%------------------------------------------------------------------------------
% 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    : Prefix_Free_Code_Combinators/0000_Prefix_Free_Code_Combinators/prob_00156_005691__11805490_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2088 ( 702 unt; 807 typ;   0 def)
%            Number of atoms       : 3013 (1508 equ;   0 cnn)
%            Maximal formula atoms :   10 (   2 avg)
%            Number of connectives : 11447 ( 165   ~;   5   |;  79   &;10130   @)
%                                         (   0 <=>;1068  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   19 (   6 avg)
%            Number of types       :   55 (  54 usr)
%            Number of type conns  : 5475 (5475   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  756 ( 753 usr;  19 con; 0-5 aty)
%            Number of variables   : 3971 ( 286   ^;3625   !;  60   ?;3971   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 09:57:19.938
%------------------------------------------------------------------------------
% Could-be-implicit typings (54)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
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thf(ty_n_t__List__Olist_I_Eo_J,type,
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thf(ty_n_tf__b,type,
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thf(ty_n_tf__a,type,
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% Explicit typings (753)
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thf(sy_c_BNF__Def_OGrp_001t__Product____Type__Oprod_Itf__b_Mtf__a_J_001tf__a,type,
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thf(sy_c_BNF__Def_OGrp_001tf__a_001t__List__Olist_I_Eo_J,type,
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thf(sy_c_BNF__Def_OGrp_001tf__a_001tf__a,type,
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thf(sy_c_BNF__Def_OGrp_001tf__a_001tf__b,type,
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thf(sy_c_BNF__Def_OGrp_001tf__b_001tf__a,type,
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thf(sy_c_BNF__Def_OGrp_001tf__b_001tf__b,type,
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thf(sy_c_BNF__Def_Oconvol_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
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    bNF_co6758072345015569366_a_b_b: ( product_prod_a_b > product_prod_a_b ) > ( product_prod_a_b > b ) > product_prod_a_b > produc3167243053651132542_a_b_b ).

thf(sy_c_BNF__Def_Oconvol_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    bNF_co2133029838456317992od_a_a: ( product_prod_a_b > a ) > ( product_prod_a_b > product_prod_a_a ) > product_prod_a_b > produc4044097585999906000od_a_a ).

thf(sy_c_BNF__Def_Oconvol_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001tf__a_001tf__a,type,
    bNF_co3724607310967375615_b_a_a: ( product_prod_a_b > a ) > ( product_prod_a_b > a ) > product_prod_a_b > product_prod_a_a ).

thf(sy_c_BNF__Def_Oconvol_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001tf__a_001tf__b,type,
    bNF_co3724607310967375616_b_a_b: ( product_prod_a_b > a ) > ( product_prod_a_b > b ) > product_prod_a_b > product_prod_a_b ).

thf(sy_c_BNF__Def_Oconvol_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001tf__b_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    bNF_co1493589538124501863od_a_a: ( product_prod_a_b > b ) > ( product_prod_a_b > product_prod_a_a ) > product_prod_a_b > produc3404657285668089871od_a_a ).

thf(sy_c_BNF__Def_Oconvol_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001tf__b_001tf__a,type,
    bNF_co937151728994699582_b_b_a: ( product_prod_a_b > b ) > ( product_prod_a_b > a ) > product_prod_a_b > product_prod_b_a ).

thf(sy_c_BNF__Def_Oconvol_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001tf__b_001tf__b,type,
    bNF_co937151728994699583_b_b_b: ( product_prod_a_b > b ) > ( product_prod_a_b > b ) > product_prod_a_b > product_prod_b_b ).

thf(sy_c_BNF__Def_Oconvol_001t__Product____Type__Oprod_Itf__b_Mtf__a_J_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_co1811873346508511455at_nat: ( product_prod_b_a > nat ) > ( product_prod_b_a > nat ) > product_prod_b_a > product_prod_nat_nat ).

thf(sy_c_BNF__Def_Oconvol_001t__Product____Type__Oprod_Itf__b_Mtf__a_J_001tf__a_001tf__a,type,
    bNF_co7991273656122196669_a_a_a: ( product_prod_b_a > a ) > ( product_prod_b_a > a ) > product_prod_b_a > product_prod_a_a ).

thf(sy_c_BNF__Def_Oconvol_001t__Product____Type__Oprod_Itf__b_Mtf__a_J_001tf__a_001tf__b,type,
    bNF_co7991273656122196670_a_a_b: ( product_prod_b_a > a ) > ( product_prod_b_a > b ) > product_prod_b_a > product_prod_a_b ).

thf(sy_c_BNF__Def_Oconvol_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_co3087766193879408736at_nat: ( product_prod_b_b > nat ) > ( product_prod_b_b > nat ) > product_prod_b_b > product_prod_nat_nat ).

thf(sy_c_BNF__Def_Oconvol_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001tf__a_001tf__a,type,
    bNF_co5212569348262174142_b_a_a: ( product_prod_b_b > a ) > ( product_prod_b_b > a ) > product_prod_b_b > product_prod_a_a ).

thf(sy_c_BNF__Def_Oconvol_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001tf__a,type,
    bNF_co6372728083680879980_a_a_a: ( a > product_prod_a_a ) > ( a > a ) > a > produc3802892049952890430_a_a_a ).

thf(sy_c_BNF__Def_Oconvol_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001tf__b,type,
    bNF_co6372728083680879981_a_a_b: ( a > product_prod_a_a ) > ( a > b ) > a > produc3802892054256119231_a_a_b ).

thf(sy_c_BNF__Def_Oconvol_001tf__a_001tf__a_001t__List__Olist_I_Eo_J,type,
    bNF_co6617811479755555573list_o: ( a > a ) > ( a > list_o ) > a > produc5884233991663340231list_o ).

thf(sy_c_BNF__Def_Oconvol_001tf__a_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    bNF_co1112036576516641918od_a_a: ( a > a ) > ( a > product_prod_a_a ) > a > produc4044097585999906000od_a_a ).

thf(sy_c_BNF__Def_Oconvol_001tf__a_001tf__a_001tf__a,type,
    bNF_convol_a_a_a: ( a > a ) > ( a > a ) > a > product_prod_a_a ).

thf(sy_c_BNF__Def_Oconvol_001tf__a_001tf__a_001tf__b,type,
    bNF_convol_a_a_b: ( a > a ) > ( a > b ) > a > product_prod_a_b ).

thf(sy_c_BNF__Def_Oconvol_001tf__a_001tf__b_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    bNF_co472596276184825789od_a_a: ( a > b ) > ( a > product_prod_a_a ) > a > produc3404657285668089871od_a_a ).

thf(sy_c_BNF__Def_Oconvol_001tf__a_001tf__b_001tf__a,type,
    bNF_convol_a_b_a: ( a > b ) > ( a > a ) > a > product_prod_b_a ).

thf(sy_c_BNF__Def_Oconvol_001tf__a_001tf__b_001tf__b,type,
    bNF_convol_a_b_b: ( a > b ) > ( a > b ) > a > product_prod_b_b ).

thf(sy_c_BNF__Def_Oconvol_001tf__b_001tf__b_001tf__b,type,
    bNF_convol_b_b_b: ( b > b ) > ( b > b ) > b > product_prod_b_b ).

thf(sy_c_BNF__Def_Ocsquare_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_I_Eo_J_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001tf__a_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_I_Eo_J_J,type,
    bNF_cs7935384970986586354list_o: set_Pr3077528766752018087list_o > ( product_prod_a_a > a ) > ( produc5884233991663340231list_o > a ) > ( produc5884233991663340231list_o > product_prod_a_a ) > ( produc5884233991663340231list_o > produc5884233991663340231list_o ) > $o ).

thf(sy_c_BNF__Def_Ocsquare_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    bNF_cs4698725948573326578od_a_a: set_Product_prod_a_a > ( product_prod_a_a > a ) > ( product_prod_a_a > a ) > ( product_prod_a_a > product_prod_a_a ) > ( product_prod_a_a > product_prod_a_a ) > $o ).

thf(sy_c_BNF__Def_Ocsquare_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001tf__b_001t__Product____Type__Oprod_Itf__b_Mtf__a_J,type,
    bNF_cs6251773214417337457od_b_a: set_Product_prod_a_a > ( product_prod_a_b > b ) > ( product_prod_b_a > b ) > ( product_prod_a_a > product_prod_a_b ) > ( product_prod_a_a > product_prod_b_a ) > $o ).

thf(sy_c_BNF__Def_Ocsquare_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
    bNF_cs5923635793044121010od_a_b: set_Product_prod_a_b > ( product_prod_a_a > a ) > ( product_prod_a_b > a ) > ( product_prod_a_b > product_prod_a_a ) > ( product_prod_a_b > product_prod_a_b ) > $o ).

thf(sy_c_BNF__Def_Ocsquare_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001tf__b_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
    bNF_cs7476683058888131889od_b_b: set_Product_prod_a_b > ( product_prod_a_b > b ) > ( product_prod_b_b > b ) > ( product_prod_a_b > product_prod_a_b ) > ( product_prod_a_b > product_prod_b_b ) > $o ).

thf(sy_c_BNF__Def_Ocsquare_001t__Product____Type__Oprod_Itf__b_Mt__List__Olist_I_Eo_J_J_001t__Product____Type__Oprod_Itf__b_Mtf__a_J_001tf__a_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_I_Eo_J_J,type,
    bNF_cs1821474949591695794list_o: set_Pr4565490804046816614list_o > ( product_prod_b_a > a ) > ( produc5884233991663340231list_o > a ) > ( produc7228699028472828166list_o > product_prod_b_a ) > ( produc7228699028472828166list_o > produc5884233991663340231list_o ) > $o ).

thf(sy_c_BNF__Def_Ocsquare_001t__Product____Type__Oprod_Itf__b_Mtf__a_J_001t__Product____Type__Oprod_Itf__b_Mtf__a_J_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    bNF_cs6158631073445945778od_a_a: set_Product_prod_b_a > ( product_prod_b_a > a ) > ( product_prod_a_a > a ) > ( product_prod_b_a > product_prod_b_a ) > ( product_prod_b_a > product_prod_a_a ) > $o ).

thf(sy_c_BNF__Def_Ocsquare_001t__Product____Type__Oprod_Itf__b_Mtf__a_J_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001tf__b_001t__Product____Type__Oprod_Itf__b_Mtf__a_J,type,
    bNF_cs7711678339289956657od_b_a: set_Product_prod_b_a > ( product_prod_b_b > b ) > ( product_prod_b_a > b ) > ( product_prod_b_a > product_prod_b_b ) > ( product_prod_b_a > product_prod_b_a ) > $o ).

thf(sy_c_BNF__Def_Ocsquare_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001t__Product____Type__Oprod_Itf__b_Mtf__a_J_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
    bNF_cs7383540917916740210od_a_b: set_Product_prod_b_b > ( product_prod_b_a > a ) > ( product_prod_a_b > a ) > ( product_prod_b_b > product_prod_b_a ) > ( product_prod_b_b > product_prod_a_b ) > $o ).

thf(sy_c_BNF__Def_Ocsquare_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001tf__b_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
    bNF_cs8936588183760751089od_b_b: set_Product_prod_b_b > ( product_prod_b_b > b ) > ( product_prod_b_b > b ) > ( product_prod_b_b > product_prod_b_b ) > ( product_prod_b_b > product_prod_b_b ) > $o ).

thf(sy_c_BNF__Def_OfstOp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_fs6799373675401686564at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_BNF__Def_OfstOp_001tf__a_001t__List__Olist_I_Eo_J_001t__List__Olist_I_Eo_J,type,
    bNF_fs1531470065415673164list_o: ( a > list_o > $o ) > ( list_o > list_o > $o ) > produc5884233991663340231list_o > produc5884233991663340231list_o ).

thf(sy_c_BNF__Def_OfstOp_001tf__a_001t__List__Olist_I_Eo_J_001tf__a,type,
    bNF_fstOp_a_list_o_a: ( a > list_o > $o ) > ( list_o > a > $o ) > product_prod_a_a > produc5884233991663340231list_o ).

thf(sy_c_BNF__Def_OfstOp_001tf__a_001t__List__Olist_I_Eo_J_001tf__b,type,
    bNF_fstOp_a_list_o_b: ( a > list_o > $o ) > ( list_o > b > $o ) > product_prod_a_b > produc5884233991663340231list_o ).

