_a_nat] :
( ( ( image_5081948215111134021_a_nat @ F @ A )
= A7 )
=> ( ( ( image_5081948215111134021_a_nat @ G @ B2 )
= B7 )
=> ( ( image_710797976182302661_a_nat @ ( produc1245767001717907318_a_nat @ F @ G )
@ ( produc5624668685630548104_a_nat @ A
@ ^ [Uu2: list_Sum_sum_a_nat] : B2 ) )
= ( produc5624668685630548104_a_nat @ A7
@ ^ [Uu2: list_Sum_sum_a_nat] : B7 ) ) ) ) ).
% map_prod_surj_on
thf(fact_311_map__prod__surj__on,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,A7: set_li6526943997496501093_a_nat,G: $o > $o,B2: set_o,B7: set_o] :
( ( ( image_5081948215111134021_a_nat @ F @ A )
= A7 )
=> ( ( ( image_o_o2 @ G @ B2 )
= B7 )
=> ( ( image_2584502784606554117_nat_o @ ( produc5492656541449503926at_o_o @ F @ G )
@ ( produc8891875240947076123_nat_o @ A
@ ^ [Uu2: list_Sum_sum_a_nat] : B2 ) )
= ( produc8891875240947076123_nat_o @ A7
@ ^ [Uu2: list_Sum_sum_a_nat] : B7 ) ) ) ) ).
% map_prod_surj_on
thf(fact_312_map__prod__surj__on,axiom,
! [F: $o > $o,A: set_o,A7: set_o,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat,B7: set_li6526943997496501093_a_nat] :
( ( ( image_o_o2 @ F @ A )
= A7 )
=> ( ( ( image_5081948215111134021_a_nat @ G @ B2 )
= B7 )
=> ( ( image_2535540365225962373_a_nat @ ( produc8883616420147373238_a_nat @ F @ G )
@ ( produc5640064037308512181_a_nat @ A
@ ^ [Uu2: $o] : B2 ) )
= ( produc5640064037308512181_a_nat @ A7
@ ^ [Uu2: $o] : B7 ) ) ) ) ).
% map_prod_surj_on
thf(fact_313_map__prod__surj__on,axiom,
! [F: $o > $o,A: set_o,A7: set_o,G: $o > $o,B2: set_o,B7: set_o] :
( ( ( image_o_o2 @ F @ A )
= A7 )
=> ( ( ( image_o_o2 @ G @ B2 )
= B7 )
=> ( ( image_9131363867636255685od_o_o @ ( produc1928436952130685430_o_o_o @ F @ G )
@ ( product_Sigma_o_o @ A
@ ^ [Uu2: $o] : B2 ) )
= ( product_Sigma_o_o @ A7
@ ^ [Uu2: $o] : B7 ) ) ) ) ).
% map_prod_surj_on
thf(fact_314_image__Collect__subsetI,axiom,
! [P: $o > $o,F: $o > $o,B2: set_o] :
( ! [X3: $o] :
( ( P @ X3 )
=> ( member_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_o_o2 @ F @ ( collect_o @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_315_image__Collect__subsetI,axiom,
! [P: list_Sum_sum_a_nat > $o,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ! [X3: list_Sum_sum_a_nat] :
( ( P @ X3 )
=> ( member408289922725080238_a_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ ( collec7555443234367654128_a_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_316_image__Collect__subsetI,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,F: ( nat > sum_sum_a_nat ) > $o,B2: set_o] :
( ! [X3: nat > sum_sum_a_nat] :
( ( P @ X3 )
=> ( member_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_2376713081370839351_nat_o @ F @ ( collec5629555741568564177_a_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_317_image__Collect__subsetI,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,F: ( nat > sum_sum_a_nat ) > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ! [X3: nat > sum_sum_a_nat] :
( ( P @ X3 )
=> ( member408289922725080238_a_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le1147066620699065093_a_nat @ ( image_6721470456781115300_a_nat @ F @ ( collec5629555741568564177_a_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_318_image__Collect__subsetI,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,F: ( nat > sum_sum_a_nat ) > sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ! [X3: nat > sum_sum_a_nat] :
( ( P @ X3 )
=> ( member_Sum_sum_a_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le1325389633284124927_a_nat @ ( image_1617820922473287582_a_nat @ F @ ( collec5629555741568564177_a_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_319_SigmaE2,axiom,
! [A2: $o,B: $o,A: set_o,B2: $o > set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ ( product_Sigma_o_o @ A @ B2 ) )
=> ~ ( ( member_o @ A2 @ A )
=> ~ ( member_o @ B @ ( B2 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_320_SigmaE2,axiom,
! [A2: $o,B: sum_sum_a_nat,A: set_o,B2: $o > set_Sum_sum_a_nat] :
( ( member6925473634427674423_a_nat @ ( produc6394072109761784702_a_nat @ A2 @ B ) @ ( produc3081175752665019439_a_nat @ A @ B2 ) )
=> ~ ( ( member_o @ A2 @ A )
=> ~ ( member_Sum_sum_a_nat @ B @ ( B2 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_321_SigmaE2,axiom,
! [A2: sum_sum_a_nat,B: $o,A: set_Sum_sum_a_nat,B2: sum_sum_a_nat > set_o] :
( ( member6686738655598238761_nat_o @ ( produc8349727785981246704_nat_o @ A2 @ B ) @ ( produc5036831428884481441_nat_o @ A @ B2 ) )
=> ~ ( ( member_Sum_sum_a_nat @ A2 @ A )
=> ~ ( member_o @ B @ ( B2 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_322_SigmaE2,axiom,
! [A2: $o,B: list_Sum_sum_a_nat,A: set_o,B2: $o > set_li6526943997496501093_a_nat] :
( ( member2772425197143436605_a_nat @ ( produc7378284081089837956_a_nat @ A2 @ B ) @ ( produc5640064037308512181_a_nat @ A @ B2 ) )
=> ~ ( ( member_o @ A2 @ A )
=> ~ ( member408289922725080238_a_nat @ B @ ( B2 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_323_SigmaE2,axiom,
! [A2: list_Sum_sum_a_nat,B: $o,A: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat > set_o] :
( ( member1161018691129610787_nat_o @ ( produc1406723247873626090_nat_o @ A2 @ B ) @ ( produc8891875240947076123_nat_o @ A @ B2 ) )
=> ~ ( ( member408289922725080238_a_nat @ A2 @ A )
=> ~ ( member_o @ B @ ( B2 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_324_SigmaE2,axiom,
! [A2: sum_sum_a_nat,B: sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: sum_sum_a_nat > set_Sum_sum_a_nat] :
( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ A2 @ B ) @ ( produc3014407214216980488_a_nat @ A @ B2 ) )
=> ~ ( ( member_Sum_sum_a_nat @ A2 @ A )
=> ~ ( member_Sum_sum_a_nat @ B @ ( B2 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_325_SigmaE2,axiom,
! [A2: nat,B: set_li6526943997496501093_a_nat,A: set_nat,B2: nat > set_se4330304633200676677_a_nat] :
( ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ A2 @ B ) @ ( produc3542699778996723019_a_nat @ A @ B2 ) )
=> ~ ( ( member_nat @ A2 @ A )
=> ~ ( member5553968465346197646_a_nat @ B @ ( B2 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_326_SigmaE2,axiom,
! [A2: list_Sum_sum_a_nat,B: sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat > set_Sum_sum_a_nat] :
( ( member6493994536747010826_a_nat @ ( produc5977302043283085393_a_nat @ A2 @ B ) @ ( produc1079312233239997058_a_nat @ A @ B2 ) )
=> ~ ( ( member408289922725080238_a_nat @ A2 @ A )
=> ~ ( member_Sum_sum_a_nat @ B @ ( B2 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_327_SigmaE2,axiom,
! [A2: sum_sum_a_nat,B: list_Sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( member312212173870163606_a_nat @ ( produc6350064662657521885_a_nat @ A2 @ B ) @ ( produc1452074852614433550_a_nat @ A @ B2 ) )
=> ~ ( ( member_Sum_sum_a_nat @ A2 @ A )
=> ~ ( member408289922725080238_a_nat @ B @ ( B2 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_328_SigmaE2,axiom,
! [A2: list_Sum_sum_a_nat,B: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ A2 @ B ) @ ( produc5624668685630548104_a_nat @ A @ B2 ) )
=> ~ ( ( member408289922725080238_a_nat @ A2 @ A )
=> ~ ( member408289922725080238_a_nat @ B @ ( B2 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_329_SigmaD2,axiom,
! [A2: set_a,B: produc5986856040925105290_a_nat,A: set_set_a,B2: set_a > set_Pr1465208939327205952_a_nat] :
( ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ A2 @ B ) @ ( produc5217640286535863200_a_nat @ A @ B2 ) )
=> ( member3071249055146688545_a_nat @ B @ ( B2 @ A2 ) ) ) ).
% SigmaD2
thf(fact_330_SigmaD2,axiom,
! [A2: nat,B: set_li6526943997496501093_a_nat,A: set_nat,B2: nat > set_se4330304633200676677_a_nat] :
( ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ A2 @ B ) @ ( produc3542699778996723019_a_nat @ A @ B2 ) )
=> ( member5553968465346197646_a_nat @ B @ ( B2 @ A2 ) ) ) ).
% SigmaD2
thf(fact_331_SigmaD1,axiom,
! [A2: set_a,B: produc5986856040925105290_a_nat,A: set_set_a,B2: set_a > set_Pr1465208939327205952_a_nat] :
( ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ A2 @ B ) @ ( produc5217640286535863200_a_nat @ A @ B2 ) )
=> ( member_set_a @ A2 @ A ) ) ).
% SigmaD1
thf(fact_332_SigmaD1,axiom,
! [A2: nat,B: set_li6526943997496501093_a_nat,A: set_nat,B2: nat > set_se4330304633200676677_a_nat] :
( ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ A2 @ B ) @ ( produc3542699778996723019_a_nat @ A @ B2 ) )
=> ( member_nat @ A2 @ A ) ) ).
% SigmaD1
thf(fact_333_SigmaE,axiom,
! [C3: product_prod_o_o,A: set_o,B2: $o > set_o] :
( ( member7466972457876170832od_o_o @ C3 @ ( product_Sigma_o_o @ A @ B2 ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ! [Y4: $o] :
( ( member_o @ Y4 @ ( B2 @ X3 ) )
=> ( C3
!= ( product_Pair_o_o @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_334_SigmaE,axiom,
! [C3: produc7898449166577977870_a_nat,A: set_o,B2: $o > set_Sum_sum_a_nat] :
( ( member6925473634427674423_a_nat @ C3 @ ( produc3081175752665019439_a_nat @ A @ B2 ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ! [Y4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Y4 @ ( B2 @ X3 ) )
=> ( C3
!= ( produc6394072109761784702_a_nat @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_335_SigmaE,axiom,
! [C3: produc7659714187748542208_nat_o,A: set_Sum_sum_a_nat,B2: sum_sum_a_nat > set_o] :
( ( member6686738655598238761_nat_o @ C3 @ ( produc5036831428884481441_nat_o @ A @ B2 ) )
=> ~ ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ A )
=> ! [Y4: $o] :
( ( member_o @ Y4 @ ( B2 @ X3 ) )
=> ( C3
!= ( produc8349727785981246704_nat_o @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_336_SigmaE,axiom,
! [C3: produc5889817346055602324_a_nat,A: set_o,B2: $o > set_li6526943997496501093_a_nat] :
( ( member2772425197143436605_a_nat @ C3 @ ( produc5640064037308512181_a_nat @ A @ B2 ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ! [Y4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Y4 @ ( B2 @ X3 ) )
=> ( C3
!= ( produc7378284081089837956_a_nat @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_337_SigmaE,axiom,
! [C3: produc4278410840041776506_nat_o,A: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat > set_o] :
( ( member1161018691129610787_nat_o @ C3 @ ( produc8891875240947076123_nat_o @ A @ B2 ) )
=> ~ ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ! [Y4: $o] :
( ( member_o @ Y4 @ ( B2 @ X3 ) )
=> ( C3
!= ( produc1406723247873626090_nat_o @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_338_SigmaE,axiom,
! [C3: produc7017002724195966439_a_nat,A: set_Sum_sum_a_nat,B2: sum_sum_a_nat > set_Sum_sum_a_nat] :
( ( member3723442691059620112_a_nat @ C3 @ ( produc3014407214216980488_a_nat @ A @ B2 ) )
=> ~ ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ A )
=> ! [Y4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Y4 @ ( B2 @ X3 ) )
=> ( C3
!= ( produc1212125651291703639_a_nat @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_339_SigmaE,axiom,
! [C3: produc5986856040925105290_a_nat,A: set_nat,B2: nat > set_se4330304633200676677_a_nat] :
( ( member3071249055146688545_a_nat @ C3 @ ( produc3542699778996723019_a_nat @ A @ B2 ) )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ! [Y4: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ Y4 @ ( B2 @ X3 ) )
=> ( C3
!= ( produc4622690550020908860_a_nat @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_340_SigmaE,axiom,
! [C3: produc1461395728222533729_a_nat,A: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat > set_Sum_sum_a_nat] :
( ( member6493994536747010826_a_nat @ C3 @ ( produc1079312233239997058_a_nat @ A @ B2 ) )
=> ~ ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ! [Y4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Y4 @ ( B2 @ X3 ) )
=> ( C3
!= ( produc5977302043283085393_a_nat @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_341_SigmaE,axiom,
! [C3: produc4502985402200462317_a_nat,A: set_Sum_sum_a_nat,B2: sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( member312212173870163606_a_nat @ C3 @ ( produc1452074852614433550_a_nat @ A @ B2 ) )
=> ~ ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ A )
=> ! [Y4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Y4 @ ( B2 @ X3 ) )
=> ( C3
!= ( produc6350064662657521885_a_nat @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_342_SigmaE,axiom,
! [C3: produc5001885624171833703_a_nat,A: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( member7457213283480048528_a_nat @ C3 @ ( produc5624668685630548104_a_nat @ A @ B2 ) )
=> ~ ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ! [Y4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Y4 @ ( B2 @ X3 ) )
=> ( C3
!= ( produc7990843422341522135_a_nat @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_343_prod__fun__imageE,axiom,
! [C3: produc8208987855206359447_a_nat,F: set_a > set_a,G: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,R2: set_Pr8537526423911375309_a_nat] :
( ( member6515367127663331886_a_nat @ C3 @ ( image_6916668542591666485_a_nat @ ( produc5812298523581686028_a_nat @ F @ G ) @ R2 ) )
=> ~ ! [X3: set_a,Y4: produc5986856040925105290_a_nat] :
( ( C3
= ( produc1626021218680649873_a_nat @ ( F @ X3 ) @ ( G @ Y4 ) ) )
=> ~ ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ X3 @ Y4 ) @ R2 ) ) ) ).
% prod_fun_imageE
thf(fact_344_prod__fun__imageE,axiom,
! [C3: produc8208987855206359447_a_nat,F: nat > set_a,G: set_li6526943997496501093_a_nat > produc5986856040925105290_a_nat,R2: set_Pr1465208939327205952_a_nat] :
( ( member6515367127663331886_a_nat @ C3 @ ( image_9052215851415248424_a_nat @ ( produc3382755674669967065_a_nat @ F @ G ) @ R2 ) )
=> ~ ! [X3: nat,Y4: set_li6526943997496501093_a_nat] :
( ( C3
= ( produc1626021218680649873_a_nat @ ( F @ X3 ) @ ( G @ Y4 ) ) )
=> ~ ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ X3 @ Y4 ) @ R2 ) ) ) ).
% prod_fun_imageE
thf(fact_345_prod__fun__imageE,axiom,
! [C3: produc5986856040925105290_a_nat,F: set_a > nat,G: produc5986856040925105290_a_nat > set_li6526943997496501093_a_nat,R2: set_Pr8537526423911375309_a_nat] :
( ( member3071249055146688545_a_nat @ C3 @ ( image_3110382862205598248_a_nat @ ( produc5294700245526393121_a_nat @ F @ G ) @ R2 ) )
=> ~ ! [X3: set_a,Y4: produc5986856040925105290_a_nat] :
( ( C3
= ( produc4622690550020908860_a_nat @ ( F @ X3 ) @ ( G @ Y4 ) ) )
=> ~ ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ X3 @ Y4 ) @ R2 ) ) ) ).
% prod_fun_imageE
thf(fact_346_prod__fun__imageE,axiom,
! [C3: produc5986856040925105290_a_nat,F: nat > nat,G: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,R2: set_Pr1465208939327205952_a_nat] :
( ( member3071249055146688545_a_nat @ C3 @ ( image_933174759805388315_a_nat @ ( produc6714918141915149656_a_nat @ F @ G ) @ R2 ) )
=> ~ ! [X3: nat,Y4: set_li6526943997496501093_a_nat] :
( ( C3
= ( produc4622690550020908860_a_nat @ ( F @ X3 ) @ ( G @ Y4 ) ) )
=> ~ ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ X3 @ Y4 ) @ R2 ) ) ) ).
% prod_fun_imageE
thf(fact_347_top__greatest,axiom,
! [A2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ A2 @ top_to795618464972521135_a_nat ) ).
% top_greatest
thf(fact_348_top__greatest,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ A2 @ top_top_set_o ) ).
% top_greatest
thf(fact_349_top_Oextremum__unique,axiom,
! [A2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ top_to795618464972521135_a_nat @ A2 )
= ( A2 = top_to795618464972521135_a_nat ) ) ).
% top.extremum_unique
thf(fact_350_top_Oextremum__unique,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ top_top_set_o @ A2 )
= ( A2 = top_top_set_o ) ) ).
% top.extremum_unique
thf(fact_351_top_Oextremum__uniqueI,axiom,
! [A2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ top_to795618464972521135_a_nat @ A2 )
=> ( A2 = top_to795618464972521135_a_nat ) ) ).
% top.extremum_uniqueI
thf(fact_352_top_Oextremum__uniqueI,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ top_top_set_o @ A2 )
=> ( A2 = top_top_set_o ) ) ).
% top.extremum_uniqueI
thf(fact_353_subset__UNIV,axiom,
! [A: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ A @ top_to795618464972521135_a_nat ) ).
% subset_UNIV
thf(fact_354_subset__UNIV,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ A @ top_top_set_o ) ).
% subset_UNIV
thf(fact_355_image__mono,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B2 )
=> ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ A ) @ ( image_5081948215111134021_a_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_356_image__mono,axiom,
! [A: set_o,B2: set_o,F: $o > $o] :
( ( ord_less_eq_set_o @ A @ B2 )
=> ( ord_less_eq_set_o @ ( image_o_o2 @ F @ A ) @ ( image_o_o2 @ F @ B2 ) ) ) ).
% image_mono
thf(fact_357_image__subsetI,axiom,
! [A: set_o,F: $o > $o,B2: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_o_o2 @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_358_image__subsetI,axiom,
! [A: set_o,F: $o > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member408289922725080238_a_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le1147066620699065093_a_nat @ ( image_733135354807394034_a_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_359_image__subsetI,axiom,
! [A: set_o,F: $o > sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le1325389633284124927_a_nat @ ( image_4139480514073730540_a_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_360_image__subsetI,axiom,
! [A: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > $o,B2: set_o] :
( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ( member_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_3984946558445957976_nat_o @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_361_image__subsetI,axiom,
! [A: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ( member408289922725080238_a_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_362_image__subsetI,axiom,
! [A: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le1325389633284124927_a_nat @ ( image_6705138127771971519_a_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_363_image__subsetI,axiom,
! [A: set_Sum_sum_a_nat,F: sum_sum_a_nat > $o,B2: set_o] :
( ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ A )
=> ( member_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_6095136190293192542_nat_o @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_364_image__subsetI,axiom,
! [A: set_Sum_sum_a_nat,F: sum_sum_a_nat > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ A )
=> ( member408289922725080238_a_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le1147066620699065093_a_nat @ ( image_7077900747146408011_a_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_365_image__subsetI,axiom,
! [A: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le1325389633284124927_a_nat @ ( image_7142520692256960453_a_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_366_subset__imageE,axiom,
! [B2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B2 @ ( image_5081948215111134021_a_nat @ F @ A ) )
=> ~ ! [C4: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ C4 @ A )
=> ( B2
!= ( image_5081948215111134021_a_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_367_subset__imageE,axiom,
! [B2: set_o,F: $o > $o,A: set_o] :
( ( ord_less_eq_set_o @ B2 @ ( image_o_o2 @ F @ A ) )
=> ~ ! [C4: set_o] :
( ( ord_less_eq_set_o @ C4 @ A )
=> ( B2
!= ( image_o_o2 @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_368_image__subset__iff,axiom,
! [F: $o > $o,A: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ ( image_o_o2 @ F @ A ) @ B2 )
= ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( member_o @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_369_image__subset__iff,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ A ) @ B2 )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
=> ( member408289922725080238_a_nat @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_370_subset__image__iff,axiom,
! [B2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B2 @ ( image_5081948215111134021_a_nat @ F @ A ) )
= ( ? [AA: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ AA @ A )
& ( B2
= ( image_5081948215111134021_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_371_subset__image__iff,axiom,
! [B2: set_o,F: $o > $o,A: set_o] :
( ( ord_less_eq_set_o @ B2 @ ( image_o_o2 @ F @ A ) )
= ( ? [AA: set_o] :
( ( ord_less_eq_set_o @ AA @ A )
& ( B2
= ( image_o_o2 @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_372_curryE,axiom,
! [F: produc8208987855206359447_a_nat > $o,A2: set_a,B: produc5986856040925105290_a_nat] :
( ( produc6259199808005150713_nat_o @ F @ A2 @ B )
=> ( F @ ( produc1626021218680649873_a_nat @ A2 @ B ) ) ) ).
% curryE
thf(fact_373_curryE,axiom,
! [F: produc5986856040925105290_a_nat > $o,A2: nat,B: set_li6526943997496501093_a_nat] :
( ( produc4568951772394601358_nat_o @ F @ A2 @ B )
=> ( F @ ( produc4622690550020908860_a_nat @ A2 @ B ) ) ) ).
% curryE
thf(fact_374_curryD,axiom,
! [F: produc8208987855206359447_a_nat > $o,A2: set_a,B: produc5986856040925105290_a_nat] :
( ( produc6259199808005150713_nat_o @ F @ A2 @ B )
=> ( F @ ( produc1626021218680649873_a_nat @ A2 @ B ) ) ) ).
% curryD
thf(fact_375_curryD,axiom,
! [F: produc5986856040925105290_a_nat > $o,A2: nat,B: set_li6526943997496501093_a_nat] :
( ( produc4568951772394601358_nat_o @ F @ A2 @ B )
=> ( F @ ( produc4622690550020908860_a_nat @ A2 @ B ) ) ) ).
% curryD
thf(fact_376_range__subsetD,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat,I2: list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ top_to6433055325616222389_a_nat ) @ B2 )
=> ( member408289922725080238_a_nat @ ( F @ I2 ) @ B2 ) ) ).
% range_subsetD
thf(fact_377_range__subsetD,axiom,
! [F: sum_sum_a_nat > $o,B2: set_o,I2: sum_sum_a_nat] :
( ( ord_less_eq_set_o @ ( image_6095136190293192542_nat_o @ F @ top_to795618464972521135_a_nat ) @ B2 )
=> ( member_o @ ( F @ I2 ) @ B2 ) ) ).
% range_subsetD
thf(fact_378_range__subsetD,axiom,
! [F: sum_sum_a_nat > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat,I2: sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( image_7077900747146408011_a_nat @ F @ top_to795618464972521135_a_nat ) @ B2 )
=> ( member408289922725080238_a_nat @ ( F @ I2 ) @ B2 ) ) ).
% range_subsetD
thf(fact_379_range__subsetD,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,B2: set_Sum_sum_a_nat,I2: sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( image_7142520692256960453_a_nat @ F @ top_to795618464972521135_a_nat ) @ B2 )
=> ( member_Sum_sum_a_nat @ ( F @ I2 ) @ B2 ) ) ).
% range_subsetD
thf(fact_380_range__subsetD,axiom,
! [F: $o > $o,B2: set_o,I2: $o] :
( ( ord_less_eq_set_o @ ( image_o_o2 @ F @ top_top_set_o ) @ B2 )
=> ( member_o @ ( F @ I2 ) @ B2 ) ) ).
% range_subsetD
thf(fact_381_range__subsetD,axiom,
! [F: $o > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat,I2: $o] :
( ( ord_le1147066620699065093_a_nat @ ( image_733135354807394034_a_nat @ F @ top_top_set_o ) @ B2 )
=> ( member408289922725080238_a_nat @ ( F @ I2 ) @ B2 ) ) ).
% range_subsetD
thf(fact_382_range__subsetD,axiom,
! [F: $o > sum_sum_a_nat,B2: set_Sum_sum_a_nat,I2: $o] :
( ( ord_le1325389633284124927_a_nat @ ( image_4139480514073730540_a_nat @ F @ top_top_set_o ) @ B2 )
=> ( member_Sum_sum_a_nat @ ( F @ I2 ) @ B2 ) ) ).
% range_subsetD
thf(fact_383_esat__UNIV__cong,axiom,
! [Phi: fo_fmla_a_b,AD2: set_a,Sigma2: nat > sum_sum_a_nat,Tau: nat > sum_sum_a_nat,I: product_prod_b_nat > set_list_a] :
( ( ad_agr_a_b_nat @ Phi @ AD2 @ Sigma2 @ Tau )
=> ( ( ord_less_eq_set_a @ ( act_edom_a_b @ Phi @ I ) @ AD2 )
=> ( ( esat_a_b @ Phi @ I @ Sigma2 @ top_to795618464972521135_a_nat )
= ( esat_a_b @ Phi @ I @ Tau @ top_to795618464972521135_a_nat ) ) ) ) ).
% esat_UNIV_cong
thf(fact_384_all__subset__image,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,P: set_li6526943997496501093_a_nat > $o] :
( ( ! [B6: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B6 @ ( image_5081948215111134021_a_nat @ F @ A ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B6 @ A )
=> ( P @ ( image_5081948215111134021_a_nat @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_385_all__subset__image,axiom,
! [F: $o > $o,A: set_o,P: set_o > $o] :
( ( ! [B6: set_o] :
( ( ord_less_eq_set_o @ B6 @ ( image_o_o2 @ F @ A ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_o] :
( ( ord_less_eq_set_o @ B6 @ A )
=> ( P @ ( image_o_o2 @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_386_DiffI,axiom,
! [C3: $o,A: set_o,B2: set_o] :
( ( member_o @ C3 @ A )
=> ( ~ ( member_o @ C3 @ B2 )
=> ( member_o @ C3 @ ( minus_minus_set_o @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_387_DiffI,axiom,
! [C3: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C3 @ A )
=> ( ~ ( member408289922725080238_a_nat @ C3 @ B2 )
=> ( member408289922725080238_a_nat @ C3 @ ( minus_7395159227704179404_a_nat @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_388_DiffI,axiom,
! [C3: sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C3 @ A )
=> ( ~ ( member_Sum_sum_a_nat @ C3 @ B2 )
=> ( member_Sum_sum_a_nat @ C3 @ ( minus_1134630996077396038_a_nat @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_389_Diff__iff,axiom,
! [C3: $o,A: set_o,B2: set_o] :
( ( member_o @ C3 @ ( minus_minus_set_o @ A @ B2 ) )
= ( ( member_o @ C3 @ A )
& ~ ( member_o @ C3 @ B2 ) ) ) ).
% Diff_iff
thf(fact_390_Diff__iff,axiom,
! [C3: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C3 @ ( minus_7395159227704179404_a_nat @ A @ B2 ) )
= ( ( member408289922725080238_a_nat @ C3 @ A )
& ~ ( member408289922725080238_a_nat @ C3 @ B2 ) ) ) ).
% Diff_iff
thf(fact_391_Diff__iff,axiom,
! [C3: sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C3 @ ( minus_1134630996077396038_a_nat @ A @ B2 ) )
= ( ( member_Sum_sum_a_nat @ C3 @ A )
& ~ ( member_Sum_sum_a_nat @ C3 @ B2 ) ) ) ).
% Diff_iff
thf(fact_392_swap__simp,axiom,
! [X2: produc5986856040925105290_a_nat,Y2: set_a] :
( ( produc4582320955043612251_set_a @ ( produc6563110914492894203_set_a @ X2 @ Y2 ) )
= ( produc1626021218680649873_a_nat @ Y2 @ X2 ) ) ).
% swap_simp
thf(fact_393_swap__simp,axiom,
! [X2: set_li6526943997496501093_a_nat,Y2: nat] :
( ( produc7974059260437611070at_nat @ ( produc3305532969823496158at_nat @ X2 @ Y2 ) )
= ( produc4622690550020908860_a_nat @ Y2 @ X2 ) ) ).
% swap_simp
thf(fact_394_swap__simp,axiom,
! [X2: set_a,Y2: produc5986856040925105290_a_nat] :
( ( produc8868603296086143729_a_nat @ ( produc1626021218680649873_a_nat @ X2 @ Y2 ) )
= ( produc6563110914492894203_set_a @ Y2 @ X2 ) ) ).
% swap_simp
thf(fact_395_swap__simp,axiom,
! [X2: nat,Y2: set_li6526943997496501093_a_nat] :
( ( produc67844803780247964_a_nat @ ( produc4622690550020908860_a_nat @ X2 @ Y2 ) )
= ( produc3305532969823496158at_nat @ Y2 @ X2 ) ) ).
% swap_simp
thf(fact_396_pair__in__swap__image,axiom,
! [Y2: produc5986856040925105290_a_nat,X2: set_a,A: set_Pr8537526423911375309_a_nat] :
( ( member5646307212991795808_set_a @ ( produc6563110914492894203_set_a @ Y2 @ X2 ) @ ( image_6047608627920130407_set_a @ produc8868603296086143729_a_nat @ A ) )
= ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ X2 @ Y2 ) @ A ) ) ).
% pair_in_swap_image
thf(fact_397_pair__in__swap__image,axiom,
! [Y2: set_li6526943997496501093_a_nat,X2: nat,A: set_Pr1465208939327205952_a_nat] :
( ( member5309323187671517307at_nat @ ( produc3305532969823496158at_nat @ Y2 @ X2 ) @ ( image_3171248892330217077at_nat @ produc67844803780247964_a_nat @ A ) )
= ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ X2 @ Y2 ) @ A ) ) ).
% pair_in_swap_image
thf(fact_398_pair__in__swap__image,axiom,
! [Y2: set_a,X2: produc5986856040925105290_a_nat,A: set_Pr319891966805635327_set_a] :
( ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ Y2 @ X2 ) @ ( image_9020305190384351591_a_nat @ produc4582320955043612251_set_a @ A ) )
= ( member5646307212991795808_set_a @ ( produc6563110914492894203_set_a @ X2 @ Y2 ) @ A ) ) ).
% pair_in_swap_image
thf(fact_399_pair__in__swap__image,axiom,
! [Y2: nat,X2: set_li6526943997496501093_a_nat,A: set_Pr4681753670034042266at_nat] :
( ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ Y2 @ X2 ) @ ( image_2804021043688564853_a_nat @ produc7974059260437611070at_nat @ A ) )
= ( member5309323187671517307at_nat @ ( produc3305532969823496158at_nat @ X2 @ Y2 ) @ A ) ) ).
% pair_in_swap_image
thf(fact_400_set__diff__eq,axiom,
( minus_minus_set_o
= ( ^ [A6: set_o,B6: set_o] :
( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A6 )
& ~ ( member_o @ X @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_401_set__diff__eq,axiom,
( minus_7395159227704179404_a_nat
= ( ^ [A6: set_li6526943997496501093_a_nat,B6: set_li6526943997496501093_a_nat] :
( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A6 )
& ~ ( member408289922725080238_a_nat @ X @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_402_set__diff__eq,axiom,
( minus_1134630996077396038_a_nat
= ( ^ [A6: set_Sum_sum_a_nat,B6: set_Sum_sum_a_nat] :
( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A6 )
& ~ ( member_Sum_sum_a_nat @ X @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_403_set__diff__eq,axiom,
( minus_5517490076408937517_a_nat
= ( ^ [A6: set_na3699693778330250182_a_nat,B6: set_na3699693778330250182_a_nat] :
( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ A6 )
& ~ ( member8690443509505302927_a_nat @ X @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_404_DiffE,axiom,
! [C3: $o,A: set_o,B2: set_o] :
( ( member_o @ C3 @ ( minus_minus_set_o @ A @ B2 ) )
=> ~ ( ( member_o @ C3 @ A )
=> ( member_o @ C3 @ B2 ) ) ) ).
% DiffE
thf(fact_405_DiffE,axiom,
! [C3: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C3 @ ( minus_7395159227704179404_a_nat @ A @ B2 ) )
=> ~ ( ( member408289922725080238_a_nat @ C3 @ A )
=> ( member408289922725080238_a_nat @ C3 @ B2 ) ) ) ).
% DiffE
thf(fact_406_DiffE,axiom,
! [C3: sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C3 @ ( minus_1134630996077396038_a_nat @ A @ B2 ) )
=> ~ ( ( member_Sum_sum_a_nat @ C3 @ A )
=> ( member_Sum_sum_a_nat @ C3 @ B2 ) ) ) ).
% DiffE
thf(fact_407_DiffD1,axiom,
! [C3: $o,A: set_o,B2: set_o] :
( ( member_o @ C3 @ ( minus_minus_set_o @ A @ B2 ) )
=> ( member_o @ C3 @ A ) ) ).
% DiffD1
thf(fact_408_DiffD1,axiom,
! [C3: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C3 @ ( minus_7395159227704179404_a_nat @ A @ B2 ) )
=> ( member408289922725080238_a_nat @ C3 @ A ) ) ).
% DiffD1
thf(fact_409_DiffD1,axiom,
! [C3: sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C3 @ ( minus_1134630996077396038_a_nat @ A @ B2 ) )
=> ( member_Sum_sum_a_nat @ C3 @ A ) ) ).
% DiffD1
thf(fact_410_DiffD2,axiom,
! [C3: $o,A: set_o,B2: set_o] :
( ( member_o @ C3 @ ( minus_minus_set_o @ A @ B2 ) )
=> ~ ( member_o @ C3 @ B2 ) ) ).
% DiffD2
thf(fact_411_DiffD2,axiom,
! [C3: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C3 @ ( minus_7395159227704179404_a_nat @ A @ B2 ) )
=> ~ ( member408289922725080238_a_nat @ C3 @ B2 ) ) ).
% DiffD2
thf(fact_412_DiffD2,axiom,
! [C3: sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C3 @ ( minus_1134630996077396038_a_nat @ A @ B2 ) )
=> ~ ( member_Sum_sum_a_nat @ C3 @ B2 ) ) ).
% DiffD2
thf(fact_413_minus__set__def,axiom,
( minus_minus_set_o
= ( ^ [A6: set_o,B6: set_o] :
( collect_o
@ ( minus_minus_o_o
@ ^ [X: $o] : ( member_o @ X @ A6 )
@ ^ [X: $o] : ( member_o @ X @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_414_minus__set__def,axiom,
( minus_7395159227704179404_a_nat
= ( ^ [A6: set_li6526943997496501093_a_nat,B6: set_li6526943997496501093_a_nat] :
( collec7555443234367654128_a_nat
@ ( minus_5799240027661600017_nat_o
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ A6 )
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_415_minus__set__def,axiom,
( minus_1134630996077396038_a_nat
= ( ^ [A6: set_Sum_sum_a_nat,B6: set_Sum_sum_a_nat] :
( collec7073057861543223018_a_nat
@ ( minus_2198590880655318551_nat_o
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ A6 )
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_416_minus__set__def,axiom,
( minus_5517490076408937517_a_nat
= ( ^ [A6: set_na3699693778330250182_a_nat,B6: set_na3699693778330250182_a_nat] :
( collec5629555741568564177_a_nat
@ ( minus_7403279044699189488_nat_o
@ ^ [X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ A6 )
@ ^ [X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_417_pred__subset__eq2,axiom,
! [R2: set_Pr8537526423911375309_a_nat,S: set_Pr8537526423911375309_a_nat] :
( ( ord_le7632391532701932844_nat_o
@ ^ [X: set_a,Y3: produc5986856040925105290_a_nat] : ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ X @ Y3 ) @ R2 )
@ ^ [X: set_a,Y3: produc5986856040925105290_a_nat] : ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ X @ Y3 ) @ S ) )
= ( ord_le8227873432414528045_a_nat @ R2 @ S ) ) ).
% pred_subset_eq2
thf(fact_418_pred__subset__eq2,axiom,
! [R2: set_Pr1465208939327205952_a_nat,S: set_Pr1465208939327205952_a_nat] :
( ( ord_le7732214529591280039_nat_o
@ ^ [X: nat,Y3: set_li6526943997496501093_a_nat] : ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ X @ Y3 ) @ R2 )
@ ^ [X: nat,Y3: set_li6526943997496501093_a_nat] : ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ X @ Y3 ) @ S ) )
= ( ord_le2863945612512301728_a_nat @ R2 @ S ) ) ).
% pred_subset_eq2
thf(fact_419_subrelI,axiom,
! [R3: set_Pr8537526423911375309_a_nat,S2: set_Pr8537526423911375309_a_nat] :
( ! [X3: set_a,Y4: produc5986856040925105290_a_nat] :
( ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ X3 @ Y4 ) @ R3 )
=> ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ X3 @ Y4 ) @ S2 ) )
=> ( ord_le8227873432414528045_a_nat @ R3 @ S2 ) ) ).
% subrelI
thf(fact_420_subrelI,axiom,
! [R3: set_Pr1465208939327205952_a_nat,S2: set_Pr1465208939327205952_a_nat] :
( ! [X3: nat,Y4: set_li6526943997496501093_a_nat] :
( ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ X3 @ Y4 ) @ R3 )
=> ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ X3 @ Y4 ) @ S2 ) )
=> ( ord_le2863945612512301728_a_nat @ R3 @ S2 ) ) ).
% subrelI
thf(fact_421_pred__subset__eq,axiom,
! [R2: set_o,S: set_o] :
( ( ord_less_eq_o_o
@ ^ [X: $o] : ( member_o @ X @ R2 )
@ ^ [X: $o] : ( member_o @ X @ S ) )
= ( ord_less_eq_set_o @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_422_pred__subset__eq,axiom,
! [R2: set_li6526943997496501093_a_nat,S: set_li6526943997496501093_a_nat] :
( ( ord_le8737610411969296920_nat_o
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ R2 )
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ S ) )
= ( ord_le1147066620699065093_a_nat @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_423_pred__subset__eq,axiom,
! [R2: set_Sum_sum_a_nat,S: set_Sum_sum_a_nat] :
( ( ord_le1477630214076318366_nat_o
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ R2 )
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ S ) )
= ( ord_le1325389633284124927_a_nat @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_424_less__eq__set__def,axiom,
( ord_less_eq_set_o
= ( ^ [A6: set_o,B6: set_o] :
( ord_less_eq_o_o
@ ^ [X: $o] : ( member_o @ X @ A6 )
@ ^ [X: $o] : ( member_o @ X @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_425_less__eq__set__def,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A6: set_li6526943997496501093_a_nat,B6: set_li6526943997496501093_a_nat] :
( ord_le8737610411969296920_nat_o
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ A6 )
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_426_less__eq__set__def,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [A6: set_Sum_sum_a_nat,B6: set_Sum_sum_a_nat] :
( ord_le1477630214076318366_nat_o
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ A6 )
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_427_proj__fmla__conj__sub,axiom,
! [Psi: fo_fmla_a_b,I: product_prod_b_nat > set_list_a,AD2: set_a,Phi: fo_fmla_a_b] :
( ( ord_less_eq_set_a @ ( act_edom_a_b @ Psi @ I ) @ AD2 )
=> ( ord_le1147066620699065093_a_nat
@ ( inf_in3249246906714053971_a_nat
@ ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ AD2 )
@ ( proj_f6037546931041448027_a_nat @ ( fo_Conj_a_b @ Phi @ Psi )
@ ( collec5629555741568564177_a_nat
@ ^ [Sigma: nat > sum_sum_a_nat] : ( esat_a_b @ Phi @ I @ Sigma @ top_to795618464972521135_a_nat ) ) ) )
@ ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ AD2 )
@ ( proj_f6037546931041448027_a_nat @ ( fo_Conj_a_b @ Phi @ Psi )
@ ( collec5629555741568564177_a_nat
@ ^ [Sigma: nat > sum_sum_a_nat] : ( esat_a_b @ Psi @ I @ Sigma @ top_to795618464972521135_a_nat ) ) ) ) )
@ ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ AD2 )
@ ( proj_f6037546931041448027_a_nat @ ( fo_Conj_a_b @ Phi @ Psi )
@ ( collec5629555741568564177_a_nat
@ ^ [Sigma: nat > sum_sum_a_nat] : ( esat_a_b @ ( fo_Conj_a_b @ Phi @ Psi ) @ I @ Sigma @ top_to795618464972521135_a_nat ) ) ) ) ) ) ).
% proj_fmla_conj_sub
thf(fact_428_subset__Collect__iff,axiom,
! [B2: set_o,A: set_o,P: $o > $o] :
( ( ord_less_eq_set_o @ B2 @ A )
=> ( ( ord_less_eq_set_o @ B2
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ B2 )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_429_subset__Collect__iff,axiom,
! [B2: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( ord_le1147066620699065093_a_nat @ B2 @ A )
=> ( ( ord_le1147066620699065093_a_nat @ B2
@ ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ B2 )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_430_subset__Collect__iff,axiom,
! [B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( ord_le1325389633284124927_a_nat @ B2 @ A )
=> ( ( ord_le1325389633284124927_a_nat @ B2
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ B2 )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_431_subset__Collect__iff,axiom,
! [B2: set_na3699693778330250182_a_nat,A: set_na3699693778330250182_a_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ( ord_le8108555184339247974_a_nat @ B2 @ A )
=> ( ( ord_le8108555184339247974_a_nat @ B2
@ ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ B2 )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_432_subset__CollectI,axiom,
! [B2: set_o,A: set_o,Q: $o > $o,P: $o > $o] :
( ( ord_less_eq_set_o @ B2 @ A )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ B2 )
& ( Q @ X ) ) )
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_433_subset__CollectI,axiom,
! [B2: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,Q: list_Sum_sum_a_nat > $o,P: list_Sum_sum_a_nat > $o] :
( ( ord_le1147066620699065093_a_nat @ B2 @ A )
=> ( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ B2 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le1147066620699065093_a_nat
@ ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ B2 )
& ( Q @ X ) ) )
@ ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_434_subset__CollectI,axiom,
! [B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,Q: sum_sum_a_nat > $o,P: sum_sum_a_nat > $o] :
( ( ord_le1325389633284124927_a_nat @ B2 @ A )
=> ( ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ B2 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le1325389633284124927_a_nat
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ B2 )
& ( Q @ X ) ) )
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_435_subset__CollectI,axiom,
! [B2: set_na3699693778330250182_a_nat,A: set_na3699693778330250182_a_nat,Q: ( nat > sum_sum_a_nat ) > $o,P: ( nat > sum_sum_a_nat ) > $o] :
( ( ord_le8108555184339247974_a_nat @ B2 @ A )
=> ( ! [X3: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X3 @ B2 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le8108555184339247974_a_nat
@ ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ B2 )
& ( Q @ X ) ) )
@ ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_436_conj__subset__def,axiom,
! [A: set_na3699693778330250182_a_nat,P: ( nat > sum_sum_a_nat ) > $o,Q: ( nat > sum_sum_a_nat ) > $o] :
( ( ord_le8108555184339247974_a_nat @ A
@ ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( P @ X )
& ( Q @ X ) ) ) )
= ( ( ord_le8108555184339247974_a_nat @ A @ ( collec5629555741568564177_a_nat @ P ) )
& ( ord_le8108555184339247974_a_nat @ A @ ( collec5629555741568564177_a_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_437_Gr__incl,axiom,
! [A: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le4837030659158318311_a_nat @ ( bNF_Gr5797238915573336290_a_nat @ A @ F )
@ ( produc5624668685630548104_a_nat @ A
@ ^ [Uu2: list_Sum_sum_a_nat] : B2 ) )
= ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ A ) @ B2 ) ) ).
% Gr_incl
thf(fact_438_Gr__incl,axiom,
! [A: set_o,F: $o > $o,B2: set_o] :
( ( ord_le192195778054999463od_o_o @ ( bNF_Gr_o_o @ A @ F )
@ ( product_Sigma_o_o @ A
@ ^ [Uu2: $o] : B2 ) )
= ( ord_less_eq_set_o @ ( image_o_o2 @ F @ A ) @ B2 ) ) ).