thf(sy_c_BNF__Def_OfstOp_001tf__a_001tf__a_001t__List__Olist_I_Eo_J,type,
    bNF_fstOp_a_a_list_o: ( a > a > $o ) > ( a > list_o > $o ) > produc5884233991663340231list_o > product_prod_a_a ).

thf(sy_c_BNF__Def_OfstOp_001tf__a_001tf__a_001tf__a,type,
    bNF_fstOp_a_a_a: ( a > a > $o ) > ( a > a > $o ) > product_prod_a_a > product_prod_a_a ).

thf(sy_c_BNF__Def_OfstOp_001tf__a_001tf__a_001tf__b,type,
    bNF_fstOp_a_a_b: ( a > a > $o ) > ( a > b > $o ) > product_prod_a_b > product_prod_a_a ).

thf(sy_c_BNF__Def_OfstOp_001tf__a_001tf__b_001t__List__Olist_I_Eo_J,type,
    bNF_fstOp_a_b_list_o: ( a > b > $o ) > ( b > list_o > $o ) > produc5884233991663340231list_o > product_prod_a_b ).

thf(sy_c_BNF__Def_OfstOp_001tf__a_001tf__b_001tf__a,type,
    bNF_fstOp_a_b_a: ( a > b > $o ) > ( b > a > $o ) > product_prod_a_a > product_prod_a_b ).

thf(sy_c_BNF__Def_OfstOp_001tf__a_001tf__b_001tf__b,type,
    bNF_fstOp_a_b_b: ( a > b > $o ) > ( b > b > $o ) > product_prod_a_b > product_prod_a_b ).

thf(sy_c_BNF__Def_OfstOp_001tf__b_001tf__a_001t__List__Olist_I_Eo_J,type,
    bNF_fstOp_b_a_list_o: ( b > a > $o ) > ( a > list_o > $o ) > produc7228699028472828166list_o > product_prod_b_a ).

thf(sy_c_BNF__Def_OfstOp_001tf__b_001tf__a_001tf__a,type,
    bNF_fstOp_b_a_a: ( b > a > $o ) > ( a > a > $o ) > product_prod_b_a > product_prod_b_a ).

thf(sy_c_BNF__Def_OfstOp_001tf__b_001tf__a_001tf__b,type,
    bNF_fstOp_b_a_b: ( b > a > $o ) > ( a > b > $o ) > product_prod_b_b > product_prod_b_a ).

thf(sy_c_BNF__Def_OfstOp_001tf__b_001tf__b_001tf__a,type,
    bNF_fstOp_b_b_a: ( b > b > $o ) > ( b > a > $o ) > product_prod_b_a > product_prod_b_b ).

thf(sy_c_BNF__Def_OfstOp_001tf__b_001tf__b_001tf__b,type,
    bNF_fstOp_b_b_b: ( b > b > $o ) > ( b > b > $o ) > product_prod_b_b > product_prod_b_b ).

thf(sy_c_BNF__Def_Opick__middlep_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_Itf__a_Mtf__b_J_J_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
    bNF_pi4789607327318135235od_b_b: ( product_prod_a_a > produc2233330766403592615od_a_b > $o ) > ( produc2233330766403592615od_a_b > product_prod_b_b > $o ) > product_prod_a_a > product_prod_b_b > produc2233330766403592615od_a_b ).

thf(sy_c_BNF__Def_Opick__middlep_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001tf__b,type,
    bNF_pi1436070092841995377_a_b_b: ( a > product_prod_a_b > $o ) > ( product_prod_a_b > b > $o ) > a > b > product_prod_a_b ).

thf(sy_c_BNF__Def_Opick__middlep_001tf__a_001tf__a_001t__List__Olist_I_Eo_J,type,
    bNF_pi4433405390038814778list_o: ( a > a > $o ) > ( a > list_o > $o ) > a > list_o > a ).

thf(sy_c_BNF__Def_Opick__middlep_001tf__a_001tf__a_001tf__a,type,
    bNF_pi1561361277288929946_a_a_a: ( a > a > $o ) > ( a > a > $o ) > a > a > a ).

thf(sy_c_BNF__Def_Opick__middlep_001tf__a_001tf__a_001tf__b,type,
    bNF_pi1561361277288929947_a_a_b: ( a > a > $o ) > ( a > b > $o ) > a > b > a ).

thf(sy_c_BNF__Def_Opick__middlep_001tf__a_001tf__b_001t__List__Olist_I_Eo_J,type,
    bNF_pi5777870426848302713list_o: ( a > b > $o ) > ( b > list_o > $o ) > a > list_o > b ).

thf(sy_c_BNF__Def_Opick__middlep_001tf__a_001tf__b_001tf__a,type,
    bNF_pi7997277732171029721_a_b_a: ( a > b > $o ) > ( b > a > $o ) > a > a > b ).

thf(sy_c_BNF__Def_Opick__middlep_001tf__a_001tf__b_001tf__b,type,
    bNF_pi7997277732171029722_a_b_b: ( a > b > $o ) > ( b > b > $o ) > a > b > b ).

thf(sy_c_BNF__Def_Opick__middlep_001tf__b_001tf__a_001tf__a,type,
    bNF_pi7157579892477060635_b_a_a: ( b > a > $o ) > ( a > a > $o ) > b > a > a ).

thf(sy_c_BNF__Def_Opick__middlep_001tf__b_001tf__a_001tf__b,type,
    bNF_pi7157579892477060636_b_a_b: ( b > a > $o ) > ( a > b > $o ) > b > b > a ).

thf(sy_c_BNF__Def_Opick__middlep_001tf__b_001tf__b_001tf__a,type,
    bNF_pi4370124310504384602_b_b_a: ( b > b > $o ) > ( b > a > $o ) > b > a > b ).

thf(sy_c_BNF__Def_Opick__middlep_001tf__b_001tf__b_001tf__b,type,
    bNF_pi4370124310504384603_b_b_b: ( b > b > $o ) > ( b > b > $o ) > b > b > b ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001tf__a_001tf__a,type,
    bNF_re2560612034516186845_b_a_a: ( product_prod_a_b > product_prod_a_b > $o ) > ( a > a > $o ) > ( product_prod_a_b > a ) > ( product_prod_a_b > a ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001tf__b_001tf__b,type,
    bNF_re8996528489398286621_b_b_b: ( product_prod_a_b > product_prod_a_b > $o ) > ( b > b > $o ) > ( product_prod_a_b > b ) > ( product_prod_a_b > b ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001tf__a_001tf__b_001_062_Itf__a_M_Eo_J_001_062_Itf__b_M_Eo_J,type,
    bNF_re5830743871565202077_o_b_o: ( a > b > $o ) > ( ( a > $o ) > ( b > $o ) > $o ) > ( a > a > $o ) > ( b > b > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001tf__a_001tf__b_001_Eo_001_Eo,type,
    bNF_rel_fun_a_b_o_o: ( a > b > $o ) > ( $o > $o > $o ) > ( a > $o ) > ( b > $o ) > $o ).

thf(sy_c_BNF__Def_OsndOp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_sn8916957246138178918at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_BNF__Def_OsndOp_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_Itf__a_Mtf__b_J_J_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
    bNF_sn6902690285988656819od_b_b: ( product_prod_a_a > produc2233330766403592615od_a_b > $o ) > ( produc2233330766403592615od_a_b > product_prod_b_b > $o ) > produc2933732385751300263od_b_b > produc500430175844901415od_b_b ).

thf(sy_c_BNF__Def_OsndOp_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
    bNF_sn3422074382465070516od_a_b: ( product_prod_a_b > product_prod_a_b > $o ) > ( product_prod_a_b > product_prod_a_b > $o ) > produc2233330766403592615od_a_b > produc2233330766403592615od_a_b ).

thf(sy_c_BNF__Def_OsndOp_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001tf__b,type,
    bNF_sn8185316101191594337_a_b_b: ( a > product_prod_a_b > $o ) > ( product_prod_a_b > b > $o ) > product_prod_a_b > produc3167243053651132542_a_b_b ).

thf(sy_c_BNF__Def_OsndOp_001tf__a_001tf__a_001t__List__Olist_I_Eo_J,type,
    bNF_sndOp_a_a_list_o: ( a > a > $o ) > ( a > list_o > $o ) > produc5884233991663340231list_o > produc5884233991663340231list_o ).

thf(sy_c_BNF__Def_OsndOp_001tf__a_001tf__a_001tf__a,type,
    bNF_sndOp_a_a_a: ( a > a > $o ) > ( a > a > $o ) > product_prod_a_a > product_prod_a_a ).

thf(sy_c_BNF__Def_OsndOp_001tf__a_001tf__a_001tf__b,type,
    bNF_sndOp_a_a_b: ( a > a > $o ) > ( a > b > $o ) > product_prod_a_b > product_prod_a_b ).

thf(sy_c_BNF__Def_OsndOp_001tf__a_001tf__b_001tf__a,type,
    bNF_sndOp_a_b_a: ( a > b > $o ) > ( b > a > $o ) > product_prod_a_a > product_prod_b_a ).

thf(sy_c_BNF__Def_OsndOp_001tf__a_001tf__b_001tf__b,type,
    bNF_sndOp_a_b_b: ( a > b > $o ) > ( b > b > $o ) > product_prod_a_b > product_prod_b_b ).

thf(sy_c_BNF__Def_OsndOp_001tf__b_001tf__a_001t__List__Olist_I_Eo_J,type,
    bNF_sndOp_b_a_list_o: ( b > a > $o ) > ( a > list_o > $o ) > produc7228699028472828166list_o > produc5884233991663340231list_o ).

thf(sy_c_BNF__Def_OsndOp_001tf__b_001tf__a_001tf__a,type,
    bNF_sndOp_b_a_a: ( b > a > $o ) > ( a > a > $o ) > product_prod_b_a > product_prod_a_a ).

thf(sy_c_BNF__Def_OsndOp_001tf__b_001tf__a_001tf__b,type,
    bNF_sndOp_b_a_b: ( b > a > $o ) > ( a > b > $o ) > product_prod_b_b > product_prod_a_b ).

thf(sy_c_BNF__Def_OsndOp_001tf__b_001tf__b_001tf__a,type,
    bNF_sndOp_b_b_a: ( b > b > $o ) > ( b > a > $o ) > product_prod_b_a > product_prod_b_a ).

thf(sy_c_BNF__Def_OsndOp_001tf__b_001tf__b_001tf__b,type,
    bNF_sndOp_b_b_b: ( b > b > $o ) > ( b > b > $o ) > product_prod_b_b > product_prod_b_b ).

thf(sy_c_BNF__Def_Ovimage2p_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
    bNF_vi6667456707345531737_nat_o: ( nat > nat ) > ( nat > nat ) > ( nat > nat > $o ) > nat > nat > $o ).

thf(sy_c_BNF__Def_Ovimage2p_001t__Nat__Onat_001tf__a_001t__Nat__Onat_001tf__a_001_Eo,type,
    bNF_vi53994321937086493at_a_o: ( nat > a ) > ( nat > a ) > ( a > a > $o ) > nat > nat > $o ).

thf(sy_c_BNF__Def_Ovimage2p_001t__Nat__Onat_001tf__a_001t__Nat__Onat_001tf__b_001_Eo,type,
    bNF_vi6378143959327467996at_b_o: ( nat > a ) > ( nat > b ) > ( a > b > $o ) > nat > nat > $o ).

thf(sy_c_BNF__Def_Ovimage2p_001tf__a_001t__Nat__Onat_001tf__a_001t__Nat__Onat_001_Eo,type,
    bNF_vi7065238521659171741_nat_o: ( a > nat ) > ( a > nat ) > ( nat > nat > $o ) > a > a > $o ).

thf(sy_c_BNF__Def_Ovimage2p_001tf__a_001tf__a_001tf__a_001tf__a_001_Eo,type,
    bNF_vi8224572084553194909_a_a_o: ( a > a ) > ( a > a ) > ( a > a > $o ) > a > a > $o ).

thf(sy_c_BNF__Def_Ovimage2p_001tf__b_001t__Nat__Onat_001tf__a_001t__Nat__Onat_001_Eo,type,
    bNF_vi3961714438943086494_nat_o: ( b > nat ) > ( a > nat ) > ( nat > nat > $o ) > b > a > $o ).

thf(sy_c_BNF__Def_Ovimage2p_001tf__b_001t__Nat__Onat_001tf__b_001t__Nat__Onat_001_Eo,type,
    bNF_vi1076643929659559261_nat_o: ( b > nat ) > ( b > nat ) > ( nat > nat > $o ) > b > b > $o ).