% Gr_incl
thf(fact_439_image__paired__Times,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( image_710797976182302661_a_nat
@ ( produc8414824516080254331_a_nat
@ ^ [X: list_Sum_sum_a_nat,Y3: list_Sum_sum_a_nat] : ( produc7990843422341522135_a_nat @ ( F @ X ) @ ( G @ Y3 ) ) )
@ ( produc5624668685630548104_a_nat @ A
@ ^ [Uu2: list_Sum_sum_a_nat] : B2 ) )
= ( produc5624668685630548104_a_nat @ ( image_5081948215111134021_a_nat @ F @ A )
@ ^ [Uu2: list_Sum_sum_a_nat] : ( image_5081948215111134021_a_nat @ G @ B2 ) ) ) ).
% image_paired_Times
thf(fact_440_image__paired__Times,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,G: $o > $o,A: set_li6526943997496501093_a_nat,B2: set_o] :
( ( image_2584502784606554117_nat_o
@ ( produc1239800662677447355_nat_o
@ ^ [X: list_Sum_sum_a_nat,Y3: $o] : ( produc1406723247873626090_nat_o @ ( F @ X ) @ ( G @ Y3 ) ) )
@ ( produc8891875240947076123_nat_o @ A
@ ^ [Uu2: list_Sum_sum_a_nat] : B2 ) )
= ( produc8891875240947076123_nat_o @ ( image_5081948215111134021_a_nat @ F @ A )
@ ^ [Uu2: list_Sum_sum_a_nat] : ( image_o_o2 @ G @ B2 ) ) ) ).
% image_paired_Times
thf(fact_441_image__paired__Times,axiom,
! [F: $o > $o,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_o,B2: set_li6526943997496501093_a_nat] :
( ( image_2535540365225962373_a_nat
@ ( produc6149260406019937723_a_nat
@ ^ [X: $o,Y3: list_Sum_sum_a_nat] : ( produc7378284081089837956_a_nat @ ( F @ X ) @ ( G @ Y3 ) ) )
@ ( produc5640064037308512181_a_nat @ A
@ ^ [Uu2: $o] : B2 ) )
= ( produc5640064037308512181_a_nat @ ( image_o_o2 @ F @ A )
@ ^ [Uu2: $o] : ( image_5081948215111134021_a_nat @ G @ B2 ) ) ) ).
% image_paired_Times
thf(fact_442_image__paired__Times,axiom,
! [F: $o > $o,G: $o > $o,A: set_o,B2: set_o] :
( ( image_9131363867636255685od_o_o
@ ( produc7436348682273225467od_o_o
@ ^ [X: $o,Y3: $o] : ( product_Pair_o_o @ ( F @ X ) @ ( G @ Y3 ) ) )
@ ( product_Sigma_o_o @ A
@ ^ [Uu2: $o] : B2 ) )
= ( product_Sigma_o_o @ ( image_o_o2 @ F @ A )
@ ^ [Uu2: $o] : ( image_o_o2 @ G @ B2 ) ) ) ).
% image_paired_Times
thf(fact_443_IntI,axiom,
! [C3: $o,A: set_o,B2: set_o] :
( ( member_o @ C3 @ A )
=> ( ( member_o @ C3 @ B2 )
=> ( member_o @ C3 @ ( inf_inf_set_o @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_444_IntI,axiom,
! [C3: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C3 @ A )
=> ( ( member408289922725080238_a_nat @ C3 @ B2 )
=> ( member408289922725080238_a_nat @ C3 @ ( inf_in3249246906714053971_a_nat @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_445_IntI,axiom,
! [C3: sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C3 @ A )
=> ( ( member_Sum_sum_a_nat @ C3 @ B2 )
=> ( member_Sum_sum_a_nat @ C3 @ ( inf_in7084830621192376909_a_nat @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_446_Int__iff,axiom,
! [C3: $o,A: set_o,B2: set_o] :
( ( member_o @ C3 @ ( inf_inf_set_o @ A @ B2 ) )
= ( ( member_o @ C3 @ A )
& ( member_o @ C3 @ B2 ) ) ) ).
% Int_iff
thf(fact_447_Int__iff,axiom,
! [C3: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C3 @ ( inf_in3249246906714053971_a_nat @ A @ B2 ) )
= ( ( member408289922725080238_a_nat @ C3 @ A )
& ( member408289922725080238_a_nat @ C3 @ B2 ) ) ) ).
% Int_iff
thf(fact_448_Int__iff,axiom,
! [C3: sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C3 @ ( inf_in7084830621192376909_a_nat @ A @ B2 ) )
= ( ( member_Sum_sum_a_nat @ C3 @ A )
& ( member_Sum_sum_a_nat @ C3 @ B2 ) ) ) ).
% Int_iff
thf(fact_449_Int__UNIV,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ( inf_in7084830621192376909_a_nat @ A @ B2 )
= top_to795618464972521135_a_nat )
= ( ( A = top_to795618464972521135_a_nat )
& ( B2 = top_to795618464972521135_a_nat ) ) ) ).
% Int_UNIV
thf(fact_450_Int__UNIV,axiom,
! [A: set_o,B2: set_o] :
( ( ( inf_inf_set_o @ A @ B2 )
= top_top_set_o )
= ( ( A = top_top_set_o )
& ( B2 = top_top_set_o ) ) ) ).
% Int_UNIV
thf(fact_451_pair__imageI,axiom,
! [A2: set_a,B: produc5986856040925105290_a_nat,A: set_Pr8537526423911375309_a_nat,F: set_a > produc5986856040925105290_a_nat > $o] :
( ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ A2 @ B ) @ A )
=> ( member_o @ ( F @ A2 @ B ) @ ( image_7140268995275227258_nat_o @ ( produc2676398950370459482_nat_o @ F ) @ A ) ) ) ).
% pair_imageI
thf(fact_452_pair__imageI,axiom,
! [A2: set_a,B: produc5986856040925105290_a_nat,A: set_Pr8537526423911375309_a_nat,F: set_a > produc5986856040925105290_a_nat > list_Sum_sum_a_nat] :
( ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ A2 @ B ) @ A )
=> ( member408289922725080238_a_nat @ ( F @ A2 @ B ) @ ( image_5074594100668181351_a_nat @ ( produc6191494036646705735_a_nat @ F ) @ A ) ) ) ).
% pair_imageI
thf(fact_453_pair__imageI,axiom,
! [A2: set_a,B: produc5986856040925105290_a_nat,A: set_Pr8537526423911375309_a_nat,F: set_a > produc5986856040925105290_a_nat > sum_sum_a_nat] :
( ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ A2 @ B ) @ A )
=> ( member_Sum_sum_a_nat @ ( F @ A2 @ B ) @ ( image_3316368612896549089_a_nat @ ( produc4880221358163306433_a_nat @ F ) @ A ) ) ) ).
% pair_imageI
thf(fact_454_pair__imageI,axiom,
! [A2: nat,B: set_li6526943997496501093_a_nat,A: set_Pr1465208939327205952_a_nat,F: nat > set_li6526943997496501093_a_nat > $o] :
( ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ A2 @ B ) @ A )
=> ( member_o @ ( F @ A2 @ B ) @ ( image_7365926897524433095_nat_o @ ( produc380675777491300335_nat_o @ F ) @ A ) ) ) ).
% pair_imageI
thf(fact_455_pair__imageI,axiom,
! [A2: nat,B: set_li6526943997496501093_a_nat,A: set_Pr1465208939327205952_a_nat,F: nat > set_li6526943997496501093_a_nat > list_Sum_sum_a_nat] :
( ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ A2 @ B ) @ A )
=> ( member408289922725080238_a_nat @ ( F @ A2 @ B ) @ ( image_4517596464259415860_a_nat @ ( produc1461104056388651612_a_nat @ F ) @ A ) ) ) ).
% pair_imageI
thf(fact_456_pair__imageI,axiom,
! [A2: nat,B: set_li6526943997496501093_a_nat,A: set_Pr1465208939327205952_a_nat,F: nat > set_li6526943997496501093_a_nat > sum_sum_a_nat] :
( ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ A2 @ B ) @ A )
=> ( member_Sum_sum_a_nat @ ( F @ A2 @ B ) @ ( image_3947232080953722670_a_nat @ ( produc2738135688281245270_a_nat @ F ) @ A ) ) ) ).
% pair_imageI
thf(fact_457_GrD1,axiom,
! [X2: set_a,Fx: produc5986856040925105290_a_nat,A: set_set_a,F: set_a > produc5986856040925105290_a_nat] :
( ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ X2 @ Fx ) @ ( bNF_Gr696196767294584518_a_nat @ A @ F ) )
=> ( member_set_a @ X2 @ A ) ) ).
% GrD1
thf(fact_458_GrD1,axiom,
! [X2: nat,Fx: set_li6526943997496501093_a_nat,A: set_nat,F: nat > set_li6526943997496501093_a_nat] :
( ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ X2 @ Fx ) @ ( bNF_Gr6830624683021809265_a_nat @ A @ F ) )
=> ( member_nat @ X2 @ A ) ) ).
% GrD1
thf(fact_459_GrD2,axiom,
! [X2: set_a,Fx: produc5986856040925105290_a_nat,A: set_set_a,F: set_a > produc5986856040925105290_a_nat] :
( ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ X2 @ Fx ) @ ( bNF_Gr696196767294584518_a_nat @ A @ F ) )
=> ( ( F @ X2 )
= Fx ) ) ).
% GrD2
thf(fact_460_GrD2,axiom,
! [X2: nat,Fx: set_li6526943997496501093_a_nat,A: set_nat,F: nat > set_li6526943997496501093_a_nat] :
( ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ X2 @ Fx ) @ ( bNF_Gr6830624683021809265_a_nat @ A @ F ) )
=> ( ( F @ X2 )
= Fx ) ) ).
% GrD2
thf(fact_461_Int__def,axiom,
( inf_inf_set_o
= ( ^ [A6: set_o,B6: set_o] :
( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A6 )
& ( member_o @ X @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_462_Int__def,axiom,
( inf_in3249246906714053971_a_nat
= ( ^ [A6: set_li6526943997496501093_a_nat,B6: set_li6526943997496501093_a_nat] :
( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A6 )
& ( member408289922725080238_a_nat @ X @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_463_Int__def,axiom,
( inf_in7084830621192376909_a_nat
= ( ^ [A6: set_Sum_sum_a_nat,B6: set_Sum_sum_a_nat] :
( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A6 )
& ( member_Sum_sum_a_nat @ X @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_464_Int__def,axiom,
( inf_in8399021836546144180_a_nat
= ( ^ [A6: set_na3699693778330250182_a_nat,B6: set_na3699693778330250182_a_nat] :
( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ A6 )
& ( member8690443509505302927_a_nat @ X @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_465_Int__Collect,axiom,
! [X2: $o,A: set_o,P: $o > $o] :
( ( member_o @ X2 @ ( inf_inf_set_o @ A @ ( collect_o @ P ) ) )
= ( ( member_o @ X2 @ A )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_466_Int__Collect,axiom,
! [X2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ X2 @ ( inf_in3249246906714053971_a_nat @ A @ ( collec7555443234367654128_a_nat @ P ) ) )
= ( ( member408289922725080238_a_nat @ X2 @ A )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_467_Int__Collect,axiom,
! [X2: sum_sum_a_nat,A: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( member_Sum_sum_a_nat @ X2 @ ( inf_in7084830621192376909_a_nat @ A @ ( collec7073057861543223018_a_nat @ P ) ) )
= ( ( member_Sum_sum_a_nat @ X2 @ A )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_468_Int__Collect,axiom,
! [X2: nat > sum_sum_a_nat,A: set_na3699693778330250182_a_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ( member8690443509505302927_a_nat @ X2 @ ( inf_in8399021836546144180_a_nat @ A @ ( collec5629555741568564177_a_nat @ P ) ) )
= ( ( member8690443509505302927_a_nat @ X2 @ A )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_469_Collect__conj__eq,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,Q: ( nat > sum_sum_a_nat ) > $o] :
( ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_in8399021836546144180_a_nat @ ( collec5629555741568564177_a_nat @ P ) @ ( collec5629555741568564177_a_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_470_IntE,axiom,
! [C3: $o,A: set_o,B2: set_o] :
( ( member_o @ C3 @ ( inf_inf_set_o @ A @ B2 ) )
=> ~ ( ( member_o @ C3 @ A )
=> ~ ( member_o @ C3 @ B2 ) ) ) ).
% IntE
thf(fact_471_IntE,axiom,
! [C3: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C3 @ ( inf_in3249246906714053971_a_nat @ A @ B2 ) )
=> ~ ( ( member408289922725080238_a_nat @ C3 @ A )
=> ~ ( member408289922725080238_a_nat @ C3 @ B2 ) ) ) ).
% IntE
thf(fact_472_IntE,axiom,
! [C3: sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C3 @ ( inf_in7084830621192376909_a_nat @ A @ B2 ) )
=> ~ ( ( member_Sum_sum_a_nat @ C3 @ A )
=> ~ ( member_Sum_sum_a_nat @ C3 @ B2 ) ) ) ).
% IntE
thf(fact_473_IntD1,axiom,
! [C3: $o,A: set_o,B2: set_o] :
( ( member_o @ C3 @ ( inf_inf_set_o @ A @ B2 ) )
=> ( member_o @ C3 @ A ) ) ).
% IntD1
thf(fact_474_IntD1,axiom,
! [C3: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C3 @ ( inf_in3249246906714053971_a_nat @ A @ B2 ) )
=> ( member408289922725080238_a_nat @ C3 @ A ) ) ).
% IntD1
thf(fact_475_IntD1,axiom,
! [C3: sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C3 @ ( inf_in7084830621192376909_a_nat @ A @ B2 ) )
=> ( member_Sum_sum_a_nat @ C3 @ A ) ) ).
% IntD1
thf(fact_476_IntD2,axiom,
! [C3: $o,A: set_o,B2: set_o] :
( ( member_o @ C3 @ ( inf_inf_set_o @ A @ B2 ) )
=> ( member_o @ C3 @ B2 ) ) ).
% IntD2
thf(fact_477_IntD2,axiom,
! [C3: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C3 @ ( inf_in3249246906714053971_a_nat @ A @ B2 ) )
=> ( member408289922725080238_a_nat @ C3 @ B2 ) ) ).
% IntD2
thf(fact_478_IntD2,axiom,
! [C3: sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C3 @ ( inf_in7084830621192376909_a_nat @ A @ B2 ) )
=> ( member_Sum_sum_a_nat @ C3 @ B2 ) ) ).
% IntD2
thf(fact_479_Int__UNIV__left,axiom,
! [B2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ top_to795618464972521135_a_nat @ B2 )
= B2 ) ).
% Int_UNIV_left
thf(fact_480_Int__UNIV__left,axiom,
! [B2: set_o] :
( ( inf_inf_set_o @ top_top_set_o @ B2 )
= B2 ) ).
% Int_UNIV_left
thf(fact_481_Int__UNIV__right,axiom,
! [A: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A @ top_to795618464972521135_a_nat )
= A ) ).
% Int_UNIV_right
thf(fact_482_Int__UNIV__right,axiom,
! [A: set_o] :
( ( inf_inf_set_o @ A @ top_top_set_o )
= A ) ).
% Int_UNIV_right
thf(fact_483_fo__fmla_Odistinct_I47_J,axiom,
! [X51: fo_fmla_a_b,X52: fo_fmla_a_b,X71: nat,X72: fo_fmla_a_b] :
( ( fo_Conj_a_b @ X51 @ X52 )
!= ( fo_Exists_a_b @ X71 @ X72 ) ) ).
% fo_fmla.distinct(47)
thf(fact_484_fo__fmla_Odistinct_I49_J,axiom,
! [X51: fo_fmla_a_b,X52: fo_fmla_a_b,X81: nat,X82: fo_fmla_a_b] :
( ( fo_Conj_a_b @ X51 @ X52 )
!= ( fo_Forall_a_b @ X81 @ X82 ) ) ).
% fo_fmla.distinct(49)
thf(fact_485_fv__fo__fmla__list__Conj,axiom,
! [Phi: fo_fmla_a_b,Psi: fo_fmla_a_b] :
( ( fv_fo_fmla_list_a_b @ ( fo_Conj_a_b @ Phi @ Psi ) )
= ( fv_fo_fmla_list_a_b @ ( fo_Conj_a_b @ Psi @ Phi ) ) ) ).
% fv_fo_fmla_list_Conj
thf(fact_486_Int__Collect__mono,axiom,
! [A: set_o,B2: set_o,P: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ A @ B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A @ ( collect_o @ P ) ) @ ( inf_inf_set_o @ B2 @ ( collect_o @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_487_Int__Collect__mono,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o,Q: list_Sum_sum_a_nat > $o] :
( ( ord_le1147066620699065093_a_nat @ A @ B2 )
=> ( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ A @ ( collec7555443234367654128_a_nat @ P ) ) @ ( inf_in3249246906714053971_a_nat @ B2 @ ( collec7555443234367654128_a_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_488_Int__Collect__mono,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ ( collec7073057861543223018_a_nat @ P ) ) @ ( inf_in7084830621192376909_a_nat @ B2 @ ( collec7073057861543223018_a_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_489_Int__Collect__mono,axiom,
! [A: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat,P: ( nat > sum_sum_a_nat ) > $o,Q: ( nat > sum_sum_a_nat ) > $o] :
( ( ord_le8108555184339247974_a_nat @ A @ B2 )
=> ( ! [X3: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le8108555184339247974_a_nat @ ( inf_in8399021836546144180_a_nat @ A @ ( collec5629555741568564177_a_nat @ P ) ) @ ( inf_in8399021836546144180_a_nat @ B2 @ ( collec5629555741568564177_a_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_490_esat_Osimps_I5_J,axiom,
! [Phi: fo_fmla_a_b,Psi: fo_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma2: nat > sum_sum_a_nat,X6: set_Sum_sum_a_nat] :
( ( esat_a_b @ ( fo_Conj_a_b @ Phi @ Psi ) @ I @ Sigma2 @ X6 )
= ( ( esat_a_b @ Phi @ I @ Sigma2 @ X6 )
& ( esat_a_b @ Psi @ I @ Sigma2 @ X6 ) ) ) ).
% esat.simps(5)
thf(fact_491_case__prod__Pair__iden,axiom,
! [P2: produc8208987855206359447_a_nat] :
( ( produc6698438707799802453_a_nat @ produc1626021218680649873_a_nat @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_492_case__prod__Pair__iden,axiom,
! [P2: produc5986856040925105290_a_nat] :
( ( produc5943727843246033139_a_nat @ produc4622690550020908860_a_nat @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_493_image__Int__subset,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ ( inf_in3249246906714053971_a_nat @ A @ B2 ) ) @ ( inf_in3249246906714053971_a_nat @ ( image_5081948215111134021_a_nat @ F @ A ) @ ( image_5081948215111134021_a_nat @ F @ B2 ) ) ) ).
% image_Int_subset
thf(fact_494_image__Int__subset,axiom,
! [F: $o > $o,A: set_o,B2: set_o] : ( ord_less_eq_set_o @ ( image_o_o2 @ F @ ( inf_inf_set_o @ A @ B2 ) ) @ ( inf_inf_set_o @ ( image_o_o2 @ F @ A ) @ ( image_o_o2 @ F @ B2 ) ) ) ).
% image_Int_subset
thf(fact_495_swap__product,axiom,
! [A: set_Pr1465208939327205952_a_nat,B2: set_set_a] :
( ( image_9020305190384351591_a_nat
@ ( produc4742807462889283007_a_nat
@ ^ [I5: produc5986856040925105290_a_nat,J: set_a] : ( produc1626021218680649873_a_nat @ J @ I5 ) )
@ ( produc931357945493331722_set_a @ A
@ ^ [Uu2: produc5986856040925105290_a_nat] : B2 ) )
= ( produc5217640286535863200_a_nat @ B2
@ ^ [Uu2: set_a] : A ) ) ).
% swap_product
thf(fact_496_swap__product,axiom,
! [A: set_se4330304633200676677_a_nat,B2: set_nat] :
( ( image_2804021043688564853_a_nat
@ ( produc6022691998427229077_a_nat
@ ^ [I5: set_li6526943997496501093_a_nat,J: nat] : ( produc4622690550020908860_a_nat @ J @ I5 ) )
@ ( produc2225542198799310317at_nat @ A
@ ^ [Uu2: set_li6526943997496501093_a_nat] : B2 ) )
= ( produc3542699778996723019_a_nat @ B2
@ ^ [Uu2: nat] : A ) ) ).
% swap_product
thf(fact_497_inf__top_Oright__neutral,axiom,
! [A2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A2 @ top_to795618464972521135_a_nat )
= A2 ) ).
% inf_top.right_neutral
thf(fact_498_inf__top_Oright__neutral,axiom,
! [A2: set_o] :
( ( inf_inf_set_o @ A2 @ top_top_set_o )
= A2 ) ).
% inf_top.right_neutral
thf(fact_499_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( top_to795618464972521135_a_nat
= ( inf_in7084830621192376909_a_nat @ A2 @ B ) )
= ( ( A2 = top_to795618464972521135_a_nat )
& ( B = top_to795618464972521135_a_nat ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_500_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_o,B: set_o] :
( ( top_top_set_o
= ( inf_inf_set_o @ A2 @ B ) )
= ( ( A2 = top_top_set_o )
& ( B = top_top_set_o ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_501_inf__top_Oleft__neutral,axiom,
! [A2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ top_to795618464972521135_a_nat @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_502_inf__top_Oleft__neutral,axiom,
! [A2: set_o] :
( ( inf_inf_set_o @ top_top_set_o @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_503_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ( inf_in7084830621192376909_a_nat @ A2 @ B )
= top_to795618464972521135_a_nat )
= ( ( A2 = top_to795618464972521135_a_nat )
& ( B = top_to795618464972521135_a_nat ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_504_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_o,B: set_o] :
( ( ( inf_inf_set_o @ A2 @ B )
= top_top_set_o )
= ( ( A2 = top_top_set_o )
& ( B = top_top_set_o ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_505_top__eq__inf__iff,axiom,
! [X2: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat] :
( ( top_to795618464972521135_a_nat
= ( inf_in7084830621192376909_a_nat @ X2 @ Y2 ) )
= ( ( X2 = top_to795618464972521135_a_nat )
& ( Y2 = top_to795618464972521135_a_nat ) ) ) ).
% top_eq_inf_iff
thf(fact_506_top__eq__inf__iff,axiom,
! [X2: set_o,Y2: set_o] :
( ( top_top_set_o
= ( inf_inf_set_o @ X2 @ Y2 ) )
= ( ( X2 = top_top_set_o )
& ( Y2 = top_top_set_o ) ) ) ).
% top_eq_inf_iff
thf(fact_507_inf__eq__top__iff,axiom,
! [X2: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat] :
( ( ( inf_in7084830621192376909_a_nat @ X2 @ Y2 )
= top_to795618464972521135_a_nat )
= ( ( X2 = top_to795618464972521135_a_nat )
& ( Y2 = top_to795618464972521135_a_nat ) ) ) ).
% inf_eq_top_iff
thf(fact_508_inf__eq__top__iff,axiom,
! [X2: set_o,Y2: set_o] :
( ( ( inf_inf_set_o @ X2 @ Y2 )
= top_top_set_o )
= ( ( X2 = top_top_set_o )
& ( Y2 = top_top_set_o ) ) ) ).
% inf_eq_top_iff
thf(fact_509_case__prodI2,axiom,
! [P2: produc8208987855206359447_a_nat,C3: set_a > produc5986856040925105290_a_nat > $o] :
( ! [A3: set_a,B3: produc5986856040925105290_a_nat] :
( ( P2
= ( produc1626021218680649873_a_nat @ A3 @ B3 ) )
=> ( C3 @ A3 @ B3 ) )
=> ( produc2676398950370459482_nat_o @ C3 @ P2 ) ) ).
% case_prodI2
thf(fact_510_case__prodI2,axiom,
! [P2: produc5986856040925105290_a_nat,C3: nat > set_li6526943997496501093_a_nat > $o] :
( ! [A3: nat,B3: set_li6526943997496501093_a_nat] :
( ( P2
= ( produc4622690550020908860_a_nat @ A3 @ B3 ) )
=> ( C3 @ A3 @ B3 ) )
=> ( produc380675777491300335_nat_o @ C3 @ P2 ) ) ).
% case_prodI2
thf(fact_511_case__prodI,axiom,
! [F: set_a > produc5986856040925105290_a_nat > $o,A2: set_a,B: produc5986856040925105290_a_nat] :
( ( F @ A2 @ B )
=> ( produc2676398950370459482_nat_o @ F @ ( produc1626021218680649873_a_nat @ A2 @ B ) ) ) ).
% case_prodI
thf(fact_512_case__prodI,axiom,
! [F: nat > set_li6526943997496501093_a_nat > $o,A2: nat,B: set_li6526943997496501093_a_nat] :
( ( F @ A2 @ B )
=> ( produc380675777491300335_nat_o @ F @ ( produc4622690550020908860_a_nat @ A2 @ B ) ) ) ).
% case_prodI
thf(fact_513_mem__case__prodI2,axiom,
! [P2: produc8208987855206359447_a_nat,Z: $o,C3: set_a > produc5986856040925105290_a_nat > set_o] :
( ! [A3: set_a,B3: produc5986856040925105290_a_nat] :
( ( P2
= ( produc1626021218680649873_a_nat @ A3 @ B3 ) )
=> ( member_o @ Z @ ( C3 @ A3 @ B3 ) ) )
=> ( member_o @ Z @ ( produc7504519220880244282_set_o @ C3 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_514_mem__case__prodI2,axiom,
! [P2: produc8208987855206359447_a_nat,Z: list_Sum_sum_a_nat,C3: set_a > produc5986856040925105290_a_nat > set_li6526943997496501093_a_nat] :
( ! [A3: set_a,B3: produc5986856040925105290_a_nat] :
( ( P2
= ( produc1626021218680649873_a_nat @ A3 @ B3 ) )
=> ( member408289922725080238_a_nat @ Z @ ( C3 @ A3 @ B3 ) ) )
=> ( member408289922725080238_a_nat @ Z @ ( produc3606380900734198311_a_nat @ C3 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_515_mem__case__prodI2,axiom,
! [P2: produc8208987855206359447_a_nat,Z: sum_sum_a_nat,C3: set_a > produc5986856040925105290_a_nat > set_Sum_sum_a_nat] :
( ! [A3: set_a,B3: produc5986856040925105290_a_nat] :
( ( P2
= ( produc1626021218680649873_a_nat @ A3 @ B3 ) )
=> ( member_Sum_sum_a_nat @ Z @ ( C3 @ A3 @ B3 ) ) )
=> ( member_Sum_sum_a_nat @ Z @ ( produc2429100861819736353_a_nat @ C3 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_516_mem__case__prodI2,axiom,
! [P2: produc5986856040925105290_a_nat,Z: $o,C3: nat > set_li6526943997496501093_a_nat > set_o] :
( ! [A3: nat,B3: set_li6526943997496501093_a_nat] :
( ( P2
= ( produc4622690550020908860_a_nat @ A3 @ B3 ) )
=> ( member_o @ Z @ ( C3 @ A3 @ B3 ) ) )
=> ( member_o @ Z @ ( produc6496079708086295247_set_o @ C3 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_517_mem__case__prodI2,axiom,
! [P2: produc5986856040925105290_a_nat,Z: list_Sum_sum_a_nat,C3: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat] :
( ! [A3: nat,B3: set_li6526943997496501093_a_nat] :
( ( P2
= ( produc4622690550020908860_a_nat @ A3 @ B3 ) )
=> ( member408289922725080238_a_nat @ Z @ ( C3 @ A3 @ B3 ) ) )
=> ( member408289922725080238_a_nat @ Z @ ( produc910888649358827068_a_nat @ C3 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_518_mem__case__prodI2,axiom,
! [P2: produc5986856040925105290_a_nat,Z: sum_sum_a_nat,C3: nat > set_li6526943997496501093_a_nat > set_Sum_sum_a_nat] :
( ! [A3: nat,B3: set_li6526943997496501093_a_nat] :
( ( P2
= ( produc4622690550020908860_a_nat @ A3 @ B3 ) )
=> ( member_Sum_sum_a_nat @ Z @ ( C3 @ A3 @ B3 ) ) )
=> ( member_Sum_sum_a_nat @ Z @ ( produc3083163451219581878_a_nat @ C3 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_519_mem__case__prodI,axiom,
! [Z: $o,C3: set_a > produc5986856040925105290_a_nat > set_o,A2: set_a,B: produc5986856040925105290_a_nat] :
( ( member_o @ Z @ ( C3 @ A2 @ B ) )
=> ( member_o @ Z @ ( produc7504519220880244282_set_o @ C3 @ ( produc1626021218680649873_a_nat @ A2 @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_520_mem__case__prodI,axiom,
! [Z: list_Sum_sum_a_nat,C3: set_a > produc5986856040925105290_a_nat > set_li6526943997496501093_a_nat,A2: set_a,B: produc5986856040925105290_a_nat] :
( ( member408289922725080238_a_nat @ Z @ ( C3 @ A2 @ B ) )
=> ( member408289922725080238_a_nat @ Z @ ( produc3606380900734198311_a_nat @ C3 @ ( produc1626021218680649873_a_nat @ A2 @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_521_mem__case__prodI,axiom,
! [Z: sum_sum_a_nat,C3: set_a > produc5986856040925105290_a_nat > set_Sum_sum_a_nat,A2: set_a,B: produc5986856040925105290_a_nat] :
( ( member_Sum_sum_a_nat @ Z @ ( C3 @ A2 @ B ) )
=> ( member_Sum_sum_a_nat @ Z @ ( produc2429100861819736353_a_nat @ C3 @ ( produc1626021218680649873_a_nat @ A2 @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_522_mem__case__prodI,axiom,
! [Z: $o,C3: nat > set_li6526943997496501093_a_nat > set_o,A2: nat,B: set_li6526943997496501093_a_nat] :
( ( member_o @ Z @ ( C3 @ A2 @ B ) )
=> ( member_o @ Z @ ( produc6496079708086295247_set_o @ C3 @ ( produc4622690550020908860_a_nat @ A2 @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_523_mem__case__prodI,axiom,
! [Z: list_Sum_sum_a_nat,C3: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,A2: nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ Z @ ( C3 @ A2 @ B ) )
=> ( member408289922725080238_a_nat @ Z @ ( produc910888649358827068_a_nat @ C3 @ ( produc4622690550020908860_a_nat @ A2 @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_524_mem__case__prodI,axiom,
! [Z: sum_sum_a_nat,C3: nat > set_li6526943997496501093_a_nat > set_Sum_sum_a_nat,A2: nat,B: set_li6526943997496501093_a_nat] :
( ( member_Sum_sum_a_nat @ Z @ ( C3 @ A2 @ B ) )
=> ( member_Sum_sum_a_nat @ Z @ ( produc3083163451219581878_a_nat @ C3 @ ( produc4622690550020908860_a_nat @ A2 @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_525_Collect__case__prod,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,Q: ( nat > sum_sum_a_nat ) > $o] :
( ( collec7485809540651541650_a_nat
@ ( produc818503465587684972_nat_o
@ ^ [A8: nat > sum_sum_a_nat,B8: nat > sum_sum_a_nat] :
( ( P @ A8 )
& ( Q @ B8 ) ) ) )
= ( produc2732871624295728584_a_nat @ ( collec5629555741568564177_a_nat @ P )
@ ^ [Uu2: nat > sum_sum_a_nat] : ( collec5629555741568564177_a_nat @ Q ) ) ) ).
% Collect_case_prod
thf(fact_526_inf__top__left,axiom,
! [X2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ top_to795618464972521135_a_nat @ X2 )
= X2 ) ).
% inf_top_left
thf(fact_527_inf__top__left,axiom,
! [X2: set_o] :
( ( inf_inf_set_o @ top_top_set_o @ X2 )
= X2 ) ).
% inf_top_left
thf(fact_528_inf__top__right,axiom,
! [X2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X2 @ top_to795618464972521135_a_nat )
= X2 ) ).
% inf_top_right
thf(fact_529_inf__top__right,axiom,
! [X2: set_o] :
( ( inf_inf_set_o @ X2 @ top_top_set_o )
= X2 ) ).
% inf_top_right
thf(fact_530_inf__Int__eq2,axiom,
! [R2: set_Pr8537526423911375309_a_nat,S: set_Pr8537526423911375309_a_nat] :
( ( inf_in5726560369174641502_nat_o
@ ^ [X: set_a,Y3: produc5986856040925105290_a_nat] : ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ X @ Y3 ) @ R2 )
@ ^ [X: set_a,Y3: produc5986856040925105290_a_nat] : ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ X @ Y3 ) @ S ) )
= ( ^ [X: set_a,Y3: produc5986856040925105290_a_nat] : ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ X @ Y3 ) @ ( inf_in2413927911095571551_a_nat @ R2 @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_531_inf__Int__eq2,axiom,
! [R2: set_Pr1465208939327205952_a_nat,S: set_Pr1465208939327205952_a_nat] :
( ( inf_in6290697450388711897_nat_o
@ ^ [X: nat,Y3: set_li6526943997496501093_a_nat] : ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ X @ Y3 ) @ R2 )
@ ^ [X: nat,Y3: set_li6526943997496501093_a_nat] : ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ X @ Y3 ) @ S ) )
= ( ^ [X: nat,Y3: set_li6526943997496501093_a_nat] : ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ X @ Y3 ) @ ( inf_in7863128716401692370_a_nat @ R2 @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_532_mem__case__prodE,axiom,
! [Z: $o,C3: set_a > produc5986856040925105290_a_nat > set_o,P2: produc8208987855206359447_a_nat] :
( ( member_o @ Z @ ( produc7504519220880244282_set_o @ C3 @ P2 ) )
=> ~ ! [X3: set_a,Y4: produc5986856040925105290_a_nat] :
( ( P2
= ( produc1626021218680649873_a_nat @ X3 @ Y4 ) )
=> ~ ( member_o @ Z @ ( C3 @ X3 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_533_mem__case__prodE,axiom,
! [Z: list_Sum_sum_a_nat,C3: set_a > produc5986856040925105290_a_nat > set_li6526943997496501093_a_nat,P2: produc8208987855206359447_a_nat] :
( ( member408289922725080238_a_nat @ Z @ ( produc3606380900734198311_a_nat @ C3 @ P2 ) )
=> ~ ! [X3: set_a,Y4: produc5986856040925105290_a_nat] :
( ( P2
= ( produc1626021218680649873_a_nat @ X3 @ Y4 ) )
=> ~ ( member408289922725080238_a_nat @ Z @ ( C3 @ X3 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_534_mem__case__prodE,axiom,
! [Z: sum_sum_a_nat,C3: set_a > produc5986856040925105290_a_nat > set_Sum_sum_a_nat,P2: produc8208987855206359447_a_nat] :
( ( member_Sum_sum_a_nat @ Z @ ( produc2429100861819736353_a_nat @ C3 @ P2 ) )
=> ~ ! [X3: set_a,Y4: produc5986856040925105290_a_nat] :
( ( P2
= ( produc1626021218680649873_a_nat @ X3 @ Y4 ) )
=> ~ ( member_Sum_sum_a_nat @ Z @ ( C3 @ X3 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_535_mem__case__prodE,axiom,
! [Z: $o,C3: nat > set_li6526943997496501093_a_nat > set_o,P2: produc5986856040925105290_a_nat] :
( ( member_o @ Z @ ( produc6496079708086295247_set_o @ C3 @ P2 ) )
=> ~ ! [X3: nat,Y4: set_li6526943997496501093_a_nat] :
( ( P2
= ( produc4622690550020908860_a_nat @ X3 @ Y4 ) )
=> ~ ( member_o @ Z @ ( C3 @ X3 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_536_mem__case__prodE,axiom,
! [Z: list_Sum_sum_a_nat,C3: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,P2: produc5986856040925105290_a_nat] :
( ( member408289922725080238_a_nat @ Z @ ( produc910888649358827068_a_nat @ C3 @ P2 ) )
=> ~ ! [X3: nat,Y4: set_li6526943997496501093_a_nat] :
( ( P2
= ( produc4622690550020908860_a_nat @ X3 @ Y4 ) )
=> ~ ( member408289922725080238_a_nat @ Z @ ( C3 @ X3 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_537_mem__case__prodE,axiom,
! [Z: sum_sum_a_nat,C3: nat > set_li6526943997496501093_a_nat > set_Sum_sum_a_nat,P2: produc5986856040925105290_a_nat] :
( ( member_Sum_sum_a_nat @ Z @ ( produc3083163451219581878_a_nat @ C3 @ P2 ) )
=> ~ ! [X3: nat,Y4: set_li6526943997496501093_a_nat] :
( ( P2
= ( produc4622690550020908860_a_nat @ X3 @ Y4 ) )
=> ~ ( member_Sum_sum_a_nat @ Z @ ( C3 @ X3 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_538_case__prodE,axiom,
! [C3: set_a > produc5986856040925105290_a_nat > $o,P2: produc8208987855206359447_a_nat] :
( ( produc2676398950370459482_nat_o @ C3 @ P2 )
=> ~ ! [X3: set_a,Y4: produc5986856040925105290_a_nat] :
( ( P2
= ( produc1626021218680649873_a_nat @ X3 @ Y4 ) )
=> ~ ( C3 @ X3 @ Y4 ) ) ) ).
% case_prodE
thf(fact_539_case__prodE,axiom,
! [C3: nat > set_li6526943997496501093_a_nat > $o,P2: produc5986856040925105290_a_nat] :
( ( produc380675777491300335_nat_o @ C3 @ P2 )
=> ~ ! [X3: nat,Y4: set_li6526943997496501093_a_nat] :
( ( P2
= ( produc4622690550020908860_a_nat @ X3 @ Y4 ) )
=> ~ ( C3 @ X3 @ Y4 ) ) ) ).
% case_prodE
thf(fact_540_case__prodD,axiom,
! [F: set_a > produc5986856040925105290_a_nat > $o,A2: set_a,B: produc5986856040925105290_a_nat] :
( ( produc2676398950370459482_nat_o @ F @ ( produc1626021218680649873_a_nat @ A2 @ B ) )
=> ( F @ A2 @ B ) ) ).
% case_prodD
thf(fact_541_case__prodD,axiom,
! [F: nat > set_li6526943997496501093_a_nat > $o,A2: nat,B: set_li6526943997496501093_a_nat] :
( ( produc380675777491300335_nat_o @ F @ ( produc4622690550020908860_a_nat @ A2 @ B ) )
=> ( F @ A2 @ B ) ) ).
% case_prodD
thf(fact_542_inf__Int__eq,axiom,
! [R2: set_o,S: set_o] :
( ( inf_inf_o_o
@ ^ [X: $o] : ( member_o @ X @ R2 )
@ ^ [X: $o] : ( member_o @ X @ S ) )
= ( ^ [X: $o] : ( member_o @ X @ ( inf_inf_set_o @ R2 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_543_inf__Int__eq,axiom,
! [R2: set_li6526943997496501093_a_nat,S: set_li6526943997496501093_a_nat] :
( ( inf_in954358986474102090_nat_o
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ R2 )
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ S ) )
= ( ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ ( inf_in3249246906714053971_a_nat @ R2 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_544_inf__Int__eq,axiom,
! [R2: set_Sum_sum_a_nat,S: set_Sum_sum_a_nat] :
( ( inf_in5242522483218605776_nat_o
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ R2 )
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ S ) )
= ( ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ ( inf_in7084830621192376909_a_nat @ R2 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_545_inf__set__def,axiom,
( inf_inf_set_o
= ( ^ [A6: set_o,B6: set_o] :
( collect_o
@ ( inf_inf_o_o
@ ^ [X: $o] : ( member_o @ X @ A6 )
@ ^ [X: $o] : ( member_o @ X @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_546_inf__set__def,axiom,
( inf_in3249246906714053971_a_nat
= ( ^ [A6: set_li6526943997496501093_a_nat,B6: set_li6526943997496501093_a_nat] :
( collec7555443234367654128_a_nat
@ ( inf_in954358986474102090_nat_o
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ A6 )
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_547_inf__set__def,axiom,
( inf_in7084830621192376909_a_nat
= ( ^ [A6: set_Sum_sum_a_nat,B6: set_Sum_sum_a_nat] :
( collec7073057861543223018_a_nat
@ ( inf_in5242522483218605776_nat_o
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ A6 )
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_548_inf__set__def,axiom,
( inf_in8399021836546144180_a_nat
= ( ^ [A6: set_na3699693778330250182_a_nat,B6: set_na3699693778330250182_a_nat] :
( collec5629555741568564177_a_nat
@ ( inf_in48813113380384809_nat_o
@ ^ [X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ A6 )
@ ^ [X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_549_Collect__case__prod__Sigma,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,Q: ( nat > sum_sum_a_nat ) > ( nat > sum_sum_a_nat ) > $o] :
( ( collec7485809540651541650_a_nat
@ ( produc818503465587684972_nat_o
@ ^ [X: nat > sum_sum_a_nat,Y3: nat > sum_sum_a_nat] :
( ( P @ X )
& ( Q @ X @ Y3 ) ) ) )
= ( produc2732871624295728584_a_nat @ ( collec5629555741568564177_a_nat @ P )
@ ^ [X: nat > sum_sum_a_nat] : ( collec5629555741568564177_a_nat @ ( Q @ X ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_550_eval__conj,axiom,
! [Phi: fo_fmla_a_b,I: product_prod_b_nat > set_list_a,T_phi: produc8208987855206359447_a_nat,Psi: fo_fmla_a_b,T_psi: produc8208987855206359447_a_nat] :
( ( fo_wf_a_b @ Phi @ I @ T_phi )
=> ( ( fo_wf_a_b @ Psi @ I @ T_psi )
=> ( fo_wf_a_b @ ( fo_Conj_a_b @ Phi @ Psi ) @ I @ ( eval_conj_a @ ( fv_fo_fmla_list_a_b @ Phi ) @ T_phi @ ( fv_fo_fmla_list_a_b @ Psi ) @ T_psi ) ) ) ) ).
% eval_conj
thf(fact_551_boolean__algebra_Oconj__one__right,axiom,
! [X2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X2 @ top_to795618464972521135_a_nat )
= X2 ) ).
% boolean_algebra.conj_one_right
thf(fact_552_boolean__algebra_Oconj__one__right,axiom,
! [X2: set_o] :
( ( inf_inf_set_o @ X2 @ top_top_set_o )
= X2 ) ).
% boolean_algebra.conj_one_right
thf(fact_553_fst__eqD,axiom,
! [X2: set_a,Y2: produc5986856040925105290_a_nat,A2: set_a] :
( ( ( produc4817319330785305509_a_nat @ ( produc1626021218680649873_a_nat @ X2 @ Y2 ) )
= A2 )
=> ( X2 = A2 ) ) ).
% fst_eqD
thf(fact_554_fst__eqD,axiom,
! [X2: nat,Y2: set_li6526943997496501093_a_nat,A2: nat] :
( ( ( produc8331164115582522960_a_nat @ ( produc4622690550020908860_a_nat @ X2 @ Y2 ) )
= A2 )
=> ( X2 = A2 ) ) ).
% fst_eqD
thf(fact_555_fst__conv,axiom,
! [X1: set_a,X22: produc5986856040925105290_a_nat] :
( ( produc4817319330785305509_a_nat @ ( produc1626021218680649873_a_nat @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_556_fst__conv,axiom,
! [X1: nat,X22: set_li6526943997496501093_a_nat] :
( ( produc8331164115582522960_a_nat @ ( produc4622690550020908860_a_nat @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_557_fstI,axiom,
! [X2: produc8208987855206359447_a_nat,Y2: set_a,Z: produc5986856040925105290_a_nat] :
( ( X2
= ( produc1626021218680649873_a_nat @ Y2 @ Z ) )
=> ( ( produc4817319330785305509_a_nat @ X2 )
= Y2 ) ) ).
% fstI
thf(fact_558_fstI,axiom,
! [X2: produc5986856040925105290_a_nat,Y2: nat,Z: set_li6526943997496501093_a_nat] :
( ( X2
= ( produc4622690550020908860_a_nat @ Y2 @ Z ) )
=> ( ( produc8331164115582522960_a_nat @ X2 )
= Y2 ) ) ).
% fstI
thf(fact_559_eq__fst__iff,axiom,
! [A2: set_a,P2: produc8208987855206359447_a_nat] :
( ( A2
= ( produc4817319330785305509_a_nat @ P2 ) )
= ( ? [B8: produc5986856040925105290_a_nat] :
( P2
= ( produc1626021218680649873_a_nat @ A2 @ B8 ) ) ) ) ).