thf(sy_c_BNF__Def_Ovimage2p_001tf__b_001tf__a_001tf__a_001tf__a_001_Eo,type,
    bNF_vi277092895069316382_a_a_o: ( b > a ) > ( a > a ) > ( a > a > $o ) > b > a > $o ).

thf(sy_c_BNF__Def_Ovimage2p_001tf__b_001tf__a_001tf__a_001tf__b_001_Eo,type,
    bNF_vi6601242532459697885_a_b_o: ( b > a ) > ( a > b ) > ( a > b > $o ) > b > a > $o ).

thf(sy_c_BNF__Def_Ovimage2p_001tf__b_001tf__a_001tf__b_001tf__a_001_Eo,type,
    bNF_vi1512537224025710879_b_a_o: ( b > a ) > ( b > a ) > ( a > a > $o ) > b > b > $o ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Set__Oset_Itf__a_J_001tf__b_001tf__a,type,
    bNF_Gr3840943654187618371_a_b_a: set_set_a > ( set_a > b ) > ( set_a > a ) > set_Product_prod_b_a ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Set__Oset_Itf__a_J_001tf__b_001tf__b,type,
    bNF_Gr3840943654187618372_a_b_b: set_set_a > ( set_a > b ) > ( set_a > b ) > set_Product_prod_b_b ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001tf__a_001tf__a_001tf__a,type,
    bNF_Gr1766759448597441700_a_a_a: set_a > ( a > a ) > ( a > a ) > set_Product_prod_a_a ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001tf__a_001tf__a_001tf__b,type,
    bNF_Gr1766759448597441701_a_a_b: set_a > ( a > a ) > ( a > b ) > set_Product_prod_a_b ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001tf__a_001tf__b_001tf__a,type,
    bNF_Gr8202675903479541475_a_b_a: set_a > ( a > b ) > ( a > a ) > set_Product_prod_b_a ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001tf__a_001tf__b_001tf__b,type,
    bNF_Gr8202675903479541476_a_b_b: set_a > ( a > b ) > ( a > b ) > set_Product_prod_b_b ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001tf__b_001tf__a_001tf__a,type,
    bNF_Gr7362978063785572389_b_a_a: set_b > ( b > a ) > ( b > a ) > set_Product_prod_a_a ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001tf__b_001tf__a_001tf__b,type,
    bNF_Gr7362978063785572390_b_a_b: set_b > ( b > a ) > ( b > b ) > set_Product_prod_a_b ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001tf__b_001tf__b_001tf__a,type,
    bNF_Gr4575522481812896356_b_b_a: set_b > ( b > b ) > ( b > a ) > set_Product_prod_b_a ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001tf__b_001tf__b_001tf__b,type,
    bNF_Gr4575522481812896357_b_b_b: set_b > ( b > b ) > ( b > b ) > set_Product_prod_b_b ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001tf__a,type,
    bNF_We1162827675446709994_rel_a: set_Product_prod_a_a > $o ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim_001tf__a,type,
    bNF_We6697304935525757620inim_a: set_Product_prod_a_a > set_a > a > $o ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2_001tf__a,type,
    bNF_We3763454674811381836max2_a: set_Product_prod_a_a > a > a > a ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim_001tf__a,type,
    bNF_We5615626441682584778inim_a: set_Product_prod_a_a > set_a > a ).

thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
    basic_1112880799063329204od_a_b: ( product_prod_a_b > nat ) > ( product_prod_a_b > nat ) > produc2233330766403592615od_a_b > nat ).

thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod_001tf__a_001tf__a,type,
    basic_3571716426759296052od_a_a: ( a > nat ) > ( a > nat ) > product_prod_a_a > nat ).

thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod_001tf__a_001tf__b,type,
    basic_3571716426759296053od_a_b: ( a > nat ) > ( b > nat ) > product_prod_a_b > nat ).

thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod_001tf__b_001tf__b,type,
    basic_784260844786620020od_b_b: ( b > nat ) > ( b > nat ) > product_prod_b_b > nat ).

thf(sy_c_Basic__BNFs_Ofsts_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
    basic_7688285234108943239od_a_b: produc2233330766403592615od_a_b > set_Product_prod_a_b ).

thf(sy_c_Basic__BNFs_Ofsts_001tf__a_001tf__a,type,
    basic_fsts_a_a: product_prod_a_a > set_a ).

thf(sy_c_Basic__BNFs_Ofsts_001tf__a_001tf__b,type,
    basic_fsts_a_b: product_prod_a_b > set_a ).

thf(sy_c_Basic__BNFs_Ofstsp_001tf__a_001t__List__Olist_I_Eo_J,type,
    basic_fstsp_a_list_o: produc5884233991663340231list_o > a > $o ).

thf(sy_c_Basic__BNFs_Ofstsp_001tf__a_001tf__a,type,
    basic_fstsp_a_a: product_prod_a_a > a > $o ).

thf(sy_c_Basic__BNFs_Ofstsp_001tf__a_001tf__b,type,
    basic_fstsp_a_b: product_prod_a_b > a > $o ).

thf(sy_c_Basic__BNFs_Ofstsp_001tf__b_001tf__a,type,
    basic_fstsp_b_a: product_prod_b_a > b > $o ).

thf(sy_c_Basic__BNFs_Ofstsp_001tf__b_001tf__b,type,
    basic_fstsp_b_b: product_prod_b_b > b > $o ).

thf(sy_c_Basic__BNFs_Opred__prod_001tf__a_001tf__a,type,
    basic_pred_prod_a_a: ( a > $o ) > ( a > $o ) > product_prod_a_a > $o ).

thf(sy_c_Basic__BNFs_Opred__prod_001tf__a_001tf__b,type,
    basic_pred_prod_a_b: ( a > $o ) > ( b > $o ) > product_prod_a_b > $o ).

thf(sy_c_Basic__BNFs_Orel__prod_001tf__a_001tf__a_001tf__a_001tf__a,type,
    basic_8447574561315602101_a_a_a: ( a > a > $o ) > ( a > a > $o ) > product_prod_a_a > product_prod_a_a > $o ).

thf(sy_c_Basic__BNFs_Orel__prod_001tf__a_001tf__a_001tf__b_001tf__b,type,
    basic_5660118979342926069_a_b_b: ( a > a > $o ) > ( b > b > $o ) > product_prod_a_b > product_prod_a_b > $o ).

thf(sy_c_Basic__BNFs_Orel__prod_001tf__a_001tf__b_001tf__a_001tf__b,type,
    basic_4820421139648956983_b_a_b: ( a > b > $o ) > ( a > b > $o ) > product_prod_a_a > product_prod_b_b > $o ).

thf(sy_c_Basic__BNFs_Osnds_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
    basic_6470750372406100165od_a_b: produc2233330766403592615od_a_b > set_Product_prod_a_b ).

thf(sy_c_Basic__BNFs_Osnds_001tf__a_001tf__a,type,
    basic_snds_a_a: product_prod_a_a > set_a ).

thf(sy_c_Basic__BNFs_Osnds_001tf__a_001tf__b,type,
    basic_snds_a_b: product_prod_a_b > set_b ).

thf(sy_c_Basic__BNFs_Osndsp_001tf__a_001tf__a,type,
    basic_sndsp_a_a: product_prod_a_a > a > $o ).

thf(sy_c_Basic__BNFs_Osndsp_001tf__a_001tf__b,type,
    basic_sndsp_a_b: product_prod_a_b > b > $o ).

thf(sy_c_Basic__BNFs_Osndsp_001tf__b_001tf__a,type,
    basic_sndsp_b_a: product_prod_b_a > a > $o ).

thf(sy_c_Basic__BNFs_Osndsp_001tf__b_001tf__b,type,
    basic_sndsp_b_b: product_prod_b_b > b > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__b_J,type,
    comple2307003614231284044_set_b: set_set_b > set_b ).

thf(sy_c_Equiv__Relations_Oequiv_001tf__a,type,
    equiv_equiv_a: set_a > set_Product_prod_a_a > $o ).

thf(sy_c_Equiv__Relations_Oquotient_001tf__a,type,
    equiv_quotient_a: set_a > set_Product_prod_a_a > set_set_a ).

thf(sy_c_Finite__Set_Ofinite_001_062_Itf__a_Mt__Option__Ooption_It__List__Olist_I_Eo_J_J_J,type,
    finite6109038453002737250list_o: set_a_option_list_o > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_I_Eo_J,type,
    finite_finite_list_o: set_list_o > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_I_Eo_J_J,type,
    finite5626129373397972976list_o: set_Pr3077528766752018087list_o > $o ).

thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
    finite_finite_a: set_a > $o ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    comp_n2711240574518091068at_nat: ( nat > nat ) > ( product_prod_nat_nat > nat ) > product_prod_nat_nat > nat ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001tf__a_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    comp_n7987077848806063030at_nat: ( nat > a ) > ( product_prod_nat_nat > nat ) > product_prod_nat_nat > a ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001tf__b_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    comp_n8333496002694486645at_nat: ( nat > b ) > ( product_prod_nat_nat > nat ) > product_prod_nat_nat > b ).

thf(sy_c_Fun_Ocomp_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_001tf__a,type,
    comp_o7170176683143658715st_o_a: ( option_list_o > option_list_o ) > ( a > option_list_o ) > a > option_list_o ).

thf(sy_c_Fun_Ocomp_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_001tf__b,type,
    comp_o7170176683143658716st_o_b: ( option_list_o > option_list_o ) > ( b > option_list_o ) > b > option_list_o ).

thf(sy_c_Fun_Ocomp_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_001tf__a_001tf__a,type,
    comp_o695885166685416001_o_a_a: ( option_list_o > a ) > ( a > option_list_o ) > a > a ).

thf(sy_c_Fun_Ocomp_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_001tf__a_001tf__b,type,
    comp_o695885166685416002_o_a_b: ( option_list_o > a ) > ( b > option_list_o ) > b > a ).

thf(sy_c_Fun_Ocomp_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_001tf__b_001tf__a,type,
    comp_o7131801621567515776_o_b_a: ( option_list_o > b ) > ( a > option_list_o ) > a > b ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__b_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
    comp_P8899913933205946077od_a_b: ( produc3802892054256119231_a_a_b > product_prod_a_a ) > ( product_prod_a_b > produc3802892054256119231_a_a_b ) > product_prod_a_b > product_prod_a_a ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_Itf__a_Mtf__b_J_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
    comp_P3398507451812940164od_a_a: ( produc2233330766403592615od_a_b > product_prod_a_a ) > ( produc1065045136403542119od_a_a > produc2233330766403592615od_a_b ) > produc1065045136403542119od_a_a > product_prod_a_a ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
    comp_P7675370900755197723od_a_b: ( produc3167243049347903741_a_b_a > product_prod_a_a ) > ( product_prod_a_b > produc3167243049347903741_a_b_a ) > product_prod_a_b > product_prod_a_a ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mtf__a_J_001t__Product____Type__Oprod_Itf__b_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
    comp_P2391947450525258140od_a_b: ( produc3167243049347903741_a_b_a > product_prod_b_a ) > ( product_prod_a_b > produc3167243049347903741_a_b_a ) > product_prod_a_b > product_prod_b_a ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_I_Eo_J_J_001tf__a_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_I_Eo_J_J,type,
    comp_P9144535235892738285list_o: ( produc5884233991663340231list_o > a ) > ( produc5884233991663340231list_o > produc5884233991663340231list_o ) > produc5884233991663340231list_o > a ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_I_Eo_J_J_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    comp_P2559030035211449165od_a_a: ( produc5884233991663340231list_o > a ) > ( product_prod_a_a > produc5884233991663340231list_o ) > product_prod_a_a > a ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_I_Eo_J_J_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
    comp_P2559030039514677966od_a_b: ( produc5884233991663340231list_o > a ) > ( product_prod_a_b > produc5884233991663340231list_o ) > product_prod_a_b > a ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_I_Eo_J_J_001tf__a_001tf__a,type,
    comp_P941169184177061668_o_a_a: ( produc5884233991663340231list_o > a ) > ( a > produc5884233991663340231list_o ) > a > a ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Nat__Onat_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_Itf__a_Mtf__b_J_J,type,
    comp_P2842972232926698961od_a_b: ( product_prod_a_a > nat ) > ( produc2233330766403592615od_a_b > product_prod_a_a ) > produc2233330766403592615od_a_b > nat ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Nat__Onat_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    comp_P2525640977308048721od_a_a: ( product_prod_a_a > nat ) > ( product_prod_a_a > product_prod_a_a ) > product_prod_a_a > nat ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mtf__a_J,type,
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thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_I_Eo_J_J_001tf__a,type,
    image_8953000392185216508st_o_a: ( produc5884233991663340231list_o > a ) > set_Pr3077528766752018087list_o > set_a ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    image_257580707435209375at_nat: ( product_prod_a_a > product_prod_nat_nat ) > set_Product_prod_a_a > set_Pr1261947904930325089at_nat ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    image_4636654165204879301od_a_a: ( product_prod_a_a > product_prod_a_a ) > set_Product_prod_a_a > set_Product_prod_a_a ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__b_Mtf__a_J,type,
    image_1849198583232203268od_b_a: ( product_prod_a_a > product_prod_b_a ) > set_Product_prod_a_a > set_Product_prod_b_a ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
    image_1849198587535432069od_b_b: ( product_prod_a_a > product_prod_b_b ) > set_Product_prod_a_a > set_Product_prod_b_b ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001tf__a,type,
    image_3437945252899457948_a_a_a: ( product_prod_a_a > a ) > set_Product_prod_a_a > set_a ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
    image_3300603549555413765od_a_b: ( product_prod_a_b > product_prod_a_b ) > set_Product_prod_a_b > set_Product_prod_a_b ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_Itf__b_Mtf__a_J,type,
    image_513147963279508931od_b_a: ( product_prod_a_b > product_prod_b_a ) > set_Product_prod_a_b > set_Product_prod_b_a ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
    image_513147967582737732od_b_b: ( product_prod_a_b > product_prod_b_b ) > set_Product_prod_a_b > set_Product_prod_b_b ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001tf__a,type,
    image_2802296252294471259_a_b_a: ( product_prod_a_b > a ) > set_Product_prod_a_b > set_a ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001tf__b,type,
    image_2802296252294471260_a_b_b: ( product_prod_a_b > b ) > set_Product_prod_a_b > set_b ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__b_Mtf__a_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    image_9012383631328784800at_nat: ( product_prod_b_a > product_prod_nat_nat ) > set_Product_prod_b_a > set_Pr1261947904930325089at_nat ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__b_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    image_8576602751829715526od_a_a: ( product_prod_b_a > product_prod_a_a ) > set_Product_prod_b_a > set_Product_prod_a_a ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__b_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
    image_8576602756132944327od_a_b: ( product_prod_b_a > product_prod_a_b ) > set_Product_prod_b_a > set_Product_prod_a_b ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__b_Mtf__a_J_001t__Product____Type__Oprod_Itf__b_Mtf__a_J,type,
    image_5789147169857039493od_b_a: ( product_prod_b_a > product_prod_b_a ) > set_Product_prod_b_a > set_Product_prod_b_a ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__b_Mtf__a_J_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
    image_5789147174160268294od_b_b: ( product_prod_b_a > product_prod_b_b ) > set_Product_prod_b_a > set_Product_prod_b_b ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__b_Mtf__a_J_001tf__a,type,
    image_9034163868087588637_b_a_a: ( product_prod_b_a > a ) > set_Product_prod_b_a > set_a ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__b_Mtf__a_J_001tf__b,type,
    image_9034163868087588638_b_a_b: ( product_prod_b_a > b ) > set_Product_prod_b_a > set_b ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    image_1030136767971877087at_nat: ( product_prod_b_b > product_prod_nat_nat ) > set_Product_prod_b_b > set_Pr1261947904930325089at_nat ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    image_7240552131877021189od_a_a: ( product_prod_b_b > product_prod_a_a ) > set_Product_prod_b_b > set_Product_prod_a_a ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
    image_7240552136180249990od_a_b: ( product_prod_b_b > product_prod_a_b ) > set_Product_prod_b_b > set_Product_prod_a_b ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001t__Product____Type__Oprod_Itf__b_Mtf__a_J,type,
    image_4453096549904345156od_b_a: ( product_prod_b_b > product_prod_b_a ) > set_Product_prod_b_b > set_Product_prod_b_a ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
    image_4453096554207573957od_b_b: ( product_prod_b_b > product_prod_b_b ) > set_Product_prod_b_b > set_Product_prod_b_b ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001tf__b,type,
    image_8398514867482601949_b_b_b: ( product_prod_b_b > b ) > set_Product_prod_b_b > set_b ).

thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001tf__a,type,
    image_set_a_a: ( set_a > a ) > set_set_a > set_a ).

thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001tf__b,type,
    image_set_a_b: ( set_a > b ) > set_set_a > set_b ).

thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__b_J_001t__Set__Oset_Itf__b_J,type,
    image_set_b_set_b: ( set_b > set_b ) > set_set_b > set_set_b ).

thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__b_J_001tf__a,type,
    image_set_b_a: ( set_b > a ) > set_set_b > set_a ).

thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__b_J_001tf__b,type,
    image_set_b_b: ( set_b > b ) > set_set_b > set_b ).

thf(sy_c_Set_Oimage_001tf__a_001t__List__Olist_I_Eo_J,type,
    image_a_list_o: ( a > list_o ) > set_a > set_list_o ).

thf(sy_c_Set_Oimage_001tf__a_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
    image_5155782636146240747list_o: ( a > option_list_o ) > set_a > set_option_list_o ).

thf(sy_c_Set_Oimage_001tf__a_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_I_Eo_J_J,type,
    image_1329905666413968462list_o: ( a > produc5884233991663340231list_o ) > set_a > set_Pr3077528766752018087list_o ).

thf(sy_c_Set_Oimage_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
    image_7400625786893224495od_a_b: ( a > product_prod_a_b ) > set_a > set_Product_prod_a_b ).

thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_Itf__a_J,type,
    image_a_set_a: ( a > set_a ) > set_a > set_set_a ).

thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_Itf__b_J,type,
    image_a_set_b: ( a > set_b ) > set_a > set_set_b ).

thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
    image_a_a2: ( a > a ) > set_a > set_a ).

thf(sy_c_Set_Oimage_001tf__a_001tf__b,type,
    image_a_b: ( a > b ) > set_a > set_b ).

thf(sy_c_Set_Oimage_001tf__b_001t__List__Olist_I_Eo_J,type,
    image_b_list_o: ( b > list_o ) > set_b > set_list_o ).

thf(sy_c_Set_Oimage_001tf__b_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
    image_3819732016193546410list_o: ( b > option_list_o ) > set_b > set_option_list_o ).

thf(sy_c_Set_Oimage_001tf__b_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
    image_6761185486561408366od_a_b: ( b > product_prod_a_b ) > set_b > set_Product_prod_a_b ).

thf(sy_c_Set_Oimage_001tf__b_001t__Set__Oset_Itf__a_J,type,
    image_b_set_a: ( b > set_a ) > set_b > set_set_a ).

thf(sy_c_Set_Oimage_001tf__b_001tf__a,type,
    image_b_a: ( b > a ) > set_b > set_a ).

thf(sy_c_Set_Oimage_001tf__b_001tf__b,type,
    image_b_b: ( b > b ) > set_b > set_b ).

thf(sy_c_Set_Oinsert_001tf__a,type,
    insert_a: a > set_a > set_a ).

thf(sy_c_Set_Oinsert_001tf__b,type,
    insert_b: b > set_b > set_b ).

thf(sy_c_Set_Ovimage_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001tf__a,type,
    vimage545595664742944273_a_b_a: ( product_prod_a_b > a ) > set_a > set_Product_prod_a_b ).

thf(sy_c_Set_Ovimage_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001tf__b,type,
    vimage545595664742944274_a_b_b: ( product_prod_a_b > b ) > set_b > set_Product_prod_a_b ).

thf(sy_c_Transitive__Closure_Ortrancl_001tf__a,type,
    transitive_rtrancl_a: set_Product_prod_a_a > set_Product_prod_a_a ).

thf(sy_c_Wellfounded_Omax__ext_001tf__a,type,
    max_ext_a: set_Product_prod_a_a > set_Pr5845495582615845127_set_a ).

thf(sy_c_Wellfounded_Owf_001t__Nat__Onat,type,
    wf_nat: set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Wellfounded_Owf_001tf__a,type,
    wf_a: set_Product_prod_a_a > $o ).

thf(sy_c_member_001_062_Itf__a_Mt__Option__Ooption_It__List__Olist_I_Eo_J_J_J,type,
    member1229624942697005410list_o: ( a > option_list_o ) > set_a_option_list_o > $o ).

thf(sy_c_member_001t__List__Olist_I_Eo_J,type,
    member_list_o: list_o > set_list_o > $o ).

thf(sy_c_member_001t__Nat__Onat,type,
    member_nat: nat > set_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_I_Eo_J_Mtf__a_J,type,
    member8444411472706555312st_o_a: produc6380261841376349063st_o_a > set_Pr5014569662245023591st_o_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
    member4368417410682638992od_a_a: produc1065045136403542119od_a_a > set_Pr294077422471562311od_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
    member3803802450123998288od_b_b: produc500430175844901415od_b_b > set_Pr423563334681122311od_b_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
    member5765840452647959760od_b_b: produc2933732385751300263od_b_b > set_Pr8729903091103688327od_b_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_Itf__a_Mtf__b_J_J,type,
    member5065438833300252112od_a_b: produc2233330766403592615od_a_b > set_Pr5493900063595384711od_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mtf__a_J,type,
    member2194267517197600294_a_b_a: produc3167243049347903741_a_b_a > set_Pr2303819121155477725_a_b_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mtf__b_J,type,
    member2194267521500829095_a_b_b: produc3167243053651132542_a_b_b > set_Pr2374853161201823710_a_b_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
    member7983343339038529360_set_a: produc1703568184450464039_set_a > set_Pr5845495582615845127_set_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_I_Eo_J_J,type,
    member7948383622993546480list_o: produc5884233991663340231list_o > set_Pr3077528766752018087list_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
    member1426531481828664017od_a_b: product_prod_a_b > set_Product_prod_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mtf__a_J,type,
    member7862447932407534991od_b_a: product_prod_b_a > set_Product_prod_b_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
    member7862447936710763792od_b_b: product_prod_b_b > set_Product_prod_b_b > $o ).

thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
    member_set_a: set_a > set_set_a > $o ).

thf(sy_c_member_001t__Set__Oset_Itf__b_J,type,
    member_set_b: set_b > set_set_b > $o ).

thf(sy_c_member_001tf__a,type,
    member_a: a > set_a > $o ).

thf(sy_c_member_001tf__b,type,
    member_b: b > set_b > $o ).

thf(sy_v_e1,type,
    e1: a > option_list_o ).

thf(sy_v_e2,type,
    e2: a > b > option_list_o ).

thf(sy_v_x____,type,
    x: product_prod_a_b ).

thf(sy_v_y____,type,
    y: product_prod_a_b ).

% Relevant facts (1275)
thf(fact_0_assms_I1_J,axiom,
    prefix7485107378405021920ding_a @ e1 ).

% assms(1)
thf(fact_1__092_060open_062fst_Ax_A_092_060in_062_Adom_Ae1_092_060close_062,axiom,
    member_a @ ( product_fst_a_b @ x ) @ ( dom_a_list_o @ e1 ) ).

% \<open>fst x \<in> dom e1\<close>
thf(fact_2__092_060open_062opt__comp_A_Ie2_A_Ifst_Ax_J_A_Isnd_Ax_J_J_A_Ie2_A_Ifst_Ax_J_A_Isnd_Ay_J_J_092_060close_062,axiom,
    prefix454693708527911765comp_o @ ( e2 @ ( product_fst_a_b @ x ) @ ( product_snd_a_b @ x ) ) @ ( e2 @ ( product_fst_a_b @ x ) @ ( product_snd_a_b @ y ) ) ).

% \<open>opt_comp (e2 (fst x) (snd x)) (e2 (fst x) (snd y))\<close>
thf(fact_3_b,axiom,
    ( ( product_fst_a_b @ x )
    = ( product_fst_a_b @ y ) ) ).

% b
thf(fact_4_a,axiom,
    prefix454693708527911765comp_o @ ( prefix6990428588352057089od_a_b @ e1 @ e2 @ x ) @ ( prefix6990428588352057089od_a_b @ e1 @ e2 @ y ) ).

% a
thf(fact_5_d,axiom,
    prefix454693708527911765comp_o @ ( e1 @ ( product_fst_a_b @ x ) ) @ ( e1 @ ( product_fst_a_b @ y ) ) ).