% eq_fst_iff
thf(fact_560_eq__fst__iff,axiom,
! [A2: nat,P2: produc5986856040925105290_a_nat] :
( ( A2
= ( produc8331164115582522960_a_nat @ P2 ) )
= ( ? [B8: set_li6526943997496501093_a_nat] :
( P2
= ( produc4622690550020908860_a_nat @ A2 @ B8 ) ) ) ) ).
% eq_fst_iff
thf(fact_561_fst__image__times,axiom,
! [B2: set_o,A: set_o] :
( ( ( B2 = bot_bot_set_o )
=> ( ( image_7896445794123959606_o_o_o @ product_fst_o_o
@ ( product_Sigma_o_o @ A
@ ^ [Uu2: $o] : B2 ) )
= bot_bot_set_o ) )
& ( ( B2 != bot_bot_set_o )
=> ( ( image_7896445794123959606_o_o_o @ product_fst_o_o
@ ( product_Sigma_o_o @ A
@ ^ [Uu2: $o] : B2 ) )
= A ) ) ) ).
% fst_image_times
thf(fact_562_empty__Collect__eq,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o] :
( ( bot_bo6441361344521902642_a_nat
= ( collec5629555741568564177_a_nat @ P ) )
= ( ! [X: nat > sum_sum_a_nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_563_empty__Collect__eq,axiom,
! [P: $o > $o] :
( ( bot_bot_set_o
= ( collect_o @ P ) )
= ( ! [X: $o] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_564_Collect__empty__eq,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o] :
( ( ( collec5629555741568564177_a_nat @ P )
= bot_bo6441361344521902642_a_nat )
= ( ! [X: nat > sum_sum_a_nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_565_Collect__empty__eq,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( ! [X: $o] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_566_all__not__in__conv,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( ! [X: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ X @ A ) )
= ( A = bot_bo1033123847703346641_a_nat ) ) ).
% all_not_in_conv
thf(fact_567_all__not__in__conv,axiom,
! [A: set_Sum_sum_a_nat] :
( ( ! [X: sum_sum_a_nat] :
~ ( member_Sum_sum_a_nat @ X @ A ) )
= ( A = bot_bo3438331934148233675_a_nat ) ) ).
% all_not_in_conv
thf(fact_568_all__not__in__conv,axiom,
! [A: set_o] :
( ( ! [X: $o] :
~ ( member_o @ X @ A ) )
= ( A = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_569_empty__iff,axiom,
! [C3: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ C3 @ bot_bo1033123847703346641_a_nat ) ).
% empty_iff
thf(fact_570_empty__iff,axiom,
! [C3: sum_sum_a_nat] :
~ ( member_Sum_sum_a_nat @ C3 @ bot_bo3438331934148233675_a_nat ) ).
% empty_iff
thf(fact_571_empty__iff,axiom,
! [C3: $o] :
~ ( member_o @ C3 @ bot_bot_set_o ) ).
% empty_iff
thf(fact_572_image__is__empty,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ( image_5081948215111134021_a_nat @ F @ A )
= bot_bo1033123847703346641_a_nat )
= ( A = bot_bo1033123847703346641_a_nat ) ) ).
% image_is_empty
thf(fact_573_image__is__empty,axiom,
! [F: $o > $o,A: set_o] :
( ( ( image_o_o2 @ F @ A )
= bot_bot_set_o )
= ( A = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_574_empty__is__image,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( bot_bo1033123847703346641_a_nat
= ( image_5081948215111134021_a_nat @ F @ A ) )
= ( A = bot_bo1033123847703346641_a_nat ) ) ).
% empty_is_image
thf(fact_575_empty__is__image,axiom,
! [F: $o > $o,A: set_o] :
( ( bot_bot_set_o
= ( image_o_o2 @ F @ A ) )
= ( A = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_576_image__empty,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( image_5081948215111134021_a_nat @ F @ bot_bo1033123847703346641_a_nat )
= bot_bo1033123847703346641_a_nat ) ).
% image_empty
thf(fact_577_image__empty,axiom,
! [F: $o > $o] :
( ( image_o_o2 @ F @ bot_bot_set_o )
= bot_bot_set_o ) ).
% image_empty
thf(fact_578_empty__subsetI,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).
% empty_subsetI
thf(fact_579_subset__empty,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
= ( A = bot_bot_set_o ) ) ).
% subset_empty
thf(fact_580_Diff__empty,axiom,
! [A: set_o] :
( ( minus_minus_set_o @ A @ bot_bot_set_o )
= A ) ).
% Diff_empty
thf(fact_581_empty__Diff,axiom,
! [A: set_o] :
( ( minus_minus_set_o @ bot_bot_set_o @ A )
= bot_bot_set_o ) ).
% empty_Diff
thf(fact_582_Diff__cancel,axiom,
! [A: set_o] :
( ( minus_minus_set_o @ A @ A )
= bot_bot_set_o ) ).
% Diff_cancel
thf(fact_583_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collec5629555741568564177_a_nat
@ ^ [S4: nat > sum_sum_a_nat] : P )
= top_to9106040778512017686_a_nat ) )
& ( ~ P
=> ( ( collec5629555741568564177_a_nat
@ ^ [S4: nat > sum_sum_a_nat] : P )
= bot_bo6441361344521902642_a_nat ) ) ) ).
% Collect_const
thf(fact_584_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collec7073057861543223018_a_nat
@ ^ [S4: sum_sum_a_nat] : P )
= top_to795618464972521135_a_nat ) )
& ( ~ P
=> ( ( collec7073057861543223018_a_nat
@ ^ [S4: sum_sum_a_nat] : P )
= bot_bo3438331934148233675_a_nat ) ) ) ).
% Collect_const
thf(fact_585_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_o
@ ^ [S4: $o] : P )
= top_top_set_o ) )
& ( ~ P
=> ( ( collect_o
@ ^ [S4: $o] : P )
= bot_bot_set_o ) ) ) ).
% Collect_const
thf(fact_586_Times__empty,axiom,
! [A: set_o,B2: set_o] :
( ( ( product_Sigma_o_o @ A
@ ^ [Uu2: $o] : B2 )
= bot_bo7073875226086086771od_o_o )
= ( ( A = bot_bot_set_o )
| ( B2 = bot_bot_set_o ) ) ) ).
% Times_empty
thf(fact_587_Diff__UNIV,axiom,
! [A: set_Sum_sum_a_nat] :
( ( minus_1134630996077396038_a_nat @ A @ top_to795618464972521135_a_nat )
= bot_bo3438331934148233675_a_nat ) ).
% Diff_UNIV
thf(fact_588_Diff__UNIV,axiom,
! [A: set_o] :
( ( minus_minus_set_o @ A @ top_top_set_o )
= bot_bot_set_o ) ).
% Diff_UNIV
thf(fact_589_Diff__eq__empty__iff,axiom,
! [A: set_o,B2: set_o] :
( ( ( minus_minus_set_o @ A @ B2 )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_590_Diff__disjoint,axiom,
! [A: set_o,B2: set_o] :
( ( inf_inf_set_o @ A @ ( minus_minus_set_o @ B2 @ A ) )
= bot_bot_set_o ) ).
% Diff_disjoint
thf(fact_591_empty__not__UNIV,axiom,
bot_bo3438331934148233675_a_nat != top_to795618464972521135_a_nat ).
% empty_not_UNIV
thf(fact_592_empty__not__UNIV,axiom,
bot_bot_set_o != top_top_set_o ).
% empty_not_UNIV
thf(fact_593_Set_Oempty__def,axiom,
( bot_bo6441361344521902642_a_nat
= ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] : $false ) ) ).
% Set.empty_def
thf(fact_594_Set_Oempty__def,axiom,
( bot_bot_set_o
= ( collect_o
@ ^ [X: $o] : $false ) ) ).
% Set.empty_def
thf(fact_595_ex__in__conv,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( ? [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ A ) )
= ( A != bot_bo1033123847703346641_a_nat ) ) ).
% ex_in_conv
thf(fact_596_ex__in__conv,axiom,
! [A: set_Sum_sum_a_nat] :
( ( ? [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ A ) )
= ( A != bot_bo3438331934148233675_a_nat ) ) ).
% ex_in_conv
thf(fact_597_ex__in__conv,axiom,
! [A: set_o] :
( ( ? [X: $o] : ( member_o @ X @ A ) )
= ( A != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_598_equals0I,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ! [Y4: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ Y4 @ A )
=> ( A = bot_bo1033123847703346641_a_nat ) ) ).
% equals0I
thf(fact_599_equals0I,axiom,
! [A: set_Sum_sum_a_nat] :
( ! [Y4: sum_sum_a_nat] :
~ ( member_Sum_sum_a_nat @ Y4 @ A )
=> ( A = bot_bo3438331934148233675_a_nat ) ) ).
% equals0I
thf(fact_600_equals0I,axiom,
! [A: set_o] :
( ! [Y4: $o] :
~ ( member_o @ Y4 @ A )
=> ( A = bot_bot_set_o ) ) ).
% equals0I
thf(fact_601_equals0D,axiom,
! [A: set_li6526943997496501093_a_nat,A2: list_Sum_sum_a_nat] :
( ( A = bot_bo1033123847703346641_a_nat )
=> ~ ( member408289922725080238_a_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_602_equals0D,axiom,
! [A: set_Sum_sum_a_nat,A2: sum_sum_a_nat] :
( ( A = bot_bo3438331934148233675_a_nat )
=> ~ ( member_Sum_sum_a_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_603_equals0D,axiom,
! [A: set_o,A2: $o] :
( ( A = bot_bot_set_o )
=> ~ ( member_o @ A2 @ A ) ) ).
% equals0D
thf(fact_604_emptyE,axiom,
! [A2: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ A2 @ bot_bo1033123847703346641_a_nat ) ).
% emptyE
thf(fact_605_emptyE,axiom,
! [A2: sum_sum_a_nat] :
~ ( member_Sum_sum_a_nat @ A2 @ bot_bo3438331934148233675_a_nat ) ).
% emptyE
thf(fact_606_emptyE,axiom,
! [A2: $o] :
~ ( member_o @ A2 @ bot_bot_set_o ) ).
% emptyE
thf(fact_607_bot_Oextremum,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).
% bot.extremum
thf(fact_608_bot_Oextremum__unique,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
= ( A2 = bot_bot_set_o ) ) ).
% bot.extremum_unique
thf(fact_609_bot_Oextremum__uniqueI,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
=> ( A2 = bot_bot_set_o ) ) ).
% bot.extremum_uniqueI
thf(fact_610_disjoint__iff__not__equal,axiom,
! [A: set_o,B2: set_o] :
( ( ( inf_inf_set_o @ A @ B2 )
= bot_bot_set_o )
= ( ! [X: $o] :
( ( member_o @ X @ A )
=> ! [Y3: $o] :
( ( member_o @ Y3 @ B2 )
=> ( X = (~ Y3) ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_611_Int__empty__right,axiom,
! [A: set_o] :
( ( inf_inf_set_o @ A @ bot_bot_set_o )
= bot_bot_set_o ) ).
% Int_empty_right
thf(fact_612_Int__empty__left,axiom,
! [B2: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ B2 )
= bot_bot_set_o ) ).
% Int_empty_left
thf(fact_613_disjoint__iff,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ( inf_in3249246906714053971_a_nat @ A @ B2 )
= bot_bo1033123847703346641_a_nat )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
=> ~ ( member408289922725080238_a_nat @ X @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_614_disjoint__iff,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ( inf_in7084830621192376909_a_nat @ A @ B2 )
= bot_bo3438331934148233675_a_nat )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A )
=> ~ ( member_Sum_sum_a_nat @ X @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_615_disjoint__iff,axiom,
! [A: set_o,B2: set_o] :
( ( ( inf_inf_set_o @ A @ B2 )
= bot_bot_set_o )
= ( ! [X: $o] :
( ( member_o @ X @ A )
=> ~ ( member_o @ X @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_616_Int__emptyI,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ~ ( member408289922725080238_a_nat @ X3 @ B2 ) )
=> ( ( inf_in3249246906714053971_a_nat @ A @ B2 )
= bot_bo1033123847703346641_a_nat ) ) ).
% Int_emptyI
thf(fact_617_Int__emptyI,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ A )
=> ~ ( member_Sum_sum_a_nat @ X3 @ B2 ) )
=> ( ( inf_in7084830621192376909_a_nat @ A @ B2 )
= bot_bo3438331934148233675_a_nat ) ) ).
% Int_emptyI
thf(fact_618_Int__emptyI,axiom,
! [A: set_o,B2: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ~ ( member_o @ X3 @ B2 ) )
=> ( ( inf_inf_set_o @ A @ B2 )
= bot_bot_set_o ) ) ).
% Int_emptyI
thf(fact_619_times__eq__iff,axiom,
! [A: set_o,B2: set_o,C: set_o,D: set_o] :
( ( ( product_Sigma_o_o @ A
@ ^ [Uu2: $o] : B2 )
= ( product_Sigma_o_o @ C
@ ^ [Uu2: $o] : D ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( ( A = bot_bot_set_o )
| ( B2 = bot_bot_set_o ) )
& ( ( C = bot_bot_set_o )
| ( D = bot_bot_set_o ) ) ) ) ) ).
% times_eq_iff
thf(fact_620_Int__Diff__disjoint,axiom,
! [A: set_o,B2: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ A @ B2 ) @ ( minus_minus_set_o @ A @ B2 ) )
= bot_bot_set_o ) ).
% Int_Diff_disjoint
thf(fact_621_Diff__triv,axiom,
! [A: set_o,B2: set_o] :
( ( ( inf_inf_set_o @ A @ B2 )
= bot_bot_set_o )
=> ( ( minus_minus_set_o @ A @ B2 )
= A ) ) ).
% Diff_triv
thf(fact_622_times__subset__iff,axiom,
! [A: set_o,C: set_o,B2: set_o,D: set_o] :
( ( ord_le192195778054999463od_o_o
@ ( product_Sigma_o_o @ A
@ ^ [Uu2: $o] : C )
@ ( product_Sigma_o_o @ B2
@ ^ [Uu2: $o] : D ) )
= ( ( A = bot_bot_set_o )
| ( C = bot_bot_set_o )
| ( ( ord_less_eq_set_o @ A @ B2 )
& ( ord_less_eq_set_o @ C @ D ) ) ) ) ).
% times_subset_iff
thf(fact_623_fst__image__Sigma,axiom,
! [A: set_o,B2: $o > set_o] :
( ( image_7896445794123959606_o_o_o @ product_fst_o_o @ ( product_Sigma_o_o @ A @ B2 ) )
= ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A )
& ( ( B2 @ X )
!= bot_bot_set_o ) ) ) ) ).
% fst_image_Sigma
thf(fact_624_fst__image__Sigma,axiom,
! [A: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat > set_o] :
( ( image_8418685337248039824_a_nat @ produc2526971070270940118_nat_o @ ( produc8891875240947076123_nat_o @ A @ B2 ) )
= ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
& ( ( B2 @ X )
!= bot_bot_set_o ) ) ) ) ).
% fst_image_Sigma
thf(fact_625_fst__image__Sigma,axiom,
! [A: set_Sum_sum_a_nat,B2: sum_sum_a_nat > set_o] :
( ( image_1219780785596053892_a_nat @ produc5210810946726950620_nat_o @ ( produc5036831428884481441_nat_o @ A @ B2 ) )
= ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A )
& ( ( B2 @ X )
!= bot_bot_set_o ) ) ) ) ).
% fst_image_Sigma
thf(fact_626_fst__image__Sigma,axiom,
! [A: set_na3699693778330250182_a_nat,B2: ( nat > sum_sum_a_nat ) > set_o] :
( ( image_7171558470974387282_a_nat @ produc8176003148060736437_nat_o @ ( produc5469918130606046970_nat_o @ A @ B2 ) )
= ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ A )
& ( ( B2 @ X )
!= bot_bot_set_o ) ) ) ) ).
% fst_image_Sigma
thf(fact_627_flip__pred,axiom,
! [A: set_Pr319891966805635327_set_a,R2: set_a > produc5986856040925105290_a_nat > $o] :
( ( ord_le10238975308788063_set_a @ A @ ( collec2075533386773609886_set_a @ ( produc7638002133863941488et_a_o @ ( conver6680288441826205363_a_nat @ R2 ) ) ) )
=> ( ord_le8227873432414528045_a_nat
@ ( image_9020305190384351591_a_nat
@ ( produc4742807462889283007_a_nat
@ ^ [X: produc5986856040925105290_a_nat,Y3: set_a] : ( produc1626021218680649873_a_nat @ Y3 @ X ) )
@ A )
@ ( collec2944593301445145964_a_nat @ ( produc2676398950370459482_nat_o @ R2 ) ) ) ) ).
% flip_pred
thf(fact_628_flip__pred,axiom,
! [A: set_Pr4681753670034042266at_nat,R2: nat > set_li6526943997496501093_a_nat > $o] :
( ( ord_le6080490343219138042at_nat @ A @ ( collec5896423582435531193at_nat @ ( produc4676069979212917453_nat_o @ ( conver4048542618835142494_a_nat @ R2 ) ) ) )
=> ( ord_le2863945612512301728_a_nat
@ ( image_2804021043688564853_a_nat
@ ( produc6022691998427229077_a_nat
@ ^ [X: set_li6526943997496501093_a_nat,Y3: nat] : ( produc4622690550020908860_a_nat @ Y3 @ X ) )
@ A )
@ ( collec3658349449910702431_a_nat @ ( produc380675777491300335_nat_o @ R2 ) ) ) ) ).
% flip_pred
thf(fact_629_antisym__Restr,axiom,
! [R3: set_Pr7343886759072863943_a_nat,A: set_Sum_sum_a_nat] :
( ( antisy1998289778759880802_a_nat @ top_to795618464972521135_a_nat @ R3 )
=> ( antisy1998289778759880802_a_nat @ top_to795618464972521135_a_nat
@ ( inf_in4755803168322931893_a_nat @ R3
@ ( produc3014407214216980488_a_nat @ A
@ ^ [Uu2: sum_sum_a_nat] : A ) ) ) ) ).
% antisym_Restr
thf(fact_630_antisym__Restr,axiom,
! [R3: set_Product_prod_o_o,A: set_o] :
( ( antisym_on_o @ top_top_set_o @ R3 )
=> ( antisym_on_o @ top_top_set_o
@ ( inf_in4898592226082374645od_o_o @ R3
@ ( product_Sigma_o_o @ A
@ ^ [Uu2: $o] : A ) ) ) ) ).
% antisym_Restr
thf(fact_631_prod_Ocollapse,axiom,
! [Prod: produc8208987855206359447_a_nat] :
( ( produc1626021218680649873_a_nat @ ( produc4817319330785305509_a_nat @ Prod ) @ ( produc4350472740430820835_a_nat @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_632_prod_Ocollapse,axiom,
! [Prod: produc5986856040925105290_a_nat] :
( ( produc4622690550020908860_a_nat @ ( produc8331164115582522960_a_nat @ Prod ) @ ( produc7763585607612862094_a_nat @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_633_less__eq__prod__simps,axiom,
! [Leq_a: set_a > set_a > $o,Less_a: set_a > set_a > $o,Leq_b: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > $o,X1: set_a,X22: produc5986856040925105290_a_nat,Y1: set_a,Y22: produc5986856040925105290_a_nat] :
( ( contai4758316629579656036_a_nat @ Leq_a @ Less_a @ Leq_b @ ( produc1626021218680649873_a_nat @ X1 @ X22 ) @ ( produc1626021218680649873_a_nat @ Y1 @ Y22 ) )
= ( ( Less_a @ X1 @ Y1 )
| ( ( Leq_a @ X1 @ Y1 )
& ( Leq_b @ X22 @ Y22 ) ) ) ) ).
% less_eq_prod_simps
thf(fact_634_less__eq__prod__simps,axiom,
! [Leq_a: nat > nat > $o,Less_a: nat > nat > $o,Leq_b: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > $o,X1: nat,X22: set_li6526943997496501093_a_nat,Y1: nat,Y22: set_li6526943997496501093_a_nat] :
( ( contai7163141967149461775_a_nat @ Leq_a @ Less_a @ Leq_b @ ( produc4622690550020908860_a_nat @ X1 @ X22 ) @ ( produc4622690550020908860_a_nat @ Y1 @ Y22 ) )
= ( ( Less_a @ X1 @ Y1 )
| ( ( Leq_a @ X1 @ Y1 )
& ( Leq_b @ X22 @ Y22 ) ) ) ) ).
% less_eq_prod_simps
thf(fact_635_snd__image__times,axiom,
! [A: set_o,B2: set_o] :
( ( ( A = bot_bot_set_o )
=> ( ( image_7896445794123959606_o_o_o @ product_snd_o_o
@ ( product_Sigma_o_o @ A
@ ^ [Uu2: $o] : B2 ) )
= bot_bot_set_o ) )
& ( ( A != bot_bot_set_o )
=> ( ( image_7896445794123959606_o_o_o @ product_snd_o_o
@ ( product_Sigma_o_o @ A
@ ^ [Uu2: $o] : B2 ) )
= B2 ) ) ) ).
% snd_image_times
thf(fact_636_eq__snd__iff,axiom,
! [B: produc5986856040925105290_a_nat,P2: produc8208987855206359447_a_nat] :
( ( B
= ( produc4350472740430820835_a_nat @ P2 ) )
= ( ? [A8: set_a] :
( P2
= ( produc1626021218680649873_a_nat @ A8 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_637_eq__snd__iff,axiom,
! [B: set_li6526943997496501093_a_nat,P2: produc5986856040925105290_a_nat] :
( ( B
= ( produc7763585607612862094_a_nat @ P2 ) )
= ( ? [A8: nat] :
( P2
= ( produc4622690550020908860_a_nat @ A8 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_638_sndI,axiom,
! [X2: produc8208987855206359447_a_nat,Y2: set_a,Z: produc5986856040925105290_a_nat] :
( ( X2
= ( produc1626021218680649873_a_nat @ Y2 @ Z ) )
=> ( ( produc4350472740430820835_a_nat @ X2 )
= Z ) ) ).
% sndI
thf(fact_639_sndI,axiom,
! [X2: produc5986856040925105290_a_nat,Y2: nat,Z: set_li6526943997496501093_a_nat] :
( ( X2
= ( produc4622690550020908860_a_nat @ Y2 @ Z ) )
=> ( ( produc7763585607612862094_a_nat @ X2 )
= Z ) ) ).
% sndI
thf(fact_640_bot__empty__eq,axiom,
( bot_bo9042073657639083596_nat_o
= ( ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ bot_bo1033123847703346641_a_nat ) ) ) ).
% bot_empty_eq
thf(fact_641_bot__empty__eq,axiom,
( bot_bo7797463397293707474_nat_o
= ( ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ bot_bo3438331934148233675_a_nat ) ) ) ).
% bot_empty_eq
thf(fact_642_bot__empty__eq,axiom,
( bot_bot_o_o
= ( ^ [X: $o] : ( member_o @ X @ bot_bot_set_o ) ) ) ).
% bot_empty_eq
thf(fact_643_bot__set__def,axiom,
( bot_bo6441361344521902642_a_nat
= ( collec5629555741568564177_a_nat @ bot_bo3382309974966529835_nat_o ) ) ).
% bot_set_def
thf(fact_644_bot__set__def,axiom,
( bot_bot_set_o
= ( collect_o @ bot_bot_o_o ) ) ).
% bot_set_def
thf(fact_645_snd__conv,axiom,
! [X1: set_a,X22: produc5986856040925105290_a_nat] :
( ( produc4350472740430820835_a_nat @ ( produc1626021218680649873_a_nat @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_646_snd__conv,axiom,
! [X1: nat,X22: set_li6526943997496501093_a_nat] :
( ( produc7763585607612862094_a_nat @ ( produc4622690550020908860_a_nat @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_647_snd__eqD,axiom,
! [X2: set_a,Y2: produc5986856040925105290_a_nat,A2: produc5986856040925105290_a_nat] :
( ( ( produc4350472740430820835_a_nat @ ( produc1626021218680649873_a_nat @ X2 @ Y2 ) )
= A2 )
=> ( Y2 = A2 ) ) ).
% snd_eqD
thf(fact_648_snd__eqD,axiom,
! [X2: nat,Y2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ( produc7763585607612862094_a_nat @ ( produc4622690550020908860_a_nat @ X2 @ Y2 ) )
= A2 )
=> ( Y2 = A2 ) ) ).
% snd_eqD
thf(fact_649_antisym__onI,axiom,
! [A: set_o,R3: set_Product_prod_o_o] :
( ! [X3: $o,Y4: $o] :
( ( member_o @ X3 @ A )
=> ( ( member_o @ Y4 @ A )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y4 ) @ R3 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y4 @ X3 ) @ R3 )
=> ( X3 = Y4 ) ) ) ) )
=> ( antisym_on_o @ A @ R3 ) ) ).
% antisym_onI
thf(fact_650_antisym__onI,axiom,
! [A: set_li6526943997496501093_a_nat,R3: set_Pr4870381170404451655_a_nat] :
( ! [X3: list_Sum_sum_a_nat,Y4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ( ( member408289922725080238_a_nat @ Y4 @ A )
=> ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ X3 @ Y4 ) @ R3 )
=> ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Y4 @ X3 ) @ R3 )
=> ( X3 = Y4 ) ) ) ) )
=> ( antisy4240500386402606696_a_nat @ A @ R3 ) ) ).
% antisym_onI
thf(fact_651_antisym__onI,axiom,
! [A: set_Sum_sum_a_nat,R3: set_Pr7343886759072863943_a_nat] :
( ! [X3: sum_sum_a_nat,Y4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ A )
=> ( ( member_Sum_sum_a_nat @ Y4 @ A )
=> ( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X3 @ Y4 ) @ R3 )
=> ( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ Y4 @ X3 ) @ R3 )
=> ( X3 = Y4 ) ) ) ) )
=> ( antisy1998289778759880802_a_nat @ A @ R3 ) ) ).
% antisym_onI
thf(fact_652_antisym__onD,axiom,
! [A: set_o,R3: set_Product_prod_o_o,X2: $o,Y2: $o] :
( ( antisym_on_o @ A @ R3 )
=> ( ( member_o @ X2 @ A )
=> ( ( member_o @ Y2 @ A )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y2 ) @ R3 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y2 @ X2 ) @ R3 )
=> ( X2 = Y2 ) ) ) ) ) ) ).
% antisym_onD
thf(fact_653_antisym__onD,axiom,
! [A: set_li6526943997496501093_a_nat,R3: set_Pr4870381170404451655_a_nat,X2: list_Sum_sum_a_nat,Y2: list_Sum_sum_a_nat] :
( ( antisy4240500386402606696_a_nat @ A @ R3 )
=> ( ( member408289922725080238_a_nat @ X2 @ A )
=> ( ( member408289922725080238_a_nat @ Y2 @ A )
=> ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ X2 @ Y2 ) @ R3 )
=> ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Y2 @ X2 ) @ R3 )
=> ( X2 = Y2 ) ) ) ) ) ) ).
% antisym_onD
thf(fact_654_antisym__onD,axiom,
! [A: set_Sum_sum_a_nat,R3: set_Pr7343886759072863943_a_nat,X2: sum_sum_a_nat,Y2: sum_sum_a_nat] :
( ( antisy1998289778759880802_a_nat @ A @ R3 )
=> ( ( member_Sum_sum_a_nat @ X2 @ A )
=> ( ( member_Sum_sum_a_nat @ Y2 @ A )
=> ( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X2 @ Y2 ) @ R3 )
=> ( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ Y2 @ X2 ) @ R3 )
=> ( X2 = Y2 ) ) ) ) ) ) ).
% antisym_onD
thf(fact_655_bot__empty__eq2,axiom,
( bot_bo2178092097334603616_nat_o
= ( ^ [X: set_a,Y3: produc5986856040925105290_a_nat] : ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ X @ Y3 ) @ bot_bo3440348358004571233_a_nat ) ) ) ).
% bot_empty_eq2
thf(fact_656_bot__empty__eq2,axiom,
( bot_bo2212031192099928027_nat_o
= ( ^ [X: nat,Y3: set_li6526943997496501093_a_nat] : ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ X @ Y3 ) @ bot_bo1546958992129757396_a_nat ) ) ) ).
% bot_empty_eq2
thf(fact_657_antisym__empty,axiom,
antisy1998289778759880802_a_nat @ top_to795618464972521135_a_nat @ bot_bo6795098209002113331_a_nat ).
% antisym_empty
thf(fact_658_antisym__empty,axiom,
antisym_on_o @ top_top_set_o @ bot_bo7073875226086086771od_o_o ).
% antisym_empty
thf(fact_659_prod_Oexhaust__sel,axiom,
! [Prod: produc8208987855206359447_a_nat] :
( Prod
= ( produc1626021218680649873_a_nat @ ( produc4817319330785305509_a_nat @ Prod ) @ ( produc4350472740430820835_a_nat @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_660_prod_Oexhaust__sel,axiom,
! [Prod: produc5986856040925105290_a_nat] :
( Prod
= ( produc4622690550020908860_a_nat @ ( produc8331164115582522960_a_nat @ Prod ) @ ( produc7763585607612862094_a_nat @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_661_surjective__pairing,axiom,
! [T2: produc8208987855206359447_a_nat] :
( T2
= ( produc1626021218680649873_a_nat @ ( produc4817319330785305509_a_nat @ T2 ) @ ( produc4350472740430820835_a_nat @ T2 ) ) ) ).
% surjective_pairing
thf(fact_662_surjective__pairing,axiom,
! [T2: produc5986856040925105290_a_nat] :
( T2
= ( produc4622690550020908860_a_nat @ ( produc8331164115582522960_a_nat @ T2 ) @ ( produc7763585607612862094_a_nat @ T2 ) ) ) ).
% surjective_pairing
thf(fact_663_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: set_a > produc5986856040925105290_a_nat > $o,X2: set_a,Y2: produc5986856040925105290_a_nat,A2: produc8208987855206359447_a_nat] :
( ( P @ X2 @ Y2 )
=> ( ( A2
= ( produc1626021218680649873_a_nat @ X2 @ Y2 ) )
=> ( P @ ( produc4817319330785305509_a_nat @ A2 ) @ ( produc4350472740430820835_a_nat @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_664_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: nat > set_li6526943997496501093_a_nat > $o,X2: nat,Y2: set_li6526943997496501093_a_nat,A2: produc5986856040925105290_a_nat] :
( ( P @ X2 @ Y2 )
=> ( ( A2
= ( produc4622690550020908860_a_nat @ X2 @ Y2 ) )
=> ( P @ ( produc8331164115582522960_a_nat @ A2 ) @ ( produc7763585607612862094_a_nat @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_665_mem__Times__iff,axiom,
! [X2: product_prod_o_o,A: set_o,B2: set_o] :
( ( member7466972457876170832od_o_o @ X2
@ ( product_Sigma_o_o @ A
@ ^ [Uu2: $o] : B2 ) )
= ( ( member_o @ ( product_fst_o_o @ X2 ) @ A )
& ( member_o @ ( product_snd_o_o @ X2 ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_666_mem__Times__iff,axiom,
! [X2: produc5889817346055602324_a_nat,A: set_o,B2: set_li6526943997496501093_a_nat] :
( ( member2772425197143436605_a_nat @ X2
@ ( produc5640064037308512181_a_nat @ A
@ ^ [Uu2: $o] : B2 ) )
= ( ( member_o @ ( produc8498531903487151984_a_nat @ X2 ) @ A )
& ( member408289922725080238_a_nat @ ( produc2362415081064084402_a_nat @ X2 ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_667_mem__Times__iff,axiom,
! [X2: produc7898449166577977870_a_nat,A: set_o,B2: set_Sum_sum_a_nat] :
( ( member6925473634427674423_a_nat @ X2
@ ( produc3081175752665019439_a_nat @ A
@ ^ [Uu2: $o] : B2 ) )
= ( ( member_o @ ( produc3255155270507488618_a_nat @ X2 ) @ A )
& ( member_Sum_sum_a_nat @ ( produc8174652519921488556_a_nat @ X2 ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_668_mem__Times__iff,axiom,
! [X2: produc4278410840041776506_nat_o,A: set_li6526943997496501093_a_nat,B2: set_o] :
( ( member1161018691129610787_nat_o @ X2
@ ( produc8891875240947076123_nat_o @ A
@ ^ [Uu2: list_Sum_sum_a_nat] : B2 ) )
= ( ( member408289922725080238_a_nat @ ( produc2526971070270940118_nat_o @ X2 ) @ A )
& ( member_o @ ( produc5614226284702648344_nat_o @ X2 ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_669_mem__Times__iff,axiom,
! [X2: produc5001885624171833703_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member7457213283480048528_a_nat @ X2
@ ( produc5624668685630548104_a_nat @ A
@ ^ [Uu2: list_Sum_sum_a_nat] : B2 ) )
= ( ( member408289922725080238_a_nat @ ( produc2015473822650161347_a_nat @ X2 ) @ A )
& ( member408289922725080238_a_nat @ ( produc2553995981594484741_a_nat @ X2 ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_670_mem__Times__iff,axiom,
! [X2: produc1461395728222533729_a_nat,A: set_li6526943997496501093_a_nat,B2: set_Sum_sum_a_nat] :
( ( member6493994536747010826_a_nat @ X2
@ ( produc1079312233239997058_a_nat @ A
@ ^ [Uu2: list_Sum_sum_a_nat] : B2 ) )
= ( ( member408289922725080238_a_nat @ ( produc1726540898437313597_a_nat @ X2 ) @ A )
& ( member_Sum_sum_a_nat @ ( produc782848267389879423_a_nat @ X2 ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_671_mem__Times__iff,axiom,
! [X2: produc7659714187748542208_nat_o,A: set_Sum_sum_a_nat,B2: set_o] :
( ( member6686738655598238761_nat_o @ X2
@ ( produc5036831428884481441_nat_o @ A
@ ^ [Uu2: sum_sum_a_nat] : B2 ) )
= ( ( member_Sum_sum_a_nat @ ( produc5210810946726950620_nat_o @ X2 ) @ A )
& ( member_o @ ( produc906936159286174750_nat_o @ X2 ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_672_mem__Times__iff,axiom,
! [X2: produc4502985402200462317_a_nat,A: set_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member312212173870163606_a_nat @ X2
@ ( produc1452074852614433550_a_nat @ A
@ ^ [Uu2: sum_sum_a_nat] : B2 ) )
= ( ( member_Sum_sum_a_nat @ ( produc2099303517811750089_a_nat @ X2 ) @ A )
& ( member408289922725080238_a_nat @ ( produc1155610886764315915_a_nat @ X2 ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_673_mem__Times__iff,axiom,
! [X2: produc7017002724195966439_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member3723442691059620112_a_nat @ X2
@ ( produc3014407214216980488_a_nat @ A
@ ^ [Uu2: sum_sum_a_nat] : B2 ) )
= ( ( member_Sum_sum_a_nat @ ( produc3940427395981429571_a_nat @ X2 ) @ A )
& ( member_Sum_sum_a_nat @ ( produc8028804885397038213_a_nat @ X2 ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_674_antisymD,axiom,
! [R3: set_Pr7343886759072863943_a_nat,X2: sum_sum_a_nat,Y2: sum_sum_a_nat] :
( ( antisy1998289778759880802_a_nat @ top_to795618464972521135_a_nat @ R3 )
=> ( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X2 @ Y2 ) @ R3 )
=> ( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ Y2 @ X2 ) @ R3 )
=> ( X2 = Y2 ) ) ) ) ).
% antisymD
thf(fact_675_antisymD,axiom,
! [R3: set_Product_prod_o_o,X2: $o,Y2: $o] :
( ( antisym_on_o @ top_top_set_o @ R3 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y2 ) @ R3 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y2 @ X2 ) @ R3 )
=> ( X2 = Y2 ) ) ) ) ).
% antisymD
thf(fact_676_antisymI,axiom,
! [R3: set_Pr7343886759072863943_a_nat] :
( ! [X3: sum_sum_a_nat,Y4: sum_sum_a_nat] :
( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X3 @ Y4 ) @ R3 )
=> ( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ Y4 @ X3 ) @ R3 )
=> ( X3 = Y4 ) ) )
=> ( antisy1998289778759880802_a_nat @ top_to795618464972521135_a_nat @ R3 ) ) ).
% antisymI
thf(fact_677_antisymI,axiom,
! [R3: set_Product_prod_o_o] :
( ! [X3: $o,Y4: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y4 ) @ R3 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y4 @ X3 ) @ R3 )
=> ( X3 = Y4 ) ) )
=> ( antisym_on_o @ top_top_set_o @ R3 ) ) ).
% antisymI
thf(fact_678_antisym__subset,axiom,
! [R3: set_Pr7343886759072863943_a_nat,S2: set_Pr7343886759072863943_a_nat] :
( ( ord_le7501488065512009319_a_nat @ R3 @ S2 )
=> ( ( antisy1998289778759880802_a_nat @ top_to795618464972521135_a_nat @ S2 )
=> ( antisy1998289778759880802_a_nat @ top_to795618464972521135_a_nat @ R3 ) ) ) ).
% antisym_subset
thf(fact_679_antisym__subset,axiom,
! [R3: set_Product_prod_o_o,S2: set_Product_prod_o_o] :
( ( ord_le192195778054999463od_o_o @ R3 @ S2 )
=> ( ( antisym_on_o @ top_top_set_o @ S2 )
=> ( antisym_on_o @ top_top_set_o @ R3 ) ) ) ).
% antisym_subset
thf(fact_680_prod_Oswap__def,axiom,
( produc4582320955043612251_set_a
= ( ^ [P3: produc7339927940534823369_set_a] : ( produc1626021218680649873_a_nat @ ( produc64190399388289357_set_a @ P3 ) @ ( produc531036989742774031_set_a @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_681_prod_Oswap__def,axiom,
( produc7974059260437611070at_nat
= ( ^ [P3: produc8224930173449934052at_nat] : ( produc4622690550020908860_a_nat @ ( produc6446428027415449392at_nat @ P3 ) @ ( produc7014006535385110258at_nat @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_682_conjI__realizer,axiom,
! [P: set_a > $o,P2: set_a,Q: produc5986856040925105290_a_nat > $o,Q3: produc5986856040925105290_a_nat] :
( ( P @ P2 )
=> ( ( Q @ Q3 )
=> ( ( P @ ( produc4817319330785305509_a_nat @ ( produc1626021218680649873_a_nat @ P2 @ Q3 ) ) )
& ( Q @ ( produc4350472740430820835_a_nat @ ( produc1626021218680649873_a_nat @ P2 @ Q3 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_683_conjI__realizer,axiom,
! [P: nat > $o,P2: nat,Q: set_li6526943997496501093_a_nat > $o,Q3: set_li6526943997496501093_a_nat] :
( ( P @ P2 )
=> ( ( Q @ Q3 )
=> ( ( P @ ( produc8331164115582522960_a_nat @ ( produc4622690550020908860_a_nat @ P2 @ Q3 ) ) )
& ( Q @ ( produc7763585607612862094_a_nat @ ( produc4622690550020908860_a_nat @ P2 @ Q3 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_684_exI__realizer,axiom,
! [P: produc5986856040925105290_a_nat > set_a > $o,Y2: produc5986856040925105290_a_nat,X2: set_a] :
( ( P @ Y2 @ X2 )
=> ( P @ ( produc4350472740430820835_a_nat @ ( produc1626021218680649873_a_nat @ X2 @ Y2 ) ) @ ( produc4817319330785305509_a_nat @ ( produc1626021218680649873_a_nat @ X2 @ Y2 ) ) ) ) ).
% exI_realizer
thf(fact_685_exI__realizer,axiom,
! [P: set_li6526943997496501093_a_nat > nat > $o,Y2: set_li6526943997496501093_a_nat,X2: nat] :
( ( P @ Y2 @ X2 )
=> ( P @ ( produc7763585607612862094_a_nat @ ( produc4622690550020908860_a_nat @ X2 @ Y2 ) ) @ ( produc8331164115582522960_a_nat @ ( produc4622690550020908860_a_nat @ X2 @ Y2 ) ) ) ) ).
% exI_realizer
thf(fact_686_less__prod__simps,axiom,
! [Leq_a: set_a > set_a > $o,Less_a: set_a > set_a > $o,Less_b: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > $o,X1: set_a,X22: produc5986856040925105290_a_nat,Y1: set_a,Y22: produc5986856040925105290_a_nat] :
( ( contai5360861433347197552_a_nat @ Leq_a @ Less_a @ Less_b @ ( produc1626021218680649873_a_nat @ X1 @ X22 ) @ ( produc1626021218680649873_a_nat @ Y1 @ Y22 ) )
= ( ( Less_a @ X1 @ Y1 )
| ( ( Leq_a @ X1 @ Y1 )
& ( Less_b @ X22 @ Y22 ) ) ) ) ).
% less_prod_simps
thf(fact_687_less__prod__simps,axiom,
! [Leq_a: nat > nat > $o,Less_a: nat > nat > $o,Less_b: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > $o,X1: nat,X22: set_li6526943997496501093_a_nat,Y1: nat,Y22: set_li6526943997496501093_a_nat] :
( ( contai9166645720955120667_a_nat @ Leq_a @ Less_a @ Less_b @ ( produc4622690550020908860_a_nat @ X1 @ X22 ) @ ( produc4622690550020908860_a_nat @ Y1 @ Y22 ) )
= ( ( Less_a @ X1 @ Y1 )
| ( ( Leq_a @ X1 @ Y1 )
& ( Less_b @ X22 @ Y22 ) ) ) ) ).
% less_prod_simps
thf(fact_688_antisymp__antisym__eq,axiom,
! [R3: set_Pr7343886759072863943_a_nat] :
( ( antisy1530855801211272108_a_nat @ top_to795618464972521135_a_nat
@ ^ [X: sum_sum_a_nat,Y3: sum_sum_a_nat] : ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X @ Y3 ) @ R3 ) )
= ( antisy1998289778759880802_a_nat @ top_to795618464972521135_a_nat @ R3 ) ) ).
% antisymp_antisym_eq
thf(fact_689_antisymp__antisym__eq,axiom,
! [R3: set_Product_prod_o_o] :
( ( antisymp_on_o @ top_top_set_o
@ ^ [X: $o,Y3: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y3 ) @ R3 ) )
= ( antisym_on_o @ top_top_set_o @ R3 ) ) ).
% antisymp_antisym_eq
thf(fact_690_snd__fst__flip,axiom,
( produc64190399388289357_set_a
= ( comp_P1959524498981142701_set_a @ produc4817319330785305509_a_nat
@ ( produc4742807462889283007_a_nat
@ ^ [X: produc5986856040925105290_a_nat,Y3: set_a] : ( produc1626021218680649873_a_nat @ Y3 @ X ) ) ) ) ).
% snd_fst_flip
thf(fact_691_snd__fst__flip,axiom,
( produc6446428027415449392at_nat
= ( comp_P8983658086972211491at_nat @ produc8331164115582522960_a_nat
@ ( produc6022691998427229077_a_nat
@ ^ [X: set_li6526943997496501093_a_nat,Y3: nat] : ( produc4622690550020908860_a_nat @ Y3 @ X ) ) ) ) ).
% snd_fst_flip
thf(fact_692_fst__snd__flip,axiom,
( produc531036989742774031_set_a
= ( comp_P3734683246922352847_set_a @ produc4350472740430820835_a_nat
@ ( produc4742807462889283007_a_nat
@ ^ [X: produc5986856040925105290_a_nat,Y3: set_a] : ( produc1626021218680649873_a_nat @ Y3 @ X ) ) ) ) ).
% fst_snd_flip
thf(fact_693_fst__snd__flip,axiom,
( produc7014006535385110258at_nat
= ( comp_P6214864338719132572at_nat @ produc7763585607612862094_a_nat
@ ( produc6022691998427229077_a_nat
@ ^ [X: set_li6526943997496501093_a_nat,Y3: nat] : ( produc4622690550020908860_a_nat @ Y3 @ X ) ) ) ) ).
% fst_snd_flip
thf(fact_694_antisymp__equality,axiom,
( antisy1530855801211272108_a_nat @ top_to795618464972521135_a_nat
@ ^ [Y6: sum_sum_a_nat,Z3: sum_sum_a_nat] : ( Y6 = Z3 ) ) ).
% antisymp_equality
thf(fact_695_antisymp__equality,axiom,
( antisymp_on_o @ top_top_set_o
@ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 ) ) ).