% d
thf(fact_6_opt__comp__sym,axiom,
    ( prefix454693708527911765comp_o
    = ( ^ [X: option_list_o,Y: option_list_o] : ( prefix454693708527911765comp_o @ Y @ X ) ) ) ).

% opt_comp_sym
thf(fact_7_prod_Oexpand,axiom,
    ! [Prod: produc5884233991663340231list_o,Prod2: produc5884233991663340231list_o] :
      ( ( ( ( product_fst_a_list_o @ Prod )
          = ( product_fst_a_list_o @ Prod2 ) )
        & ( ( product_snd_a_list_o @ Prod )
          = ( product_snd_a_list_o @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_8_prod_Oexpand,axiom,
    ! [Prod: product_prod_b_a,Prod2: product_prod_b_a] :
      ( ( ( ( product_fst_b_a @ Prod )
          = ( product_fst_b_a @ Prod2 ) )
        & ( ( product_snd_b_a @ Prod )
          = ( product_snd_b_a @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_9_prod_Oexpand,axiom,
    ! [Prod: product_prod_b_b,Prod2: product_prod_b_b] :
      ( ( ( ( product_fst_b_b @ Prod )
          = ( product_fst_b_b @ Prod2 ) )
        & ( ( product_snd_b_b @ Prod )
          = ( product_snd_b_b @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_10_prod_Oexpand,axiom,
    ! [Prod: product_prod_a_a,Prod2: product_prod_a_a] :
      ( ( ( ( product_fst_a_a @ Prod )
          = ( product_fst_a_a @ Prod2 ) )
        & ( ( product_snd_a_a @ Prod )
          = ( product_snd_a_a @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_11_prod_Oexpand,axiom,
    ! [Prod: product_prod_a_b,Prod2: product_prod_a_b] :
      ( ( ( ( product_fst_a_b @ Prod )
          = ( product_fst_a_b @ Prod2 ) )
        & ( ( product_snd_a_b @ Prod )
          = ( product_snd_a_b @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_12_prod__eqI,axiom,
    ! [P: produc5884233991663340231list_o,Q: produc5884233991663340231list_o] :
      ( ( ( product_fst_a_list_o @ P )
        = ( product_fst_a_list_o @ Q ) )
     => ( ( ( product_snd_a_list_o @ P )
          = ( product_snd_a_list_o @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_13_prod__eqI,axiom,
    ! [P: product_prod_b_a,Q: product_prod_b_a] :
      ( ( ( product_fst_b_a @ P )
        = ( product_fst_b_a @ Q ) )
     => ( ( ( product_snd_b_a @ P )
          = ( product_snd_b_a @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_14_prod__eqI,axiom,
    ! [P: product_prod_b_b,Q: product_prod_b_b] :
      ( ( ( product_fst_b_b @ P )
        = ( product_fst_b_b @ Q ) )
     => ( ( ( product_snd_b_b @ P )
          = ( product_snd_b_b @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_15_prod__eqI,axiom,
    ! [P: product_prod_a_a,Q: product_prod_a_a] :
      ( ( ( product_fst_a_a @ P )
        = ( product_fst_a_a @ Q ) )
     => ( ( ( product_snd_a_a @ P )
          = ( product_snd_a_a @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_16_prod__eqI,axiom,
    ! [P: product_prod_a_b,Q: product_prod_a_b] :
      ( ( ( product_fst_a_b @ P )
        = ( product_fst_a_b @ Q ) )
     => ( ( ( product_snd_a_b @ P )
          = ( product_snd_a_b @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_17_exE__realizer_H,axiom,
    ! [P2: list_o > a > $o,P: produc5884233991663340231list_o] :
      ( ( P2 @ ( product_snd_a_list_o @ P ) @ ( product_fst_a_list_o @ P ) )
     => ~ ! [X2: a,Y2: list_o] :
            ~ ( P2 @ Y2 @ X2 ) ) ).

% exE_realizer'
thf(fact_18_exE__realizer_H,axiom,
    ! [P2: a > b > $o,P: product_prod_b_a] :
      ( ( P2 @ ( product_snd_b_a @ P ) @ ( product_fst_b_a @ P ) )
     => ~ ! [X2: b,Y2: a] :
            ~ ( P2 @ Y2 @ X2 ) ) ).

% exE_realizer'
thf(fact_19_exE__realizer_H,axiom,
    ! [P2: b > b > $o,P: product_prod_b_b] :
      ( ( P2 @ ( product_snd_b_b @ P ) @ ( product_fst_b_b @ P ) )
     => ~ ! [X2: b,Y2: b] :
            ~ ( P2 @ Y2 @ X2 ) ) ).

% exE_realizer'
thf(fact_20_exE__realizer_H,axiom,
    ! [P2: a > a > $o,P: product_prod_a_a] :
      ( ( P2 @ ( product_snd_a_a @ P ) @ ( product_fst_a_a @ P ) )
     => ~ ! [X2: a,Y2: a] :
            ~ ( P2 @ Y2 @ X2 ) ) ).

% exE_realizer'
thf(fact_21_exE__realizer_H,axiom,
    ! [P2: b > a > $o,P: product_prod_a_b] :
      ( ( P2 @ ( product_snd_a_b @ P ) @ ( product_fst_a_b @ P ) )
     => ~ ! [X2: a,Y2: b] :
            ~ ( P2 @ Y2 @ X2 ) ) ).

% exE_realizer'
thf(fact_22_prod__eq__iff,axiom,
    ( ( ^ [Y3: produc5884233991663340231list_o,Z: produc5884233991663340231list_o] : ( Y3 = Z ) )
    = ( ^ [S: produc5884233991663340231list_o,T: produc5884233991663340231list_o] :
          ( ( ( product_fst_a_list_o @ S )
            = ( product_fst_a_list_o @ T ) )
          & ( ( product_snd_a_list_o @ S )
            = ( product_snd_a_list_o @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_23_prod__eq__iff,axiom,
    ( ( ^ [Y3: product_prod_b_a,Z: product_prod_b_a] : ( Y3 = Z ) )
    = ( ^ [S: product_prod_b_a,T: product_prod_b_a] :
          ( ( ( product_fst_b_a @ S )
            = ( product_fst_b_a @ T ) )
          & ( ( product_snd_b_a @ S )
            = ( product_snd_b_a @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_24_prod__eq__iff,axiom,
    ( ( ^ [Y3: product_prod_b_b,Z: product_prod_b_b] : ( Y3 = Z ) )
    = ( ^ [S: product_prod_b_b,T: product_prod_b_b] :
          ( ( ( product_fst_b_b @ S )
            = ( product_fst_b_b @ T ) )
          & ( ( product_snd_b_b @ S )
            = ( product_snd_b_b @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_25_prod__eq__iff,axiom,
    ( ( ^ [Y3: product_prod_a_a,Z: product_prod_a_a] : ( Y3 = Z ) )
    = ( ^ [S: product_prod_a_a,T: product_prod_a_a] :
          ( ( ( product_fst_a_a @ S )
            = ( product_fst_a_a @ T ) )
          & ( ( product_snd_a_a @ S )
            = ( product_snd_a_a @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_26_prod__eq__iff,axiom,
    ( ( ^ [Y3: product_prod_a_b,Z: product_prod_a_b] : ( Y3 = Z ) )
    = ( ^ [S: product_prod_a_b,T: product_prod_a_b] :
          ( ( ( product_fst_a_b @ S )
            = ( product_fst_a_b @ T ) )
          & ( ( product_snd_a_b @ S )
            = ( product_snd_a_b @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_27_sndsp_Ocases,axiom,
    ! [P: product_prod_b_a,A: a] :
      ( ( basic_sndsp_b_a @ P @ A )
     => ( A
        = ( product_snd_b_a @ P ) ) ) ).

% sndsp.cases
thf(fact_28_sndsp_Ocases,axiom,
    ! [P: product_prod_b_b,A: b] :
      ( ( basic_sndsp_b_b @ P @ A )
     => ( A
        = ( product_snd_b_b @ P ) ) ) ).

% sndsp.cases
thf(fact_29_sndsp_Ocases,axiom,
    ! [P: product_prod_a_a,A: a] :
      ( ( basic_sndsp_a_a @ P @ A )
     => ( A
        = ( product_snd_a_a @ P ) ) ) ).

% sndsp.cases
thf(fact_30_sndsp_Ocases,axiom,
    ! [P: product_prod_a_b,A: b] :
      ( ( basic_sndsp_a_b @ P @ A )
     => ( A
        = ( product_snd_a_b @ P ) ) ) ).

% sndsp.cases
thf(fact_31_sndsp_Osimps,axiom,
    ( basic_sndsp_b_a
    = ( ^ [P3: product_prod_b_a,A2: a] :
          ( A2
          = ( product_snd_b_a @ P3 ) ) ) ) ).

% sndsp.simps
thf(fact_32_sndsp_Osimps,axiom,
    ( basic_sndsp_b_b
    = ( ^ [P3: product_prod_b_b,A2: b] :
          ( A2
          = ( product_snd_b_b @ P3 ) ) ) ) ).

% sndsp.simps
thf(fact_33_sndsp_Osimps,axiom,
    ( basic_sndsp_a_a
    = ( ^ [P3: product_prod_a_a,A2: a] :
          ( A2
          = ( product_snd_a_a @ P3 ) ) ) ) ).

% sndsp.simps
thf(fact_34_sndsp_Osimps,axiom,
    ( basic_sndsp_a_b
    = ( ^ [P3: product_prod_a_b,A2: b] :
          ( A2
          = ( product_snd_a_b @ P3 ) ) ) ) ).

% sndsp.simps
thf(fact_35_sndsp_Ointros,axiom,
    ! [P: product_prod_b_a] : ( basic_sndsp_b_a @ P @ ( product_snd_b_a @ P ) ) ).

% sndsp.intros
thf(fact_36_sndsp_Ointros,axiom,
    ! [P: product_prod_b_b] : ( basic_sndsp_b_b @ P @ ( product_snd_b_b @ P ) ) ).

% sndsp.intros
thf(fact_37_sndsp_Ointros,axiom,
    ! [P: product_prod_a_a] : ( basic_sndsp_a_a @ P @ ( product_snd_a_a @ P ) ) ).

% sndsp.intros
thf(fact_38_sndsp_Ointros,axiom,
    ! [P: product_prod_a_b] : ( basic_sndsp_a_b @ P @ ( product_snd_a_b @ P ) ) ).

% sndsp.intros
thf(fact_39_encode__dependent__prod__def,axiom,
    ( prefix213913222920965280list_o
    = ( ^ [E: a > option_list_o,F: a > list_o > option_list_o,X: produc5884233991663340231list_o] : ( prefix5314359684614007693append @ ( E @ ( product_fst_a_list_o @ X ) ) @ ( F @ ( product_fst_a_list_o @ X ) @ ( product_snd_a_list_o @ X ) ) ) ) ) ).

% encode_dependent_prod_def
thf(fact_40_encode__dependent__prod__def,axiom,
    ( prefix4202973006379381055od_b_a
    = ( ^ [E: b > option_list_o,F: b > a > option_list_o,X: product_prod_b_a] : ( prefix5314359684614007693append @ ( E @ ( product_fst_b_a @ X ) ) @ ( F @ ( product_fst_b_a @ X ) @ ( product_snd_b_a @ X ) ) ) ) ) ).

% encode_dependent_prod_def
thf(fact_41_encode__dependent__prod__def,axiom,
    ( prefix4202973006379381056od_b_b
    = ( ^ [E: b > option_list_o,F: b > b > option_list_o,X: product_prod_b_b] : ( prefix5314359684614007693append @ ( E @ ( product_fst_b_b @ X ) ) @ ( F @ ( product_fst_b_b @ X ) @ ( product_snd_b_b @ X ) ) ) ) ) ).

% encode_dependent_prod_def
thf(fact_42_encode__dependent__prod__def,axiom,
    ( prefix6990428588352057088od_a_a
    = ( ^ [E: a > option_list_o,F: a > a > option_list_o,X: product_prod_a_a] : ( prefix5314359684614007693append @ ( E @ ( product_fst_a_a @ X ) ) @ ( F @ ( product_fst_a_a @ X ) @ ( product_snd_a_a @ X ) ) ) ) ) ).

% encode_dependent_prod_def
thf(fact_43_encode__dependent__prod__def,axiom,
    ( prefix6990428588352057089od_a_b
    = ( ^ [E: a > option_list_o,F: a > b > option_list_o,X: product_prod_a_b] : ( prefix5314359684614007693append @ ( E @ ( product_fst_a_b @ X ) ) @ ( F @ ( product_fst_a_b @ X ) @ ( product_snd_a_b @ X ) ) ) ) ) ).