% antisymp_equality
thf(fact_696_antisympI,axiom,
! [R2: sum_sum_a_nat > sum_sum_a_nat > $o] :
( ! [X3: sum_sum_a_nat,Y4: sum_sum_a_nat] :
( ( R2 @ X3 @ Y4 )
=> ( ( R2 @ Y4 @ X3 )
=> ( X3 = Y4 ) ) )
=> ( antisy1530855801211272108_a_nat @ top_to795618464972521135_a_nat @ R2 ) ) ).
% antisympI
thf(fact_697_antisympI,axiom,
! [R2: $o > $o > $o] :
( ! [X3: $o,Y4: $o] :
( ( R2 @ X3 @ Y4 )
=> ( ( R2 @ Y4 @ X3 )
=> ( X3 = Y4 ) ) )
=> ( antisymp_on_o @ top_top_set_o @ R2 ) ) ).
% antisympI
thf(fact_698_antisympD,axiom,
! [R2: sum_sum_a_nat > sum_sum_a_nat > $o,X2: sum_sum_a_nat,Y2: sum_sum_a_nat] :
( ( antisy1530855801211272108_a_nat @ top_to795618464972521135_a_nat @ R2 )
=> ( ( R2 @ X2 @ Y2 )
=> ( ( R2 @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ) ).
% antisympD
thf(fact_699_antisympD,axiom,
! [R2: $o > $o > $o,X2: $o,Y2: $o] :
( ( antisymp_on_o @ top_top_set_o @ R2 )
=> ( ( R2 @ X2 @ Y2 )
=> ( ( R2 @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ) ).
% antisympD
thf(fact_700_image__eq__imp__comp,axiom,
! [F: $o > list_Sum_sum_a_nat,A: set_o,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat,H: list_Sum_sum_a_nat > $o] :
( ( ( image_733135354807394034_a_nat @ F @ A )
= ( image_5081948215111134021_a_nat @ G @ B2 ) )
=> ( ( image_o_o2 @ ( comp_l142836649995191522at_o_o @ H @ F ) @ A )
= ( image_3984946558445957976_nat_o @ ( comp_l7931591902798318543_a_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_701_image__eq__imp__comp,axiom,
! [F: list_Sum_sum_a_nat > $o,A: set_li6526943997496501093_a_nat,G: $o > $o,B2: set_o,H: $o > list_Sum_sum_a_nat] :
( ( ( image_3984946558445957976_nat_o @ F @ A )
= ( image_o_o2 @ G @ B2 ) )
=> ( ( image_5081948215111134021_a_nat @ ( comp_o1741750288207193781_a_nat @ H @ F ) @ A )
= ( image_733135354807394034_a_nat @ ( comp_o7361543398147360456_nat_o @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_702_image__eq__imp__comp,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,G: $o > list_Sum_sum_a_nat,B2: set_o,H: list_Sum_sum_a_nat > $o] :
( ( ( image_5081948215111134021_a_nat @ F @ A )
= ( image_733135354807394034_a_nat @ G @ B2 ) )
=> ( ( image_3984946558445957976_nat_o @ ( comp_l7931591902798318543_a_nat @ H @ F ) @ A )
= ( image_o_o2 @ ( comp_l142836649995191522at_o_o @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_703_image__eq__imp__comp,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat,H: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( ( image_5081948215111134021_a_nat @ F @ A )
= ( image_5081948215111134021_a_nat @ G @ B2 ) )
=> ( ( image_5081948215111134021_a_nat @ ( comp_l3689981812903912354_a_nat @ H @ F ) @ A )
= ( image_5081948215111134021_a_nat @ ( comp_l3689981812903912354_a_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_704_image__eq__imp__comp,axiom,
! [F: $o > $o,A: set_o,G: list_Sum_sum_a_nat > $o,B2: set_li6526943997496501093_a_nat,H: $o > list_Sum_sum_a_nat] :
( ( ( image_o_o2 @ F @ A )
= ( image_3984946558445957976_nat_o @ G @ B2 ) )
=> ( ( image_733135354807394034_a_nat @ ( comp_o7361543398147360456_nat_o @ H @ F ) @ A )
= ( image_5081948215111134021_a_nat @ ( comp_o1741750288207193781_a_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_705_image__eq__imp__comp,axiom,
! [F: $o > $o,A: set_o,G: $o > $o,B2: set_o,H: $o > $o] :
( ( ( image_o_o2 @ F @ A )
= ( image_o_o2 @ G @ B2 ) )
=> ( ( image_o_o2 @ ( comp_o_o_o @ H @ F ) @ A )
= ( image_o_o2 @ ( comp_o_o_o @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_706_image__comp,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,R3: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat @ F @ ( image_5081948215111134021_a_nat @ G @ R3 ) )
= ( image_5081948215111134021_a_nat @ ( comp_l3689981812903912354_a_nat @ F @ G ) @ R3 ) ) ).
% image_comp
thf(fact_707_image__comp,axiom,
! [F: $o > $o,G: $o > $o,R3: set_o] :
( ( image_o_o2 @ F @ ( image_o_o2 @ G @ R3 ) )
= ( image_o_o2 @ ( comp_o_o_o @ F @ G ) @ R3 ) ) ).
% image_comp
thf(fact_708_Sup_OSUP__image,axiom,
! [Sup: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( Sup @ ( image_5081948215111134021_a_nat @ G @ ( image_5081948215111134021_a_nat @ F @ A ) ) )
= ( Sup @ ( image_5081948215111134021_a_nat @ ( comp_l3689981812903912354_a_nat @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_709_Sup_OSUP__image,axiom,
! [Sup: set_o > $o,G: $o > $o,F: $o > $o,A: set_o] :
( ( Sup @ ( image_o_o2 @ G @ ( image_o_o2 @ F @ A ) ) )
= ( Sup @ ( image_o_o2 @ ( comp_o_o_o @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_710_Inf_OINF__image,axiom,
! [Inf: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( Inf @ ( image_5081948215111134021_a_nat @ G @ ( image_5081948215111134021_a_nat @ F @ A ) ) )
= ( Inf @ ( image_5081948215111134021_a_nat @ ( comp_l3689981812903912354_a_nat @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_711_Inf_OINF__image,axiom,
! [Inf: set_o > $o,G: $o > $o,F: $o > $o,A: set_o] :
( ( Inf @ ( image_o_o2 @ G @ ( image_o_o2 @ F @ A ) ) )
= ( Inf @ ( image_o_o2 @ ( comp_o_o_o @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_712_antisymp__onD,axiom,
! [A: set_o,R2: $o > $o > $o,X2: $o,Y2: $o] :
( ( antisymp_on_o @ A @ R2 )
=> ( ( member_o @ X2 @ A )
=> ( ( member_o @ Y2 @ A )
=> ( ( R2 @ X2 @ Y2 )
=> ( ( R2 @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ) ) ) ).
% antisymp_onD
thf(fact_713_antisymp__onD,axiom,
! [A: set_li6526943997496501093_a_nat,R2: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o,X2: list_Sum_sum_a_nat,Y2: list_Sum_sum_a_nat] :
( ( antisy6532343992071249074_a_nat @ A @ R2 )
=> ( ( member408289922725080238_a_nat @ X2 @ A )
=> ( ( member408289922725080238_a_nat @ Y2 @ A )
=> ( ( R2 @ X2 @ Y2 )
=> ( ( R2 @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ) ) ) ).
% antisymp_onD
thf(fact_714_antisymp__onD,axiom,
! [A: set_Sum_sum_a_nat,R2: sum_sum_a_nat > sum_sum_a_nat > $o,X2: sum_sum_a_nat,Y2: sum_sum_a_nat] :
( ( antisy1530855801211272108_a_nat @ A @ R2 )
=> ( ( member_Sum_sum_a_nat @ X2 @ A )
=> ( ( member_Sum_sum_a_nat @ Y2 @ A )
=> ( ( R2 @ X2 @ Y2 )
=> ( ( R2 @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ) ) ) ).
% antisymp_onD
thf(fact_715_antisymp__onI,axiom,
! [A: set_o,R2: $o > $o > $o] :
( ! [X3: $o,Y4: $o] :
( ( member_o @ X3 @ A )
=> ( ( member_o @ Y4 @ A )
=> ( ( R2 @ X3 @ Y4 )
=> ( ( R2 @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ) )
=> ( antisymp_on_o @ A @ R2 ) ) ).
% antisymp_onI
thf(fact_716_antisymp__onI,axiom,
! [A: set_li6526943997496501093_a_nat,R2: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
( ! [X3: list_Sum_sum_a_nat,Y4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ( ( member408289922725080238_a_nat @ Y4 @ A )
=> ( ( R2 @ X3 @ Y4 )
=> ( ( R2 @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ) )
=> ( antisy6532343992071249074_a_nat @ A @ R2 ) ) ).
% antisymp_onI
thf(fact_717_antisymp__onI,axiom,
! [A: set_Sum_sum_a_nat,R2: sum_sum_a_nat > sum_sum_a_nat > $o] :
( ! [X3: sum_sum_a_nat,Y4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ A )
=> ( ( member_Sum_sum_a_nat @ Y4 @ A )
=> ( ( R2 @ X3 @ Y4 )
=> ( ( R2 @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ) )
=> ( antisy1530855801211272108_a_nat @ A @ R2 ) ) ).
% antisymp_onI
thf(fact_718_comp__surj,axiom,
! [F: $o > $o,G: $o > $o] :
( ( ( image_o_o2 @ F @ top_top_set_o )
= top_top_set_o )
=> ( ( ( image_o_o2 @ G @ top_top_set_o )
= top_top_set_o )
=> ( ( image_o_o2 @ ( comp_o_o_o @ G @ F ) @ top_top_set_o )
= top_top_set_o ) ) ) ).
% comp_surj
thf(fact_719_comp__surj,axiom,
! [F: sum_sum_a_nat > $o,G: $o > $o] :
( ( ( image_6095136190293192542_nat_o @ F @ top_to795618464972521135_a_nat )
= top_top_set_o )
=> ( ( ( image_o_o2 @ G @ top_top_set_o )
= top_top_set_o )
=> ( ( image_6095136190293192542_nat_o @ ( comp_o835585038542465372_a_nat @ G @ F ) @ top_to795618464972521135_a_nat )
= top_top_set_o ) ) ) ).
% comp_surj
thf(fact_720_comp__surj,axiom,
! [F: $o > sum_sum_a_nat,G: sum_sum_a_nat > $o] :
( ( ( image_4139480514073730540_a_nat @ F @ top_top_set_o )
= top_to795618464972521135_a_nat )
=> ( ( ( image_6095136190293192542_nat_o @ G @ top_to795618464972521135_a_nat )
= top_top_set_o )
=> ( ( image_o_o2 @ ( comp_S5928190372961538268at_o_o @ G @ F ) @ top_top_set_o )
= top_top_set_o ) ) ) ).
% comp_surj
thf(fact_721_comp__surj,axiom,
! [F: $o > $o,G: $o > sum_sum_a_nat] :
( ( ( image_o_o2 @ F @ top_top_set_o )
= top_top_set_o )
=> ( ( ( image_4139480514073730540_a_nat @ G @ top_top_set_o )
= top_to795618464972521135_a_nat )
=> ( ( image_4139480514073730540_a_nat @ ( comp_o2791240714761927374_nat_o @ G @ F ) @ top_top_set_o )
= top_to795618464972521135_a_nat ) ) ) ).
% comp_surj
thf(fact_722_comp__surj,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: sum_sum_a_nat > $o] :
( ( ( image_7142520692256960453_a_nat @ F @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat )
=> ( ( ( image_6095136190293192542_nat_o @ G @ top_to795618464972521135_a_nat )
= top_top_set_o )
=> ( ( image_6095136190293192542_nat_o @ ( comp_S2923948626727118147_a_nat @ G @ F ) @ top_to795618464972521135_a_nat )
= top_top_set_o ) ) ) ).
% comp_surj
thf(fact_723_comp__surj,axiom,
! [F: sum_sum_a_nat > $o,G: $o > sum_sum_a_nat] :
( ( ( image_6095136190293192542_nat_o @ F @ top_to795618464972521135_a_nat )
= top_top_set_o )
=> ( ( ( image_4139480514073730540_a_nat @ G @ top_top_set_o )
= top_to795618464972521135_a_nat )
=> ( ( image_7142520692256960453_a_nat @ ( comp_o8471783577450470197_a_nat @ G @ F ) @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat ) ) ) ).
% comp_surj
thf(fact_724_comp__surj,axiom,
! [F: $o > sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat] :
( ( ( image_4139480514073730540_a_nat @ F @ top_top_set_o )
= top_to795618464972521135_a_nat )
=> ( ( ( image_7142520692256960453_a_nat @ G @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat )
=> ( ( image_4139480514073730540_a_nat @ ( comp_S4879604302946580149_nat_o @ G @ F ) @ top_top_set_o )
= top_to795618464972521135_a_nat ) ) ) ).
% comp_surj
thf(fact_725_comp__surj,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,G: list_Sum_sum_a_nat > $o] :
( ( ( image_5081948215111134021_a_nat @ F @ top_to6433055325616222389_a_nat )
= top_to6433055325616222389_a_nat )
=> ( ( ( image_3984946558445957976_nat_o @ G @ top_to6433055325616222389_a_nat )
= top_top_set_o )
=> ( ( image_3984946558445957976_nat_o @ ( comp_l7931591902798318543_a_nat @ G @ F ) @ top_to6433055325616222389_a_nat )
= top_top_set_o ) ) ) ).
% comp_surj
thf(fact_726_comp__surj,axiom,
! [F: list_Sum_sum_a_nat > $o,G: $o > list_Sum_sum_a_nat] :
( ( ( image_3984946558445957976_nat_o @ F @ top_to6433055325616222389_a_nat )
= top_top_set_o )
=> ( ( ( image_733135354807394034_a_nat @ G @ top_top_set_o )
= top_to6433055325616222389_a_nat )
=> ( ( image_5081948215111134021_a_nat @ ( comp_o1741750288207193781_a_nat @ G @ F ) @ top_to6433055325616222389_a_nat )
= top_to6433055325616222389_a_nat ) ) ) ).
% comp_surj
thf(fact_727_comp__surj,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat] :
( ( ( image_7142520692256960453_a_nat @ F @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat )
=> ( ( ( image_7142520692256960453_a_nat @ G @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat )
=> ( ( image_7142520692256960453_a_nat @ ( comp_S4540598949873307932_a_nat @ G @ F ) @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat ) ) ) ).
% comp_surj
thf(fact_728_fun_Oset__map,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,V: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( image_5081948215111134021_a_nat @ ( comp_l3689981812903912354_a_nat @ F @ V ) @ top_to6433055325616222389_a_nat )
= ( image_5081948215111134021_a_nat @ F @ ( image_5081948215111134021_a_nat @ V @ top_to6433055325616222389_a_nat ) ) ) ).
% fun.set_map
thf(fact_729_fun_Oset__map,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,V: sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( image_7077900747146408011_a_nat @ ( comp_l4830707110231227036_a_nat @ F @ V ) @ top_to795618464972521135_a_nat )
= ( image_5081948215111134021_a_nat @ F @ ( image_7077900747146408011_a_nat @ V @ top_to795618464972521135_a_nat ) ) ) ).
% fun.set_map
thf(fact_730_fun_Oset__map,axiom,
! [F: $o > $o,V: sum_sum_a_nat > $o] :
( ( image_6095136190293192542_nat_o @ ( comp_o835585038542465372_a_nat @ F @ V ) @ top_to795618464972521135_a_nat )
= ( image_o_o2 @ F @ ( image_6095136190293192542_nat_o @ V @ top_to795618464972521135_a_nat ) ) ) ).
% fun.set_map
thf(fact_731_fun_Oset__map,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,V: $o > list_Sum_sum_a_nat] :
( ( image_733135354807394034_a_nat @ ( comp_l1960031069582106677_nat_o @ F @ V ) @ top_top_set_o )
= ( image_5081948215111134021_a_nat @ F @ ( image_733135354807394034_a_nat @ V @ top_top_set_o ) ) ) ).
% fun.set_map
thf(fact_732_fun_Oset__map,axiom,
! [F: $o > $o,V: $o > $o] :
( ( image_o_o2 @ ( comp_o_o_o @ F @ V ) @ top_top_set_o )
= ( image_o_o2 @ F @ ( image_o_o2 @ V @ top_top_set_o ) ) ) ).
% fun.set_map
thf(fact_733_fun_Omap__ident__strong,axiom,
! [T2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ! [Z4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Z4 @ ( image_5081948215111134021_a_nat @ T2 @ top_to6433055325616222389_a_nat ) )
=> ( ( F @ Z4 )
= Z4 ) )
=> ( ( comp_l3689981812903912354_a_nat @ F @ T2 )
= T2 ) ) ).
% fun.map_ident_strong
thf(fact_734_fun_Omap__ident__strong,axiom,
! [T2: sum_sum_a_nat > $o,F: $o > $o] :
( ! [Z4: $o] :
( ( member_o @ Z4 @ ( image_6095136190293192542_nat_o @ T2 @ top_to795618464972521135_a_nat ) )
=> ( ( F @ Z4 )
= Z4 ) )
=> ( ( comp_o835585038542465372_a_nat @ F @ T2 )
= T2 ) ) ).
% fun.map_ident_strong
thf(fact_735_fun_Omap__ident__strong,axiom,
! [T2: sum_sum_a_nat > list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ! [Z4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Z4 @ ( image_7077900747146408011_a_nat @ T2 @ top_to795618464972521135_a_nat ) )
=> ( ( F @ Z4 )
= Z4 ) )
=> ( ( comp_l4830707110231227036_a_nat @ F @ T2 )
= T2 ) ) ).
% fun.map_ident_strong
thf(fact_736_fun_Omap__ident__strong,axiom,
! [T2: sum_sum_a_nat > sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ! [Z4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Z4 @ ( image_7142520692256960453_a_nat @ T2 @ top_to795618464972521135_a_nat ) )
=> ( ( F @ Z4 )
= Z4 ) )
=> ( ( comp_S4540598949873307932_a_nat @ F @ T2 )
= T2 ) ) ).
% fun.map_ident_strong
thf(fact_737_fun_Omap__ident__strong,axiom,
! [T2: $o > $o,F: $o > $o] :
( ! [Z4: $o] :
( ( member_o @ Z4 @ ( image_o_o2 @ T2 @ top_top_set_o ) )
=> ( ( F @ Z4 )
= Z4 ) )
=> ( ( comp_o_o_o @ F @ T2 )
= T2 ) ) ).
% fun.map_ident_strong
thf(fact_738_fun_Omap__ident__strong,axiom,
! [T2: $o > list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ! [Z4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Z4 @ ( image_733135354807394034_a_nat @ T2 @ top_top_set_o ) )
=> ( ( F @ Z4 )
= Z4 ) )
=> ( ( comp_l1960031069582106677_nat_o @ F @ T2 )
= T2 ) ) ).
% fun.map_ident_strong
thf(fact_739_fun_Omap__ident__strong,axiom,
! [T2: $o > sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ! [Z4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Z4 @ ( image_4139480514073730540_a_nat @ T2 @ top_top_set_o ) )
=> ( ( F @ Z4 )
= Z4 ) )
=> ( ( comp_S4879604302946580149_nat_o @ F @ T2 )
= T2 ) ) ).
% fun.map_ident_strong
thf(fact_740_antisymp__less__eq,axiom,
! [R3: sum_sum_a_nat > sum_sum_a_nat > $o,S2: sum_sum_a_nat > sum_sum_a_nat > $o] :
( ( ord_le5975651393758295374_nat_o @ R3 @ S2 )
=> ( ( antisy1530855801211272108_a_nat @ top_to795618464972521135_a_nat @ S2 )
=> ( antisy1530855801211272108_a_nat @ top_to795618464972521135_a_nat @ R3 ) ) ) ).
% antisymp_less_eq
thf(fact_741_antisymp__less__eq,axiom,
! [R3: $o > $o > $o,S2: $o > $o > $o] :
( ( ord_less_eq_o_o_o @ R3 @ S2 )
=> ( ( antisymp_on_o @ top_top_set_o @ S2 )
=> ( antisymp_on_o @ top_top_set_o @ R3 ) ) ) ).
% antisymp_less_eq
thf(fact_742_antisym__bot,axiom,
antisy1530855801211272108_a_nat @ top_to795618464972521135_a_nat @ bot_bo6002550079725213722_nat_o ).
% antisym_bot
thf(fact_743_antisym__bot,axiom,
antisymp_on_o @ top_top_set_o @ bot_bot_o_o_o ).
% antisym_bot
thf(fact_744_apfst__conv,axiom,
! [F: set_a > set_a,X2: set_a,Y2: produc5986856040925105290_a_nat] :
( ( produc3618268529120100806_a_nat @ F @ ( produc1626021218680649873_a_nat @ X2 @ Y2 ) )
= ( produc1626021218680649873_a_nat @ ( F @ X2 ) @ Y2 ) ) ).
% apfst_conv
thf(fact_745_apfst__conv,axiom,
! [F: nat > nat,X2: nat,Y2: set_li6526943997496501093_a_nat] :
( ( produc3126380199051534471_a_nat @ F @ ( produc4622690550020908860_a_nat @ X2 @ Y2 ) )
= ( produc4622690550020908860_a_nat @ ( F @ X2 ) @ Y2 ) ) ).
% apfst_conv
thf(fact_746_apsnd__conv,axiom,
! [F: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,X2: set_a,Y2: produc5986856040925105290_a_nat] :
( ( produc6023321650250846794_set_a @ F @ ( produc1626021218680649873_a_nat @ X2 @ Y2 ) )
= ( produc1626021218680649873_a_nat @ X2 @ ( F @ Y2 ) ) ) ).
% apsnd_conv
thf(fact_747_apsnd__conv,axiom,
! [F: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,X2: nat,Y2: set_li6526943997496501093_a_nat] :
( ( produc7736357154574597134at_nat @ F @ ( produc4622690550020908860_a_nat @ X2 @ Y2 ) )
= ( produc4622690550020908860_a_nat @ X2 @ ( F @ Y2 ) ) ) ).
% apsnd_conv
thf(fact_748_fun_Orel__refl__strong,axiom,
! [X2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,Ra: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
( ! [Z4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Z4 @ ( image_5081948215111134021_a_nat @ X2 @ top_to6433055325616222389_a_nat ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( bNF_re5874602293210315485_a_nat
@ ^ [Y6: list_Sum_sum_a_nat,Z3: list_Sum_sum_a_nat] : ( Y6 = Z3 )
@ Ra
@ X2
@ X2 ) ) ).
% fun.rel_refl_strong
thf(fact_749_fun_Orel__refl__strong,axiom,
! [X2: sum_sum_a_nat > $o,Ra: $o > $o > $o] :
( ! [Z4: $o] :
( ( member_o @ Z4 @ ( image_6095136190293192542_nat_o @ X2 @ top_to795618464972521135_a_nat ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( bNF_re6132504398938703773at_o_o
@ ^ [Y6: sum_sum_a_nat,Z3: sum_sum_a_nat] : ( Y6 = Z3 )
@ Ra
@ X2
@ X2 ) ) ).
% fun.rel_refl_strong
thf(fact_750_fun_Orel__refl__strong,axiom,
! [X2: sum_sum_a_nat > list_Sum_sum_a_nat,Ra: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
( ! [Z4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Z4 @ ( image_7077900747146408011_a_nat @ X2 @ top_to795618464972521135_a_nat ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( bNF_re221791634285057885_a_nat
@ ^ [Y6: sum_sum_a_nat,Z3: sum_sum_a_nat] : ( Y6 = Z3 )
@ Ra
@ X2
@ X2 ) ) ).
% fun.rel_refl_strong
thf(fact_751_fun_Orel__refl__strong,axiom,
! [X2: sum_sum_a_nat > sum_sum_a_nat,Ra: sum_sum_a_nat > sum_sum_a_nat > $o] :
( ! [Z4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Z4 @ ( image_7142520692256960453_a_nat @ X2 @ top_to795618464972521135_a_nat ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( bNF_re4546881453962818525_a_nat
@ ^ [Y6: sum_sum_a_nat,Z3: sum_sum_a_nat] : ( Y6 = Z3 )
@ Ra
@ X2
@ X2 ) ) ).
% fun.rel_refl_strong
thf(fact_752_fun_Orel__refl__strong,axiom,
! [X2: $o > $o,Ra: $o > $o > $o] :
( ! [Z4: $o] :
( ( member_o @ Z4 @ ( image_o_o2 @ X2 @ top_top_set_o ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( bNF_rel_fun_o_o_o_o
@ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
@ Ra
@ X2
@ X2 ) ) ).
% fun.rel_refl_strong
thf(fact_753_fun_Orel__refl__strong,axiom,
! [X2: $o > list_Sum_sum_a_nat,Ra: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
( ! [Z4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Z4 @ ( image_733135354807394034_a_nat @ X2 @ top_top_set_o ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( bNF_re235242773155154717_a_nat
@ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
@ Ra
@ X2
@ X2 ) ) ).
% fun.rel_refl_strong
thf(fact_754_fun_Orel__refl__strong,axiom,
! [X2: $o > sum_sum_a_nat,Ra: sum_sum_a_nat > sum_sum_a_nat > $o] :
( ! [Z4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Z4 @ ( image_4139480514073730540_a_nat @ X2 @ top_top_set_o ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( bNF_re6999100887211370397_a_nat
@ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
@ Ra
@ X2
@ X2 ) ) ).
% fun.rel_refl_strong
thf(fact_755_fun_Orel__mono__strong,axiom,
! [R2: $o > $o > $o,X2: $o > $o,Y2: $o > $o,Ra: $o > $o > $o] :
( ( bNF_rel_fun_o_o_o_o
@ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
@ R2
@ X2
@ Y2 )
=> ( ! [Z4: $o,Yb: $o] :
( ( member_o @ Z4 @ ( image_o_o2 @ X2 @ top_top_set_o ) )
=> ( ( member_o @ Yb @ ( image_o_o2 @ Y2 @ top_top_set_o ) )
=> ( ( R2 @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( bNF_rel_fun_o_o_o_o
@ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
@ Ra
@ X2
@ Y2 ) ) ) ).
% fun.rel_mono_strong
thf(fact_756_fun_Orel__mono__strong,axiom,
! [R2: $o > $o > $o,X2: sum_sum_a_nat > $o,Y2: sum_sum_a_nat > $o,Ra: $o > $o > $o] :
( ( bNF_re6132504398938703773at_o_o
@ ^ [Y6: sum_sum_a_nat,Z3: sum_sum_a_nat] : ( Y6 = Z3 )
@ R2
@ X2
@ Y2 )
=> ( ! [Z4: $o,Yb: $o] :
( ( member_o @ Z4 @ ( image_6095136190293192542_nat_o @ X2 @ top_to795618464972521135_a_nat ) )
=> ( ( member_o @ Yb @ ( image_6095136190293192542_nat_o @ Y2 @ top_to795618464972521135_a_nat ) )
=> ( ( R2 @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( bNF_re6132504398938703773at_o_o
@ ^ [Y6: sum_sum_a_nat,Z3: sum_sum_a_nat] : ( Y6 = Z3 )
@ Ra
@ X2
@ Y2 ) ) ) ).
% fun.rel_mono_strong
thf(fact_757_fun_Orel__mono__strong,axiom,
! [R2: $o > sum_sum_a_nat > $o,X2: $o > $o,Y2: $o > sum_sum_a_nat,Ra: $o > sum_sum_a_nat > $o] :
( ( bNF_re411022188895864260_a_nat
@ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
@ R2
@ X2
@ Y2 )
=> ( ! [Z4: $o,Yb: sum_sum_a_nat] :
( ( member_o @ Z4 @ ( image_o_o2 @ X2 @ top_top_set_o ) )
=> ( ( member_Sum_sum_a_nat @ Yb @ ( image_4139480514073730540_a_nat @ Y2 @ top_top_set_o ) )
=> ( ( R2 @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( bNF_re411022188895864260_a_nat
@ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
@ Ra
@ X2
@ Y2 ) ) ) ).
% fun.rel_mono_strong
thf(fact_758_fun_Orel__mono__strong,axiom,
! [R2: sum_sum_a_nat > $o > $o,X2: $o > sum_sum_a_nat,Y2: $o > $o,Ra: sum_sum_a_nat > $o > $o] :
( ( bNF_re2366677865115326262_nat_o
@ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
@ R2
@ X2
@ Y2 )
=> ( ! [Z4: sum_sum_a_nat,Yb: $o] :
( ( member_Sum_sum_a_nat @ Z4 @ ( image_4139480514073730540_a_nat @ X2 @ top_top_set_o ) )
=> ( ( member_o @ Yb @ ( image_o_o2 @ Y2 @ top_top_set_o ) )
=> ( ( R2 @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( bNF_re2366677865115326262_nat_o
@ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
@ Ra
@ X2
@ Y2 ) ) ) ).
% fun.rel_mono_strong
thf(fact_759_fun_Orel__mono__strong,axiom,
! [R2: $o > list_Sum_sum_a_nat > $o,X2: $o > $o,Y2: $o > list_Sum_sum_a_nat,Ra: $o > list_Sum_sum_a_nat > $o] :
( ( bNF_re6050751611158805194_a_nat
@ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
@ R2
@ X2
@ Y2 )
=> ( ! [Z4: $o,Yb: list_Sum_sum_a_nat] :
( ( member_o @ Z4 @ ( image_o_o2 @ X2 @ top_top_set_o ) )
=> ( ( member408289922725080238_a_nat @ Yb @ ( image_733135354807394034_a_nat @ Y2 @ top_top_set_o ) )
=> ( ( R2 @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( bNF_re6050751611158805194_a_nat
@ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
@ Ra
@ X2
@ Y2 ) ) ) ).
% fun.rel_mono_strong
thf(fact_760_fun_Orel__mono__strong,axiom,
! [R2: list_Sum_sum_a_nat > $o > $o,X2: $o > list_Sum_sum_a_nat,Y2: $o > $o,Ra: list_Sum_sum_a_nat > $o > $o] :
( ( bNF_re79190777942593328_nat_o
@ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
@ R2
@ X2
@ Y2 )
=> ( ! [Z4: list_Sum_sum_a_nat,Yb: $o] :
( ( member408289922725080238_a_nat @ Z4 @ ( image_733135354807394034_a_nat @ X2 @ top_top_set_o ) )
=> ( ( member_o @ Yb @ ( image_o_o2 @ Y2 @ top_top_set_o ) )
=> ( ( R2 @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( bNF_re79190777942593328_nat_o
@ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
@ Ra
@ X2
@ Y2 ) ) ) ).
% fun.rel_mono_strong
thf(fact_761_fun_Orel__mono__strong,axiom,
! [R2: $o > sum_sum_a_nat > $o,X2: sum_sum_a_nat > $o,Y2: sum_sum_a_nat > sum_sum_a_nat,Ra: $o > sum_sum_a_nat > $o] :
( ( bNF_re1683418894890734084_a_nat
@ ^ [Y6: sum_sum_a_nat,Z3: sum_sum_a_nat] : ( Y6 = Z3 )
@ R2
@ X2
@ Y2 )
=> ( ! [Z4: $o,Yb: sum_sum_a_nat] :
( ( member_o @ Z4 @ ( image_6095136190293192542_nat_o @ X2 @ top_to795618464972521135_a_nat ) )
=> ( ( member_Sum_sum_a_nat @ Yb @ ( image_7142520692256960453_a_nat @ Y2 @ top_to795618464972521135_a_nat ) )
=> ( ( R2 @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( bNF_re1683418894890734084_a_nat
@ ^ [Y6: sum_sum_a_nat,Z3: sum_sum_a_nat] : ( Y6 = Z3 )
@ Ra
@ X2
@ Y2 ) ) ) ).
% fun.rel_mono_strong
thf(fact_762_fun_Orel__mono__strong,axiom,
! [R2: sum_sum_a_nat > $o > $o,X2: sum_sum_a_nat > sum_sum_a_nat,Y2: sum_sum_a_nat > $o,Ra: sum_sum_a_nat > $o > $o] :
( ( bNF_re3639074571110196086_nat_o
@ ^ [Y6: sum_sum_a_nat,Z3: sum_sum_a_nat] : ( Y6 = Z3 )
@ R2
@ X2
@ Y2 )
=> ( ! [Z4: sum_sum_a_nat,Yb: $o] :
( ( member_Sum_sum_a_nat @ Z4 @ ( image_7142520692256960453_a_nat @ X2 @ top_to795618464972521135_a_nat ) )
=> ( ( member_o @ Yb @ ( image_6095136190293192542_nat_o @ Y2 @ top_to795618464972521135_a_nat ) )
=> ( ( R2 @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( bNF_re3639074571110196086_nat_o
@ ^ [Y6: sum_sum_a_nat,Z3: sum_sum_a_nat] : ( Y6 = Z3 )
@ Ra
@ X2
@ Y2 ) ) ) ).
% fun.rel_mono_strong
thf(fact_763_fun_Orel__mono__strong,axiom,
! [R2: sum_sum_a_nat > sum_sum_a_nat > $o,X2: $o > sum_sum_a_nat,Y2: $o > sum_sum_a_nat,Ra: sum_sum_a_nat > sum_sum_a_nat > $o] :
( ( bNF_re6999100887211370397_a_nat
@ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
@ R2
@ X2
@ Y2 )
=> ( ! [Z4: sum_sum_a_nat,Yb: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Z4 @ ( image_4139480514073730540_a_nat @ X2 @ top_top_set_o ) )
=> ( ( member_Sum_sum_a_nat @ Yb @ ( image_4139480514073730540_a_nat @ Y2 @ top_top_set_o ) )
=> ( ( R2 @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( bNF_re6999100887211370397_a_nat
@ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
@ Ra
@ X2
@ Y2 ) ) ) ).
% fun.rel_mono_strong
thf(fact_764_fun_Orel__mono__strong,axiom,
! [R2: $o > list_Sum_sum_a_nat > $o,X2: sum_sum_a_nat > $o,Y2: sum_sum_a_nat > list_Sum_sum_a_nat,Ra: $o > list_Sum_sum_a_nat > $o] :
( ( bNF_re4093294260564961034_a_nat
@ ^ [Y6: sum_sum_a_nat,Z3: sum_sum_a_nat] : ( Y6 = Z3 )
@ R2
@ X2
@ Y2 )
=> ( ! [Z4: $o,Yb: list_Sum_sum_a_nat] :
( ( member_o @ Z4 @ ( image_6095136190293192542_nat_o @ X2 @ top_to795618464972521135_a_nat ) )
=> ( ( member408289922725080238_a_nat @ Yb @ ( image_7077900747146408011_a_nat @ Y2 @ top_to795618464972521135_a_nat ) )
=> ( ( R2 @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( bNF_re4093294260564961034_a_nat
@ ^ [Y6: sum_sum_a_nat,Z3: sum_sum_a_nat] : ( Y6 = Z3 )
@ Ra
@ X2
@ Y2 ) ) ) ).
% fun.rel_mono_strong
thf(fact_765_fun_Orel__cong,axiom,
! [X2: $o > $o,Ya: $o > $o,Y2: $o > $o,Xa: $o > $o,R2: $o > $o > $o,Ra: $o > $o > $o] :
( ( X2 = Ya )
=> ( ( Y2 = Xa )
=> ( ! [Z4: $o,Yb: $o] :
( ( member_o @ Z4 @ ( image_o_o2 @ Ya @ top_top_set_o ) )
=> ( ( member_o @ Yb @ ( image_o_o2 @ Xa @ top_top_set_o ) )
=> ( ( R2 @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( bNF_rel_fun_o_o_o_o
@ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
@ R2
@ X2
@ Y2 )
= ( bNF_rel_fun_o_o_o_o
@ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
@ Ra
@ Ya
@ Xa ) ) ) ) ) ).
% fun.rel_cong
thf(fact_766_fun_Orel__cong,axiom,
! [X2: sum_sum_a_nat > $o,Ya: sum_sum_a_nat > $o,Y2: sum_sum_a_nat > $o,Xa: sum_sum_a_nat > $o,R2: $o > $o > $o,Ra: $o > $o > $o] :
( ( X2 = Ya )
=> ( ( Y2 = Xa )
=> ( ! [Z4: $o,Yb: $o] :
( ( member_o @ Z4 @ ( image_6095136190293192542_nat_o @ Ya @ top_to795618464972521135_a_nat ) )
=> ( ( member_o @ Yb @ ( image_6095136190293192542_nat_o @ Xa @ top_to795618464972521135_a_nat ) )
=> ( ( R2 @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( bNF_re6132504398938703773at_o_o
@ ^ [Y6: sum_sum_a_nat,Z3: sum_sum_a_nat] : ( Y6 = Z3 )
@ R2
@ X2
@ Y2 )
= ( bNF_re6132504398938703773at_o_o
@ ^ [Y6: sum_sum_a_nat,Z3: sum_sum_a_nat] : ( Y6 = Z3 )
@ Ra
@ Ya
@ Xa ) ) ) ) ) ).
% fun.rel_cong
thf(fact_767_fun_Orel__cong,axiom,
! [X2: $o > $o,Ya: $o > $o,Y2: $o > sum_sum_a_nat,Xa: $o > sum_sum_a_nat,R2: $o > sum_sum_a_nat > $o,Ra: $o > sum_sum_a_nat > $o] :
( ( X2 = Ya )
=> ( ( Y2 = Xa )
=> ( ! [Z4: $o,Yb: sum_sum_a_nat] :
( ( member_o @ Z4 @ ( image_o_o2 @ Ya @ top_top_set_o ) )
=> ( ( member_Sum_sum_a_nat @ Yb @ ( image_4139480514073730540_a_nat @ Xa @ top_top_set_o ) )
=> ( ( R2 @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( bNF_re411022188895864260_a_nat
@ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
@ R2
@ X2
@ Y2 )
= ( bNF_re411022188895864260_a_nat
@ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
@ Ra
@ Ya
@ Xa ) ) ) ) ) ).
% fun.rel_cong
thf(fact_768_fun_Orel__cong,axiom,
! [X2: $o > sum_sum_a_nat,Ya: $o > sum_sum_a_nat,Y2: $o > $o,Xa: $o > $o,R2: sum_sum_a_nat > $o > $o,Ra: sum_sum_a_nat > $o > $o] :
( ( X2 = Ya )
=> ( ( Y2 = Xa )
=> ( ! [Z4: sum_sum_a_nat,Yb: $o] :
( ( member_Sum_sum_a_nat @ Z4 @ ( image_4139480514073730540_a_nat @ Ya @ top_top_set_o ) )
=> ( ( member_o @ Yb @ ( image_o_o2 @ Xa @ top_top_set_o ) )
=> ( ( R2 @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( bNF_re2366677865115326262_nat_o
@ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
@ R2
@ X2
@ Y2 )
= ( bNF_re2366677865115326262_nat_o
@ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
@ Ra
@ Ya
@ Xa ) ) ) ) ) ).
% fun.rel_cong
thf(fact_769_fun_Orel__cong,axiom,
! [X2: $o > $o,Ya: $o > $o,Y2: $o > list_Sum_sum_a_nat,Xa: $o > list_Sum_sum_a_nat,R2: $o > list_Sum_sum_a_nat > $o,Ra: $o > list_Sum_sum_a_nat > $o] :
( ( X2 = Ya )
=> ( ( Y2 = Xa )
=> ( ! [Z4: $o,Yb: list_Sum_sum_a_nat] :
( ( member_o @ Z4 @ ( image_o_o2 @ Ya @ top_top_set_o ) )
=> ( ( member408289922725080238_a_nat @ Yb @ ( image_733135354807394034_a_nat @ Xa @ top_top_set_o ) )
=> ( ( R2 @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( bNF_re6050751611158805194_a_nat
@ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
@ R2
@ X2
@ Y2 )
= ( bNF_re6050751611158805194_a_nat
@ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
@ Ra
@ Ya
@ Xa ) ) ) ) ) ).
% fun.rel_cong
thf(fact_770_fun_Orel__cong,axiom,
! [X2: $o > list_Sum_sum_a_nat,Ya: $o > list_Sum_sum_a_nat,Y2: $o > $o,Xa: $o > $o,R2: list_Sum_sum_a_nat > $o > $o,Ra: list_Sum_sum_a_nat > $o > $o] :
( ( X2 = Ya )
=> ( ( Y2 = Xa )
=> ( ! [Z4: list_Sum_sum_a_nat,Yb: $o] :
( ( member408289922725080238_a_nat @ Z4 @ ( image_733135354807394034_a_nat @ Ya @ top_top_set_o ) )
=> ( ( member_o @ Yb @ ( image_o_o2 @ Xa @ top_top_set_o ) )
=> ( ( R2 @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( bNF_re79190777942593328_nat_o
@ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
@ R2
@ X2
@ Y2 )
= ( bNF_re79190777942593328_nat_o
@ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
@ Ra
@ Ya
@ Xa ) ) ) ) ) ).
% fun.rel_cong
thf(fact_771_fun_Orel__cong,axiom,
! [X2: sum_sum_a_nat > $o,Ya: sum_sum_a_nat > $o,Y2: sum_sum_a_nat > sum_sum_a_nat,Xa: sum_sum_a_nat > sum_sum_a_nat,R2: $o > sum_sum_a_nat > $o,Ra: $o > sum_sum_a_nat > $o] :
( ( X2 = Ya )
=> ( ( Y2 = Xa )
=> ( ! [Z4: $o,Yb: sum_sum_a_nat] :
( ( member_o @ Z4 @ ( image_6095136190293192542_nat_o @ Ya @ top_to795618464972521135_a_nat ) )
=> ( ( member_Sum_sum_a_nat @ Yb @ ( image_7142520692256960453_a_nat @ Xa @ top_to795618464972521135_a_nat ) )
=> ( ( R2 @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( bNF_re1683418894890734084_a_nat
@ ^ [Y6: sum_sum_a_nat,Z3: sum_sum_a_nat] : ( Y6 = Z3 )
@ R2
@ X2
@ Y2 )
= ( bNF_re1683418894890734084_a_nat
@ ^ [Y6: sum_sum_a_nat,Z3: sum_sum_a_nat] : ( Y6 = Z3 )
@ Ra
@ Ya
@ Xa ) ) ) ) ) ).
% fun.rel_cong
thf(fact_772_fun_Orel__cong,axiom,
! [X2: sum_sum_a_nat > sum_sum_a_nat,Ya: sum_sum_a_nat > sum_sum_a_nat,Y2: sum_sum_a_nat > $o,Xa: sum_sum_a_nat > $o,R2: sum_sum_a_nat > $o > $o,Ra: sum_sum_a_nat > $o > $o] :
( ( X2 = Ya )
=> ( ( Y2 = Xa )
=> ( ! [Z4: sum_sum_a_nat,Yb: $o] :
( ( member_Sum_sum_a_nat @ Z4 @ ( image_7142520692256960453_a_nat @ Ya @ top_to795618464972521135_a_nat ) )
=> ( ( member_o @ Yb @ ( image_6095136190293192542_nat_o @ Xa @ top_to795618464972521135_a_nat ) )
=> ( ( R2 @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( bNF_re3639074571110196086_nat_o
@ ^ [Y6: sum_sum_a_nat,Z3: sum_sum_a_nat] : ( Y6 = Z3 )
@ R2
@ X2
@ Y2 )
= ( bNF_re3639074571110196086_nat_o
@ ^ [Y6: sum_sum_a_nat,Z3: sum_sum_a_nat] : ( Y6 = Z3 )
@ Ra
@ Ya
@ Xa ) ) ) ) ) ).
% fun.rel_cong
thf(fact_773_fun_Orel__cong,axiom,
! [X2: $o > sum_sum_a_nat,Ya: $o > sum_sum_a_nat,Y2: $o > sum_sum_a_nat,Xa: $o > sum_sum_a_nat,R2: sum_sum_a_nat > sum_sum_a_nat > $o,Ra: sum_sum_a_nat > sum_sum_a_nat > $o] :
( ( X2 = Ya )
=> ( ( Y2 = Xa )
=> ( ! [Z4: sum_sum_a_nat,Yb: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Z4 @ ( image_4139480514073730540_a_nat @ Ya @ top_top_set_o ) )
=> ( ( member_Sum_sum_a_nat @ Yb @ ( image_4139480514073730540_a_nat @ Xa @ top_top_set_o ) )
=> ( ( R2 @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( bNF_re6999100887211370397_a_nat
@ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
@ R2
@ X2
@ Y2 )
= ( bNF_re6999100887211370397_a_nat
@ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
@ Ra
@ Ya
@ Xa ) ) ) ) ) ).
% fun.rel_cong
thf(fact_774_fun_Orel__cong,axiom,
! [X2: sum_sum_a_nat > $o,Ya: sum_sum_a_nat > $o,Y2: sum_sum_a_nat > list_Sum_sum_a_nat,Xa: sum_sum_a_nat > list_Sum_sum_a_nat,R2: $o > list_Sum_sum_a_nat > $o,Ra: $o > list_Sum_sum_a_nat > $o] :
( ( X2 = Ya )
=> ( ( Y2 = Xa )
=> ( ! [Z4: $o,Yb: list_Sum_sum_a_nat] :
( ( member_o @ Z4 @ ( image_6095136190293192542_nat_o @ Ya @ top_to795618464972521135_a_nat ) )
=> ( ( member408289922725080238_a_nat @ Yb @ ( image_7077900747146408011_a_nat @ Xa @ top_to795618464972521135_a_nat ) )
=> ( ( R2 @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( bNF_re4093294260564961034_a_nat
@ ^ [Y6: sum_sum_a_nat,Z3: sum_sum_a_nat] : ( Y6 = Z3 )
@ R2
@ X2
@ Y2 )
= ( bNF_re4093294260564961034_a_nat
@ ^ [Y6: sum_sum_a_nat,Z3: sum_sum_a_nat] : ( Y6 = Z3 )
@ Ra
@ Ya
@ Xa ) ) ) ) ) ).
% fun.rel_cong
thf(fact_775_range__constant,axiom,
! [X2: list_Sum_sum_a_nat] :
( ( image_5081948215111134021_a_nat
@ ^ [Uu2: list_Sum_sum_a_nat] : X2
@ top_to6433055325616222389_a_nat )
= ( insert2950094090816004437_a_nat @ X2 @ bot_bo1033123847703346641_a_nat ) ) ).
% range_constant
thf(fact_776_range__constant,axiom,
! [X2: $o] :
( ( image_6095136190293192542_nat_o
@ ^ [Uu2: sum_sum_a_nat] : X2
@ top_to795618464972521135_a_nat )
= ( insert_o @ X2 @ bot_bot_set_o ) ) ).
% range_constant
thf(fact_777_range__constant,axiom,
! [X2: $o] :
( ( image_o_o2
@ ^ [Uu2: $o] : X2
@ top_top_set_o )
= ( insert_o @ X2 @ bot_bot_set_o ) ) ).
% range_constant
thf(fact_778_insertCI,axiom,
! [A2: $o,B2: set_o,B: $o] :
( ( ~ ( member_o @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_o @ A2 @ ( insert_o @ B @ B2 ) ) ) ).
% insertCI
thf(fact_779_insertCI,axiom,
! [A2: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat,B: list_Sum_sum_a_nat] :
( ( ~ ( member408289922725080238_a_nat @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member408289922725080238_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_780_insertCI,axiom,
! [A2: sum_sum_a_nat,B2: set_Sum_sum_a_nat,B: sum_sum_a_nat] :
( ( ~ ( member_Sum_sum_a_nat @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_Sum_sum_a_nat @ A2 @ ( insert_Sum_sum_a_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_781_insert__iff,axiom,
! [A2: $o,B: $o,A: set_o] :
( ( member_o @ A2 @ ( insert_o @ B @ A ) )
= ( ( A2 = B )
| ( member_o @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_782_insert__iff,axiom,
! [A2: list_Sum_sum_a_nat,B: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ B @ A ) )
= ( ( A2 = B )
| ( member408289922725080238_a_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_783_insert__iff,axiom,
! [A2: sum_sum_a_nat,B: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A2 @ ( insert_Sum_sum_a_nat @ B @ A ) )
= ( ( A2 = B )
| ( member_Sum_sum_a_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_784_insert__absorb2,axiom,
! [X2: $o,A: set_o] :
( ( insert_o @ X2 @ ( insert_o @ X2 @ A ) )
= ( insert_o @ X2 @ A ) ) ).
% insert_absorb2
thf(fact_785_image__insert,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat @ F @ ( insert2950094090816004437_a_nat @ A2 @ B2 ) )
= ( insert2950094090816004437_a_nat @ ( F @ A2 ) @ ( image_5081948215111134021_a_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_786_image__insert,axiom,
! [F: $o > $o,A2: $o,B2: set_o] :
( ( image_o_o2 @ F @ ( insert_o @ A2 @ B2 ) )
= ( insert_o @ ( F @ A2 ) @ ( image_o_o2 @ F @ B2 ) ) ) ).
% image_insert
thf(fact_787_insert__image,axiom,
! [X2: $o,A: set_o,F: $o > $o] :
( ( member_o @ X2 @ A )
=> ( ( insert_o @ ( F @ X2 ) @ ( image_o_o2 @ F @ A ) )
= ( image_o_o2 @ F @ A ) ) ) ).
% insert_image
thf(fact_788_insert__image,axiom,
! [X2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A )
=> ( ( insert2950094090816004437_a_nat @ ( F @ X2 ) @ ( image_5081948215111134021_a_nat @ F @ A ) )
= ( image_5081948215111134021_a_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_789_insert__image,axiom,
! [X2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ X2 @ A )
=> ( ( insert_o @ ( F @ X2 ) @ ( image_3984946558445957976_nat_o @ F @ A ) )
= ( image_3984946558445957976_nat_o @ F @ A ) ) ) ).
% insert_image
thf(fact_790_insert__image,axiom,
! [X2: sum_sum_a_nat,A: set_Sum_sum_a_nat,F: sum_sum_a_nat > $o] :
( ( member_Sum_sum_a_nat @ X2 @ A )
=> ( ( insert_o @ ( F @ X2 ) @ ( image_6095136190293192542_nat_o @ F @ A ) )
= ( image_6095136190293192542_nat_o @ F @ A ) ) ) ).
% insert_image
thf(fact_791_singletonI,axiom,
! [A2: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ A2 @ bot_bo1033123847703346641_a_nat ) ) ).
% singletonI
thf(fact_792_singletonI,axiom,
! [A2: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ A2 @ ( insert_Sum_sum_a_nat @ A2 @ bot_bo3438331934148233675_a_nat ) ) ).
% singletonI
thf(fact_793_singletonI,axiom,
! [A2: $o] : ( member_o @ A2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) ).
% singletonI
thf(fact_794_insert__subset,axiom,
! [X2: $o,A: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ ( insert_o @ X2 @ A ) @ B2 )
= ( ( member_o @ X2 @ B2 )
& ( ord_less_eq_set_o @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_795_insert__subset,axiom,
! [X2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( insert2950094090816004437_a_nat @ X2 @ A ) @ B2 )
= ( ( member408289922725080238_a_nat @ X2 @ B2 )
& ( ord_le1147066620699065093_a_nat @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_796_insert__subset,axiom,
! [X2: sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( insert_Sum_sum_a_nat @ X2 @ A ) @ B2 )
= ( ( member_Sum_sum_a_nat @ X2 @ B2 )
& ( ord_le1325389633284124927_a_nat @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_797_case__prod__Pair,axiom,
( ( produc6698438707799802453_a_nat @ produc1626021218680649873_a_nat )
= id_Pro81263520388867046_a_nat ) ).
% case_prod_Pair
thf(fact_798_case__prod__Pair,axiom,
( ( produc5943727843246033139_a_nat @ produc4622690550020908860_a_nat )
= id_Pro6126725435659366873_a_nat ) ).
% case_prod_Pair
thf(fact_799_Int__insert__right__if1,axiom,
! [A2: $o,A: set_o,B2: set_o] :
( ( member_o @ A2 @ A )
=> ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B2 ) )
= ( insert_o @ A2 @ ( inf_inf_set_o @ A @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_800_Int__insert__right__if1,axiom,
! [A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A2 @ A )
=> ( ( inf_in3249246906714053971_a_nat @ A @ ( insert2950094090816004437_a_nat @ A2 @ B2 ) )
= ( insert2950094090816004437_a_nat @ A2 @ ( inf_in3249246906714053971_a_nat @ A @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_801_Int__insert__right__if1,axiom,
! [A2: sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A2 @ A )
=> ( ( inf_in7084830621192376909_a_nat @ A @ ( insert_Sum_sum_a_nat @ A2 @ B2 ) )
= ( insert_Sum_sum_a_nat @ A2 @ ( inf_in7084830621192376909_a_nat @ A @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_802_Int__insert__right__if0,axiom,
! [A2: $o,A: set_o,B2: set_o] :
( ~ ( member_o @ A2 @ A )
=> ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B2 ) )
= ( inf_inf_set_o @ A @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_803_Int__insert__right__if0,axiom,
! [A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ A2 @ A )
=> ( ( inf_in3249246906714053971_a_nat @ A @ ( insert2950094090816004437_a_nat @ A2 @ B2 ) )
= ( inf_in3249246906714053971_a_nat @ A @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_804_Int__insert__right__if0,axiom,
! [A2: sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ~ ( member_Sum_sum_a_nat @ A2 @ A )
=> ( ( inf_in7084830621192376909_a_nat @ A @ ( insert_Sum_sum_a_nat @ A2 @ B2 ) )
= ( inf_in7084830621192376909_a_nat @ A @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_805_insert__inter__insert,axiom,
! [A2: $o,A: set_o,B2: set_o] :
( ( inf_inf_set_o @ ( insert_o @ A2 @ A ) @ ( insert_o @ A2 @ B2 ) )
= ( insert_o @ A2 @ ( inf_inf_set_o @ A @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_806_Int__insert__left__if1,axiom,
! [A2: $o,C: set_o,B2: set_o] :
( ( member_o @ A2 @ C )
=> ( ( inf_inf_set_o @ ( insert_o @ A2 @ B2 ) @ C )
= ( insert_o @ A2 @ ( inf_inf_set_o @ B2 @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_807_Int__insert__left__if1,axiom,
! [A2: list_Sum_sum_a_nat,C: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A2 @ C )
=> ( ( inf_in3249246906714053971_a_nat @ ( insert2950094090816004437_a_nat @ A2 @ B2 ) @ C )
= ( insert2950094090816004437_a_nat @ A2 @ ( inf_in3249246906714053971_a_nat @ B2 @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_808_Int__insert__left__if1,axiom,
! [A2: sum_sum_a_nat,C: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A2 @ C )
=> ( ( inf_in7084830621192376909_a_nat @ ( insert_Sum_sum_a_nat @ A2 @ B2 ) @ C )
= ( insert_Sum_sum_a_nat @ A2 @ ( inf_in7084830621192376909_a_nat @ B2 @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_809_Int__insert__left__if0,axiom,
! [A2: $o,C: set_o,B2: set_o] :
( ~ ( member_o @ A2 @ C )
=> ( ( inf_inf_set_o @ ( insert_o @ A2 @ B2 ) @ C )
= ( inf_inf_set_o @ B2 @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_810_Int__insert__left__if0,axiom,
! [A2: list_Sum_sum_a_nat,C: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ A2 @ C )
=> ( ( inf_in3249246906714053971_a_nat @ ( insert2950094090816004437_a_nat @ A2 @ B2 ) @ C )
= ( inf_in3249246906714053971_a_nat @ B2 @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_811_Int__insert__left__if0,axiom,
! [A2: sum_sum_a_nat,C: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ~ ( member_Sum_sum_a_nat @ A2 @ C )
=> ( ( inf_in7084830621192376909_a_nat @ ( insert_Sum_sum_a_nat @ A2 @ B2 ) @ C )
= ( inf_in7084830621192376909_a_nat @ B2 @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_812_image__id,axiom,
( ( image_5081948215111134021_a_nat @ id_lis6021530241265049846_a_nat )
= id_set2110540503428828374_a_nat ) ).
% image_id
thf(fact_813_image__id,axiom,
( ( image_o_o2 @ id_o )
= id_set_o ) ).
% image_id
thf(fact_814_Diff__insert0,axiom,
! [X2: $o,A: set_o,B2: set_o] :
( ~ ( member_o @ X2 @ A )
=> ( ( minus_minus_set_o @ A @ ( insert_o @ X2 @ B2 ) )
= ( minus_minus_set_o @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_815_Diff__insert0,axiom,
! [X2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ X2 @ A )
=> ( ( minus_7395159227704179404_a_nat @ A @ ( insert2950094090816004437_a_nat @ X2 @ B2 ) )
= ( minus_7395159227704179404_a_nat @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_816_Diff__insert0,axiom,
! [X2: sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ~ ( member_Sum_sum_a_nat @ X2 @ A )
=> ( ( minus_1134630996077396038_a_nat @ A @ ( insert_Sum_sum_a_nat @ X2 @ B2 ) )
= ( minus_1134630996077396038_a_nat @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_817_insert__Diff1,axiom,
! [X2: $o,B2: set_o,A: set_o] :
( ( member_o @ X2 @ B2 )
=> ( ( minus_minus_set_o @ ( insert_o @ X2 @ A ) @ B2 )
= ( minus_minus_set_o @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_818_insert__Diff1,axiom,
! [X2: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ B2 )
=> ( ( minus_7395159227704179404_a_nat @ ( insert2950094090816004437_a_nat @ X2 @ A ) @ B2 )
= ( minus_7395159227704179404_a_nat @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_819_insert__Diff1,axiom,
! [X2: sum_sum_a_nat,B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ B2 )
=> ( ( minus_1134630996077396038_a_nat @ ( insert_Sum_sum_a_nat @ X2 @ A ) @ B2 )
= ( minus_1134630996077396038_a_nat @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_820_singleton__conv,axiom,
! [A2: nat > sum_sum_a_nat] :
( ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] : ( X = A2 ) )
= ( insert5265011953798106934_a_nat @ A2 @ bot_bo6441361344521902642_a_nat ) ) ).
% singleton_conv
thf(fact_821_singleton__conv,axiom,
! [A2: $o] :
( ( collect_o
@ ^ [X: $o] : ( X = A2 ) )
= ( insert_o @ A2 @ bot_bot_set_o ) ) ).
% singleton_conv
thf(fact_822_singleton__conv2,axiom,
! [A2: nat > sum_sum_a_nat] :
( ( collec5629555741568564177_a_nat
@ ( ^ [Y6: nat > sum_sum_a_nat,Z3: nat > sum_sum_a_nat] : ( Y6 = Z3 )
@ A2 ) )
= ( insert5265011953798106934_a_nat @ A2 @ bot_bo6441361344521902642_a_nat ) ) ).
% singleton_conv2
thf(fact_823_singleton__conv2,axiom,
! [A2: $o] :
( ( collect_o
@ ( ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
@ A2 ) )
= ( insert_o @ A2 @ bot_bot_set_o ) ) ).
% singleton_conv2
thf(fact_824_singleton__insert__inj__eq_H,axiom,
! [A2: $o,A: set_o,B: $o] :
( ( ( insert_o @ A2 @ A )
= ( insert_o @ B @ bot_bot_set_o ) )
= ( ( A2 = B )
& ( ord_less_eq_set_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_825_singleton__insert__inj__eq,axiom,
! [B: $o,A2: $o,A: set_o] :
( ( ( insert_o @ B @ bot_bot_set_o )
= ( insert_o @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_less_eq_set_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_826_insert__disjoint_I1_J,axiom,
! [A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ( inf_in3249246906714053971_a_nat @ ( insert2950094090816004437_a_nat @ A2 @ A ) @ B2 )
= bot_bo1033123847703346641_a_nat )
= ( ~ ( member408289922725080238_a_nat @ A2 @ B2 )
& ( ( inf_in3249246906714053971_a_nat @ A @ B2 )
= bot_bo1033123847703346641_a_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_827_insert__disjoint_I1_J,axiom,
! [A2: sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ( inf_in7084830621192376909_a_nat @ ( insert_Sum_sum_a_nat @ A2 @ A ) @ B2 )
= bot_bo3438331934148233675_a_nat )
= ( ~ ( member_Sum_sum_a_nat @ A2 @ B2 )
& ( ( inf_in7084830621192376909_a_nat @ A @ B2 )
= bot_bo3438331934148233675_a_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_828_insert__disjoint_I1_J,axiom,
! [A2: $o,A: set_o,B2: set_o] :
( ( ( inf_inf_set_o @ ( insert_o @ A2 @ A ) @ B2 )
= bot_bot_set_o )
= ( ~ ( member_o @ A2 @ B2 )
& ( ( inf_inf_set_o @ A @ B2 )
= bot_bot_set_o ) ) ) ).
% insert_disjoint(1)
thf(fact_829_insert__disjoint_I2_J,axiom,
! [A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( bot_bo1033123847703346641_a_nat
= ( inf_in3249246906714053971_a_nat @ ( insert2950094090816004437_a_nat @ A2 @ A ) @ B2 ) )
= ( ~ ( member408289922725080238_a_nat @ A2 @ B2 )
& ( bot_bo1033123847703346641_a_nat
= ( inf_in3249246906714053971_a_nat @ A @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_830_insert__disjoint_I2_J,axiom,
! [A2: sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( bot_bo3438331934148233675_a_nat
= ( inf_in7084830621192376909_a_nat @ ( insert_Sum_sum_a_nat @ A2 @ A ) @ B2 ) )
= ( ~ ( member_Sum_sum_a_nat @ A2 @ B2 )
& ( bot_bo3438331934148233675_a_nat
= ( inf_in7084830621192376909_a_nat @ A @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_831_insert__disjoint_I2_J,axiom,
! [A2: $o,A: set_o,B2: set_o] :
( ( bot_bot_set_o
= ( inf_inf_set_o @ ( insert_o @ A2 @ A ) @ B2 ) )
= ( ~ ( member_o @ A2 @ B2 )
& ( bot_bot_set_o
= ( inf_inf_set_o @ A @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_832_disjoint__insert_I1_J,axiom,
! [B2: set_li6526943997496501093_a_nat,A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ( inf_in3249246906714053971_a_nat @ B2 @ ( insert2950094090816004437_a_nat @ A2 @ A ) )
= bot_bo1033123847703346641_a_nat )
= ( ~ ( member408289922725080238_a_nat @ A2 @ B2 )
& ( ( inf_in3249246906714053971_a_nat @ B2 @ A )
= bot_bo1033123847703346641_a_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_833_disjoint__insert_I1_J,axiom,
! [B2: set_Sum_sum_a_nat,A2: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ( inf_in7084830621192376909_a_nat @ B2 @ ( insert_Sum_sum_a_nat @ A2 @ A ) )
= bot_bo3438331934148233675_a_nat )
= ( ~ ( member_Sum_sum_a_nat @ A2 @ B2 )
& ( ( inf_in7084830621192376909_a_nat @ B2 @ A )
= bot_bo3438331934148233675_a_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_834_disjoint__insert_I1_J,axiom,
! [B2: set_o,A2: $o,A: set_o] :
( ( ( inf_inf_set_o @ B2 @ ( insert_o @ A2 @ A ) )
= bot_bot_set_o )
= ( ~ ( member_o @ A2 @ B2 )
& ( ( inf_inf_set_o @ B2 @ A )
= bot_bot_set_o ) ) ) ).
% disjoint_insert(1)
thf(fact_835_disjoint__insert_I2_J,axiom,
! [A: set_li6526943997496501093_a_nat,B: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( bot_bo1033123847703346641_a_nat
= ( inf_in3249246906714053971_a_nat @ A @ ( insert2950094090816004437_a_nat @ B @ B2 ) ) )
= ( ~ ( member408289922725080238_a_nat @ B @ A )
& ( bot_bo1033123847703346641_a_nat
= ( inf_in3249246906714053971_a_nat @ A @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_836_disjoint__insert_I2_J,axiom,
! [A: set_Sum_sum_a_nat,B: sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( bot_bo3438331934148233675_a_nat
= ( inf_in7084830621192376909_a_nat @ A @ ( insert_Sum_sum_a_nat @ B @ B2 ) ) )
= ( ~ ( member_Sum_sum_a_nat @ B @ A )
& ( bot_bo3438331934148233675_a_nat
= ( inf_in7084830621192376909_a_nat @ A @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_837_disjoint__insert_I2_J,axiom,
! [A: set_o,B: $o,B2: set_o] :
( ( bot_bot_set_o
= ( inf_inf_set_o @ A @ ( insert_o @ B @ B2 ) ) )
= ( ~ ( member_o @ B @ A )
& ( bot_bot_set_o
= ( inf_inf_set_o @ A @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_838_insert__Diff__single,axiom,
! [A2: $o,A: set_o] :
( ( insert_o @ A2 @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
= ( insert_o @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_839_Inf_OINF__id__eq,axiom,
! [Inf: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( Inf @ ( image_5081948215111134021_a_nat @ id_lis6021530241265049846_a_nat @ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_id_eq
thf(fact_840_Inf_OINF__id__eq,axiom,
! [Inf: set_o > $o,A: set_o] :
( ( Inf @ ( image_o_o2 @ id_o @ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_id_eq
thf(fact_841_Sup_OSUP__id__eq,axiom,
! [Sup: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( Sup @ ( image_5081948215111134021_a_nat @ id_lis6021530241265049846_a_nat @ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_id_eq
thf(fact_842_Sup_OSUP__id__eq,axiom,
! [Sup: set_o > $o,A: set_o] :
( ( Sup @ ( image_o_o2 @ id_o @ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_id_eq
thf(fact_843_singleton__inject,axiom,
! [A2: $o,B: $o] :
( ( ( insert_o @ A2 @ bot_bot_set_o )
= ( insert_o @ B @ bot_bot_set_o ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_844_insert__not__empty,axiom,
! [A2: $o,A: set_o] :
( ( insert_o @ A2 @ A )
!= bot_bot_set_o ) ).
% insert_not_empty
thf(fact_845_doubleton__eq__iff,axiom,
! [A2: $o,B: $o,C3: $o,D2: $o] :
( ( ( insert_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) )
= ( insert_o @ C3 @ ( insert_o @ D2 @ bot_bot_set_o ) ) )
= ( ( ( A2 = C3 )
& ( B = D2 ) )
| ( ( A2 = D2 )
& ( B = C3 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_846_singleton__iff,axiom,
! [B: list_Sum_sum_a_nat,A2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ B @ ( insert2950094090816004437_a_nat @ A2 @ bot_bo1033123847703346641_a_nat ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_847_singleton__iff,axiom,
! [B: sum_sum_a_nat,A2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ B @ ( insert_Sum_sum_a_nat @ A2 @ bot_bo3438331934148233675_a_nat ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_848_singleton__iff,axiom,
! [B: $o,A2: $o] :
( ( member_o @ B @ ( insert_o @ A2 @ bot_bot_set_o ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_849_singletonD,axiom,
! [B: list_Sum_sum_a_nat,A2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ B @ ( insert2950094090816004437_a_nat @ A2 @ bot_bo1033123847703346641_a_nat ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_850_singletonD,axiom,
! [B: sum_sum_a_nat,A2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ B @ ( insert_Sum_sum_a_nat @ A2 @ bot_bo3438331934148233675_a_nat ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_851_singletonD,axiom,
! [B: $o,A2: $o] :
( ( member_o @ B @ ( insert_o @ A2 @ bot_bot_set_o ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_852_Collect__conv__if2,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,A2: nat > sum_sum_a_nat] :
( ( ( P @ A2 )
=> ( ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( A2 = X )
& ( P @ X ) ) )
= ( insert5265011953798106934_a_nat @ A2 @ bot_bo6441361344521902642_a_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( A2 = X )
& ( P @ X ) ) )
= bot_bo6441361344521902642_a_nat ) ) ) ).
% Collect_conv_if2
thf(fact_853_Collect__conv__if2,axiom,
! [P: $o > $o,A2: $o] :
( ( ( P @ A2 )
=> ( ( collect_o
@ ^ [X: $o] :
( ( A2 = X )
& ( P @ X ) ) )
= ( insert_o @ A2 @ bot_bot_set_o ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_o
@ ^ [X: $o] :
( ( A2 = X )
& ( P @ X ) ) )
= bot_bot_set_o ) ) ) ).
% Collect_conv_if2
thf(fact_854_Collect__conv__if,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,A2: nat > sum_sum_a_nat] :
( ( ( P @ A2 )
=> ( ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( X = A2 )
& ( P @ X ) ) )
= ( insert5265011953798106934_a_nat @ A2 @ bot_bo6441361344521902642_a_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( X = A2 )
& ( P @ X ) ) )
= bot_bo6441361344521902642_a_nat ) ) ) ).
% Collect_conv_if
thf(fact_855_Collect__conv__if,axiom,
! [P: $o > $o,A2: $o] :
( ( ( P @ A2 )
=> ( ( collect_o
@ ^ [X: $o] :
( ( X = A2 )
& ( P @ X ) ) )
= ( insert_o @ A2 @ bot_bot_set_o ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_o
@ ^ [X: $o] :
( ( X = A2 )
& ( P @ X ) ) )
= bot_bot_set_o ) ) ) ).
% Collect_conv_if
thf(fact_856_DEADID_Oin__rel,axiom,
( ( ^ [Y6: list_Sum_sum_a_nat,Z3: list_Sum_sum_a_nat] : ( Y6 = Z3 ) )
= ( ^ [A8: list_Sum_sum_a_nat,B8: list_Sum_sum_a_nat] :
? [Z5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Z5 @ top_to6433055325616222389_a_nat )
& ( ( id_lis6021530241265049846_a_nat @ Z5 )
= A8 )
& ( ( id_lis6021530241265049846_a_nat @ Z5 )
= B8 ) ) ) ) ).
% DEADID.in_rel
thf(fact_857_DEADID_Oin__rel,axiom,
( ( ^ [Y6: sum_sum_a_nat,Z3: sum_sum_a_nat] : ( Y6 = Z3 ) )
= ( ^ [A8: sum_sum_a_nat,B8: sum_sum_a_nat] :
? [Z5: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Z5 @ top_to795618464972521135_a_nat )
& ( ( id_Sum_sum_a_nat @ Z5 )
= A8 )
& ( ( id_Sum_sum_a_nat @ Z5 )
= B8 ) ) ) ) ).
% DEADID.in_rel
thf(fact_858_DEADID_Oin__rel,axiom,
( ( ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 ) )
= ( ^ [A8: $o,B8: $o] :
? [Z5: $o] :
( ( member_o @ Z5 @ top_top_set_o )
& ( ( id_o @ Z5 )
= A8 )
& ( ( id_o @ Z5 )
= B8 ) ) ) ) ).
% DEADID.in_rel
thf(fact_859_insert__UNIV,axiom,
! [X2: sum_sum_a_nat] :
( ( insert_Sum_sum_a_nat @ X2 @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat ) ).
% insert_UNIV
thf(fact_860_insert__UNIV,axiom,
! [X2: $o] :
( ( insert_o @ X2 @ top_top_set_o )
= top_top_set_o ) ).
% insert_UNIV
thf(fact_861_subset__insertI2,axiom,
! [A: set_o,B2: set_o,B: $o] :
( ( ord_less_eq_set_o @ A @ B2 )
=> ( ord_less_eq_set_o @ A @ ( insert_o @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_862_subset__insertI,axiom,
! [B2: set_o,A2: $o] : ( ord_less_eq_set_o @ B2 @ ( insert_o @ A2 @ B2 ) ) ).
% subset_insertI
thf(fact_863_subset__insert,axiom,
! [X2: $o,A: set_o,B2: set_o] :
( ~ ( member_o @ X2 @ A )
=> ( ( ord_less_eq_set_o @ A @ ( insert_o @ X2 @ B2 ) )
= ( ord_less_eq_set_o @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_864_subset__insert,axiom,
! [X2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ X2 @ A )
=> ( ( ord_le1147066620699065093_a_nat @ A @ ( insert2950094090816004437_a_nat @ X2 @ B2 ) )
= ( ord_le1147066620699065093_a_nat @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_865_subset__insert,axiom,
! [X2: sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ~ ( member_Sum_sum_a_nat @ X2 @ A )
=> ( ( ord_le1325389633284124927_a_nat @ A @ ( insert_Sum_sum_a_nat @ X2 @ B2 ) )
= ( ord_le1325389633284124927_a_nat @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_866_insert__mono,axiom,
! [C: set_o,D: set_o,A2: $o] :
( ( ord_less_eq_set_o @ C @ D )
=> ( ord_less_eq_set_o @ ( insert_o @ A2 @ C ) @ ( insert_o @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_867_insert__compr,axiom,
( insert_o
= ( ^ [A8: $o,B6: set_o] :
( collect_o
@ ^ [X: $o] :
( ( X = A8 )
| ( member_o @ X @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_868_insert__compr,axiom,
( insert2950094090816004437_a_nat
= ( ^ [A8: list_Sum_sum_a_nat,B6: set_li6526943997496501093_a_nat] :
( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( X = A8 )
| ( member408289922725080238_a_nat @ X @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_869_insert__compr,axiom,
( insert_Sum_sum_a_nat
= ( ^ [A8: sum_sum_a_nat,B6: set_Sum_sum_a_nat] :
( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( X = A8 )
| ( member_Sum_sum_a_nat @ X @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_870_insert__compr,axiom,
( insert5265011953798106934_a_nat
= ( ^ [A8: nat > sum_sum_a_nat,B6: set_na3699693778330250182_a_nat] :
( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( X = A8 )
| ( member8690443509505302927_a_nat @ X @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_871_insert__Collect,axiom,
! [A2: $o,P: $o > $o] :
( ( insert_o @ A2 @ ( collect_o @ P ) )
= ( collect_o
@ ^ [U: $o] :
( ( U != A2 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_872_insert__Collect,axiom,
! [A2: nat > sum_sum_a_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ( insert5265011953798106934_a_nat @ A2 @ ( collec5629555741568564177_a_nat @ P ) )
= ( collec5629555741568564177_a_nat
@ ^ [U: nat > sum_sum_a_nat] :
( ( U != A2 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_873_insertE,axiom,
! [A2: $o,B: $o,A: set_o] :
( ( member_o @ A2 @ ( insert_o @ B @ A ) )
=> ( ( A2 = (~ B) )
=> ( member_o @ A2 @ A ) ) ) ).
% insertE
thf(fact_874_insertE,axiom,
! [A2: list_Sum_sum_a_nat,B: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ B @ A ) )
=> ( ( A2 != B )
=> ( member408289922725080238_a_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_875_insertE,axiom,
! [A2: sum_sum_a_nat,B: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A2 @ ( insert_Sum_sum_a_nat @ B @ A ) )
=> ( ( A2 != B )
=> ( member_Sum_sum_a_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_876_insertI1,axiom,
! [A2: $o,B2: set_o] : ( member_o @ A2 @ ( insert_o @ A2 @ B2 ) ) ).
% insertI1
thf(fact_877_insertI1,axiom,
! [A2: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] : ( member408289922725080238_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ A2 @ B2 ) ) ).
% insertI1
thf(fact_878_insertI1,axiom,
! [A2: sum_sum_a_nat,B2: set_Sum_sum_a_nat] : ( member_Sum_sum_a_nat @ A2 @ ( insert_Sum_sum_a_nat @ A2 @ B2 ) ) ).
% insertI1
thf(fact_879_insertI2,axiom,
! [A2: $o,B2: set_o,B: $o] :
( ( member_o @ A2 @ B2 )
=> ( member_o @ A2 @ ( insert_o @ B @ B2 ) ) ) ).
% insertI2
thf(fact_880_insertI2,axiom,
! [A2: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat,B: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ A2 @ B2 )
=> ( member408289922725080238_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_881_insertI2,axiom,
! [A2: sum_sum_a_nat,B2: set_Sum_sum_a_nat,B: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A2 @ B2 )
=> ( member_Sum_sum_a_nat @ A2 @ ( insert_Sum_sum_a_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_882_Set_Oset__insert,axiom,
! [X2: $o,A: set_o] :
( ( member_o @ X2 @ A )
=> ~ ! [B9: set_o] :
( ( A
= ( insert_o @ X2 @ B9 ) )
=> ( member_o @ X2 @ B9 ) ) ) ).
% Set.set_insert
thf(fact_883_Set_Oset__insert,axiom,
! [X2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A )
=> ~ ! [B9: set_li6526943997496501093_a_nat] :
( ( A
= ( insert2950094090816004437_a_nat @ X2 @ B9 ) )
=> ( member408289922725080238_a_nat @ X2 @ B9 ) ) ) ).
% Set.set_insert
thf(fact_884_Set_Oset__insert,axiom,
! [X2: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ A )
=> ~ ! [B9: set_Sum_sum_a_nat] :
( ( A
= ( insert_Sum_sum_a_nat @ X2 @ B9 ) )
=> ( member_Sum_sum_a_nat @ X2 @ B9 ) ) ) ).
% Set.set_insert
thf(fact_885_insert__ident,axiom,
! [X2: $o,A: set_o,B2: set_o] :
( ~ ( member_o @ X2 @ A )
=> ( ~ ( member_o @ X2 @ B2 )
=> ( ( ( insert_o @ X2 @ A )
= ( insert_o @ X2 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_886_insert__ident,axiom,
! [X2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ X2 @ A )
=> ( ~ ( member408289922725080238_a_nat @ X2 @ B2 )
=> ( ( ( insert2950094090816004437_a_nat @ X2 @ A )
= ( insert2950094090816004437_a_nat @ X2 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_887_insert__ident,axiom,
! [X2: sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ~ ( member_Sum_sum_a_nat @ X2 @ A )
=> ( ~ ( member_Sum_sum_a_nat @ X2 @ B2 )
=> ( ( ( insert_Sum_sum_a_nat @ X2 @ A )
= ( insert_Sum_sum_a_nat @ X2 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_888_insert__absorb,axiom,
! [A2: $o,A: set_o] :
( ( member_o @ A2 @ A )
=> ( ( insert_o @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_889_insert__absorb,axiom,
! [A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A2 @ A )
=> ( ( insert2950094090816004437_a_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_890_insert__absorb,axiom,
! [A2: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A2 @ A )
=> ( ( insert_Sum_sum_a_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_891_insert__eq__iff,axiom,
! [A2: $o,A: set_o,B: $o,B2: set_o] :
( ~ ( member_o @ A2 @ A )
=> ( ~ ( member_o @ B @ B2 )
=> ( ( ( insert_o @ A2 @ A )
= ( insert_o @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 = (~ B) )
=> ? [C5: set_o] :
( ( A
= ( insert_o @ B @ C5 ) )
& ~ ( member_o @ B @ C5 )
& ( B2
= ( insert_o @ A2 @ C5 ) )
& ~ ( member_o @ A2 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_892_insert__eq__iff,axiom,
! [A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ A2 @ A )
=> ( ~ ( member408289922725080238_a_nat @ B @ B2 )
=> ( ( ( insert2950094090816004437_a_nat @ A2 @ A )
= ( insert2950094090816004437_a_nat @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C5: set_li6526943997496501093_a_nat] :
( ( A
= ( insert2950094090816004437_a_nat @ B @ C5 ) )
& ~ ( member408289922725080238_a_nat @ B @ C5 )
& ( B2
= ( insert2950094090816004437_a_nat @ A2 @ C5 ) )
& ~ ( member408289922725080238_a_nat @ A2 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_893_insert__eq__iff,axiom,
! [A2: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ~ ( member_Sum_sum_a_nat @ A2 @ A )
=> ( ~ ( member_Sum_sum_a_nat @ B @ B2 )
=> ( ( ( insert_Sum_sum_a_nat @ A2 @ A )
= ( insert_Sum_sum_a_nat @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C5: set_Sum_sum_a_nat] :
( ( A
= ( insert_Sum_sum_a_nat @ B @ C5 ) )
& ~ ( member_Sum_sum_a_nat @ B @ C5 )
& ( B2
= ( insert_Sum_sum_a_nat @ A2 @ C5 ) )
& ~ ( member_Sum_sum_a_nat @ A2 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_894_insert__commute,axiom,
! [X2: $o,Y2: $o,A: set_o] :
( ( insert_o @ X2 @ ( insert_o @ Y2 @ A ) )
= ( insert_o @ Y2 @ ( insert_o @ X2 @ A ) ) ) ).
% insert_commute
thf(fact_895_mk__disjoint__insert,axiom,
! [A2: $o,A: set_o] :
( ( member_o @ A2 @ A )
=> ? [B9: set_o] :
( ( A
= ( insert_o @ A2 @ B9 ) )
& ~ ( member_o @ A2 @ B9 ) ) ) ).
% mk_disjoint_insert
thf(fact_896_mk__disjoint__insert,axiom,
! [A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A2 @ A )
=> ? [B9: set_li6526943997496501093_a_nat] :
( ( A
= ( insert2950094090816004437_a_nat @ A2 @ B9 ) )
& ~ ( member408289922725080238_a_nat @ A2 @ B9 ) ) ) ).
% mk_disjoint_insert
thf(fact_897_mk__disjoint__insert,axiom,
! [A2: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A2 @ A )
=> ? [B9: set_Sum_sum_a_nat] :
( ( A
= ( insert_Sum_sum_a_nat @ A2 @ B9 ) )
& ~ ( member_Sum_sum_a_nat @ A2 @ B9 ) ) ) ).
% mk_disjoint_insert
thf(fact_898_insert__Diff__if,axiom,
! [X2: $o,B2: set_o,A: set_o] :
( ( ( member_o @ X2 @ B2 )
=> ( ( minus_minus_set_o @ ( insert_o @ X2 @ A ) @ B2 )
= ( minus_minus_set_o @ A @ B2 ) ) )
& ( ~ ( member_o @ X2 @ B2 )
=> ( ( minus_minus_set_o @ ( insert_o @ X2 @ A ) @ B2 )
= ( insert_o @ X2 @ ( minus_minus_set_o @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_899_insert__Diff__if,axiom,
! [X2: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ( member408289922725080238_a_nat @ X2 @ B2 )
=> ( ( minus_7395159227704179404_a_nat @ ( insert2950094090816004437_a_nat @ X2 @ A ) @ B2 )
= ( minus_7395159227704179404_a_nat @ A @ B2 ) ) )
& ( ~ ( member408289922725080238_a_nat @ X2 @ B2 )
=> ( ( minus_7395159227704179404_a_nat @ ( insert2950094090816004437_a_nat @ X2 @ A ) @ B2 )
= ( insert2950094090816004437_a_nat @ X2 @ ( minus_7395159227704179404_a_nat @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_900_insert__Diff__if,axiom,
! [X2: sum_sum_a_nat,B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ( member_Sum_sum_a_nat @ X2 @ B2 )
=> ( ( minus_1134630996077396038_a_nat @ ( insert_Sum_sum_a_nat @ X2 @ A ) @ B2 )
= ( minus_1134630996077396038_a_nat @ A @ B2 ) ) )
& ( ~ ( member_Sum_sum_a_nat @ X2 @ B2 )
=> ( ( minus_1134630996077396038_a_nat @ ( insert_Sum_sum_a_nat @ X2 @ A ) @ B2 )
= ( insert_Sum_sum_a_nat @ X2 @ ( minus_1134630996077396038_a_nat @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_901_Int__insert__right,axiom,
! [A2: $o,A: set_o,B2: set_o] :
( ( ( member_o @ A2 @ A )
=> ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B2 ) )
= ( insert_o @ A2 @ ( inf_inf_set_o @ A @ B2 ) ) ) )
& ( ~ ( member_o @ A2 @ A )
=> ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B2 ) )
= ( inf_inf_set_o @ A @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_902_Int__insert__right,axiom,
! [A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ( member408289922725080238_a_nat @ A2 @ A )
=> ( ( inf_in3249246906714053971_a_nat @ A @ ( insert2950094090816004437_a_nat @ A2 @ B2 ) )
= ( insert2950094090816004437_a_nat @ A2 @ ( inf_in3249246906714053971_a_nat @ A @ B2 ) ) ) )
& ( ~ ( member408289922725080238_a_nat @ A2 @ A )
=> ( ( inf_in3249246906714053971_a_nat @ A @ ( insert2950094090816004437_a_nat @ A2 @ B2 ) )
= ( inf_in3249246906714053971_a_nat @ A @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_903_Int__insert__right,axiom,
! [A2: sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ( member_Sum_sum_a_nat @ A2 @ A )
=> ( ( inf_in7084830621192376909_a_nat @ A @ ( insert_Sum_sum_a_nat @ A2 @ B2 ) )
= ( insert_Sum_sum_a_nat @ A2 @ ( inf_in7084830621192376909_a_nat @ A @ B2 ) ) ) )
& ( ~ ( member_Sum_sum_a_nat @ A2 @ A )
=> ( ( inf_in7084830621192376909_a_nat @ A @ ( insert_Sum_sum_a_nat @ A2 @ B2 ) )
= ( inf_in7084830621192376909_a_nat @ A @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_904_Int__insert__left,axiom,
! [A2: $o,C: set_o,B2: set_o] :
( ( ( member_o @ A2 @ C )
=> ( ( inf_inf_set_o @ ( insert_o @ A2 @ B2 ) @ C )
= ( insert_o @ A2 @ ( inf_inf_set_o @ B2 @ C ) ) ) )
& ( ~ ( member_o @ A2 @ C )
=> ( ( inf_inf_set_o @ ( insert_o @ A2 @ B2 ) @ C )
= ( inf_inf_set_o @ B2 @ C ) ) ) ) ).
% Int_insert_left
thf(fact_905_Int__insert__left,axiom,
! [A2: list_Sum_sum_a_nat,C: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ( member408289922725080238_a_nat @ A2 @ C )
=> ( ( inf_in3249246906714053971_a_nat @ ( insert2950094090816004437_a_nat @ A2 @ B2 ) @ C )
= ( insert2950094090816004437_a_nat @ A2 @ ( inf_in3249246906714053971_a_nat @ B2 @ C ) ) ) )
& ( ~ ( member408289922725080238_a_nat @ A2 @ C )
=> ( ( inf_in3249246906714053971_a_nat @ ( insert2950094090816004437_a_nat @ A2 @ B2 ) @ C )
= ( inf_in3249246906714053971_a_nat @ B2 @ C ) ) ) ) ).
% Int_insert_left
thf(fact_906_Int__insert__left,axiom,
! [A2: sum_sum_a_nat,C: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ( member_Sum_sum_a_nat @ A2 @ C )
=> ( ( inf_in7084830621192376909_a_nat @ ( insert_Sum_sum_a_nat @ A2 @ B2 ) @ C )
= ( insert_Sum_sum_a_nat @ A2 @ ( inf_in7084830621192376909_a_nat @ B2 @ C ) ) ) )
& ( ~ ( member_Sum_sum_a_nat @ A2 @ C )
=> ( ( inf_in7084830621192376909_a_nat @ ( insert_Sum_sum_a_nat @ A2 @ B2 ) @ C )
= ( inf_in7084830621192376909_a_nat @ B2 @ C ) ) ) ) ).
% Int_insert_left
thf(fact_907_surj__id,axiom,
( ( image_5081948215111134021_a_nat @ id_lis6021530241265049846_a_nat @ top_to6433055325616222389_a_nat )
= top_to6433055325616222389_a_nat ) ).
% surj_id
thf(fact_908_surj__id,axiom,
( ( image_7142520692256960453_a_nat @ id_Sum_sum_a_nat @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat ) ).
% surj_id
thf(fact_909_surj__id,axiom,
( ( image_o_o2 @ id_o @ top_top_set_o )
= top_top_set_o ) ).
% surj_id
thf(fact_910_subset__singleton__iff,axiom,
! [X6: set_o,A2: $o] :
( ( ord_less_eq_set_o @ X6 @ ( insert_o @ A2 @ bot_bot_set_o ) )
= ( ( X6 = bot_bot_set_o )
| ( X6
= ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ).
% subset_singleton_iff
thf(fact_911_subset__singletonD,axiom,
! [A: set_o,X2: $o] :
( ( ord_less_eq_set_o @ A @ ( insert_o @ X2 @ bot_bot_set_o ) )
=> ( ( A = bot_bot_set_o )
| ( A
= ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ).