% encode_dependent_prod_def
thf(fact_44_snd__apsnd,axiom,
    ! [F2: a > b,X3: product_prod_a_a] :
      ( ( product_snd_a_b @ ( product_apsnd_a_b_a @ F2 @ X3 ) )
      = ( F2 @ ( product_snd_a_a @ X3 ) ) ) ).

% snd_apsnd
thf(fact_45_snd__apsnd,axiom,
    ! [F2: a > a,X3: product_prod_b_a] :
      ( ( product_snd_b_a @ ( product_apsnd_a_a_b @ F2 @ X3 ) )
      = ( F2 @ ( product_snd_b_a @ X3 ) ) ) ).

% snd_apsnd
thf(fact_46_snd__apsnd,axiom,
    ! [F2: b > a,X3: product_prod_b_b] :
      ( ( product_snd_b_a @ ( product_apsnd_b_a_b @ F2 @ X3 ) )
      = ( F2 @ ( product_snd_b_b @ X3 ) ) ) ).

% snd_apsnd
thf(fact_47_snd__apsnd,axiom,
    ! [F2: a > b,X3: product_prod_b_a] :
      ( ( product_snd_b_b @ ( product_apsnd_a_b_b @ F2 @ X3 ) )
      = ( F2 @ ( product_snd_b_a @ X3 ) ) ) ).

% snd_apsnd
thf(fact_48_snd__apsnd,axiom,
    ! [F2: b > b,X3: product_prod_b_b] :
      ( ( product_snd_b_b @ ( product_apsnd_b_b_b @ F2 @ X3 ) )
      = ( F2 @ ( product_snd_b_b @ X3 ) ) ) ).

% snd_apsnd
thf(fact_49_snd__apsnd,axiom,
    ! [F2: b > a,X3: product_prod_a_b] :
      ( ( product_snd_a_a @ ( product_apsnd_b_a_a @ F2 @ X3 ) )
      = ( F2 @ ( product_snd_a_b @ X3 ) ) ) ).

% snd_apsnd
thf(fact_50_snd__apsnd,axiom,
    ! [F2: a > a,X3: product_prod_a_a] :
      ( ( product_snd_a_a @ ( product_apsnd_a_a_a @ F2 @ X3 ) )
      = ( F2 @ ( product_snd_a_a @ X3 ) ) ) ).

% snd_apsnd
thf(fact_51_snd__apsnd,axiom,
    ! [F2: b > b,X3: product_prod_a_b] :
      ( ( product_snd_a_b @ ( product_apsnd_b_b_a @ F2 @ X3 ) )
      = ( F2 @ ( product_snd_a_b @ X3 ) ) ) ).

% snd_apsnd
thf(fact_52_fst__apsnd,axiom,
    ! [F2: b > b,X3: product_prod_a_b] :
      ( ( product_fst_a_b @ ( product_apsnd_b_b_a @ F2 @ X3 ) )
      = ( product_fst_a_b @ X3 ) ) ).

% fst_apsnd
thf(fact_53_fst__apsnd,axiom,
    ! [F2: a > b,X3: product_prod_a_a] :
      ( ( product_fst_a_b @ ( product_apsnd_a_b_a @ F2 @ X3 ) )
      = ( product_fst_a_a @ X3 ) ) ).

% fst_apsnd
thf(fact_54_fst__apsnd,axiom,
    ! [F2: a > a,X3: product_prod_b_a] :
      ( ( product_fst_b_a @ ( product_apsnd_a_a_b @ F2 @ X3 ) )
      = ( product_fst_b_a @ X3 ) ) ).

% fst_apsnd
thf(fact_55_fst__apsnd,axiom,
    ! [F2: b > a,X3: product_prod_b_b] :
      ( ( product_fst_b_a @ ( product_apsnd_b_a_b @ F2 @ X3 ) )
      = ( product_fst_b_b @ X3 ) ) ).

% fst_apsnd
thf(fact_56_fst__apsnd,axiom,
    ! [F2: b > a,X3: product_prod_a_b] :
      ( ( product_fst_a_a @ ( product_apsnd_b_a_a @ F2 @ X3 ) )
      = ( product_fst_a_b @ X3 ) ) ).

% fst_apsnd
thf(fact_57_fst__apsnd,axiom,
    ! [F2: a > a,X3: product_prod_a_a] :
      ( ( product_fst_a_a @ ( product_apsnd_a_a_a @ F2 @ X3 ) )
      = ( product_fst_a_a @ X3 ) ) ).

% fst_apsnd
thf(fact_58_fst__apsnd,axiom,
    ! [F2: a > b,X3: product_prod_b_a] :
      ( ( product_fst_b_b @ ( product_apsnd_a_b_b @ F2 @ X3 ) )
      = ( product_fst_b_a @ X3 ) ) ).

% fst_apsnd
thf(fact_59_fst__apsnd,axiom,
    ! [F2: b > b,X3: product_prod_b_b] :
      ( ( product_fst_b_b @ ( product_apsnd_b_b_b @ F2 @ X3 ) )
      = ( product_fst_b_b @ X3 ) ) ).

% fst_apsnd
thf(fact_60_fst__apsnd,axiom,
    ! [F2: list_o > b,X3: produc5884233991663340231list_o] :
      ( ( product_fst_a_b @ ( produc3255047714182728159_o_b_a @ F2 @ X3 ) )
      = ( product_fst_a_list_o @ X3 ) ) ).

% fst_apsnd
thf(fact_61_fst__apsnd,axiom,
    ! [F2: list_o > a,X3: produc5884233991663340231list_o] :
      ( ( product_fst_a_a @ ( produc6042503296155404192_o_a_a @ F2 @ X3 ) )
      = ( product_fst_a_list_o @ X3 ) ) ).

% fst_apsnd
thf(fact_62_apsnd__eq__conv,axiom,
    ! [F2: b > b,X3: product_prod_a_b,G: b > b] :
      ( ( ( product_apsnd_b_b_a @ F2 @ X3 )
        = ( product_apsnd_b_b_a @ G @ X3 ) )
      = ( ( F2 @ ( product_snd_a_b @ X3 ) )
        = ( G @ ( product_snd_a_b @ X3 ) ) ) ) ).

% apsnd_eq_conv
thf(fact_63_assms_I2_J,axiom,
    ! [X3: a] :
      ( ( member_a @ X3 @ ( dom_a_list_o @ e1 ) )
     => ( prefix7485107378405021921ding_b @ ( e2 @ X3 ) ) ) ).

% assms(2)
thf(fact_64_opt__comp__append,axiom,
    ! [X3: option_list_o,Y4: option_list_o,Z2: option_list_o] :
      ( ( prefix454693708527911765comp_o @ ( prefix5314359684614007693append @ X3 @ Y4 ) @ Z2 )
     => ( prefix454693708527911765comp_o @ X3 @ Z2 ) ) ).

% opt_comp_append
thf(fact_65_opt__comp__append__2,axiom,
    ! [X3: option_list_o,Y4: option_list_o,Z2: option_list_o] :
      ( ( prefix454693708527911765comp_o @ X3 @ ( prefix5314359684614007693append @ Y4 @ Z2 ) )
     => ( prefix454693708527911765comp_o @ X3 @ Y4 ) ) ).

% opt_comp_append_2
thf(fact_66_opt__comp__append__3,axiom,
    ! [X3: option_list_o,Y4: option_list_o,Z2: option_list_o] :
      ( ( prefix454693708527911765comp_o @ ( prefix5314359684614007693append @ X3 @ Y4 ) @ ( prefix5314359684614007693append @ X3 @ Z2 ) )
     => ( prefix454693708527911765comp_o @ Y4 @ Z2 ) ) ).

% opt_comp_append_3
thf(fact_67_is__encodingI__2,axiom,
    ! [E2: a > option_list_o] :
      ( ! [X2: a,Y2: a] :
          ( ( prefix454693708527911765comp_o @ ( E2 @ X2 ) @ ( E2 @ Y2 ) )
         => ( X2 = Y2 ) )
     => ( prefix7485107378405021920ding_a @ E2 ) ) ).

% is_encodingI_2
thf(fact_68_is__encodingI__2,axiom,
    ! [E2: b > option_list_o] :
      ( ! [X2: b,Y2: b] :
          ( ( prefix454693708527911765comp_o @ ( E2 @ X2 ) @ ( E2 @ Y2 ) )
         => ( X2 = Y2 ) )
     => ( prefix7485107378405021921ding_b @ E2 ) ) ).

% is_encodingI_2
thf(fact_69_is__encodingD,axiom,
    ! [E2: a > option_list_o,X3: a,Y4: a] :
      ( ( prefix7485107378405021920ding_a @ E2 )
     => ( ( prefix454693708527911765comp_o @ ( E2 @ X3 ) @ ( E2 @ Y4 ) )
       => ( X3 = Y4 ) ) ) ).

% is_encodingD
thf(fact_70_is__encodingD,axiom,
    ! [E2: b > option_list_o,X3: b,Y4: b] :
      ( ( prefix7485107378405021921ding_b @ E2 )
     => ( ( prefix454693708527911765comp_o @ ( E2 @ X3 ) @ ( E2 @ Y4 ) )
       => ( X3 = Y4 ) ) ) ).

% is_encodingD
thf(fact_71_snd__swap,axiom,
    ! [X3: produc5884233991663340231list_o] :
      ( ( product_snd_list_o_a @ ( produc370354279364378199list_o @ X3 ) )
      = ( product_fst_a_list_o @ X3 ) ) ).

% snd_swap
thf(fact_72_snd__swap,axiom,
    ! [X3: product_prod_b_b] :
      ( ( product_snd_b_b @ ( product_swap_b_b @ X3 ) )
      = ( product_fst_b_b @ X3 ) ) ).

% snd_swap
thf(fact_73_snd__swap,axiom,
    ! [X3: product_prod_a_a] :
      ( ( product_snd_a_a @ ( product_swap_a_a @ X3 ) )
      = ( product_fst_a_a @ X3 ) ) ).

% snd_swap
thf(fact_74_snd__swap,axiom,
    ! [X3: product_prod_a_b] :
      ( ( product_snd_b_a @ ( product_swap_a_b @ X3 ) )
      = ( product_fst_a_b @ X3 ) ) ).

% snd_swap
thf(fact_75_snd__swap,axiom,
    ! [X3: product_prod_b_a] :
      ( ( product_snd_a_b @ ( product_swap_b_a @ X3 ) )
      = ( product_fst_b_a @ X3 ) ) ).

% snd_swap
thf(fact_76_fst__swap,axiom,
    ! [X3: produc6380261841376349063st_o_a] :
      ( ( product_fst_a_list_o @ ( produc84879911036722967st_o_a @ X3 ) )
      = ( product_snd_list_o_a @ X3 ) ) ).

% fst_swap
thf(fact_77_fst__swap,axiom,
    ! [X3: product_prod_b_b] :
      ( ( product_fst_b_b @ ( product_swap_b_b @ X3 ) )
      = ( product_snd_b_b @ X3 ) ) ).

% fst_swap
thf(fact_78_fst__swap,axiom,
    ! [X3: product_prod_a_a] :
      ( ( product_fst_a_a @ ( product_swap_a_a @ X3 ) )
      = ( product_snd_a_a @ X3 ) ) ).

% fst_swap
thf(fact_79_fst__swap,axiom,
    ! [X3: product_prod_b_a] :
      ( ( product_fst_a_b @ ( product_swap_b_a @ X3 ) )
      = ( product_snd_b_a @ X3 ) ) ).

% fst_swap
thf(fact_80_fst__swap,axiom,
    ! [X3: product_prod_a_b] :
      ( ( product_fst_b_a @ ( product_swap_a_b @ X3 ) )
      = ( product_snd_a_b @ X3 ) ) ).

% fst_swap
thf(fact_81_encoding__imp__inj,axiom,
    ! [F2: a > option_list_o] :
      ( ( prefix7485107378405021920ding_a @ F2 )
     => ( inj_on374126998980950615list_o @ F2 @ ( dom_a_list_o @ F2 ) ) ) ).

% encoding_imp_inj
thf(fact_82_encoding__imp__inj,axiom,
    ! [F2: b > option_list_o] :
      ( ( prefix7485107378405021921ding_b @ F2 )
     => ( inj_on8261448415883032086list_o @ F2 @ ( dom_b_list_o @ F2 ) ) ) ).