% subset_singletonD
thf(fact_912_in__image__insert__iff,axiom,
! [B2: set_se4330304633200676677_a_nat,X2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ! [C4: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ C4 @ B2 )
=> ~ ( member408289922725080238_a_nat @ X2 @ C4 ) )
=> ( ( member5553968465346197646_a_nat @ A @ ( image_3472601871771700037_a_nat @ ( insert2950094090816004437_a_nat @ X2 ) @ B2 ) )
= ( ( member408289922725080238_a_nat @ X2 @ A )
& ( member5553968465346197646_a_nat @ ( minus_7395159227704179404_a_nat @ A @ ( insert2950094090816004437_a_nat @ X2 @ bot_bo1033123847703346641_a_nat ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_913_in__image__insert__iff,axiom,
! [B2: set_se4904748513628223167_a_nat,X2: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ! [C4: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ C4 @ B2 )
=> ~ ( member_Sum_sum_a_nat @ X2 @ C4 ) )
=> ( ( member8098812455498974984_a_nat @ A @ ( image_5599399343138760645_a_nat @ ( insert_Sum_sum_a_nat @ X2 ) @ B2 ) )
= ( ( member_Sum_sum_a_nat @ X2 @ A )
& ( member8098812455498974984_a_nat @ ( minus_1134630996077396038_a_nat @ A @ ( insert_Sum_sum_a_nat @ X2 @ bot_bo3438331934148233675_a_nat ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_914_in__image__insert__iff,axiom,
! [B2: set_set_o,X2: $o,A: set_o] :
( ! [C4: set_o] :
( ( member_set_o @ C4 @ B2 )
=> ~ ( member_o @ X2 @ C4 ) )
=> ( ( member_set_o @ A @ ( image_set_o_set_o @ ( insert_o @ X2 ) @ B2 ) )
= ( ( member_o @ X2 @ A )
& ( member_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X2 @ bot_bot_set_o ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_915_Diff__insert__absorb,axiom,
! [X2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ X2 @ A )
=> ( ( minus_7395159227704179404_a_nat @ ( insert2950094090816004437_a_nat @ X2 @ A ) @ ( insert2950094090816004437_a_nat @ X2 @ bot_bo1033123847703346641_a_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_916_Diff__insert__absorb,axiom,
! [X2: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ~ ( member_Sum_sum_a_nat @ X2 @ A )
=> ( ( minus_1134630996077396038_a_nat @ ( insert_Sum_sum_a_nat @ X2 @ A ) @ ( insert_Sum_sum_a_nat @ X2 @ bot_bo3438331934148233675_a_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_917_Diff__insert__absorb,axiom,
! [X2: $o,A: set_o] :
( ~ ( member_o @ X2 @ A )
=> ( ( minus_minus_set_o @ ( insert_o @ X2 @ A ) @ ( insert_o @ X2 @ bot_bot_set_o ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_918_Diff__insert2,axiom,
! [A: set_o,A2: $o,B2: set_o] :
( ( minus_minus_set_o @ A @ ( insert_o @ A2 @ B2 ) )
= ( minus_minus_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_919_insert__Diff,axiom,
! [A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A2 @ A )
=> ( ( insert2950094090816004437_a_nat @ A2 @ ( minus_7395159227704179404_a_nat @ A @ ( insert2950094090816004437_a_nat @ A2 @ bot_bo1033123847703346641_a_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_920_insert__Diff,axiom,
! [A2: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A2 @ A )
=> ( ( insert_Sum_sum_a_nat @ A2 @ ( minus_1134630996077396038_a_nat @ A @ ( insert_Sum_sum_a_nat @ A2 @ bot_bo3438331934148233675_a_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_921_insert__Diff,axiom,
! [A2: $o,A: set_o] :
( ( member_o @ A2 @ A )
=> ( ( insert_o @ A2 @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
= A ) ) ).
% insert_Diff
thf(fact_922_Diff__insert,axiom,
! [A: set_o,A2: $o,B2: set_o] :
( ( minus_minus_set_o @ A @ ( insert_o @ A2 @ B2 ) )
= ( minus_minus_set_o @ ( minus_minus_set_o @ A @ B2 ) @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ).
% Diff_insert
thf(fact_923_subset__Diff__insert,axiom,
! [A: set_o,B2: set_o,X2: $o,C: set_o] :
( ( ord_less_eq_set_o @ A @ ( minus_minus_set_o @ B2 @ ( insert_o @ X2 @ C ) ) )
= ( ( ord_less_eq_set_o @ A @ ( minus_minus_set_o @ B2 @ C ) )
& ~ ( member_o @ X2 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_924_subset__Diff__insert,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,X2: list_Sum_sum_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ ( minus_7395159227704179404_a_nat @ B2 @ ( insert2950094090816004437_a_nat @ X2 @ C ) ) )
= ( ( ord_le1147066620699065093_a_nat @ A @ ( minus_7395159227704179404_a_nat @ B2 @ C ) )
& ~ ( member408289922725080238_a_nat @ X2 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_925_subset__Diff__insert,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,X2: sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ ( minus_1134630996077396038_a_nat @ B2 @ ( insert_Sum_sum_a_nat @ X2 @ C ) ) )
= ( ( ord_le1325389633284124927_a_nat @ A @ ( minus_1134630996077396038_a_nat @ B2 @ C ) )
& ~ ( member_Sum_sum_a_nat @ X2 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_926_image__constant,axiom,
! [X2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,C3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A )
=> ( ( image_5081948215111134021_a_nat
@ ^ [X: list_Sum_sum_a_nat] : C3
@ A )
= ( insert2950094090816004437_a_nat @ C3 @ bot_bo1033123847703346641_a_nat ) ) ) ).
% image_constant
thf(fact_927_image__constant,axiom,
! [X2: $o,A: set_o,C3: $o] :
( ( member_o @ X2 @ A )
=> ( ( image_o_o2
@ ^ [X: $o] : C3
@ A )
= ( insert_o @ C3 @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_928_image__constant,axiom,
! [X2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,C3: $o] :
( ( member408289922725080238_a_nat @ X2 @ A )
=> ( ( image_3984946558445957976_nat_o
@ ^ [X: list_Sum_sum_a_nat] : C3
@ A )
= ( insert_o @ C3 @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_929_image__constant,axiom,
! [X2: sum_sum_a_nat,A: set_Sum_sum_a_nat,C3: $o] :
( ( member_Sum_sum_a_nat @ X2 @ A )
=> ( ( image_6095136190293192542_nat_o
@ ^ [X: sum_sum_a_nat] : C3
@ A )
= ( insert_o @ C3 @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_930_image__constant__conv,axiom,
! [A: set_li6526943997496501093_a_nat,C3: list_Sum_sum_a_nat] :
( ( ( A = bot_bo1033123847703346641_a_nat )
=> ( ( image_5081948215111134021_a_nat
@ ^ [X: list_Sum_sum_a_nat] : C3
@ A )
= bot_bo1033123847703346641_a_nat ) )
& ( ( A != bot_bo1033123847703346641_a_nat )
=> ( ( image_5081948215111134021_a_nat
@ ^ [X: list_Sum_sum_a_nat] : C3
@ A )
= ( insert2950094090816004437_a_nat @ C3 @ bot_bo1033123847703346641_a_nat ) ) ) ) ).
% image_constant_conv
thf(fact_931_image__constant__conv,axiom,
! [A: set_o,C3: $o] :
( ( ( A = bot_bot_set_o )
=> ( ( image_o_o2
@ ^ [X: $o] : C3
@ A )
= bot_bot_set_o ) )
& ( ( A != bot_bot_set_o )
=> ( ( image_o_o2
@ ^ [X: $o] : C3
@ A )
= ( insert_o @ C3 @ bot_bot_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_932_range__eq__singletonD,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: list_Sum_sum_a_nat,X2: list_Sum_sum_a_nat] :
( ( ( image_5081948215111134021_a_nat @ F @ top_to6433055325616222389_a_nat )
= ( insert2950094090816004437_a_nat @ A2 @ bot_bo1033123847703346641_a_nat ) )
=> ( ( F @ X2 )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_933_range__eq__singletonD,axiom,
! [F: sum_sum_a_nat > $o,A2: $o,X2: sum_sum_a_nat] :
( ( ( image_6095136190293192542_nat_o @ F @ top_to795618464972521135_a_nat )
= ( insert_o @ A2 @ bot_bot_set_o ) )
=> ( ( F @ X2 )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_934_range__eq__singletonD,axiom,
! [F: $o > $o,A2: $o,X2: $o] :
( ( ( image_o_o2 @ F @ top_top_set_o )
= ( insert_o @ A2 @ bot_bot_set_o ) )
=> ( ( F @ X2 )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_935_subset__insert__iff,axiom,
! [A: set_li6526943997496501093_a_nat,X2: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ ( insert2950094090816004437_a_nat @ X2 @ B2 ) )
= ( ( ( member408289922725080238_a_nat @ X2 @ A )
=> ( ord_le1147066620699065093_a_nat @ ( minus_7395159227704179404_a_nat @ A @ ( insert2950094090816004437_a_nat @ X2 @ bot_bo1033123847703346641_a_nat ) ) @ B2 ) )
& ( ~ ( member408289922725080238_a_nat @ X2 @ A )
=> ( ord_le1147066620699065093_a_nat @ A @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_936_subset__insert__iff,axiom,
! [A: set_Sum_sum_a_nat,X2: sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ ( insert_Sum_sum_a_nat @ X2 @ B2 ) )
= ( ( ( member_Sum_sum_a_nat @ X2 @ A )
=> ( ord_le1325389633284124927_a_nat @ ( minus_1134630996077396038_a_nat @ A @ ( insert_Sum_sum_a_nat @ X2 @ bot_bo3438331934148233675_a_nat ) ) @ B2 ) )
& ( ~ ( member_Sum_sum_a_nat @ X2 @ A )
=> ( ord_le1325389633284124927_a_nat @ A @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_937_subset__insert__iff,axiom,
! [A: set_o,X2: $o,B2: set_o] :
( ( ord_less_eq_set_o @ A @ ( insert_o @ X2 @ B2 ) )
= ( ( ( member_o @ X2 @ A )
=> ( ord_less_eq_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X2 @ bot_bot_set_o ) ) @ B2 ) )
& ( ~ ( member_o @ X2 @ A )
=> ( ord_less_eq_set_o @ A @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_938_Diff__single__insert,axiom,
! [A: set_o,X2: $o,B2: set_o] :
( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X2 @ bot_bot_set_o ) ) @ B2 )
=> ( ord_less_eq_set_o @ A @ ( insert_o @ X2 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_939_antisym__singleton,axiom,
! [X2: produc7017002724195966439_a_nat] : ( antisy1998289778759880802_a_nat @ top_to795618464972521135_a_nat @ ( insert900367560037198775_a_nat @ X2 @ bot_bo6795098209002113331_a_nat ) ) ).
% antisym_singleton
thf(fact_940_antisym__singleton,axiom,
! [X2: product_prod_o_o] : ( antisym_on_o @ top_top_set_o @ ( insert6201435330877294327od_o_o @ X2 @ bot_bo7073875226086086771od_o_o ) ) ).
% antisym_singleton
thf(fact_941_is__singletonI,axiom,
! [X2: $o] : ( is_singleton_o @ ( insert_o @ X2 @ bot_bot_set_o ) ) ).
% is_singletonI
thf(fact_942_image__split__eq__Sigma,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( image_8390800116927144231_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( produc7990843422341522135_a_nat @ ( F @ X ) @ ( G @ X ) )
@ A )
= ( produc5624668685630548104_a_nat @ ( image_5081948215111134021_a_nat @ F @ A )
@ ^ [X: list_Sum_sum_a_nat] : ( image_5081948215111134021_a_nat @ G @ ( inf_in3249246906714053971_a_nat @ ( vimage7538740559230589179_a_nat @ F @ ( insert2950094090816004437_a_nat @ X @ bot_bo1033123847703346641_a_nat ) ) @ A ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_943_image__split__eq__Sigma,axiom,
! [F: list_Sum_sum_a_nat > $o,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( image_6144275418292344916_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( produc7378284081089837956_a_nat @ ( F @ X ) @ ( G @ X ) )
@ A )
= ( produc5640064037308512181_a_nat @ ( image_3984946558445957976_nat_o @ F @ A )
@ ^ [X: $o] : ( image_5081948215111134021_a_nat @ G @ ( inf_in3249246906714053971_a_nat @ ( vimage6136348398256869390_nat_o @ F @ ( insert_o @ X @ bot_bot_set_o ) ) @ A ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_944_image__split__eq__Sigma,axiom,
! [F: $o > $o,G: $o > $o,A: set_o] :
( ( image_4057150146340385428od_o_o
@ ^ [X: $o] : ( product_Pair_o_o @ ( F @ X ) @ ( G @ X ) )
@ A )
= ( product_Sigma_o_o @ ( image_o_o2 @ F @ A )
@ ^ [X: $o] : ( image_o_o2 @ G @ ( inf_inf_set_o @ ( vimage_o_o @ F @ ( insert_o @ X @ bot_bot_set_o ) ) @ A ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_945_vimage__eq,axiom,
! [A2: $o,F: $o > $o,B2: set_o] :
( ( member_o @ A2 @ ( vimage_o_o @ F @ B2 ) )
= ( member_o @ ( F @ A2 ) @ B2 ) ) ).
% vimage_eq
thf(fact_946_vimage__eq,axiom,
! [A2: $o,F: $o > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member_o @ A2 @ ( vimage2884537194618305448_a_nat @ F @ B2 ) )
= ( member408289922725080238_a_nat @ ( F @ A2 ) @ B2 ) ) ).
% vimage_eq
thf(fact_947_vimage__eq,axiom,
! [A2: $o,F: $o > sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member_o @ A2 @ ( vimage1882779926522203554_a_nat @ F @ B2 ) )
= ( member_Sum_sum_a_nat @ ( F @ A2 ) @ B2 ) ) ).
% vimage_eq
thf(fact_948_vimage__eq,axiom,
! [A2: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > $o,B2: set_o] :
( ( member408289922725080238_a_nat @ A2 @ ( vimage6136348398256869390_nat_o @ F @ B2 ) )
= ( member_o @ ( F @ A2 ) @ B2 ) ) ).
% vimage_eq
thf(fact_949_vimage__eq,axiom,
! [A2: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A2 @ ( vimage7538740559230589179_a_nat @ F @ B2 ) )
= ( member408289922725080238_a_nat @ ( F @ A2 ) @ B2 ) ) ).
% vimage_eq
thf(fact_950_vimage__eq,axiom,
! [A2: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ A2 @ ( vimage7679724146056310901_a_nat @ F @ B2 ) )
= ( member_Sum_sum_a_nat @ ( F @ A2 ) @ B2 ) ) ).
% vimage_eq
thf(fact_951_vimage__eq,axiom,
! [A2: sum_sum_a_nat,F: sum_sum_a_nat > $o,B2: set_o] :
( ( member_Sum_sum_a_nat @ A2 @ ( vimage3838435602741665556_nat_o @ F @ B2 ) )
= ( member_o @ ( F @ A2 ) @ B2 ) ) ).
% vimage_eq
thf(fact_952_vimage__eq,axiom,
! [A2: sum_sum_a_nat,F: sum_sum_a_nat > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member_Sum_sum_a_nat @ A2 @ ( vimage8052486765430747393_a_nat @ F @ B2 ) )
= ( member408289922725080238_a_nat @ ( F @ A2 ) @ B2 ) ) ).
% vimage_eq
thf(fact_953_vimage__eq,axiom,
! [A2: sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A2 @ ( vimage6545432446589551483_a_nat @ F @ B2 ) )
= ( member_Sum_sum_a_nat @ ( F @ A2 ) @ B2 ) ) ).
% vimage_eq
thf(fact_954_vimageI,axiom,
! [F: $o > $o,A2: $o,B: $o,B2: set_o] :
( ( ( F @ A2 )
= B )
=> ( ( member_o @ B @ B2 )
=> ( member_o @ A2 @ ( vimage_o_o @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_955_vimageI,axiom,
! [F: list_Sum_sum_a_nat > $o,A2: list_Sum_sum_a_nat,B: $o,B2: set_o] :
( ( ( F @ A2 )
= B )
=> ( ( member_o @ B @ B2 )
=> ( member408289922725080238_a_nat @ A2 @ ( vimage6136348398256869390_nat_o @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_956_vimageI,axiom,
! [F: sum_sum_a_nat > $o,A2: sum_sum_a_nat,B: $o,B2: set_o] :
( ( ( F @ A2 )
= B )
=> ( ( member_o @ B @ B2 )
=> ( member_Sum_sum_a_nat @ A2 @ ( vimage3838435602741665556_nat_o @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_957_vimageI,axiom,
! [F: $o > list_Sum_sum_a_nat,A2: $o,B: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ( F @ A2 )
= B )
=> ( ( member408289922725080238_a_nat @ B @ B2 )
=> ( member_o @ A2 @ ( vimage2884537194618305448_a_nat @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_958_vimageI,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: list_Sum_sum_a_nat,B: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ( F @ A2 )
= B )
=> ( ( member408289922725080238_a_nat @ B @ B2 )
=> ( member408289922725080238_a_nat @ A2 @ ( vimage7538740559230589179_a_nat @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_959_vimageI,axiom,
! [F: sum_sum_a_nat > list_Sum_sum_a_nat,A2: sum_sum_a_nat,B: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ( F @ A2 )
= B )
=> ( ( member408289922725080238_a_nat @ B @ B2 )
=> ( member_Sum_sum_a_nat @ A2 @ ( vimage8052486765430747393_a_nat @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_960_vimageI,axiom,
! [F: $o > sum_sum_a_nat,A2: $o,B: sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ( F @ A2 )
= B )
=> ( ( member_Sum_sum_a_nat @ B @ B2 )
=> ( member_o @ A2 @ ( vimage1882779926522203554_a_nat @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_961_vimageI,axiom,
! [F: list_Sum_sum_a_nat > sum_sum_a_nat,A2: list_Sum_sum_a_nat,B: sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ( F @ A2 )
= B )
=> ( ( member_Sum_sum_a_nat @ B @ B2 )
=> ( member408289922725080238_a_nat @ A2 @ ( vimage7679724146056310901_a_nat @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_962_vimageI,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A2: sum_sum_a_nat,B: sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ( F @ A2 )
= B )
=> ( ( member_Sum_sum_a_nat @ B @ B2 )
=> ( member_Sum_sum_a_nat @ A2 @ ( vimage6545432446589551483_a_nat @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_963_vimage__Collect__eq,axiom,
! [F: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ( vimage8281748387122106427_a_nat @ F @ ( collec5629555741568564177_a_nat @ P ) )
= ( collec5629555741568564177_a_nat
@ ^ [Y3: nat > sum_sum_a_nat] : ( P @ ( F @ Y3 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_964_vimage__UNIV,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat] :
( ( vimage6545432446589551483_a_nat @ F @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat ) ).
% vimage_UNIV
thf(fact_965_vimage__UNIV,axiom,
! [F: $o > sum_sum_a_nat] :
( ( vimage1882779926522203554_a_nat @ F @ top_to795618464972521135_a_nat )
= top_top_set_o ) ).
% vimage_UNIV
thf(fact_966_vimage__UNIV,axiom,
! [F: sum_sum_a_nat > $o] :
( ( vimage3838435602741665556_nat_o @ F @ top_top_set_o )
= top_to795618464972521135_a_nat ) ).
% vimage_UNIV
thf(fact_967_vimage__UNIV,axiom,
! [F: $o > $o] :
( ( vimage_o_o @ F @ top_top_set_o )
= top_top_set_o ) ).
% vimage_UNIV
thf(fact_968_vimage__empty,axiom,
! [F: $o > $o] :
( ( vimage_o_o @ F @ bot_bot_set_o )
= bot_bot_set_o ) ).
% vimage_empty
thf(fact_969_vimage__const,axiom,
! [C3: $o,A: set_o] :
( ( ( member_o @ C3 @ A )
=> ( ( vimage3838435602741665556_nat_o
@ ^ [X: sum_sum_a_nat] : C3
@ A )
= top_to795618464972521135_a_nat ) )
& ( ~ ( member_o @ C3 @ A )
=> ( ( vimage3838435602741665556_nat_o
@ ^ [X: sum_sum_a_nat] : C3
@ A )
= bot_bo3438331934148233675_a_nat ) ) ) ).
% vimage_const
thf(fact_970_vimage__const,axiom,
! [C3: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ( member408289922725080238_a_nat @ C3 @ A )
=> ( ( vimage8052486765430747393_a_nat
@ ^ [X: sum_sum_a_nat] : C3
@ A )
= top_to795618464972521135_a_nat ) )
& ( ~ ( member408289922725080238_a_nat @ C3 @ A )
=> ( ( vimage8052486765430747393_a_nat
@ ^ [X: sum_sum_a_nat] : C3
@ A )
= bot_bo3438331934148233675_a_nat ) ) ) ).
% vimage_const
thf(fact_971_vimage__const,axiom,
! [C3: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ( member_Sum_sum_a_nat @ C3 @ A )
=> ( ( vimage6545432446589551483_a_nat
@ ^ [X: sum_sum_a_nat] : C3
@ A )
= top_to795618464972521135_a_nat ) )
& ( ~ ( member_Sum_sum_a_nat @ C3 @ A )
=> ( ( vimage6545432446589551483_a_nat
@ ^ [X: sum_sum_a_nat] : C3
@ A )
= bot_bo3438331934148233675_a_nat ) ) ) ).
% vimage_const
thf(fact_972_vimage__const,axiom,
! [C3: $o,A: set_o] :
( ( ( member_o @ C3 @ A )
=> ( ( vimage_o_o
@ ^ [X: $o] : C3
@ A )
= top_top_set_o ) )
& ( ~ ( member_o @ C3 @ A )
=> ( ( vimage_o_o
@ ^ [X: $o] : C3
@ A )
= bot_bot_set_o ) ) ) ).
% vimage_const
thf(fact_973_vimage__const,axiom,
! [C3: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ( member408289922725080238_a_nat @ C3 @ A )
=> ( ( vimage2884537194618305448_a_nat
@ ^ [X: $o] : C3
@ A )
= top_top_set_o ) )
& ( ~ ( member408289922725080238_a_nat @ C3 @ A )
=> ( ( vimage2884537194618305448_a_nat
@ ^ [X: $o] : C3
@ A )
= bot_bot_set_o ) ) ) ).
% vimage_const
thf(fact_974_vimage__const,axiom,
! [C3: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ( member_Sum_sum_a_nat @ C3 @ A )
=> ( ( vimage1882779926522203554_a_nat
@ ^ [X: $o] : C3
@ A )
= top_top_set_o ) )
& ( ~ ( member_Sum_sum_a_nat @ C3 @ A )
=> ( ( vimage1882779926522203554_a_nat
@ ^ [X: $o] : C3
@ A )
= bot_bot_set_o ) ) ) ).
% vimage_const
thf(fact_975_image__vimage__eq,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat @ F @ ( vimage7538740559230589179_a_nat @ F @ A ) )
= ( inf_in3249246906714053971_a_nat @ A @ ( image_5081948215111134021_a_nat @ F @ top_to6433055325616222389_a_nat ) ) ) ).
% image_vimage_eq
thf(fact_976_image__vimage__eq,axiom,
! [F: $o > $o,A: set_o] :
( ( image_o_o2 @ F @ ( vimage_o_o @ F @ A ) )
= ( inf_inf_set_o @ A @ ( image_o_o2 @ F @ top_top_set_o ) ) ) ).
% image_vimage_eq
thf(fact_977_vimage__singleton__eq,axiom,
! [A2: $o,F: $o > $o,B: $o] :
( ( member_o @ A2 @ ( vimage_o_o @ F @ ( insert_o @ B @ bot_bot_set_o ) ) )
= ( ( F @ A2 )
= B ) ) ).
% vimage_singleton_eq
thf(fact_978_vimage__singleton__eq,axiom,
! [A2: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > $o,B: $o] :
( ( member408289922725080238_a_nat @ A2 @ ( vimage6136348398256869390_nat_o @ F @ ( insert_o @ B @ bot_bot_set_o ) ) )
= ( ( F @ A2 )
= B ) ) ).
% vimage_singleton_eq
thf(fact_979_vimage__singleton__eq,axiom,
! [A2: sum_sum_a_nat,F: sum_sum_a_nat > $o,B: $o] :
( ( member_Sum_sum_a_nat @ A2 @ ( vimage3838435602741665556_nat_o @ F @ ( insert_o @ B @ bot_bot_set_o ) ) )
= ( ( F @ A2 )
= B ) ) ).
% vimage_singleton_eq
thf(fact_980_surj__image__vimage__eq,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ( image_5081948215111134021_a_nat @ F @ top_to6433055325616222389_a_nat )
= top_to6433055325616222389_a_nat )
=> ( ( image_5081948215111134021_a_nat @ F @ ( vimage7538740559230589179_a_nat @ F @ A ) )
= A ) ) ).
% surj_image_vimage_eq
thf(fact_981_surj__image__vimage__eq,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ( image_7142520692256960453_a_nat @ F @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat )
=> ( ( image_7142520692256960453_a_nat @ F @ ( vimage6545432446589551483_a_nat @ F @ A ) )
= A ) ) ).
% surj_image_vimage_eq
thf(fact_982_surj__image__vimage__eq,axiom,
! [F: sum_sum_a_nat > $o,A: set_o] :
( ( ( image_6095136190293192542_nat_o @ F @ top_to795618464972521135_a_nat )
= top_top_set_o )
=> ( ( image_6095136190293192542_nat_o @ F @ ( vimage3838435602741665556_nat_o @ F @ A ) )
= A ) ) ).
% surj_image_vimage_eq
thf(fact_983_surj__image__vimage__eq,axiom,
! [F: $o > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ( image_4139480514073730540_a_nat @ F @ top_top_set_o )
= top_to795618464972521135_a_nat )
=> ( ( image_4139480514073730540_a_nat @ F @ ( vimage1882779926522203554_a_nat @ F @ A ) )
= A ) ) ).
% surj_image_vimage_eq
thf(fact_984_surj__image__vimage__eq,axiom,
! [F: $o > $o,A: set_o] :
( ( ( image_o_o2 @ F @ top_top_set_o )
= top_top_set_o )
=> ( ( image_o_o2 @ F @ ( vimage_o_o @ F @ A ) )
= A ) ) ).
% surj_image_vimage_eq
thf(fact_985_image__vimage__subset,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ ( vimage7538740559230589179_a_nat @ F @ A ) ) @ A ) ).
% image_vimage_subset
thf(fact_986_image__vimage__subset,axiom,
! [F: $o > $o,A: set_o] : ( ord_less_eq_set_o @ ( image_o_o2 @ F @ ( vimage_o_o @ F @ A ) ) @ A ) ).
% image_vimage_subset
thf(fact_987_image__subset__iff__subset__vimage,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ A ) @ B2 )
= ( ord_le1147066620699065093_a_nat @ A @ ( vimage7538740559230589179_a_nat @ F @ B2 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_988_image__subset__iff__subset__vimage,axiom,
! [F: $o > $o,A: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ ( image_o_o2 @ F @ A ) @ B2 )
= ( ord_less_eq_set_o @ A @ ( vimage_o_o @ F @ B2 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_989_vimage__Collect,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,F: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat,Q: ( nat > sum_sum_a_nat ) > $o] :
( ! [X3: nat > sum_sum_a_nat] :
( ( P @ ( F @ X3 ) )
= ( Q @ X3 ) )
=> ( ( vimage8281748387122106427_a_nat @ F @ ( collec5629555741568564177_a_nat @ P ) )
= ( collec5629555741568564177_a_nat @ Q ) ) ) ).
% vimage_Collect
thf(fact_990_vimageI2,axiom,
! [F: $o > $o,A2: $o,A: set_o] :
( ( member_o @ ( F @ A2 ) @ A )
=> ( member_o @ A2 @ ( vimage_o_o @ F @ A ) ) ) ).
% vimageI2
thf(fact_991_vimageI2,axiom,
! [F: list_Sum_sum_a_nat > $o,A2: list_Sum_sum_a_nat,A: set_o] :
( ( member_o @ ( F @ A2 ) @ A )
=> ( member408289922725080238_a_nat @ A2 @ ( vimage6136348398256869390_nat_o @ F @ A ) ) ) ).
% vimageI2
thf(fact_992_vimageI2,axiom,
! [F: sum_sum_a_nat > $o,A2: sum_sum_a_nat,A: set_o] :
( ( member_o @ ( F @ A2 ) @ A )
=> ( member_Sum_sum_a_nat @ A2 @ ( vimage3838435602741665556_nat_o @ F @ A ) ) ) ).
% vimageI2
thf(fact_993_vimageI2,axiom,
! [F: $o > list_Sum_sum_a_nat,A2: $o,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ ( F @ A2 ) @ A )
=> ( member_o @ A2 @ ( vimage2884537194618305448_a_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_994_vimageI2,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ ( F @ A2 ) @ A )
=> ( member408289922725080238_a_nat @ A2 @ ( vimage7538740559230589179_a_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_995_vimageI2,axiom,
! [F: sum_sum_a_nat > list_Sum_sum_a_nat,A2: sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ ( F @ A2 ) @ A )
=> ( member_Sum_sum_a_nat @ A2 @ ( vimage8052486765430747393_a_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_996_vimageI2,axiom,
! [F: $o > sum_sum_a_nat,A2: $o,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ ( F @ A2 ) @ A )
=> ( member_o @ A2 @ ( vimage1882779926522203554_a_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_997_vimageI2,axiom,
! [F: list_Sum_sum_a_nat > sum_sum_a_nat,A2: list_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ ( F @ A2 ) @ A )
=> ( member408289922725080238_a_nat @ A2 @ ( vimage7679724146056310901_a_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_998_vimageI2,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A2: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ ( F @ A2 ) @ A )
=> ( member_Sum_sum_a_nat @ A2 @ ( vimage6545432446589551483_a_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_999_vimageE,axiom,
! [A2: $o,F: $o > $o,B2: set_o] :
( ( member_o @ A2 @ ( vimage_o_o @ F @ B2 ) )
=> ( member_o @ ( F @ A2 ) @ B2 ) ) ).
% vimageE
thf(fact_1000_vimageE,axiom,
! [A2: $o,F: $o > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member_o @ A2 @ ( vimage2884537194618305448_a_nat @ F @ B2 ) )
=> ( member408289922725080238_a_nat @ ( F @ A2 ) @ B2 ) ) ).
% vimageE
thf(fact_1001_vimageE,axiom,
! [A2: $o,F: $o > sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member_o @ A2 @ ( vimage1882779926522203554_a_nat @ F @ B2 ) )
=> ( member_Sum_sum_a_nat @ ( F @ A2 ) @ B2 ) ) ).
% vimageE
thf(fact_1002_vimageE,axiom,
! [A2: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > $o,B2: set_o] :
( ( member408289922725080238_a_nat @ A2 @ ( vimage6136348398256869390_nat_o @ F @ B2 ) )
=> ( member_o @ ( F @ A2 ) @ B2 ) ) ).
% vimageE
thf(fact_1003_vimageE,axiom,
! [A2: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A2 @ ( vimage7538740559230589179_a_nat @ F @ B2 ) )
=> ( member408289922725080238_a_nat @ ( F @ A2 ) @ B2 ) ) ).
% vimageE
thf(fact_1004_vimageE,axiom,
! [A2: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ A2 @ ( vimage7679724146056310901_a_nat @ F @ B2 ) )
=> ( member_Sum_sum_a_nat @ ( F @ A2 ) @ B2 ) ) ).
% vimageE
thf(fact_1005_vimageE,axiom,
! [A2: sum_sum_a_nat,F: sum_sum_a_nat > $o,B2: set_o] :
( ( member_Sum_sum_a_nat @ A2 @ ( vimage3838435602741665556_nat_o @ F @ B2 ) )
=> ( member_o @ ( F @ A2 ) @ B2 ) ) ).
% vimageE
thf(fact_1006_vimageE,axiom,
! [A2: sum_sum_a_nat,F: sum_sum_a_nat > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member_Sum_sum_a_nat @ A2 @ ( vimage8052486765430747393_a_nat @ F @ B2 ) )
=> ( member408289922725080238_a_nat @ ( F @ A2 ) @ B2 ) ) ).
% vimageE
thf(fact_1007_vimageE,axiom,
! [A2: sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A2 @ ( vimage6545432446589551483_a_nat @ F @ B2 ) )
=> ( member_Sum_sum_a_nat @ ( F @ A2 ) @ B2 ) ) ).
% vimageE
thf(fact_1008_vimageD,axiom,
! [A2: $o,F: $o > $o,A: set_o] :
( ( member_o @ A2 @ ( vimage_o_o @ F @ A ) )
=> ( member_o @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_1009_vimageD,axiom,
! [A2: $o,F: $o > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member_o @ A2 @ ( vimage2884537194618305448_a_nat @ F @ A ) )
=> ( member408289922725080238_a_nat @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_1010_vimageD,axiom,
! [A2: $o,F: $o > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_o @ A2 @ ( vimage1882779926522203554_a_nat @ F @ A ) )
=> ( member_Sum_sum_a_nat @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_1011_vimageD,axiom,
! [A2: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > $o,A: set_o] :
( ( member408289922725080238_a_nat @ A2 @ ( vimage6136348398256869390_nat_o @ F @ A ) )
=> ( member_o @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_1012_vimageD,axiom,
! [A2: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A2 @ ( vimage7538740559230589179_a_nat @ F @ A ) )
=> ( member408289922725080238_a_nat @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_1013_vimageD,axiom,
! [A2: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ A2 @ ( vimage7679724146056310901_a_nat @ F @ A ) )
=> ( member_Sum_sum_a_nat @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_1014_vimageD,axiom,
! [A2: sum_sum_a_nat,F: sum_sum_a_nat > $o,A: set_o] :
( ( member_Sum_sum_a_nat @ A2 @ ( vimage3838435602741665556_nat_o @ F @ A ) )
=> ( member_o @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_1015_vimageD,axiom,
! [A2: sum_sum_a_nat,F: sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member_Sum_sum_a_nat @ A2 @ ( vimage8052486765430747393_a_nat @ F @ A ) )
=> ( member408289922725080238_a_nat @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_1016_vimageD,axiom,
! [A2: sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A2 @ ( vimage6545432446589551483_a_nat @ F @ A ) )
=> ( member_Sum_sum_a_nat @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_1017_vimage__def,axiom,
( vimage9018520469004791789_nat_o
= ( ^ [F3: ( nat > sum_sum_a_nat ) > $o,B6: set_o] :
( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] : ( member_o @ ( F3 @ X ) @ B6 ) ) ) ) ).
% vimage_def
thf(fact_1018_vimage__def,axiom,
( vimage8538096104420067162_a_nat
= ( ^ [F3: ( nat > sum_sum_a_nat ) > list_Sum_sum_a_nat,B6: set_li6526943997496501093_a_nat] :
( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] : ( member408289922725080238_a_nat @ ( F3 @ X ) @ B6 ) ) ) ) ).
% vimage_def
thf(fact_1019_vimage__def,axiom,
( vimage5925955513428286036_a_nat
= ( ^ [F3: ( nat > sum_sum_a_nat ) > sum_sum_a_nat,B6: set_Sum_sum_a_nat] :
( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] : ( member_Sum_sum_a_nat @ ( F3 @ X ) @ B6 ) ) ) ) ).
% vimage_def
thf(fact_1020_surj__vimage__empty,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ( image_5081948215111134021_a_nat @ F @ top_to6433055325616222389_a_nat )
= top_to6433055325616222389_a_nat )
=> ( ( ( vimage7538740559230589179_a_nat @ F @ A )
= bot_bo1033123847703346641_a_nat )
= ( A = bot_bo1033123847703346641_a_nat ) ) ) ).
% surj_vimage_empty
thf(fact_1021_surj__vimage__empty,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ( image_7142520692256960453_a_nat @ F @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat )
=> ( ( ( vimage6545432446589551483_a_nat @ F @ A )
= bot_bo3438331934148233675_a_nat )
= ( A = bot_bo3438331934148233675_a_nat ) ) ) ).
% surj_vimage_empty
thf(fact_1022_surj__vimage__empty,axiom,
! [F: sum_sum_a_nat > $o,A: set_o] :
( ( ( image_6095136190293192542_nat_o @ F @ top_to795618464972521135_a_nat )
= top_top_set_o )
=> ( ( ( vimage3838435602741665556_nat_o @ F @ A )
= bot_bo3438331934148233675_a_nat )
= ( A = bot_bot_set_o ) ) ) ).
% surj_vimage_empty
thf(fact_1023_surj__vimage__empty,axiom,
! [F: $o > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ( image_4139480514073730540_a_nat @ F @ top_top_set_o )
= top_to795618464972521135_a_nat )
=> ( ( ( vimage1882779926522203554_a_nat @ F @ A )
= bot_bot_set_o )
= ( A = bot_bo3438331934148233675_a_nat ) ) ) ).
% surj_vimage_empty
thf(fact_1024_surj__vimage__empty,axiom,
! [F: $o > $o,A: set_o] :
( ( ( image_o_o2 @ F @ top_top_set_o )
= top_top_set_o )
=> ( ( ( vimage_o_o @ F @ A )
= bot_bot_set_o )
= ( A = bot_bot_set_o ) ) ) ).
% surj_vimage_empty
thf(fact_1025_vimage__subsetD,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ( image_5081948215111134021_a_nat @ F @ top_to6433055325616222389_a_nat )
= top_to6433055325616222389_a_nat )
=> ( ( ord_le1147066620699065093_a_nat @ ( vimage7538740559230589179_a_nat @ F @ B2 ) @ A )
=> ( ord_le1147066620699065093_a_nat @ B2 @ ( image_5081948215111134021_a_nat @ F @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_1026_vimage__subsetD,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ( image_7142520692256960453_a_nat @ F @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat )
=> ( ( ord_le1325389633284124927_a_nat @ ( vimage6545432446589551483_a_nat @ F @ B2 ) @ A )
=> ( ord_le1325389633284124927_a_nat @ B2 @ ( image_7142520692256960453_a_nat @ F @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_1027_vimage__subsetD,axiom,
! [F: sum_sum_a_nat > $o,B2: set_o,A: set_Sum_sum_a_nat] :
( ( ( image_6095136190293192542_nat_o @ F @ top_to795618464972521135_a_nat )
= top_top_set_o )
=> ( ( ord_le1325389633284124927_a_nat @ ( vimage3838435602741665556_nat_o @ F @ B2 ) @ A )
=> ( ord_less_eq_set_o @ B2 @ ( image_6095136190293192542_nat_o @ F @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_1028_vimage__subsetD,axiom,
! [F: $o > sum_sum_a_nat,B2: set_Sum_sum_a_nat,A: set_o] :
( ( ( image_4139480514073730540_a_nat @ F @ top_top_set_o )
= top_to795618464972521135_a_nat )
=> ( ( ord_less_eq_set_o @ ( vimage1882779926522203554_a_nat @ F @ B2 ) @ A )
=> ( ord_le1325389633284124927_a_nat @ B2 @ ( image_4139480514073730540_a_nat @ F @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_1029_vimage__subsetD,axiom,
! [F: $o > $o,B2: set_o,A: set_o] :
( ( ( image_o_o2 @ F @ top_top_set_o )
= top_top_set_o )
=> ( ( ord_less_eq_set_o @ ( vimage_o_o @ F @ B2 ) @ A )
=> ( ord_less_eq_set_o @ B2 @ ( image_o_o2 @ F @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_1030_Pair__vimage__Sigma,axiom,
! [X2: set_a,A: set_set_a,F: set_a > set_Pr1465208939327205952_a_nat] :
( ( ( member_set_a @ X2 @ A )
=> ( ( vimage350067387867788510_a_nat @ ( produc1626021218680649873_a_nat @ X2 ) @ ( produc5217640286535863200_a_nat @ A @ F ) )
= ( F @ X2 ) ) )
& ( ~ ( member_set_a @ X2 @ A )
=> ( ( vimage350067387867788510_a_nat @ ( produc1626021218680649873_a_nat @ X2 ) @ ( produc5217640286535863200_a_nat @ A @ F ) )
= bot_bo1546958992129757396_a_nat ) ) ) ).
% Pair_vimage_Sigma
thf(fact_1031_Pair__vimage__Sigma,axiom,
! [X2: nat,A: set_nat,F: nat > set_se4330304633200676677_a_nat] :
( ( ( member_nat @ X2 @ A )
=> ( ( vimage9041054357004555124_a_nat @ ( produc4622690550020908860_a_nat @ X2 ) @ ( produc3542699778996723019_a_nat @ A @ F ) )
= ( F @ X2 ) ) )
& ( ~ ( member_nat @ X2 @ A )
=> ( ( vimage9041054357004555124_a_nat @ ( produc4622690550020908860_a_nat @ X2 ) @ ( produc3542699778996723019_a_nat @ A @ F ) )
= bot_bo3237059034911209905_a_nat ) ) ) ).
% Pair_vimage_Sigma
thf(fact_1032_Pair__vimage__Sigma,axiom,
! [X2: $o,A: set_o,F: $o > set_o] :
( ( ( member_o @ X2 @ A )
=> ( ( vimage8945963521958007626od_o_o @ ( product_Pair_o_o @ X2 ) @ ( product_Sigma_o_o @ A @ F ) )
= ( F @ X2 ) ) )
& ( ~ ( member_o @ X2 @ A )
=> ( ( vimage8945963521958007626od_o_o @ ( product_Pair_o_o @ X2 ) @ ( product_Sigma_o_o @ A @ F ) )
= bot_bot_set_o ) ) ) ).
% Pair_vimage_Sigma
thf(fact_1033_Pair__vimage__Sigma,axiom,
! [X2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_o] :
( ( ( member408289922725080238_a_nat @ X2 @ A )
=> ( ( vimage3541326180493997085_nat_o @ ( produc1406723247873626090_nat_o @ X2 ) @ ( produc8891875240947076123_nat_o @ A @ F ) )
= ( F @ X2 ) ) )
& ( ~ ( member408289922725080238_a_nat @ X2 @ A )
=> ( ( vimage3541326180493997085_nat_o @ ( produc1406723247873626090_nat_o @ X2 ) @ ( produc8891875240947076123_nat_o @ A @ F ) )
= bot_bot_set_o ) ) ) ).
% Pair_vimage_Sigma
thf(fact_1034_Pair__vimage__Sigma,axiom,
! [X2: sum_sum_a_nat,A: set_Sum_sum_a_nat,F: sum_sum_a_nat > set_o] :
( ( ( member_Sum_sum_a_nat @ X2 @ A )
=> ( ( vimage1019114229048754979_nat_o @ ( produc8349727785981246704_nat_o @ X2 ) @ ( produc5036831428884481441_nat_o @ A @ F ) )
= ( F @ X2 ) ) )
& ( ~ ( member_Sum_sum_a_nat @ X2 @ A )
=> ( ( vimage1019114229048754979_nat_o @ ( produc8349727785981246704_nat_o @ X2 ) @ ( produc5036831428884481441_nat_o @ A @ F ) )
= bot_bot_set_o ) ) ) ).
% Pair_vimage_Sigma
thf(fact_1035_is__singletonI_H,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( A != bot_bo1033123847703346641_a_nat )
=> ( ! [X3: list_Sum_sum_a_nat,Y4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ( ( member408289922725080238_a_nat @ Y4 @ A )
=> ( X3 = Y4 ) ) )
=> ( is_sin2231188923920309881_a_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_1036_is__singletonI_H,axiom,
! [A: set_Sum_sum_a_nat] :
( ( A != bot_bo3438331934148233675_a_nat )
=> ( ! [X3: sum_sum_a_nat,Y4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ A )
=> ( ( member_Sum_sum_a_nat @ Y4 @ A )
=> ( X3 = Y4 ) ) )
=> ( is_sin5176708635568246003_a_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_1037_is__singletonI_H,axiom,
! [A: set_o] :
( ( A != bot_bot_set_o )
=> ( ! [X3: $o,Y4: $o] :
( ( member_o @ X3 @ A )
=> ( ( member_o @ Y4 @ A )
=> ( X3 = Y4 ) ) )
=> ( is_singleton_o @ A ) ) ) ).
% is_singletonI'
thf(fact_1038_is__singletonE,axiom,
! [A: set_o] :
( ( is_singleton_o @ A )
=> ~ ! [X3: $o] :
( A
!= ( insert_o @ X3 @ bot_bot_set_o ) ) ) ).
% is_singletonE
thf(fact_1039_is__singleton__def,axiom,
( is_singleton_o
= ( ^ [A6: set_o] :
? [X: $o] :
( A6
= ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).
% is_singleton_def
thf(fact_1040_image__Fpow__mono,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ A ) @ B2 )
=> ( ord_le8138476598237931237_a_nat @ ( image_3472601871771700037_a_nat @ ( image_5081948215111134021_a_nat @ F ) @ ( finite3225531020566593003_a_nat @ A ) ) @ ( finite3225531020566593003_a_nat @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_1041_image__Fpow__mono,axiom,
! [F: $o > $o,A: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ ( image_o_o2 @ F @ A ) @ B2 )
=> ( ord_le4374716579403074808_set_o @ ( image_set_o_set_o @ ( image_o_o2 @ F ) @ ( finite_Fpow_o @ A ) ) @ ( finite_Fpow_o @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_1042_is__singleton__the__elem,axiom,
( is_singleton_o
= ( ^ [A6: set_o] :
( A6
= ( insert_o @ ( the_elem_o @ A6 ) @ bot_bot_set_o ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1043_the__elem__eq,axiom,
! [X2: $o] :
( ( the_elem_o @ ( insert_o @ X2 @ bot_bot_set_o ) )
= X2 ) ).
% the_elem_eq
thf(fact_1044_the__elem__image__unique,axiom,
! [A: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,X2: list_Sum_sum_a_nat] :
( ( A != bot_bo1033123847703346641_a_nat )
=> ( ! [Y4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Y4 @ A )
=> ( ( F @ Y4 )
= ( F @ X2 ) ) )
=> ( ( the_el2583442515771010938_a_nat @ ( image_5081948215111134021_a_nat @ F @ A ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_1045_the__elem__image__unique,axiom,
! [A: set_o,F: $o > $o,X2: $o] :
( ( A != bot_bot_set_o )
=> ( ! [Y4: $o] :
( ( member_o @ Y4 @ A )
=> ( ( F @ Y4 )
= ( F @ X2 ) ) )
=> ( ( the_elem_o @ ( image_o_o2 @ F @ A ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_1046_type__definition_ORep__range,axiom,
! [Rep: list_Sum_sum_a_nat > list_Sum_sum_a_nat,Abs: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( type_d3408157338959482604_a_nat @ Rep @ Abs @ A )
=> ( ( image_5081948215111134021_a_nat @ Rep @ top_to6433055325616222389_a_nat )
= A ) ) ).
% type_definition.Rep_range
thf(fact_1047_type__definition_ORep__range,axiom,
! [Rep: $o > $o,Abs: $o > $o,A: set_o] :
( ( type_definition_o_o @ Rep @ Abs @ A )
=> ( ( image_o_o2 @ Rep @ top_top_set_o )
= A ) ) ).
% type_definition.Rep_range
thf(fact_1048_type__definition_OAbs__image,axiom,
! [Rep: list_Sum_sum_a_nat > list_Sum_sum_a_nat,Abs: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( type_d3408157338959482604_a_nat @ Rep @ Abs @ A )
=> ( ( image_5081948215111134021_a_nat @ Abs @ A )
= top_to6433055325616222389_a_nat ) ) ).
% type_definition.Abs_image
thf(fact_1049_type__definition_OAbs__image,axiom,
! [Rep: $o > $o,Abs: $o > $o,A: set_o] :
( ( type_definition_o_o @ Rep @ Abs @ A )
=> ( ( image_o_o2 @ Abs @ A )
= top_top_set_o ) ) ).
% type_definition.Abs_image
thf(fact_1050_type__definition_Ouniv,axiom,
! [Rep: list_Sum_sum_a_nat > list_Sum_sum_a_nat,Abs: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( type_d3408157338959482604_a_nat @ Rep @ Abs @ A )
=> ( top_to6433055325616222389_a_nat
= ( image_5081948215111134021_a_nat @ Abs @ A ) ) ) ).
% type_definition.univ
thf(fact_1051_type__definition_Ouniv,axiom,
! [Rep: $o > $o,Abs: $o > $o,A: set_o] :
( ( type_definition_o_o @ Rep @ Abs @ A )
=> ( top_top_set_o
= ( image_o_o2 @ Abs @ A ) ) ) ).
% type_definition.univ
thf(fact_1052_insert__Times__insert,axiom,
! [A2: $o,A: set_o,B: $o,B2: set_o] :
( ( product_Sigma_o_o @ ( insert_o @ A2 @ A )
@ ^ [Uu2: $o] : ( insert_o @ B @ B2 ) )
= ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A2 @ B )
@ ( sup_su5769328420594410459od_o_o
@ ( product_Sigma_o_o @ A
@ ^ [Uu2: $o] : ( insert_o @ B @ B2 ) )
@ ( product_Sigma_o_o @ ( insert_o @ A2 @ A )
@ ^ [Uu2: $o] : B2 ) ) ) ) ).
% insert_Times_insert
thf(fact_1053_insert__Times__insert,axiom,
! [A2: set_a,A: set_set_a,B: produc5986856040925105290_a_nat,B2: set_Pr1465208939327205952_a_nat] :
( ( produc5217640286535863200_a_nat @ ( insert_set_a @ A2 @ A )
@ ^ [Uu2: set_a] : ( insert9175641148397066938_a_nat @ B @ B2 ) )
= ( insert5623531327003279303_a_nat @ ( produc1626021218680649873_a_nat @ A2 @ B )
@ ( sup_su7876432514636195577_a_nat
@ ( produc5217640286535863200_a_nat @ A
@ ^ [Uu2: set_a] : ( insert9175641148397066938_a_nat @ B @ B2 ) )
@ ( produc5217640286535863200_a_nat @ ( insert_set_a @ A2 @ A )
@ ^ [Uu2: set_a] : B2 ) ) ) ) ).
% insert_Times_insert
thf(fact_1054_insert__Times__insert,axiom,
! [A2: nat,A: set_nat,B: set_li6526943997496501093_a_nat,B2: set_se4330304633200676677_a_nat] :
( ( produc3542699778996723019_a_nat @ ( insert_nat @ A2 @ A )
@ ^ [Uu2: nat] : ( insert8442372787808178485_a_nat @ B @ B2 ) )
= ( insert9175641148397066938_a_nat @ ( produc4622690550020908860_a_nat @ A2 @ B )
@ ( sup_su9161890791271854956_a_nat
@ ( produc3542699778996723019_a_nat @ A
@ ^ [Uu2: nat] : ( insert8442372787808178485_a_nat @ B @ B2 ) )
@ ( produc3542699778996723019_a_nat @ ( insert_nat @ A2 @ A )
@ ^ [Uu2: nat] : B2 ) ) ) ) ).
% insert_Times_insert
thf(fact_1055_refl__on__singleton,axiom,
! [X2: $o] : ( refl_on_o @ ( insert_o @ X2 @ bot_bot_set_o ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X2 @ X2 ) @ bot_bo7073875226086086771od_o_o ) ) ).
% refl_on_singleton
thf(fact_1056_Un__iff,axiom,
! [C3: $o,A: set_o,B2: set_o] :
( ( member_o @ C3 @ ( sup_sup_set_o @ A @ B2 ) )
= ( ( member_o @ C3 @ A )
| ( member_o @ C3 @ B2 ) ) ) ).
% Un_iff
thf(fact_1057_Un__iff,axiom,
! [C3: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C3 @ ( sup_su4083067149120280889_a_nat @ A @ B2 ) )
= ( ( member408289922725080238_a_nat @ C3 @ A )
| ( member408289922725080238_a_nat @ C3 @ B2 ) ) ) ).
% Un_iff
thf(fact_1058_Un__iff,axiom,
! [C3: sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C3 @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) )
= ( ( member_Sum_sum_a_nat @ C3 @ A )
| ( member_Sum_sum_a_nat @ C3 @ B2 ) ) ) ).
% Un_iff
thf(fact_1059_UnCI,axiom,
! [C3: $o,B2: set_o,A: set_o] :
( ( ~ ( member_o @ C3 @ B2 )
=> ( member_o @ C3 @ A ) )
=> ( member_o @ C3 @ ( sup_sup_set_o @ A @ B2 ) ) ) ).
% UnCI
thf(fact_1060_UnCI,axiom,
! [C3: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ~ ( member408289922725080238_a_nat @ C3 @ B2 )
=> ( member408289922725080238_a_nat @ C3 @ A ) )
=> ( member408289922725080238_a_nat @ C3 @ ( sup_su4083067149120280889_a_nat @ A @ B2 ) ) ) ).
% UnCI
thf(fact_1061_UnCI,axiom,
! [C3: sum_sum_a_nat,B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ~ ( member_Sum_sum_a_nat @ C3 @ B2 )
=> ( member_Sum_sum_a_nat @ C3 @ A ) )
=> ( member_Sum_sum_a_nat @ C3 @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) ) ) ).
% UnCI
thf(fact_1062_boolean__algebra_Odisj__one__right,axiom,
! [X2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X2 @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat ) ).
% boolean_algebra.disj_one_right
thf(fact_1063_boolean__algebra_Odisj__one__right,axiom,
! [X2: set_o] :
( ( sup_sup_set_o @ X2 @ top_top_set_o )
= top_top_set_o ) ).
% boolean_algebra.disj_one_right
thf(fact_1064_boolean__algebra_Odisj__one__left,axiom,
! [X2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ top_to795618464972521135_a_nat @ X2 )
= top_to795618464972521135_a_nat ) ).
% boolean_algebra.disj_one_left
thf(fact_1065_boolean__algebra_Odisj__one__left,axiom,
! [X2: set_o] :
( ( sup_sup_set_o @ top_top_set_o @ X2 )
= top_top_set_o ) ).
% boolean_algebra.disj_one_left
thf(fact_1066_sup__top__right,axiom,
! [X2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X2 @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat ) ).
% sup_top_right
thf(fact_1067_sup__top__right,axiom,
! [X2: set_o] :
( ( sup_sup_set_o @ X2 @ top_top_set_o )
= top_top_set_o ) ).
% sup_top_right
thf(fact_1068_sup__top__left,axiom,
! [X2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ top_to795618464972521135_a_nat @ X2 )
= top_to795618464972521135_a_nat ) ).
% sup_top_left
thf(fact_1069_sup__top__left,axiom,
! [X2: set_o] :
( ( sup_sup_set_o @ top_top_set_o @ X2 )
= top_top_set_o ) ).
% sup_top_left
thf(fact_1070_Un__empty,axiom,
! [A: set_o,B2: set_o] :
( ( ( sup_sup_set_o @ A @ B2 )
= bot_bot_set_o )
= ( ( A = bot_bot_set_o )
& ( B2 = bot_bot_set_o ) ) ) ).
% Un_empty
thf(fact_1071_Un__insert__right,axiom,
! [A: set_o,A2: $o,B2: set_o] :
( ( sup_sup_set_o @ A @ ( insert_o @ A2 @ B2 ) )
= ( insert_o @ A2 @ ( sup_sup_set_o @ A @ B2 ) ) ) ).
% Un_insert_right
thf(fact_1072_Un__insert__left,axiom,
! [A2: $o,B2: set_o,C: set_o] :
( ( sup_sup_set_o @ ( insert_o @ A2 @ B2 ) @ C )
= ( insert_o @ A2 @ ( sup_sup_set_o @ B2 @ C ) ) ) ).
% Un_insert_left
thf(fact_1073_if__image__distrib,axiom,
! [P: list_Sum_sum_a_nat > $o,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,S: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( if_lis4685338526944683083_a_nat @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
@ S )
= ( sup_su4083067149120280889_a_nat @ ( image_5081948215111134021_a_nat @ F @ ( inf_in3249246906714053971_a_nat @ S @ ( collec7555443234367654128_a_nat @ P ) ) )
@ ( image_5081948215111134021_a_nat @ G
@ ( inf_in3249246906714053971_a_nat @ S
@ ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
~ ( P @ X ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1074_if__image__distrib,axiom,
! [P: $o > $o,F: $o > $o,G: $o > $o,S: set_o] :
( ( image_o_o2
@ ^ [X: $o] :
( ( ( P @ X )
=> ( F @ X ) )
& ( ~ ( P @ X )
=> ( G @ X ) ) )
@ S )
= ( sup_sup_set_o @ ( image_o_o2 @ F @ ( inf_inf_set_o @ S @ ( collect_o @ P ) ) )
@ ( image_o_o2 @ G
@ ( inf_inf_set_o @ S
@ ( collect_o
@ ^ [X: $o] :
~ ( P @ X ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1075_refl__on__empty,axiom,
refl_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).
% refl_on_empty
thf(fact_1076_act__edom_Osimps_I5_J,axiom,
! [Phi: fo_fmla_a_b,Psi: fo_fmla_a_b,I: product_prod_b_nat > set_list_a] :
( ( act_edom_a_b @ ( fo_Conj_a_b @ Phi @ Psi ) @ I )
= ( sup_sup_set_a @ ( act_edom_a_b @ Phi @ I ) @ ( act_edom_a_b @ Psi @ I ) ) ) ).
% act_edom.simps(5)
thf(fact_1077_UnI2,axiom,
! [C3: $o,B2: set_o,A: set_o] :
( ( member_o @ C3 @ B2 )
=> ( member_o @ C3 @ ( sup_sup_set_o @ A @ B2 ) ) ) ).
% UnI2
thf(fact_1078_UnI2,axiom,
! [C3: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C3 @ B2 )
=> ( member408289922725080238_a_nat @ C3 @ ( sup_su4083067149120280889_a_nat @ A @ B2 ) ) ) ).
% UnI2
thf(fact_1079_UnI2,axiom,
! [C3: sum_sum_a_nat,B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C3 @ B2 )
=> ( member_Sum_sum_a_nat @ C3 @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) ) ) ).
% UnI2
thf(fact_1080_UnI1,axiom,
! [C3: $o,A: set_o,B2: set_o] :
( ( member_o @ C3 @ A )
=> ( member_o @ C3 @ ( sup_sup_set_o @ A @ B2 ) ) ) ).
% UnI1
thf(fact_1081_UnI1,axiom,
! [C3: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C3 @ A )
=> ( member408289922725080238_a_nat @ C3 @ ( sup_su4083067149120280889_a_nat @ A @ B2 ) ) ) ).
% UnI1
thf(fact_1082_UnI1,axiom,
! [C3: sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C3 @ A )
=> ( member_Sum_sum_a_nat @ C3 @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) ) ) ).
% UnI1
thf(fact_1083_UnE,axiom,
! [C3: $o,A: set_o,B2: set_o] :
( ( member_o @ C3 @ ( sup_sup_set_o @ A @ B2 ) )
=> ( ~ ( member_o @ C3 @ A )
=> ( member_o @ C3 @ B2 ) ) ) ).
% UnE
thf(fact_1084_UnE,axiom,
! [C3: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C3 @ ( sup_su4083067149120280889_a_nat @ A @ B2 ) )
=> ( ~ ( member408289922725080238_a_nat @ C3 @ A )
=> ( member408289922725080238_a_nat @ C3 @ B2 ) ) ) ).
% UnE
thf(fact_1085_UnE,axiom,
! [C3: sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C3 @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) )
=> ( ~ ( member_Sum_sum_a_nat @ C3 @ A )
=> ( member_Sum_sum_a_nat @ C3 @ B2 ) ) ) ).
% UnE
thf(fact_1086_Collect__disj__eq,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,Q: ( nat > sum_sum_a_nat ) > $o] :
( ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_su3329769938372955546_a_nat @ ( collec5629555741568564177_a_nat @ P ) @ ( collec5629555741568564177_a_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_1087_Un__def,axiom,
( sup_sup_set_o
= ( ^ [A6: set_o,B6: set_o] :
( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A6 )
| ( member_o @ X @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_1088_Un__def,axiom,
( sup_su4083067149120280889_a_nat
= ( ^ [A6: set_li6526943997496501093_a_nat,B6: set_li6526943997496501093_a_nat] :
( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A6 )
| ( member408289922725080238_a_nat @ X @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_1089_Un__def,axiom,
( sup_su6804446743777130803_a_nat
= ( ^ [A6: set_Sum_sum_a_nat,B6: set_Sum_sum_a_nat] :
( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A6 )
| ( member_Sum_sum_a_nat @ X @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_1090_Un__def,axiom,
( sup_su3329769938372955546_a_nat
= ( ^ [A6: set_na3699693778330250182_a_nat,B6: set_na3699693778330250182_a_nat] :
( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ A6 )
| ( member8690443509505302927_a_nat @ X @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_1091_Un__UNIV__left,axiom,
! [B2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ top_to795618464972521135_a_nat @ B2 )
= top_to795618464972521135_a_nat ) ).
% Un_UNIV_left
thf(fact_1092_Un__UNIV__left,axiom,
! [B2: set_o] :
( ( sup_sup_set_o @ top_top_set_o @ B2 )
= top_top_set_o ) ).
% Un_UNIV_left
thf(fact_1093_Un__UNIV__right,axiom,
! [A: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat ) ).
% Un_UNIV_right
thf(fact_1094_Un__UNIV__right,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ A @ top_top_set_o )
= top_top_set_o ) ).
% Un_UNIV_right
thf(fact_1095_refl__onD,axiom,
! [A: set_o,R3: set_Product_prod_o_o,A2: $o] :
( ( refl_on_o @ A @ R3 )
=> ( ( member_o @ A2 @ A )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ A2 ) @ R3 ) ) ) ).
% refl_onD
thf(fact_1096_refl__onD,axiom,
! [A: set_li6526943997496501093_a_nat,R3: set_Pr4870381170404451655_a_nat,A2: list_Sum_sum_a_nat] :
( ( refl_o8238942462679651204_a_nat @ A @ R3 )
=> ( ( member408289922725080238_a_nat @ A2 @ A )
=> ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ A2 @ A2 ) @ R3 ) ) ) ).
% refl_onD
thf(fact_1097_refl__onD,axiom,
! [A: set_Sum_sum_a_nat,R3: set_Pr7343886759072863943_a_nat,A2: sum_sum_a_nat] :
( ( refl_o828724780208292734_a_nat @ A @ R3 )
=> ( ( member_Sum_sum_a_nat @ A2 @ A )
=> ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ A2 @ A2 ) @ R3 ) ) ) ).
% refl_onD
thf(fact_1098_refl__onD1,axiom,
! [A: set_o,R3: set_Product_prod_o_o,X2: $o,Y2: $o] :
( ( refl_on_o @ A @ R3 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y2 ) @ R3 )
=> ( member_o @ X2 @ A ) ) ) ).
% refl_onD1
thf(fact_1099_refl__onD1,axiom,
! [A: set_li6526943997496501093_a_nat,R3: set_Pr4870381170404451655_a_nat,X2: list_Sum_sum_a_nat,Y2: list_Sum_sum_a_nat] :
( ( refl_o8238942462679651204_a_nat @ A @ R3 )
=> ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ X2 @ Y2 ) @ R3 )
=> ( member408289922725080238_a_nat @ X2 @ A ) ) ) ).
% refl_onD1
thf(fact_1100_refl__onD1,axiom,
! [A: set_Sum_sum_a_nat,R3: set_Pr7343886759072863943_a_nat,X2: sum_sum_a_nat,Y2: sum_sum_a_nat] :
( ( refl_o828724780208292734_a_nat @ A @ R3 )
=> ( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X2 @ Y2 ) @ R3 )
=> ( member_Sum_sum_a_nat @ X2 @ A ) ) ) ).
% refl_onD1
thf(fact_1101_refl__onD2,axiom,
! [A: set_o,R3: set_Product_prod_o_o,X2: $o,Y2: $o] :
( ( refl_on_o @ A @ R3 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y2 ) @ R3 )
=> ( member_o @ Y2 @ A ) ) ) ).
% refl_onD2
thf(fact_1102_refl__onD2,axiom,
! [A: set_li6526943997496501093_a_nat,R3: set_Pr4870381170404451655_a_nat,X2: list_Sum_sum_a_nat,Y2: list_Sum_sum_a_nat] :
( ( refl_o8238942462679651204_a_nat @ A @ R3 )
=> ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ X2 @ Y2 ) @ R3 )
=> ( member408289922725080238_a_nat @ Y2 @ A ) ) ) ).
% refl_onD2
thf(fact_1103_refl__onD2,axiom,
! [A: set_Sum_sum_a_nat,R3: set_Pr7343886759072863943_a_nat,X2: sum_sum_a_nat,Y2: sum_sum_a_nat] :
( ( refl_o828724780208292734_a_nat @ A @ R3 )
=> ( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X2 @ Y2 ) @ R3 )
=> ( member_Sum_sum_a_nat @ Y2 @ A ) ) ) ).
% refl_onD2
thf(fact_1104_Un__empty__left,axiom,
! [B2: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_1105_Un__empty__right,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ A @ bot_bot_set_o )
= A ) ).
% Un_empty_right
thf(fact_1106_reflD,axiom,
! [R3: set_Pr7343886759072863943_a_nat,A2: sum_sum_a_nat] :
( ( refl_o828724780208292734_a_nat @ top_to795618464972521135_a_nat @ R3 )
=> ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ A2 @ A2 ) @ R3 ) ) ).
% reflD
thf(fact_1107_reflD,axiom,
! [R3: set_Product_prod_o_o,A2: $o] :
( ( refl_on_o @ top_top_set_o @ R3 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ A2 ) @ R3 ) ) ).
% reflD
thf(fact_1108_reflI,axiom,
! [R3: set_Pr7343886759072863943_a_nat] :
( ! [X3: sum_sum_a_nat] : ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X3 @ X3 ) @ R3 )
=> ( refl_o828724780208292734_a_nat @ top_to795618464972521135_a_nat @ R3 ) ) ).
% reflI
thf(fact_1109_reflI,axiom,
! [R3: set_Product_prod_o_o] :
( ! [X3: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ X3 ) @ R3 )
=> ( refl_on_o @ top_top_set_o @ R3 ) ) ).
% reflI
thf(fact_1110_singleton__Un__iff,axiom,
! [X2: $o,A: set_o,B2: set_o] :
( ( ( insert_o @ X2 @ bot_bot_set_o )
= ( sup_sup_set_o @ A @ B2 ) )
= ( ( ( A = bot_bot_set_o )
& ( B2
= ( insert_o @ X2 @ bot_bot_set_o ) ) )
| ( ( A
= ( insert_o @ X2 @ bot_bot_set_o ) )
& ( B2 = bot_bot_set_o ) )
| ( ( A
= ( insert_o @ X2 @ bot_bot_set_o ) )
& ( B2
= ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1111_Un__singleton__iff,axiom,
! [A: set_o,B2: set_o,X2: $o] :
( ( ( sup_sup_set_o @ A @ B2 )
= ( insert_o @ X2 @ bot_bot_set_o ) )
= ( ( ( A = bot_bot_set_o )
& ( B2
= ( insert_o @ X2 @ bot_bot_set_o ) ) )
| ( ( A
= ( insert_o @ X2 @ bot_bot_set_o ) )
& ( B2 = bot_bot_set_o ) )
| ( ( A
= ( insert_o @ X2 @ bot_bot_set_o ) )
& ( B2
= ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1112_insert__is__Un,axiom,
( insert_o
= ( ^ [A8: $o] : ( sup_sup_set_o @ ( insert_o @ A8 @ bot_bot_set_o ) ) ) ) ).
% insert_is_Un
thf(fact_1113_image__Un,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat @ F @ ( sup_su4083067149120280889_a_nat @ A @ B2 ) )
= ( sup_su4083067149120280889_a_nat @ ( image_5081948215111134021_a_nat @ F @ A ) @ ( image_5081948215111134021_a_nat @ F @ B2 ) ) ) ).
% image_Un
thf(fact_1114_image__Un,axiom,
! [F: $o > $o,A: set_o,B2: set_o] :
( ( image_o_o2 @ F @ ( sup_sup_set_o @ A @ B2 ) )
= ( sup_sup_set_o @ ( image_o_o2 @ F @ A ) @ ( image_o_o2 @ F @ B2 ) ) ) ).
% image_Un
thf(fact_1115_Set_Oinsert__def,axiom,
( insert_o
= ( ^ [A8: $o] :
( sup_sup_set_o
@ ( collect_o
@ ^ [X: $o] : ( X = A8 ) ) ) ) ) ).
% Set.insert_def
thf(fact_1116_Set_Oinsert__def,axiom,
( insert5265011953798106934_a_nat
= ( ^ [A8: nat > sum_sum_a_nat] :
( sup_su3329769938372955546_a_nat
@ ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] : ( X = A8 ) ) ) ) ) ).
% Set.insert_def
thf(fact_1117_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_Sum_sum_a_nat,X2: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat] :
( ( ( inf_in7084830621192376909_a_nat @ A2 @ X2 )
= bot_bo3438331934148233675_a_nat )
=> ( ( ( sup_su6804446743777130803_a_nat @ A2 @ X2 )
= top_to795618464972521135_a_nat )
=> ( ( ( inf_in7084830621192376909_a_nat @ A2 @ Y2 )
= bot_bo3438331934148233675_a_nat )
=> ( ( ( sup_su6804446743777130803_a_nat @ A2 @ Y2 )
= top_to795618464972521135_a_nat )
=> ( X2 = Y2 ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1118_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_o,X2: set_o,Y2: set_o] :
( ( ( inf_inf_set_o @ A2 @ X2 )
= bot_bot_set_o )
=> ( ( ( sup_sup_set_o @ A2 @ X2 )
= top_top_set_o )
=> ( ( ( inf_inf_set_o @ A2 @ Y2 )
= bot_bot_set_o )
=> ( ( ( sup_sup_set_o @ A2 @ Y2 )
= top_top_set_o )
=> ( X2 = Y2 ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1119_refl__onI,axiom,
! [R3: set_Product_prod_o_o,A: set_o] :
( ( ord_le192195778054999463od_o_o @ R3
@ ( product_Sigma_o_o @ A
@ ^ [Uu2: $o] : A ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ X3 ) @ R3 ) )
=> ( refl_on_o @ A @ R3 ) ) ) ).
% refl_onI
thf(fact_1120_refl__onI,axiom,
! [R3: set_Pr4870381170404451655_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ord_le4837030659158318311_a_nat @ R3
@ ( produc5624668685630548104_a_nat @ A
@ ^ [Uu2: list_Sum_sum_a_nat] : A ) )
=> ( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ X3 @ X3 ) @ R3 ) )
=> ( refl_o8238942462679651204_a_nat @ A @ R3 ) ) ) ).
% refl_onI
thf(fact_1121_refl__onI,axiom,
! [R3: set_Pr7343886759072863943_a_nat,A: set_Sum_sum_a_nat] :
( ( ord_le7501488065512009319_a_nat @ R3
@ ( produc3014407214216980488_a_nat @ A
@ ^ [Uu2: sum_sum_a_nat] : A ) )
=> ( ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ A )
=> ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X3 @ X3 ) @ R3 ) )
=> ( refl_o828724780208292734_a_nat @ A @ R3 ) ) ) ).
% refl_onI
thf(fact_1122_refl__on__domain,axiom,
! [A: set_o,R3: set_Product_prod_o_o,A2: $o,B: $o] :
( ( refl_on_o @ A @ R3 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R3 )
=> ( ( member_o @ A2 @ A )
& ( member_o @ B @ A ) ) ) ) ).
% refl_on_domain
thf(fact_1123_refl__on__domain,axiom,
! [A: set_li6526943997496501093_a_nat,R3: set_Pr4870381170404451655_a_nat,A2: list_Sum_sum_a_nat,B: list_Sum_sum_a_nat] :
( ( refl_o8238942462679651204_a_nat @ A @ R3 )
=> ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ A2 @ B ) @ R3 )
=> ( ( member408289922725080238_a_nat @ A2 @ A )
& ( member408289922725080238_a_nat @ B @ A ) ) ) ) ).
% refl_on_domain
thf(fact_1124_refl__on__domain,axiom,
! [A: set_Sum_sum_a_nat,R3: set_Pr7343886759072863943_a_nat,A2: sum_sum_a_nat,B: sum_sum_a_nat] :
( ( refl_o828724780208292734_a_nat @ A @ R3 )
=> ( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ A2 @ B ) @ R3 )
=> ( ( member_Sum_sum_a_nat @ A2 @ A )
& ( member_Sum_sum_a_nat @ B @ A ) ) ) ) ).
% refl_on_domain
thf(fact_1125_Field__insert,axiom,
! [A2: $o,B: $o,R3: set_Product_prod_o_o] :
( ( field_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R3 ) )
= ( sup_sup_set_o @ ( insert_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( field_o @ R3 ) ) ) ).
% Field_insert
thf(fact_1126_Field__empty,axiom,
( ( field_o @ bot_bo7073875226086086771od_o_o )
= bot_bot_set_o ) ).
% Field_empty
thf(fact_1127_sup__Un__eq,axiom,
! [R2: set_o,S: set_o] :
( ( sup_sup_o_o
@ ^ [X: $o] : ( member_o @ X @ R2 )
@ ^ [X: $o] : ( member_o @ X @ S ) )
= ( ^ [X: $o] : ( member_o @ X @ ( sup_sup_set_o @ R2 @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_1128_sup__Un__eq,axiom,
! [R2: set_li6526943997496501093_a_nat,S: set_li6526943997496501093_a_nat] :
( ( sup_su1334248866174809316_nat_o
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ R2 )
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ S ) )
= ( ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ ( sup_su4083067149120280889_a_nat @ R2 @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_1129_sup__Un__eq,axiom,
! [R2: set_Sum_sum_a_nat,S: set_Sum_sum_a_nat] :
( ( sup_su491480579010597738_nat_o
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ R2 )
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ S ) )
= ( ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ R2 @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_1130_sup__set__def,axiom,
( sup_sup_set_o
= ( ^ [A6: set_o,B6: set_o] :
( collect_o
@ ( sup_sup_o_o
@ ^ [X: $o] : ( member_o @ X @ A6 )
@ ^ [X: $o] : ( member_o @ X @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_1131_sup__set__def,axiom,
( sup_su4083067149120280889_a_nat
= ( ^ [A6: set_li6526943997496501093_a_nat,B6: set_li6526943997496501093_a_nat] :
( collec7555443234367654128_a_nat
@ ( sup_su1334248866174809316_nat_o
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ A6 )
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_1132_sup__set__def,axiom,
( sup_su6804446743777130803_a_nat
= ( ^ [A6: set_Sum_sum_a_nat,B6: set_Sum_sum_a_nat] :
( collec7073057861543223018_a_nat
@ ( sup_su491480579010597738_nat_o
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ A6 )
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_1133_sup__set__def,axiom,
( sup_su3329769938372955546_a_nat
= ( ^ [A6: set_na3699693778330250182_a_nat,B6: set_na3699693778330250182_a_nat] :
( collec5629555741568564177_a_nat
@ ( sup_su3226716170639437251_nat_o
@ ^ [X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ A6 )
@ ^ [X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_1134_sup__Un__eq2,axiom,
! [R2: set_Pr8537526423911375309_a_nat,S: set_Pr8537526423911375309_a_nat] :
( ( sup_su1343304085748361720_nat_o
@ ^ [X: set_a,Y3: produc5986856040925105290_a_nat] : ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ X @ Y3 ) @ R2 )
@ ^ [X: set_a,Y3: produc5986856040925105290_a_nat] : ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ X @ Y3 ) @ S ) )
= ( ^ [X: set_a,Y3: produc5986856040925105290_a_nat] : ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ X @ Y3 ) @ ( sup_su7876432514636195577_a_nat @ R2 @ S ) ) ) ) ).
% sup_Un_eq2
thf(fact_1135_sup__Un__eq2,axiom,
! [R2: set_Pr1465208939327205952_a_nat,S: set_Pr1465208939327205952_a_nat] :
( ( sup_su5657261288608606835_nat_o
@ ^ [X: nat,Y3: set_li6526943997496501093_a_nat] : ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ X @ Y3 ) @ R2 )
@ ^ [X: nat,Y3: set_li6526943997496501093_a_nat] : ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ X @ Y3 ) @ S ) )
= ( ^ [X: nat,Y3: set_li6526943997496501093_a_nat] : ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ X @ Y3 ) @ ( sup_su9161890791271854956_a_nat @ R2 @ S ) ) ) ) ).
% sup_Un_eq2
thf(fact_1136_FieldI2,axiom,
! [I2: $o,J2: $o,R2: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I2 @ J2 ) @ R2 )
=> ( member_o @ J2 @ ( field_o @ R2 ) ) ) ).
% FieldI2
thf(fact_1137_FieldI2,axiom,
! [I2: list_Sum_sum_a_nat,J2: list_Sum_sum_a_nat,R2: set_Pr4870381170404451655_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ I2 @ J2 ) @ R2 )
=> ( member408289922725080238_a_nat @ J2 @ ( field_8091184615201973458_a_nat @ R2 ) ) ) ).
% FieldI2
thf(fact_1138_FieldI2,axiom,
! [I2: sum_sum_a_nat,J2: sum_sum_a_nat,R2: set_Pr7343886759072863943_a_nat] :
( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ I2 @ J2 ) @ R2 )
=> ( member_Sum_sum_a_nat @ J2 @ ( field_Sum_sum_a_nat @ R2 ) ) ) ).
% FieldI2
thf(fact_1139_FieldI1,axiom,
! [I2: $o,J2: $o,R2: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I2 @ J2 ) @ R2 )
=> ( member_o @ I2 @ ( field_o @ R2 ) ) ) ).
% FieldI1
thf(fact_1140_FieldI1,axiom,
! [I2: list_Sum_sum_a_nat,J2: list_Sum_sum_a_nat,R2: set_Pr4870381170404451655_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ I2 @ J2 ) @ R2 )
=> ( member408289922725080238_a_nat @ I2 @ ( field_8091184615201973458_a_nat @ R2 ) ) ) ).
% FieldI1
thf(fact_1141_FieldI1,axiom,
! [I2: sum_sum_a_nat,J2: sum_sum_a_nat,R2: set_Pr7343886759072863943_a_nat] :
( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ I2 @ J2 ) @ R2 )
=> ( member_Sum_sum_a_nat @ I2 @ ( field_Sum_sum_a_nat @ R2 ) ) ) ).
% FieldI1
thf(fact_1142_linear__order__on__singleton,axiom,
! [X2: $o] : ( order_6238756238976269133r_on_o @ ( insert_o @ X2 @ bot_bot_set_o ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X2 @ X2 ) @ bot_bo7073875226086086771od_o_o ) ) ).
% linear_order_on_singleton
thf(fact_1143_linear__order,axiom,
order_6238756238976269133r_on_o @ top_top_set_o @ ( collec3167064739498627218od_o_o @ ( produc6197397395684419436_o_o_o @ ord_less_eq_o ) ) ).
% linear_order
thf(fact_1144_above__def,axiom,
( order_6051292016123231334_a_nat
= ( ^ [R4: set_Pr3078437455524598791_a_nat,A8: nat > sum_sum_a_nat] :
( collec5629555741568564177_a_nat
@ ^ [B8: nat > sum_sum_a_nat] : ( member1563331007001999952_a_nat @ ( produc1519640808906236439_a_nat @ A8 @ B8 ) @ R4 ) ) ) ) ).
% above_def
thf(fact_1145_set__update,axiom,
! [K: set_a,V: produc5986856040925105290_a_nat,Al: alist_5504510941512245929_a_nat] :
( ( set_se6394840760456946669_a_nat @ ( update8026975735590707640_a_nat @ K @ V @ Al ) )
= ( sup_su7876432514636195577_a_nat
@ ( minus_8912216149059716262_a_nat @ ( set_se6394840760456946669_a_nat @ Al )
@ ( produc5217640286535863200_a_nat @ ( insert_set_a @ K @ bot_bot_set_set_a )
@ ^ [Uu2: set_a] : top_to6009278897662198000_a_nat ) )
@ ( insert5623531327003279303_a_nat @ ( produc1626021218680649873_a_nat @ K @ V ) @ bot_bo3440348358004571233_a_nat ) ) ) ).
% set_update
thf(fact_1146_set__update,axiom,
! [K: nat,V: set_li6526943997496501093_a_nat,Al: alist_7015014170102249372_a_nat] :
( ( set_na5559583468520259224_a_nat @ ( update4397653113450408803_a_nat @ K @ V @ Al ) )
= ( sup_su9161890791271854956_a_nat
@ ( minus_8204380250167179801_a_nat @ ( set_na5559583468520259224_a_nat @ Al )
@ ( produc3542699778996723019_a_nat @ ( insert_nat @ K @ bot_bot_set_nat )
@ ^ [Uu2: nat] : top_to3073311637947473045_a_nat ) )
@ ( insert9175641148397066938_a_nat @ ( produc4622690550020908860_a_nat @ K @ V ) @ bot_bo1546958992129757396_a_nat ) ) ) ).
% set_update
thf(fact_1147_set__update,axiom,
! [K: $o,V: sum_sum_a_nat,Al: alist_3839350496853122428_a_nat] :
( ( set_o_Sum_sum_a_nat @ ( update4371090251427669015_a_nat @ K @ V @ Al ) )
= ( sup_su2285096139650635586_a_nat
@ ( minus_566666840696630485_a_nat @ ( set_o_Sum_sum_a_nat @ Al )
@ ( produc3081175752665019439_a_nat @ ( insert_o @ K @ bot_bot_set_o )
@ ^ [Uu2: $o] : top_to795618464972521135_a_nat ) )
@ ( insert590505920034879454_a_nat @ ( produc6394072109761784702_a_nat @ K @ V ) @ bot_bo5022905905639185370_a_nat ) ) ) ).
% set_update
thf(fact_1148_set__update,axiom,
! [K: $o,V: $o,Al: alist_o_o] :
( ( set_o_o @ ( update_o_o @ K @ V @ Al ) )
= ( sup_su5769328420594410459od_o_o
@ ( minus_7052513330764642158od_o_o @ ( set_o_o @ Al )
@ ( product_Sigma_o_o @ ( insert_o @ K @ bot_bot_set_o )
@ ^ [Uu2: $o] : top_top_set_o ) )
@ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ K @ V ) @ bot_bo7073875226086086771od_o_o ) ) ) ).
% set_update
thf(fact_1149_set__delete,axiom,
! [K: $o,Al: alist_3839350496853122428_a_nat] :
( ( set_o_Sum_sum_a_nat @ ( delete2500342324469489945_a_nat @ K @ Al ) )
= ( minus_566666840696630485_a_nat @ ( set_o_Sum_sum_a_nat @ Al )
@ ( produc3081175752665019439_a_nat @ ( insert_o @ K @ bot_bot_set_o )
@ ^ [Uu2: $o] : top_to795618464972521135_a_nat ) ) ) ).
% set_delete
thf(fact_1150_set__delete,axiom,
! [K: $o,Al: alist_o_o] :
( ( set_o_o @ ( delete_o_o @ K @ Al ) )
= ( minus_7052513330764642158od_o_o @ ( set_o_o @ Al )
@ ( product_Sigma_o_o @ ( insert_o @ K @ bot_bot_set_o )
@ ^ [Uu2: $o] : top_top_set_o ) ) ) ).
% set_delete
thf(fact_1151_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [R3: set_Pr4870381170404451655_a_nat,A2: list_Sum_sum_a_nat,B: list_Sum_sum_a_nat] :
( ( refl_o8238942462679651204_a_nat @ ( field_8091184615201973458_a_nat @ R3 ) @ R3 )
=> ( ( antisy4240500386402606696_a_nat @ top_to6433055325616222389_a_nat @ R3 )
=> ( ( member408289922725080238_a_nat @ A2 @ ( field_8091184615201973458_a_nat @ R3 ) )
=> ( ( member408289922725080238_a_nat @ B @ ( field_8091184615201973458_a_nat @ R3 ) )
=> ( ( ( image_3954144318686624531_a_nat @ R3 @ ( insert2950094090816004437_a_nat @ A2 @ bot_bo1033123847703346641_a_nat ) )
= ( image_3954144318686624531_a_nat @ R3 @ ( insert2950094090816004437_a_nat @ B @ bot_bo1033123847703346641_a_nat ) ) )
= ( A2 = B ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
thf(fact_1152_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [R3: set_Pr7343886759072863943_a_nat,A2: sum_sum_a_nat,B: sum_sum_a_nat] :
( ( refl_o828724780208292734_a_nat @ ( field_Sum_sum_a_nat @ R3 ) @ R3 )
=> ( ( antisy1998289778759880802_a_nat @ top_to795618464972521135_a_nat @ R3 )
=> ( ( member_Sum_sum_a_nat @ A2 @ ( field_Sum_sum_a_nat @ R3 ) )
=> ( ( member_Sum_sum_a_nat @ B @ ( field_Sum_sum_a_nat @ R3 ) )
=> ( ( ( image_7217254153141108115_a_nat @ R3 @ ( insert_Sum_sum_a_nat @ A2 @ bot_bo3438331934148233675_a_nat ) )
= ( image_7217254153141108115_a_nat @ R3 @ ( insert_Sum_sum_a_nat @ B @ bot_bo3438331934148233675_a_nat ) ) )
= ( A2 = B ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
thf(fact_1153_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [R3: set_Product_prod_o_o,A2: $o,B: $o] :
( ( refl_on_o @ ( field_o @ R3 ) @ R3 )
=> ( ( antisym_on_o @ top_top_set_o @ R3 )
=> ( ( member_o @ A2 @ ( field_o @ R3 ) )
=> ( ( member_o @ B @ ( field_o @ R3 ) )
=> ( ( ( image_o_o @ R3 @ ( insert_o @ A2 @ bot_bot_set_o ) )
= ( image_o_o @ R3 @ ( insert_o @ B @ bot_bot_set_o ) ) )
= ( A2 = B ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
thf(fact_1154_ImageI,axiom,
! [A2: $o,B: $o,R3: set_Product_prod_o_o,A: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R3 )
=> ( ( member_o @ A2 @ A )
=> ( member_o @ B @ ( image_o_o @ R3 @ A ) ) ) ) ).
% ImageI
thf(fact_1155_ImageI,axiom,
! [A2: $o,B: sum_sum_a_nat,R3: set_Pr6116151503928652142_a_nat,A: set_o] :
( ( member6925473634427674423_a_nat @ ( produc6394072109761784702_a_nat @ A2 @ B ) @ R3 )
=> ( ( member_o @ A2 @ A )
=> ( member_Sum_sum_a_nat @ B @ ( image_8761407235119216570_a_nat @ R3 @ A ) ) ) ) ).
% ImageI
thf(fact_1156_ImageI,axiom,
! [A2: sum_sum_a_nat,B: $o,R3: set_Pr7317283733329499488_nat_o,A: set_Sum_sum_a_nat] :
( ( member6686738655598238761_nat_o @ ( produc8349727785981246704_nat_o @ A2 @ B ) @ R3 )
=> ( ( member_Sum_sum_a_nat @ A2 @ A )
=> ( member_o @ B @ ( image_1493690874483902764_nat_o @ R3 @ A ) ) ) ) ).
% ImageI
thf(fact_1157_ImageI,axiom,
! [A2: $o,B: list_Sum_sum_a_nat,R3: set_Pr7721736691719792244_a_nat,A: set_o] :
( ( member2772425197143436605_a_nat @ ( produc7378284081089837956_a_nat @ A2 @ B ) @ R3 )
=> ( ( member_o @ A2 @ A )
=> ( member408289922725080238_a_nat @ B @ ( image_7687085000819358144_a_nat @ R3 @ A ) ) ) ) ).
% ImageI
thf(fact_1158_ImageI,axiom,
! [A2: list_Sum_sum_a_nat,B: $o,R3: set_Pr2164324812932626522_nat_o,A: set_li6526943997496501093_a_nat] :
( ( member1161018691129610787_nat_o @ ( produc1406723247873626090_nat_o @ A2 @ B ) @ R3 )
=> ( ( member408289922725080238_a_nat @ A2 @ A )
=> ( member_o @ B @ ( image_1715524167603146278_nat_o @ R3 @ A ) ) ) ) ).
% ImageI
thf(fact_1159_ImageI,axiom,
! [A2: sum_sum_a_nat,B: sum_sum_a_nat,R3: set_Pr7343886759072863943_a_nat,A: set_Sum_sum_a_nat] :
( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ A2 @ B ) @ R3 )
=> ( ( member_Sum_sum_a_nat @ A2 @ A )
=> ( member_Sum_sum_a_nat @ B @ ( image_7217254153141108115_a_nat @ R3 @ A ) ) ) ) ).
% ImageI
thf(fact_1160_ImageI,axiom,
! [A2: nat,B: set_li6526943997496501093_a_nat,R3: set_Pr1465208939327205952_a_nat,A: set_nat] :
( ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ A2 @ B ) @ R3 )
=> ( ( member_nat @ A2 @ A )
=> ( member5553968465346197646_a_nat @ B @ ( image_6068982090326499328_a_nat @ R3 @ A ) ) ) ) ).
% ImageI
thf(fact_1161_ImageI,axiom,
! [A2: list_Sum_sum_a_nat,B: sum_sum_a_nat,R3: set_Pr1075064412021831105_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member6493994536747010826_a_nat @ ( produc5977302043283085393_a_nat @ A2 @ B ) @ R3 )
=> ( ( member408289922725080238_a_nat @ A2 @ A )
=> ( member_Sum_sum_a_nat @ B @ ( image_4893485636860001933_a_nat @ R3 @ A ) ) ) ) ).
% ImageI
thf(fact_1162_ImageI,axiom,
! [A2: sum_sum_a_nat,B: list_Sum_sum_a_nat,R3: set_Pr1143021518387678029_a_nat,A: set_Sum_sum_a_nat] :
( ( member312212173870163606_a_nat @ ( produc6350064662657521885_a_nat @ A2 @ B ) @ R3 )
=> ( ( member_Sum_sum_a_nat @ A2 @ A )
=> ( member408289922725080238_a_nat @ B @ ( image_5266248256234438425_a_nat @ R3 @ A ) ) ) ) ).