% encoding_imp_inj
thf(fact_83_fstsp_Ocases,axiom,
    ! [P: product_prod_b_a,A: b] :
      ( ( basic_fstsp_b_a @ P @ A )
     => ( A
        = ( product_fst_b_a @ P ) ) ) ).

% fstsp.cases
thf(fact_84_fstsp_Ocases,axiom,
    ! [P: product_prod_a_a,A: a] :
      ( ( basic_fstsp_a_a @ P @ A )
     => ( A
        = ( product_fst_a_a @ P ) ) ) ).

% fstsp.cases
thf(fact_85_fstsp_Ocases,axiom,
    ! [P: product_prod_b_b,A: b] :
      ( ( basic_fstsp_b_b @ P @ A )
     => ( A
        = ( product_fst_b_b @ P ) ) ) ).

% fstsp.cases
thf(fact_86_fstsp_Ocases,axiom,
    ! [P: produc5884233991663340231list_o,A: a] :
      ( ( basic_fstsp_a_list_o @ P @ A )
     => ( A
        = ( product_fst_a_list_o @ P ) ) ) ).

% fstsp.cases
thf(fact_87_fstsp_Ocases,axiom,
    ! [P: product_prod_a_b,A: a] :
      ( ( basic_fstsp_a_b @ P @ A )
     => ( A
        = ( product_fst_a_b @ P ) ) ) ).

% fstsp.cases
thf(fact_88_fstsp_Osimps,axiom,
    ( basic_fstsp_b_a
    = ( ^ [P3: product_prod_b_a,A2: b] :
          ( A2
          = ( product_fst_b_a @ P3 ) ) ) ) ).

% fstsp.simps
thf(fact_89_fstsp_Osimps,axiom,
    ( basic_fstsp_a_a
    = ( ^ [P3: product_prod_a_a,A2: a] :
          ( A2
          = ( product_fst_a_a @ P3 ) ) ) ) ).

% fstsp.simps
thf(fact_90_fstsp_Osimps,axiom,
    ( basic_fstsp_b_b
    = ( ^ [P3: product_prod_b_b,A2: b] :
          ( A2
          = ( product_fst_b_b @ P3 ) ) ) ) ).

% fstsp.simps
thf(fact_91_fstsp_Osimps,axiom,
    ( basic_fstsp_a_list_o
    = ( ^ [P3: produc5884233991663340231list_o,A2: a] :
          ( A2
          = ( product_fst_a_list_o @ P3 ) ) ) ) ).

% fstsp.simps
thf(fact_92_fstsp_Osimps,axiom,
    ( basic_fstsp_a_b
    = ( ^ [P3: product_prod_a_b,A2: a] :
          ( A2
          = ( product_fst_a_b @ P3 ) ) ) ) ).

% fstsp.simps
thf(fact_93_fstsp_Ointros,axiom,
    ! [P: product_prod_b_a] : ( basic_fstsp_b_a @ P @ ( product_fst_b_a @ P ) ) ).

% fstsp.intros
thf(fact_94_fstsp_Ointros,axiom,
    ! [P: product_prod_a_a] : ( basic_fstsp_a_a @ P @ ( product_fst_a_a @ P ) ) ).

% fstsp.intros
thf(fact_95_fstsp_Ointros,axiom,
    ! [P: product_prod_b_b] : ( basic_fstsp_b_b @ P @ ( product_fst_b_b @ P ) ) ).

% fstsp.intros
thf(fact_96_fstsp_Ointros,axiom,
    ! [P: produc5884233991663340231list_o] : ( basic_fstsp_a_list_o @ P @ ( product_fst_a_list_o @ P ) ) ).

% fstsp.intros
thf(fact_97_fstsp_Ointros,axiom,
    ! [P: product_prod_a_b] : ( basic_fstsp_a_b @ P @ ( product_fst_a_b @ P ) ) ).

% fstsp.intros
thf(fact_98_is__encoding__def,axiom,
    ( prefix7485107378405021920ding_a
    = ( ^ [F: a > option_list_o] :
        ! [X: a,Y: a] :
          ( ( prefix8824957607401505554efix_o @ ( F @ X ) @ ( F @ Y ) )
         => ( X = Y ) ) ) ) ).

% is_encoding_def
thf(fact_99_is__encoding__def,axiom,
    ( prefix7485107378405021921ding_b
    = ( ^ [F: b > option_list_o] :
        ! [X: b,Y: b] :
          ( ( prefix8824957607401505554efix_o @ ( F @ X ) @ ( F @ Y ) )
         => ( X = Y ) ) ) ) ).

% is_encoding_def
thf(fact_100_prod_Ocollapse,axiom,
    ! [Prod: produc5884233991663340231list_o] :
      ( ( produc6899623729582506167list_o @ ( product_fst_a_list_o @ Prod ) @ ( product_snd_a_list_o @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_101_prod_Ocollapse,axiom,
    ! [Prod: product_prod_b_a] :
      ( ( product_Pair_b_a @ ( product_fst_b_a @ Prod ) @ ( product_snd_b_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_102_prod_Ocollapse,axiom,
    ! [Prod: product_prod_b_b] :
      ( ( product_Pair_b_b @ ( product_fst_b_b @ Prod ) @ ( product_snd_b_b @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_103_prod_Ocollapse,axiom,
    ! [Prod: product_prod_a_a] :
      ( ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_104_prod_Ocollapse,axiom,
    ! [Prod: product_prod_a_b] :
      ( ( product_Pair_a_b @ ( product_fst_a_b @ Prod ) @ ( product_snd_a_b @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_105_fst__comp__apsnd,axiom,
    ! [F2: b > b] :
      ( ( comp_P2648956292766268207od_a_b @ product_fst_a_b @ ( product_apsnd_b_b_a @ F2 ) )
      = product_fst_a_b ) ).

% fst_comp_apsnd
thf(fact_106_fst__comp__apsnd,axiom,
    ! [F2: a > b] :
      ( ( comp_P2648956288463039406od_a_a @ product_fst_a_b @ ( product_apsnd_a_b_a @ F2 ) )
      = product_fst_a_a ) ).

% fst_comp_apsnd
thf(fact_107_fst__comp__apsnd,axiom,
    ! [F2: a > a] :
      ( ( comp_P7171963437664642794od_b_a @ product_fst_b_a @ ( product_apsnd_a_a_b @ F2 ) )
      = product_fst_b_a ) ).

% fst_comp_apsnd
thf(fact_108_fst__comp__apsnd,axiom,
    ! [F2: b > a] :
      ( ( comp_P7171963441967871595od_b_b @ product_fst_b_a @ ( product_apsnd_b_a_b @ F2 ) )
      = product_fst_b_b ) ).

% fst_comp_apsnd
thf(fact_109_fst__comp__apsnd,axiom,
    ! [F2: b > a] :
      ( ( comp_P6892385181472540974od_a_b @ product_fst_a_a @ ( product_apsnd_b_a_a @ F2 ) )
      = product_fst_a_b ) ).

% fst_comp_apsnd
thf(fact_110_fst__comp__apsnd,axiom,
    ! [F2: a > a] :
      ( ( comp_P6892385177169312173od_a_a @ product_fst_a_a @ ( product_apsnd_a_a_a @ F2 ) )
      = product_fst_a_a ) ).

% fst_comp_apsnd
thf(fact_111_fst__comp__apsnd,axiom,
    ! [F2: a > b] :
      ( ( comp_P2928534548958370027od_b_a @ product_fst_b_b @ ( product_apsnd_a_b_b @ F2 ) )
      = product_fst_b_a ) ).

% fst_comp_apsnd
thf(fact_112_fst__comp__apsnd,axiom,
    ! [F2: b > b] :
      ( ( comp_P2928534553261598828od_b_b @ product_fst_b_b @ ( product_apsnd_b_b_b @ F2 ) )
      = product_fst_b_b ) ).

% fst_comp_apsnd
thf(fact_113_fst__comp__apsnd,axiom,
    ! [F2: list_o > b] :
      ( ( comp_P6832527368215522638list_o @ product_fst_a_b @ ( produc3255047714182728159_o_b_a @ F2 ) )
      = product_fst_a_list_o ) ).

% fst_comp_apsnd
thf(fact_114_fst__comp__apsnd,axiom,
    ! [F2: list_o > a] :
      ( ( comp_P7669339814283919181list_o @ product_fst_a_a @ ( produc6042503296155404192_o_a_a @ F2 ) )
      = product_fst_a_list_o ) ).

% fst_comp_apsnd
thf(fact_115_opt__comp__def,axiom,
    ( prefix454693708527911765comp_o
    = ( ^ [X: option_list_o,Y: option_list_o] :
          ( ( prefix8824957607401505554efix_o @ X @ Y )
          | ( prefix8824957607401505554efix_o @ Y @ X ) ) ) ) ).

% opt_comp_def
thf(fact_116_case__prod__beta,axiom,
    ( produc4348216232050026237od_a_b
    = ( ^ [F: b > a > product_prod_a_b,P3: product_prod_b_a] : ( F @ ( product_fst_b_a @ P3 ) @ ( product_snd_b_a @ P3 ) ) ) ) ).

% case_prod_beta
thf(fact_117_case__prod__beta,axiom,
    ( produc6204743795672244857od_b_a
    = ( ^ [F: a > b > product_prod_b_a,P3: product_prod_a_b] : ( F @ ( product_fst_a_b @ P3 ) @ ( product_snd_a_b @ P3 ) ) ) ) ).

% case_prod_beta
thf(fact_118_case__prod__beta,axiom,
    ( produc3537405659489547051_a_b_o
    = ( ^ [F: a > b > $o,P3: product_prod_a_b] : ( F @ ( product_fst_a_b @ P3 ) @ ( product_snd_a_b @ P3 ) ) ) ) ).

% case_prod_beta
thf(fact_119_case__prod__beta,axiom,
    ( produc6028431345588019474_a_b_b
    = ( ^ [F: a > b > b,P3: product_prod_a_b] : ( F @ ( product_fst_a_b @ P3 ) @ ( product_snd_a_b @ P3 ) ) ) ) ).

% case_prod_beta
thf(fact_120_case__prod__beta,axiom,
    ( produc6028431345588019473_a_b_a
    = ( ^ [F: a > b > a,P3: product_prod_a_b] : ( F @ ( product_fst_a_b @ P3 ) @ ( product_snd_a_b @ P3 ) ) ) ) ).

% case_prod_beta
thf(fact_121_prod_Oinject,axiom,
    ! [X1: b,X22: b,Y1: b,Y22: b] :
      ( ( ( product_Pair_b_b @ X1 @ X22 )
        = ( product_Pair_b_b @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_122_prod_Oinject,axiom,
    ! [X1: b,X22: a,Y1: b,Y22: a] :
      ( ( ( product_Pair_b_a @ X1 @ X22 )
        = ( product_Pair_b_a @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_123_prod_Oinject,axiom,
    ! [X1: a,X22: list_o,Y1: a,Y22: list_o] :
      ( ( ( produc6899623729582506167list_o @ X1 @ X22 )
        = ( produc6899623729582506167list_o @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_124_prod_Oinject,axiom,
    ! [X1: a,X22: b,Y1: a,Y22: b] :
      ( ( ( product_Pair_a_b @ X1 @ X22 )
        = ( product_Pair_a_b @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_125_prod_Oinject,axiom,
    ! [X1: a,X22: a,Y1: a,Y22: a] :
      ( ( ( product_Pair_a_a @ X1 @ X22 )
        = ( product_Pair_a_a @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_126_old_Oprod_Oinject,axiom,
    ! [A: b,B: b,A3: b,B2: b] :
      ( ( ( product_Pair_b_b @ A @ B )
        = ( product_Pair_b_b @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_127_old_Oprod_Oinject,axiom,
    ! [A: b,B: a,A3: b,B2: a] :
      ( ( ( product_Pair_b_a @ A @ B )
        = ( product_Pair_b_a @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_128_old_Oprod_Oinject,axiom,
    ! [A: a,B: list_o,A3: a,B2: list_o] :
      ( ( ( produc6899623729582506167list_o @ A @ B )
        = ( produc6899623729582506167list_o @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_129_old_Oprod_Oinject,axiom,
    ! [A: a,B: b,A3: a,B2: b] :
      ( ( ( product_Pair_a_b @ A @ B )
        = ( product_Pair_a_b @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_130_old_Oprod_Oinject,axiom,
    ! [A: a,B: a,A3: a,B2: a] :
      ( ( ( product_Pair_a_a @ A @ B )
        = ( product_Pair_a_a @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_131_swap__swap,axiom,
    ! [P: product_prod_a_b] :
      ( ( product_swap_b_a @ ( product_swap_a_b @ P ) )
      = P ) ).