% ImageI
thf(fact_1163_ImageI,axiom,
! [A2: list_Sum_sum_a_nat,B: list_Sum_sum_a_nat,R3: set_Pr4870381170404451655_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ A2 @ B ) @ R3 )
=> ( ( member408289922725080238_a_nat @ A2 @ A )
=> ( member408289922725080238_a_nat @ B @ ( image_3954144318686624531_a_nat @ R3 @ A ) ) ) ) ).
% ImageI
thf(fact_1164_Image__empty2,axiom,
! [R2: set_Product_prod_o_o] :
( ( image_o_o @ R2 @ bot_bot_set_o )
= bot_bot_set_o ) ).
% Image_empty2
thf(fact_1165_Image__singleton__iff,axiom,
! [B: produc5986856040925105290_a_nat,R3: set_Pr8537526423911375309_a_nat,A2: set_a] :
( ( member3071249055146688545_a_nat @ B @ ( image_6838867273210920277_a_nat @ R3 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) )
= ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ A2 @ B ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_1166_Image__singleton__iff,axiom,
! [B: set_li6526943997496501093_a_nat,R3: set_Pr1465208939327205952_a_nat,A2: nat] :
( ( member5553968465346197646_a_nat @ B @ ( image_6068982090326499328_a_nat @ R3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ A2 @ B ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_1167_Image__singleton__iff,axiom,
! [B: $o,R3: set_Product_prod_o_o,A2: $o] :
( ( member_o @ B @ ( image_o_o @ R3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
= ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_1168_Image__singleton__iff,axiom,
! [B: list_Sum_sum_a_nat,R3: set_Pr7721736691719792244_a_nat,A2: $o] :
( ( member408289922725080238_a_nat @ B @ ( image_7687085000819358144_a_nat @ R3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
= ( member2772425197143436605_a_nat @ ( produc7378284081089837956_a_nat @ A2 @ B ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_1169_Image__singleton__iff,axiom,
! [B: sum_sum_a_nat,R3: set_Pr6116151503928652142_a_nat,A2: $o] :
( ( member_Sum_sum_a_nat @ B @ ( image_8761407235119216570_a_nat @ R3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
= ( member6925473634427674423_a_nat @ ( produc6394072109761784702_a_nat @ A2 @ B ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_1170_rev__ImageI,axiom,
! [A2: $o,A: set_o,B: $o,R3: set_Product_prod_o_o] :
( ( member_o @ A2 @ A )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R3 )
=> ( member_o @ B @ ( image_o_o @ R3 @ A ) ) ) ) ).
% rev_ImageI
thf(fact_1171_rev__ImageI,axiom,
! [A2: $o,A: set_o,B: sum_sum_a_nat,R3: set_Pr6116151503928652142_a_nat] :
( ( member_o @ A2 @ A )
=> ( ( member6925473634427674423_a_nat @ ( produc6394072109761784702_a_nat @ A2 @ B ) @ R3 )
=> ( member_Sum_sum_a_nat @ B @ ( image_8761407235119216570_a_nat @ R3 @ A ) ) ) ) ).
% rev_ImageI
thf(fact_1172_rev__ImageI,axiom,
! [A2: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: $o,R3: set_Pr7317283733329499488_nat_o] :
( ( member_Sum_sum_a_nat @ A2 @ A )
=> ( ( member6686738655598238761_nat_o @ ( produc8349727785981246704_nat_o @ A2 @ B ) @ R3 )
=> ( member_o @ B @ ( image_1493690874483902764_nat_o @ R3 @ A ) ) ) ) ).
% rev_ImageI
thf(fact_1173_rev__ImageI,axiom,
! [A2: $o,A: set_o,B: list_Sum_sum_a_nat,R3: set_Pr7721736691719792244_a_nat] :
( ( member_o @ A2 @ A )
=> ( ( member2772425197143436605_a_nat @ ( produc7378284081089837956_a_nat @ A2 @ B ) @ R3 )
=> ( member408289922725080238_a_nat @ B @ ( image_7687085000819358144_a_nat @ R3 @ A ) ) ) ) ).
% rev_ImageI
thf(fact_1174_rev__ImageI,axiom,
! [A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: $o,R3: set_Pr2164324812932626522_nat_o] :
( ( member408289922725080238_a_nat @ A2 @ A )
=> ( ( member1161018691129610787_nat_o @ ( produc1406723247873626090_nat_o @ A2 @ B ) @ R3 )
=> ( member_o @ B @ ( image_1715524167603146278_nat_o @ R3 @ A ) ) ) ) ).
% rev_ImageI
thf(fact_1175_rev__ImageI,axiom,
! [A2: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: sum_sum_a_nat,R3: set_Pr7343886759072863943_a_nat] :
( ( member_Sum_sum_a_nat @ A2 @ A )
=> ( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ A2 @ B ) @ R3 )
=> ( member_Sum_sum_a_nat @ B @ ( image_7217254153141108115_a_nat @ R3 @ A ) ) ) ) ).
% rev_ImageI
thf(fact_1176_rev__ImageI,axiom,
! [A2: nat,A: set_nat,B: set_li6526943997496501093_a_nat,R3: set_Pr1465208939327205952_a_nat] :
( ( member_nat @ A2 @ A )
=> ( ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ A2 @ B ) @ R3 )
=> ( member5553968465346197646_a_nat @ B @ ( image_6068982090326499328_a_nat @ R3 @ A ) ) ) ) ).
% rev_ImageI
thf(fact_1177_rev__ImageI,axiom,
! [A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: sum_sum_a_nat,R3: set_Pr1075064412021831105_a_nat] :
( ( member408289922725080238_a_nat @ A2 @ A )
=> ( ( member6493994536747010826_a_nat @ ( produc5977302043283085393_a_nat @ A2 @ B ) @ R3 )
=> ( member_Sum_sum_a_nat @ B @ ( image_4893485636860001933_a_nat @ R3 @ A ) ) ) ) ).
% rev_ImageI
thf(fact_1178_rev__ImageI,axiom,
! [A2: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: list_Sum_sum_a_nat,R3: set_Pr1143021518387678029_a_nat] :
( ( member_Sum_sum_a_nat @ A2 @ A )
=> ( ( member312212173870163606_a_nat @ ( produc6350064662657521885_a_nat @ A2 @ B ) @ R3 )
=> ( member408289922725080238_a_nat @ B @ ( image_5266248256234438425_a_nat @ R3 @ A ) ) ) ) ).
% rev_ImageI
thf(fact_1179_rev__ImageI,axiom,
! [A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: list_Sum_sum_a_nat,R3: set_Pr4870381170404451655_a_nat] :
( ( member408289922725080238_a_nat @ A2 @ A )
=> ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ A2 @ B ) @ R3 )
=> ( member408289922725080238_a_nat @ B @ ( image_3954144318686624531_a_nat @ R3 @ A ) ) ) ) ).
% rev_ImageI
thf(fact_1180_Image__iff,axiom,
! [B: produc5986856040925105290_a_nat,R3: set_Pr8537526423911375309_a_nat,A: set_set_a] :
( ( member3071249055146688545_a_nat @ B @ ( image_6838867273210920277_a_nat @ R3 @ A ) )
= ( ? [X: set_a] :
( ( member_set_a @ X @ A )
& ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ X @ B ) @ R3 ) ) ) ) ).
% Image_iff
thf(fact_1181_Image__iff,axiom,
! [B: set_li6526943997496501093_a_nat,R3: set_Pr1465208939327205952_a_nat,A: set_nat] :
( ( member5553968465346197646_a_nat @ B @ ( image_6068982090326499328_a_nat @ R3 @ A ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A )
& ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ X @ B ) @ R3 ) ) ) ) ).
% Image_iff
thf(fact_1182_ImageE,axiom,
! [B: $o,R3: set_Product_prod_o_o,A: set_o] :
( ( member_o @ B @ ( image_o_o @ R3 @ A ) )
=> ~ ! [X3: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ B ) @ R3 )
=> ~ ( member_o @ X3 @ A ) ) ) ).
% ImageE
thf(fact_1183_ImageE,axiom,
! [B: $o,R3: set_Pr7317283733329499488_nat_o,A: set_Sum_sum_a_nat] :
( ( member_o @ B @ ( image_1493690874483902764_nat_o @ R3 @ A ) )
=> ~ ! [X3: sum_sum_a_nat] :
( ( member6686738655598238761_nat_o @ ( produc8349727785981246704_nat_o @ X3 @ B ) @ R3 )
=> ~ ( member_Sum_sum_a_nat @ X3 @ A ) ) ) ).
% ImageE
thf(fact_1184_ImageE,axiom,
! [B: sum_sum_a_nat,R3: set_Pr6116151503928652142_a_nat,A: set_o] :
( ( member_Sum_sum_a_nat @ B @ ( image_8761407235119216570_a_nat @ R3 @ A ) )
=> ~ ! [X3: $o] :
( ( member6925473634427674423_a_nat @ ( produc6394072109761784702_a_nat @ X3 @ B ) @ R3 )
=> ~ ( member_o @ X3 @ A ) ) ) ).
% ImageE
thf(fact_1185_ImageE,axiom,
! [B: $o,R3: set_Pr2164324812932626522_nat_o,A: set_li6526943997496501093_a_nat] :
( ( member_o @ B @ ( image_1715524167603146278_nat_o @ R3 @ A ) )
=> ~ ! [X3: list_Sum_sum_a_nat] :
( ( member1161018691129610787_nat_o @ ( produc1406723247873626090_nat_o @ X3 @ B ) @ R3 )
=> ~ ( member408289922725080238_a_nat @ X3 @ A ) ) ) ).
% ImageE
thf(fact_1186_ImageE,axiom,
! [B: list_Sum_sum_a_nat,R3: set_Pr7721736691719792244_a_nat,A: set_o] :
( ( member408289922725080238_a_nat @ B @ ( image_7687085000819358144_a_nat @ R3 @ A ) )
=> ~ ! [X3: $o] :
( ( member2772425197143436605_a_nat @ ( produc7378284081089837956_a_nat @ X3 @ B ) @ R3 )
=> ~ ( member_o @ X3 @ A ) ) ) ).
% ImageE
thf(fact_1187_ImageE,axiom,
! [B: sum_sum_a_nat,R3: set_Pr7343886759072863943_a_nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ B @ ( image_7217254153141108115_a_nat @ R3 @ A ) )
=> ~ ! [X3: sum_sum_a_nat] :
( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X3 @ B ) @ R3 )
=> ~ ( member_Sum_sum_a_nat @ X3 @ A ) ) ) ).
% ImageE
thf(fact_1188_ImageE,axiom,
! [B: set_li6526943997496501093_a_nat,R3: set_Pr1465208939327205952_a_nat,A: set_nat] :
( ( member5553968465346197646_a_nat @ B @ ( image_6068982090326499328_a_nat @ R3 @ A ) )
=> ~ ! [X3: nat] :
( ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ X3 @ B ) @ R3 )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% ImageE
thf(fact_1189_ImageE,axiom,
! [B: list_Sum_sum_a_nat,R3: set_Pr1143021518387678029_a_nat,A: set_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ B @ ( image_5266248256234438425_a_nat @ R3 @ A ) )
=> ~ ! [X3: sum_sum_a_nat] :
( ( member312212173870163606_a_nat @ ( produc6350064662657521885_a_nat @ X3 @ B ) @ R3 )
=> ~ ( member_Sum_sum_a_nat @ X3 @ A ) ) ) ).
% ImageE
thf(fact_1190_ImageE,axiom,
! [B: sum_sum_a_nat,R3: set_Pr1075064412021831105_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member_Sum_sum_a_nat @ B @ ( image_4893485636860001933_a_nat @ R3 @ A ) )
=> ~ ! [X3: list_Sum_sum_a_nat] :
( ( member6493994536747010826_a_nat @ ( produc5977302043283085393_a_nat @ X3 @ B ) @ R3 )
=> ~ ( member408289922725080238_a_nat @ X3 @ A ) ) ) ).
% ImageE
thf(fact_1191_ImageE,axiom,
! [B: list_Sum_sum_a_nat,R3: set_Pr4870381170404451655_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ B @ ( image_3954144318686624531_a_nat @ R3 @ A ) )
=> ~ ! [X3: list_Sum_sum_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ X3 @ B ) @ R3 )
=> ~ ( member408289922725080238_a_nat @ X3 @ A ) ) ) ).
% ImageE
thf(fact_1192_Image__singleton,axiom,
! [R3: set_Pr8537526423911375309_a_nat,A2: set_a] :
( ( image_6838867273210920277_a_nat @ R3 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
= ( collec3658349449910702431_a_nat
@ ^ [B8: produc5986856040925105290_a_nat] : ( member6515367127663331886_a_nat @ ( produc1626021218680649873_a_nat @ A2 @ B8 ) @ R3 ) ) ) ).
% Image_singleton
thf(fact_1193_Image__singleton,axiom,
! [R3: set_Pr1465208939327205952_a_nat,A2: nat] :
( ( image_6068982090326499328_a_nat @ R3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( collec7528627406912015568_a_nat
@ ^ [B8: set_li6526943997496501093_a_nat] : ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ A2 @ B8 ) @ R3 ) ) ) ).
% Image_singleton
thf(fact_1194_Image__singleton,axiom,
! [R3: set_Pr729842707005549141_a_nat,A2: $o] :
( ( image_2576641571370245281_a_nat @ R3 @ ( insert_o @ A2 @ bot_bot_set_o ) )
= ( collec5629555741568564177_a_nat
@ ^ [B8: nat > sum_sum_a_nat] : ( member8592244091583864222_a_nat @ ( produc6532394071696349797_a_nat @ A2 @ B8 ) @ R3 ) ) ) ).
% Image_singleton
thf(fact_1195_subset__Image1__Image1__iff,axiom,
! [R3: set_Pr4870381170404451655_a_nat,A2: list_Sum_sum_a_nat,B: list_Sum_sum_a_nat] :
( ( order_1401692832302633372_a_nat @ ( field_8091184615201973458_a_nat @ R3 ) @ R3 )
=> ( ( member408289922725080238_a_nat @ A2 @ ( field_8091184615201973458_a_nat @ R3 ) )
=> ( ( member408289922725080238_a_nat @ B @ ( field_8091184615201973458_a_nat @ R3 ) )
=> ( ( ord_le1147066620699065093_a_nat @ ( image_3954144318686624531_a_nat @ R3 @ ( insert2950094090816004437_a_nat @ A2 @ bot_bo1033123847703346641_a_nat ) ) @ ( image_3954144318686624531_a_nat @ R3 @ ( insert2950094090816004437_a_nat @ B @ bot_bo1033123847703346641_a_nat ) ) )
= ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ B @ A2 ) @ R3 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1196_subset__Image1__Image1__iff,axiom,
! [R3: set_Pr7343886759072863943_a_nat,A2: sum_sum_a_nat,B: sum_sum_a_nat] :
( ( order_838039882648382870_a_nat @ ( field_Sum_sum_a_nat @ R3 ) @ R3 )
=> ( ( member_Sum_sum_a_nat @ A2 @ ( field_Sum_sum_a_nat @ R3 ) )
=> ( ( member_Sum_sum_a_nat @ B @ ( field_Sum_sum_a_nat @ R3 ) )
=> ( ( ord_le1325389633284124927_a_nat @ ( image_7217254153141108115_a_nat @ R3 @ ( insert_Sum_sum_a_nat @ A2 @ bot_bo3438331934148233675_a_nat ) ) @ ( image_7217254153141108115_a_nat @ R3 @ ( insert_Sum_sum_a_nat @ B @ bot_bo3438331934148233675_a_nat ) ) )
= ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ B @ A2 ) @ R3 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1197_subset__Image1__Image1__iff,axiom,
! [R3: set_Product_prod_o_o,A2: $o,B: $o] :
( ( order_preorder_on_o @ ( field_o @ R3 ) @ R3 )
=> ( ( member_o @ A2 @ ( field_o @ R3 ) )
=> ( ( member_o @ B @ ( field_o @ R3 ) )
=> ( ( ord_less_eq_set_o @ ( image_o_o @ R3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) @ ( image_o_o @ R3 @ ( insert_o @ B @ bot_bot_set_o ) ) )
= ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A2 ) @ R3 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1198_INT__simps_I3_J,axiom,
! [C: set_o,A: $o > set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ( C = bot_bot_set_o )
=> ( ( comple1528121977673479270_a_nat
@ ( image_3365592128754359116_a_nat
@ ^ [X: $o] : ( minus_1134630996077396038_a_nat @ ( A @ X ) @ B2 )
@ C ) )
= top_to795618464972521135_a_nat ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple1528121977673479270_a_nat
@ ( image_3365592128754359116_a_nat
@ ^ [X: $o] : ( minus_1134630996077396038_a_nat @ ( A @ X ) @ B2 )
@ C ) )
= ( minus_1134630996077396038_a_nat @ ( comple1528121977673479270_a_nat @ ( image_3365592128754359116_a_nat @ A @ C ) ) @ B2 ) ) ) ) ).
% INT_simps(3)
thf(fact_1199_INT__simps_I3_J,axiom,
! [C: set_o,A: $o > set_o,B2: set_o] :
( ( ( C = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( minus_minus_set_o @ ( A @ X ) @ B2 )
@ C ) )
= top_top_set_o ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( minus_minus_set_o @ ( A @ X ) @ B2 )
@ C ) )
= ( minus_minus_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ A @ C ) ) @ B2 ) ) ) ) ).
% INT_simps(3)
thf(fact_1200_INT__simps_I1_J,axiom,
! [C: set_o,A: $o > set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ( C = bot_bot_set_o )
=> ( ( comple1528121977673479270_a_nat
@ ( image_3365592128754359116_a_nat
@ ^ [X: $o] : ( inf_in7084830621192376909_a_nat @ ( A @ X ) @ B2 )
@ C ) )
= top_to795618464972521135_a_nat ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple1528121977673479270_a_nat
@ ( image_3365592128754359116_a_nat
@ ^ [X: $o] : ( inf_in7084830621192376909_a_nat @ ( A @ X ) @ B2 )
@ C ) )
= ( inf_in7084830621192376909_a_nat @ ( comple1528121977673479270_a_nat @ ( image_3365592128754359116_a_nat @ A @ C ) ) @ B2 ) ) ) ) ).
% INT_simps(1)
thf(fact_1201_INT__simps_I1_J,axiom,
! [C: set_o,A: $o > set_o,B2: set_o] :
( ( ( C = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( inf_inf_set_o @ ( A @ X ) @ B2 )
@ C ) )
= top_top_set_o ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( inf_inf_set_o @ ( A @ X ) @ B2 )
@ C ) )
= ( inf_inf_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ A @ C ) ) @ B2 ) ) ) ) ).
% INT_simps(1)
thf(fact_1202_Inter__iff,axiom,
! [A: $o,C: set_set_o] :
( ( member_o @ A @ ( comple3063163877087187839_set_o @ C ) )
= ( ! [X: set_o] :
( ( member_set_o @ X @ C )
=> ( member_o @ A @ X ) ) ) ) ).
% Inter_iff
thf(fact_1203_Inter__iff,axiom,
! [A: list_Sum_sum_a_nat,C: set_se4330304633200676677_a_nat] :
( ( member408289922725080238_a_nat @ A @ ( comple852897431680229100_a_nat @ C ) )
= ( ! [X: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X @ C )
=> ( member408289922725080238_a_nat @ A @ X ) ) ) ) ).
% Inter_iff
thf(fact_1204_Inter__iff,axiom,
! [A: sum_sum_a_nat,C: set_se4904748513628223167_a_nat] :
( ( member_Sum_sum_a_nat @ A @ ( comple1528121977673479270_a_nat @ C ) )
= ( ! [X: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X @ C )
=> ( member_Sum_sum_a_nat @ A @ X ) ) ) ) ).
% Inter_iff
thf(fact_1205_InterI,axiom,
! [C: set_set_o,A: $o] :
( ! [X5: set_o] :
( ( member_set_o @ X5 @ C )
=> ( member_o @ A @ X5 ) )
=> ( member_o @ A @ ( comple3063163877087187839_set_o @ C ) ) ) ).
% InterI
thf(fact_1206_InterI,axiom,
! [C: set_se4330304633200676677_a_nat,A: list_Sum_sum_a_nat] :
( ! [X5: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X5 @ C )
=> ( member408289922725080238_a_nat @ A @ X5 ) )
=> ( member408289922725080238_a_nat @ A @ ( comple852897431680229100_a_nat @ C ) ) ) ).
% InterI
thf(fact_1207_InterI,axiom,
! [C: set_se4904748513628223167_a_nat,A: sum_sum_a_nat] :
( ! [X5: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X5 @ C )
=> ( member_Sum_sum_a_nat @ A @ X5 ) )
=> ( member_Sum_sum_a_nat @ A @ ( comple1528121977673479270_a_nat @ C ) ) ) ).
% InterI
thf(fact_1208_Inf__top__conv_I2_J,axiom,
! [A: set_se4904748513628223167_a_nat] :
( ( top_to795618464972521135_a_nat
= ( comple1528121977673479270_a_nat @ A ) )
= ( ! [X: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X @ A )
=> ( X = top_to795618464972521135_a_nat ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_1209_Inf__top__conv_I2_J,axiom,
! [A: set_set_o] :
( ( top_top_set_o
= ( comple3063163877087187839_set_o @ A ) )
= ( ! [X: set_o] :
( ( member_set_o @ X @ A )
=> ( X = top_top_set_o ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_1210_Inf__top__conv_I2_J,axiom,
! [A: set_o] :
( ( top_top_o
= ( complete_Inf_Inf_o @ A ) )
= ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( X = top_top_o ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_1211_Inf__top__conv_I1_J,axiom,
! [A: set_se4904748513628223167_a_nat] :
( ( ( comple1528121977673479270_a_nat @ A )
= top_to795618464972521135_a_nat )
= ( ! [X: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X @ A )
=> ( X = top_to795618464972521135_a_nat ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_1212_Inf__top__conv_I1_J,axiom,
! [A: set_set_o] :
( ( ( comple3063163877087187839_set_o @ A )
= top_top_set_o )
= ( ! [X: set_o] :
( ( member_set_o @ X @ A )
=> ( X = top_top_set_o ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_1213_Inf__top__conv_I1_J,axiom,
! [A: set_o] :
( ( ( complete_Inf_Inf_o @ A )
= top_top_o )
= ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( X = top_top_o ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_1214_Inter__UNIV__conv_I2_J,axiom,
! [A: set_se4904748513628223167_a_nat] :
( ( top_to795618464972521135_a_nat
= ( comple1528121977673479270_a_nat @ A ) )
= ( ! [X: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X @ A )
=> ( X = top_to795618464972521135_a_nat ) ) ) ) ).
% Inter_UNIV_conv(2)
thf(fact_1215_Inter__UNIV__conv_I2_J,axiom,
! [A: set_set_o] :
( ( top_top_set_o
= ( comple3063163877087187839_set_o @ A ) )
= ( ! [X: set_o] :
( ( member_set_o @ X @ A )
=> ( X = top_top_set_o ) ) ) ) ).
% Inter_UNIV_conv(2)
thf(fact_1216_Inter__UNIV__conv_I1_J,axiom,
! [A: set_se4904748513628223167_a_nat] :
( ( ( comple1528121977673479270_a_nat @ A )
= top_to795618464972521135_a_nat )
= ( ! [X: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X @ A )
=> ( X = top_to795618464972521135_a_nat ) ) ) ) ).
% Inter_UNIV_conv(1)
thf(fact_1217_Inter__UNIV__conv_I1_J,axiom,
! [A: set_set_o] :
( ( ( comple3063163877087187839_set_o @ A )
= top_top_set_o )
= ( ! [X: set_o] :
( ( member_set_o @ X @ A )
=> ( X = top_top_set_o ) ) ) ) ).
% Inter_UNIV_conv(1)
thf(fact_1218_INF__identity__eq,axiom,
! [A: set_o] :
( ( complete_Inf_Inf_o
@ ( image_o_o2
@ ^ [X: $o] : X
@ A ) )
= ( complete_Inf_Inf_o @ A ) ) ).
% INF_identity_eq
thf(fact_1219_INT__I,axiom,
! [A: set_o,B: $o,B2: $o > set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member_o @ B @ ( B2 @ X3 ) ) )
=> ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_1220_INT__I,axiom,
! [A: set_o,B: list_Sum_sum_a_nat,B2: $o > set_li6526943997496501093_a_nat] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member408289922725080238_a_nat @ B @ ( B2 @ X3 ) ) )
=> ( member408289922725080238_a_nat @ B @ ( comple852897431680229100_a_nat @ ( image_2951602039182858962_a_nat @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_1221_INT__I,axiom,
! [A: set_o,B: sum_sum_a_nat,B2: $o > set_Sum_sum_a_nat] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member_Sum_sum_a_nat @ B @ ( B2 @ X3 ) ) )
=> ( member_Sum_sum_a_nat @ B @ ( comple1528121977673479270_a_nat @ ( image_3365592128754359116_a_nat @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_1222_INT__I,axiom,
! [A: set_li6526943997496501093_a_nat,B: $o,B2: list_Sum_sum_a_nat > set_o] :
( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ( member_o @ B @ ( B2 @ X3 ) ) )
=> ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_4041951406191573560_set_o @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_1223_INT__I,axiom,
! [A: set_li6526943997496501093_a_nat,B: list_Sum_sum_a_nat,B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ( member408289922725080238_a_nat @ B @ ( B2 @ X3 ) ) )
=> ( member408289922725080238_a_nat @ B @ ( comple852897431680229100_a_nat @ ( image_7676043921908783909_a_nat @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_1224_INT__I,axiom,
! [A: set_li6526943997496501093_a_nat,B: sum_sum_a_nat,B2: list_Sum_sum_a_nat > set_Sum_sum_a_nat] :
( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ( member_Sum_sum_a_nat @ B @ ( B2 @ X3 ) ) )
=> ( member_Sum_sum_a_nat @ B @ ( comple1528121977673479270_a_nat @ ( image_3940260845811589407_a_nat @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_1225_INT__I,axiom,
! [A: set_Sum_sum_a_nat,B: $o,B2: sum_sum_a_nat > set_o] :
( ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ A )
=> ( member_o @ B @ ( B2 @ X3 ) ) )
=> ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_3342437468262948414_set_o @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_1226_INT__I,axiom,
! [A: set_Sum_sum_a_nat,B: list_Sum_sum_a_nat,B2: sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ A )
=> ( member408289922725080238_a_nat @ B @ ( B2 @ X3 ) ) )
=> ( member408289922725080238_a_nat @ B @ ( comple852897431680229100_a_nat @ ( image_7788113019651049515_a_nat @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_1227_INT__I,axiom,
! [A: set_Sum_sum_a_nat,B: sum_sum_a_nat,B2: sum_sum_a_nat > set_Sum_sum_a_nat] :
( ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ A )
=> ( member_Sum_sum_a_nat @ B @ ( B2 @ X3 ) ) )
=> ( member_Sum_sum_a_nat @ B @ ( comple1528121977673479270_a_nat @ ( image_7877458644602423589_a_nat @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_1228_Inf__UNIV,axiom,
( ( comple3063163877087187839_set_o @ top_top_set_set_o )
= bot_bot_set_o ) ).
% Inf_UNIV
thf(fact_1229_Inf__UNIV,axiom,
( ( complete_Inf_Inf_o @ top_top_set_o )
= bot_bot_o ) ).
% Inf_UNIV
thf(fact_1230_Inf__empty,axiom,
( ( comple1528121977673479270_a_nat @ bot_bo2635121477170169643_a_nat )
= top_to795618464972521135_a_nat ) ).
% Inf_empty
thf(fact_1231_Inf__empty,axiom,
( ( comple3063163877087187839_set_o @ bot_bot_set_set_o )
= top_top_set_o ) ).
% Inf_empty
thf(fact_1232_Inf__empty,axiom,
( ( complete_Inf_Inf_o @ bot_bot_set_o )
= top_top_o ) ).
% Inf_empty
thf(fact_1233_Inf__insert,axiom,
! [A2: $o,A: set_o] :
( ( complete_Inf_Inf_o @ ( insert_o @ A2 @ A ) )
= ( inf_inf_o @ A2 @ ( complete_Inf_Inf_o @ A ) ) ) ).
% Inf_insert
thf(fact_1234_INF__id__eq,axiom,
! [A: set_o] :
( ( complete_Inf_Inf_o @ ( image_o_o2 @ id_o @ A ) )
= ( complete_Inf_Inf_o @ A ) ) ).
% INF_id_eq
thf(fact_1235_INF__top__conv_I2_J,axiom,
! [B2: $o > $o,A: set_o] :
( ( top_top_o
= ( complete_Inf_Inf_o @ ( image_o_o2 @ B2 @ A ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ( B2 @ X )
= top_top_o ) ) ) ) ).
% INF_top_conv(2)
thf(fact_1236_INF__top__conv_I1_J,axiom,
! [B2: $o > $o,A: set_o] :
( ( ( complete_Inf_Inf_o @ ( image_o_o2 @ B2 @ A ) )
= top_top_o )
= ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ( B2 @ X )
= top_top_o ) ) ) ) ).
% INF_top_conv(1)
thf(fact_1237_INF__top,axiom,
! [A: set_o] :
( ( complete_Inf_Inf_o
@ ( image_o_o2
@ ^ [X: $o] : top_top_o
@ A ) )
= top_top_o ) ).
% INF_top
thf(fact_1238_INF__const,axiom,
! [A: set_o,F: $o] :
( ( A != bot_bot_set_o )
=> ( ( complete_Inf_Inf_o
@ ( image_o_o2
@ ^ [I5: $o] : F
@ A ) )
= F ) ) ).
% INF_const
thf(fact_1239_INT__constant,axiom,
! [A: set_o,C3: set_Sum_sum_a_nat] :
( ( ( A = bot_bot_set_o )
=> ( ( comple1528121977673479270_a_nat
@ ( image_3365592128754359116_a_nat
@ ^ [Y3: $o] : C3
@ A ) )
= top_to795618464972521135_a_nat ) )
& ( ( A != bot_bot_set_o )
=> ( ( comple1528121977673479270_a_nat
@ ( image_3365592128754359116_a_nat
@ ^ [Y3: $o] : C3
@ A ) )
= C3 ) ) ) ).
% INT_constant
thf(fact_1240_INT__constant,axiom,
! [A: set_o,C3: set_o] :
( ( ( A = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [Y3: $o] : C3
@ A ) )
= top_top_set_o ) )
& ( ( A != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [Y3: $o] : C3
@ A ) )
= C3 ) ) ) ).
% INT_constant
thf(fact_1241_INT__simps_I2_J,axiom,
! [C: set_o,A: set_Sum_sum_a_nat,B2: $o > set_Sum_sum_a_nat] :
( ( ( C = bot_bot_set_o )
=> ( ( comple1528121977673479270_a_nat
@ ( image_3365592128754359116_a_nat
@ ^ [X: $o] : ( inf_in7084830621192376909_a_nat @ A @ ( B2 @ X ) )
@ C ) )
= top_to795618464972521135_a_nat ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple1528121977673479270_a_nat
@ ( image_3365592128754359116_a_nat
@ ^ [X: $o] : ( inf_in7084830621192376909_a_nat @ A @ ( B2 @ X ) )
@ C ) )
= ( inf_in7084830621192376909_a_nat @ A @ ( comple1528121977673479270_a_nat @ ( image_3365592128754359116_a_nat @ B2 @ C ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_1242_INT__simps_I2_J,axiom,
! [C: set_o,A: set_o,B2: $o > set_o] :
( ( ( C = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( inf_inf_set_o @ A @ ( B2 @ X ) )
@ C ) )
= top_top_set_o ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( inf_inf_set_o @ A @ ( B2 @ X ) )
@ C ) )
= ( inf_inf_set_o @ A @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ C ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_1243_INF__eq__iff,axiom,
! [I: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > $o,C3: $o] :
( ( I != bot_bo1033123847703346641_a_nat )
=> ( ! [I4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ I4 @ I )
=> ( ord_less_eq_o @ ( F @ I4 ) @ C3 ) )
=> ( ( ( complete_Inf_Inf_o @ ( image_3984946558445957976_nat_o @ F @ I ) )
= C3 )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ I )
=> ( ( F @ X )
= C3 ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1244_INF__eq__iff,axiom,
! [I: set_Sum_sum_a_nat,F: sum_sum_a_nat > $o,C3: $o] :
( ( I != bot_bo3438331934148233675_a_nat )
=> ( ! [I4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ I4 @ I )
=> ( ord_less_eq_o @ ( F @ I4 ) @ C3 ) )
=> ( ( ( complete_Inf_Inf_o @ ( image_6095136190293192542_nat_o @ F @ I ) )
= C3 )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ I )
=> ( ( F @ X )
= C3 ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1245_INF__eq__iff,axiom,
! [I: set_o,F: $o > $o,C3: $o] :
( ( I != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I )
=> ( ord_less_eq_o @ ( F @ I4 ) @ C3 ) )
=> ( ( ( complete_Inf_Inf_o @ ( image_o_o2 @ F @ I ) )
= C3 )
= ( ! [X: $o] :
( ( member_o @ X @ I )
=> ( ( F @ X )
= C3 ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1246_INT__insert__distrib,axiom,
! [U2: $o,A: set_o,A2: $o,B2: $o > set_o] :
( ( member_o @ U2 @ A )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( insert_o @ A2 @ ( B2 @ X ) )
@ A ) )
= ( insert_o @ A2 @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ A ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_1247_INT__insert__distrib,axiom,
! [U2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,A2: $o,B2: list_Sum_sum_a_nat > set_o] :
( ( member408289922725080238_a_nat @ U2 @ A )
=> ( ( comple3063163877087187839_set_o
@ ( image_4041951406191573560_set_o
@ ^ [X: list_Sum_sum_a_nat] : ( insert_o @ A2 @ ( B2 @ X ) )
@ A ) )
= ( insert_o @ A2 @ ( comple3063163877087187839_set_o @ ( image_4041951406191573560_set_o @ B2 @ A ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_1248_INT__insert__distrib,axiom,
! [U2: sum_sum_a_nat,A: set_Sum_sum_a_nat,A2: $o,B2: sum_sum_a_nat > set_o] :
( ( member_Sum_sum_a_nat @ U2 @ A )
=> ( ( comple3063163877087187839_set_o
@ ( image_3342437468262948414_set_o
@ ^ [X: sum_sum_a_nat] : ( insert_o @ A2 @ ( B2 @ X ) )
@ A ) )
= ( insert_o @ A2 @ ( comple3063163877087187839_set_o @ ( image_3342437468262948414_set_o @ B2 @ A ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_1249_INF__eq,axiom,
! [A: set_o,B2: set_o,G: $o > $o,F: $o > $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ? [X4: $o] :
( ( member_o @ X4 @ B2 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I4 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o2 @ F @ A ) )
= ( complete_Inf_Inf_o @ ( image_o_o2 @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1250_INF__eq,axiom,
! [A: set_o,B2: set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > $o,F: $o > $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ? [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ B2 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I4 ) ) ) )
=> ( ! [J3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ J3 @ B2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o2 @ F @ A ) )
= ( complete_Inf_Inf_o @ ( image_3984946558445957976_nat_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1251_INF__eq,axiom,
! [A: set_o,B2: set_Sum_sum_a_nat,G: sum_sum_a_nat > $o,F: $o > $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ? [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ B2 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I4 ) ) ) )
=> ( ! [J3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ J3 @ B2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o2 @ F @ A ) )
= ( complete_Inf_Inf_o @ ( image_6095136190293192542_nat_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1252_INF__eq,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_o,G: $o > $o,F: list_Sum_sum_a_nat > $o] :
( ! [I4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ I4 @ A )
=> ? [X4: $o] :
( ( member_o @ X4 @ B2 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I4 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B2 )
=> ? [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_3984946558445957976_nat_o @ F @ A ) )
= ( complete_Inf_Inf_o @ ( image_o_o2 @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1253_INF__eq,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > $o,F: list_Sum_sum_a_nat > $o] :
( ! [I4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ I4 @ A )
=> ? [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ B2 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I4 ) ) ) )
=> ( ! [J3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ J3 @ B2 )
=> ? [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_3984946558445957976_nat_o @ F @ A ) )
= ( complete_Inf_Inf_o @ ( image_3984946558445957976_nat_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1254_INF__eq,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_Sum_sum_a_nat,G: sum_sum_a_nat > $o,F: list_Sum_sum_a_nat > $o] :
( ! [I4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ I4 @ A )
=> ? [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ B2 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I4 ) ) ) )
=> ( ! [J3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ J3 @ B2 )
=> ? [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_3984946558445957976_nat_o @ F @ A ) )
= ( complete_Inf_Inf_o @ ( image_6095136190293192542_nat_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1255_INF__eq,axiom,
! [A: set_Sum_sum_a_nat,B2: set_o,G: $o > $o,F: sum_sum_a_nat > $o] :
( ! [I4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ I4 @ A )
=> ? [X4: $o] :
( ( member_o @ X4 @ B2 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I4 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B2 )
=> ? [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ A )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_6095136190293192542_nat_o @ F @ A ) )
= ( complete_Inf_Inf_o @ ( image_o_o2 @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1256_INF__eq,axiom,
! [A: set_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > $o,F: sum_sum_a_nat > $o] :
( ! [I4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ I4 @ A )
=> ? [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ B2 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I4 ) ) ) )
=> ( ! [J3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ J3 @ B2 )
=> ? [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ A )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_6095136190293192542_nat_o @ F @ A ) )
= ( complete_Inf_Inf_o @ ( image_3984946558445957976_nat_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1257_INF__eq,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,G: sum_sum_a_nat > $o,F: sum_sum_a_nat > $o] :
( ! [I4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ I4 @ A )
=> ? [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ B2 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I4 ) ) ) )
=> ( ! [J3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ J3 @ B2 )
=> ? [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ A )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_6095136190293192542_nat_o @ F @ A ) )
= ( complete_Inf_Inf_o @ ( image_6095136190293192542_nat_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1258_Inf__less__eq,axiom,
! [A: set_o,U2: $o] :
( ! [V2: $o] :
( ( member_o @ V2 @ A )
=> ( ord_less_eq_o @ V2 @ U2 ) )
=> ( ( A != bot_bot_set_o )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ U2 ) ) ) ).
% Inf_less_eq
thf(fact_1259_Inf__superset__mono,axiom,
! [B2: set_o,A: set_o] :
( ( ord_less_eq_set_o @ B2 @ A )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ ( complete_Inf_Inf_o @ B2 ) ) ) ).
% Inf_superset_mono
thf(fact_1260_INF__image,axiom,
! [G: list_Sum_sum_a_nat > $o,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( complete_Inf_Inf_o @ ( image_3984946558445957976_nat_o @ G @ ( image_5081948215111134021_a_nat @ F @ A ) ) )
= ( complete_Inf_Inf_o @ ( image_3984946558445957976_nat_o @ ( comp_l7931591902798318543_a_nat @ G @ F ) @ A ) ) ) ).
% INF_image
thf(fact_1261_INF__image,axiom,
! [G: $o > $o,F: $o > $o,A: set_o] :
( ( complete_Inf_Inf_o @ ( image_o_o2 @ G @ ( image_o_o2 @ F @ A ) ) )
= ( complete_Inf_Inf_o @ ( image_o_o2 @ ( comp_o_o_o @ G @ F ) @ A ) ) ) ).
% INF_image
thf(fact_1262_Inf__greatest,axiom,
! [A: set_o,Z: $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_o @ Z @ X3 ) )
=> ( ord_less_eq_o @ Z @ ( complete_Inf_Inf_o @ A ) ) ) ).
% Inf_greatest
thf(fact_1263_le__Inf__iff,axiom,
! [B: $o,A: set_o] :
( ( ord_less_eq_o @ B @ ( complete_Inf_Inf_o @ A ) )
= ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_o @ B @ X ) ) ) ) ).
% le_Inf_iff
thf(fact_1264_Inf__lower2,axiom,
! [U2: $o,A: set_o,V: $o] :
( ( member_o @ U2 @ A )
=> ( ( ord_less_eq_o @ U2 @ V )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ V ) ) ) ).
% Inf_lower2
thf(fact_1265_Inf__lower,axiom,
! [X2: $o,A: set_o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ X2 ) ) ).
% Inf_lower
thf(fact_1266_Inf__mono,axiom,
! [B2: set_o,A: set_o] :
( ! [B3: $o] :
( ( member_o @ B3 @ B2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A )
& ( ord_less_eq_o @ X4 @ B3 ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ ( complete_Inf_Inf_o @ B2 ) ) ) ).
% Inf_mono
thf(fact_1267_Inf__eqI,axiom,
! [A: set_o,X2: $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_less_eq_o @ X2 @ I4 ) )
=> ( ! [Y4: $o] :
( ! [I6: $o] :
( ( member_o @ I6 @ A )
=> ( ord_less_eq_o @ Y4 @ I6 ) )
=> ( ord_less_eq_o @ Y4 @ X2 ) )
=> ( ( complete_Inf_Inf_o @ A )
= X2 ) ) ) ).
% Inf_eqI
thf(fact_1268_INF__greatest,axiom,
! [A: set_o,U2: $o,F: $o > $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_less_eq_o @ U2 @ ( F @ I4 ) ) )
=> ( ord_less_eq_o @ U2 @ ( complete_Inf_Inf_o @ ( image_o_o2 @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_1269_INF__greatest,axiom,
! [A: set_li6526943997496501093_a_nat,U2: $o,F: list_Sum_sum_a_nat > $o] :
( ! [I4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ I4 @ A )
=> ( ord_less_eq_o @ U2 @ ( F @ I4 ) ) )
=> ( ord_less_eq_o @ U2 @ ( complete_Inf_Inf_o @ ( image_3984946558445957976_nat_o @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_1270_INF__greatest,axiom,
! [A: set_Sum_sum_a_nat,U2: $o,F: sum_sum_a_nat > $o] :
( ! [I4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ I4 @ A )
=> ( ord_less_eq_o @ U2 @ ( F @ I4 ) ) )
=> ( ord_less_eq_o @ U2 @ ( complete_Inf_Inf_o @ ( image_6095136190293192542_nat_o @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_1271_le__INF__iff,axiom,
! [U2: $o,F: $o > $o,A: set_o] :
( ( ord_less_eq_o @ U2 @ ( complete_Inf_Inf_o @ ( image_o_o2 @ F @ A ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_o @ U2 @ ( F @ X ) ) ) ) ) ).
% le_INF_iff
thf(fact_1272_INF__lower2,axiom,
! [I2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > $o,U2: $o] :
( ( member408289922725080238_a_nat @ I2 @ A )
=> ( ( ord_less_eq_o @ ( F @ I2 ) @ U2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_3984946558445957976_nat_o @ F @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_1273_INF__lower2,axiom,
! [I2: sum_sum_a_nat,A: set_Sum_sum_a_nat,F: sum_sum_a_nat > $o,U2: $o] :
( ( member_Sum_sum_a_nat @ I2 @ A )
=> ( ( ord_less_eq_o @ ( F @ I2 ) @ U2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_6095136190293192542_nat_o @ F @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_1274_UNIV__bool,axiom,
( top_top_set_o
= ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% UNIV_bool
thf(fact_1275_Inf__bool__def,axiom,
( complete_Inf_Inf_o
= ( ^ [A6: set_o] :
~ ( member_o @ $false @ A6 ) ) ) ).
% Inf_bool_def
% Helper facts (3)
thf(help_If_3_1_If_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_T,axiom,
! [X2: list_Sum_sum_a_nat,Y2: list_Sum_sum_a_nat] :
( ( if_lis4685338526944683083_a_nat @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_T,axiom,
! [X2: list_Sum_sum_a_nat,Y2: list_Sum_sum_a_nat] :
( ( if_lis4685338526944683083_a_nat @ $true @ X2 @ Y2 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ ad )
@ ( proj_f6037546931041448027_a_nat @ ( fo_Exists_a_b @ i2 @ phi )
@ ( collec5629555741568564177_a_nat
@ ^ [Sigma: nat > sum_sum_a_nat] : ( esat_a_b @ phi @ i @ Sigma @ top_to795618464972521135_a_nat ) ) ) )
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ ad )
@ ( proj_f6037546931041448027_a_nat @ ( fo_Exists_a_b @ i2 @ phi )
@ ( collec5629555741568564177_a_nat
@ ^ [Sigma: nat > sum_sum_a_nat] : ( esat_a_b @ ( fo_Exists_a_b @ i2 @ phi ) @ i @ Sigma @ top_to795618464972521135_a_nat ) ) ) ) ) ).
%------------------------------------------------------------------------------