% swap_swap
thf(fact_132_swap__swap,axiom,
    ! [P: product_prod_b_a] :
      ( ( product_swap_a_b @ ( product_swap_b_a @ P ) )
      = P ) ).

% swap_swap
thf(fact_133_apsnd__conv,axiom,
    ! [F2: b > b,X3: b,Y4: b] :
      ( ( product_apsnd_b_b_b @ F2 @ ( product_Pair_b_b @ X3 @ Y4 ) )
      = ( product_Pair_b_b @ X3 @ ( F2 @ Y4 ) ) ) ).

% apsnd_conv
thf(fact_134_apsnd__conv,axiom,
    ! [F2: b > a,X3: b,Y4: b] :
      ( ( product_apsnd_b_a_b @ F2 @ ( product_Pair_b_b @ X3 @ Y4 ) )
      = ( product_Pair_b_a @ X3 @ ( F2 @ Y4 ) ) ) ).

% apsnd_conv
thf(fact_135_apsnd__conv,axiom,
    ! [F2: a > b,X3: b,Y4: a] :
      ( ( product_apsnd_a_b_b @ F2 @ ( product_Pair_b_a @ X3 @ Y4 ) )
      = ( product_Pair_b_b @ X3 @ ( F2 @ Y4 ) ) ) ).

% apsnd_conv
thf(fact_136_apsnd__conv,axiom,
    ! [F2: a > a,X3: b,Y4: a] :
      ( ( product_apsnd_a_a_b @ F2 @ ( product_Pair_b_a @ X3 @ Y4 ) )
      = ( product_Pair_b_a @ X3 @ ( F2 @ Y4 ) ) ) ).

% apsnd_conv
thf(fact_137_apsnd__conv,axiom,
    ! [F2: b > a,X3: a,Y4: b] :
      ( ( product_apsnd_b_a_a @ F2 @ ( product_Pair_a_b @ X3 @ Y4 ) )
      = ( product_Pair_a_a @ X3 @ ( F2 @ Y4 ) ) ) ).

% apsnd_conv
thf(fact_138_apsnd__conv,axiom,
    ! [F2: a > b,X3: a,Y4: a] :
      ( ( product_apsnd_a_b_a @ F2 @ ( product_Pair_a_a @ X3 @ Y4 ) )
      = ( product_Pair_a_b @ X3 @ ( F2 @ Y4 ) ) ) ).

% apsnd_conv
thf(fact_139_apsnd__conv,axiom,
    ! [F2: a > a,X3: a,Y4: a] :
      ( ( product_apsnd_a_a_a @ F2 @ ( product_Pair_a_a @ X3 @ Y4 ) )
      = ( product_Pair_a_a @ X3 @ ( F2 @ Y4 ) ) ) ).

% apsnd_conv
thf(fact_140_apsnd__conv,axiom,
    ! [F2: b > b,X3: a,Y4: b] :
      ( ( product_apsnd_b_b_a @ F2 @ ( product_Pair_a_b @ X3 @ Y4 ) )
      = ( product_Pair_a_b @ X3 @ ( F2 @ Y4 ) ) ) ).

% apsnd_conv
thf(fact_141_apsnd__conv,axiom,
    ! [F2: list_o > b,X3: a,Y4: list_o] :
      ( ( produc3255047714182728159_o_b_a @ F2 @ ( produc6899623729582506167list_o @ X3 @ Y4 ) )
      = ( product_Pair_a_b @ X3 @ ( F2 @ Y4 ) ) ) ).

% apsnd_conv
thf(fact_142_apsnd__conv,axiom,
    ! [F2: list_o > a,X3: a,Y4: list_o] :
      ( ( produc6042503296155404192_o_a_a @ F2 @ ( produc6899623729582506167list_o @ X3 @ Y4 ) )
      = ( product_Pair_a_a @ X3 @ ( F2 @ Y4 ) ) ) ).

% apsnd_conv
thf(fact_143_swap__simp,axiom,
    ! [X3: list_o,Y4: a] :
      ( ( produc84879911036722967st_o_a @ ( produc6614149361254850935st_o_a @ X3 @ Y4 ) )
      = ( produc6899623729582506167list_o @ Y4 @ X3 ) ) ).

% swap_simp
thf(fact_144_swap__simp,axiom,
    ! [X3: b,Y4: b] :
      ( ( product_swap_b_b @ ( product_Pair_b_b @ X3 @ Y4 ) )
      = ( product_Pair_b_b @ Y4 @ X3 ) ) ).

% swap_simp
thf(fact_145_swap__simp,axiom,
    ! [X3: a,Y4: list_o] :
      ( ( produc370354279364378199list_o @ ( produc6899623729582506167list_o @ X3 @ Y4 ) )
      = ( produc6614149361254850935st_o_a @ Y4 @ X3 ) ) ).

% swap_simp
thf(fact_146_swap__simp,axiom,
    ! [X3: a,Y4: a] :
      ( ( product_swap_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) )
      = ( product_Pair_a_a @ Y4 @ X3 ) ) ).

% swap_simp
thf(fact_147_swap__simp,axiom,
    ! [X3: b,Y4: a] :
      ( ( product_swap_b_a @ ( product_Pair_b_a @ X3 @ Y4 ) )
      = ( product_Pair_a_b @ Y4 @ X3 ) ) ).

% swap_simp
thf(fact_148_swap__simp,axiom,
    ! [X3: a,Y4: b] :
      ( ( product_swap_a_b @ ( product_Pair_a_b @ X3 @ Y4 ) )
      = ( product_Pair_b_a @ Y4 @ X3 ) ) ).

% swap_simp
thf(fact_149_snd__comp__apsnd,axiom,
    ! [F2: a > b] :
      ( ( comp_P2009515988131223277od_a_a @ product_snd_a_b @ ( product_apsnd_a_b_a @ F2 ) )
      = ( comp_a9170378074801158467od_a_a @ F2 @ product_snd_a_a ) ) ).

% snd_comp_apsnd
thf(fact_150_snd__comp__apsnd,axiom,
    ! [F2: a > a] :
      ( ( comp_P7811403737996458923od_b_a @ product_snd_b_a @ ( product_apsnd_a_a_b @ F2 ) )
      = ( comp_a7022362793160298563od_b_a @ F2 @ product_snd_b_a ) ) ).

% snd_comp_apsnd
thf(fact_151_snd__comp__apsnd,axiom,
    ! [F2: b > a] :
      ( ( comp_P7811403742299687724od_b_b @ product_snd_b_a @ ( product_apsnd_b_a_b @ F2 ) )
      = ( comp_b1738939347233587781od_b_b @ F2 @ product_snd_b_b ) ) ).

% snd_comp_apsnd
thf(fact_152_snd__comp__apsnd,axiom,
    ! [F2: a > b] :
      ( ( comp_P2928534548958370027od_b_a @ product_snd_b_b @ ( product_apsnd_a_b_b @ F2 ) )
      = ( comp_a6382922492828482434od_b_a @ F2 @ product_snd_b_a ) ) ).

% snd_comp_apsnd
thf(fact_153_snd__comp__apsnd,axiom,
    ! [F2: b > b] :
      ( ( comp_P2928534553261598828od_b_b @ product_snd_b_b @ ( product_apsnd_b_b_b @ F2 ) )
      = ( comp_b1099499046901771652od_b_b @ F2 @ product_snd_b_b ) ) ).

% snd_comp_apsnd
thf(fact_154_snd__comp__apsnd,axiom,
    ! [F2: b > a] :
      ( ( comp_P6892385181472540974od_a_b @ product_snd_a_a @ ( product_apsnd_b_a_a @ F2 ) )
      = ( comp_b4526394929206263814od_a_b @ F2 @ product_snd_a_b ) ) ).

% snd_comp_apsnd
thf(fact_155_snd__comp__apsnd,axiom,
    ! [F2: a > a] :
      ( ( comp_P6892385177169312173od_a_a @ product_snd_a_a @ ( product_apsnd_a_a_a @ F2 ) )
      = ( comp_a586446338278198788od_a_a @ F2 @ product_snd_a_a ) ) ).

% snd_comp_apsnd
thf(fact_156_snd__comp__apsnd,axiom,
    ! [F2: b > b] :
      ( ( comp_P2009515992434452078od_a_b @ product_snd_a_b @ ( product_apsnd_b_b_a @ F2 ) )
      = ( comp_b3886954628874447685od_a_b @ F2 @ product_snd_a_b ) ) ).

% snd_comp_apsnd
thf(fact_157_fun_Omap__comp,axiom,
    ! [G: a > a,F2: a > a,V: a > a] :
      ( ( comp_a_a_a @ G @ ( comp_a_a_a @ F2 @ V ) )
      = ( comp_a_a_a @ ( comp_a_a_a @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_158_fun_Omap__comp,axiom,
    ! [G: b > a,F2: a > b,V: product_prod_a_b > a] :
      ( ( comp_b4526394929206263814od_a_b @ G @ ( comp_a9170378079104387268od_a_b @ F2 @ V ) )
      = ( comp_a586446342581427589od_a_b @ ( comp_b_a_a @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_159_fun_Omap__comp,axiom,
    ! [G: b > b,F2: b > b,V: product_prod_a_b > b] :
      ( ( comp_b3886954628874447685od_a_b @ G @ ( comp_b3886954628874447685od_a_b @ F2 @ V ) )
      = ( comp_b3886954628874447685od_a_b @ ( comp_b_b_b @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_160_fun_Omap__comp,axiom,
    ! [G: b > b,F2: a > b,V: product_prod_a_b > a] :
      ( ( comp_b3886954628874447685od_a_b @ G @ ( comp_a9170378079104387268od_a_b @ F2 @ V ) )
      = ( comp_a9170378079104387268od_a_b @ ( comp_b_b_a @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_161_fun_Omap__comp,axiom,
    ! [G: a > b,F2: b > a,V: product_prod_a_b > b] :
      ( ( comp_a9170378079104387268od_a_b @ G @ ( comp_b4526394929206263814od_a_b @ F2 @ V ) )
      = ( comp_b3886954628874447685od_a_b @ ( comp_a_b_b @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_162_fun_Omap__comp,axiom,
    ! [G: a > b,F2: a > a,V: product_prod_a_b > a] :
      ( ( comp_a9170378079104387268od_a_b @ G @ ( comp_a586446342581427589od_a_b @ F2 @ V ) )
      = ( comp_a9170378079104387268od_a_b @ ( comp_a_b_a @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_163_fun_Omap__comp,axiom,
    ! [G: a > a,F2: a > a,V: product_prod_a_b > a] :
      ( ( comp_a586446342581427589od_a_b @ G @ ( comp_a586446342581427589od_a_b @ F2 @ V ) )
      = ( comp_a586446342581427589od_a_b @ ( comp_a_a_a @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_164_fun_Omap__comp,axiom,
    ! [G: a > a,F2: product_prod_a_b > a,V: a > product_prod_a_b] :
      ( ( comp_a_a_a @ G @ ( comp_P3199017072728932485_b_a_a @ F2 @ V ) )
      = ( comp_P3199017072728932485_b_a_a @ ( comp_a586446342581427589od_a_b @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_165_fun_Omap__comp,axiom,
    ! [G: product_prod_a_a > b,F2: a > product_prod_a_a,V: product_prod_a_b > a] :
      ( ( comp_P6252944881140724845od_a_b @ G @ ( comp_a1036870397537576092od_a_b @ F2 @ V ) )
      = ( comp_a9170378079104387268od_a_b @ ( comp_P3190265798616278979_a_b_a @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_166_fun_Omap__comp,axiom,
    ! [G: product_prod_a_a > a,F2: a > product_prod_a_a,V: product_prod_a_b > a] :
      ( ( comp_P6892385181472540974od_a_b @ G @ ( comp_a1036870397537576092od_a_b @ F2 @ V ) )
      = ( comp_a586446342581427589od_a_b @ ( comp_P5977721380588955012_a_a_a @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_167_old_Oprod_Ocase,axiom,
    ! [F2: b > a > product_prod_a_b,X1: b,X22: a] :
      ( ( produc4348216232050026237od_a_b @ F2 @ ( product_Pair_b_a @ X1 @ X22 ) )
      = ( F2 @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_168_old_Oprod_Ocase,axiom,
    ! [F2: a > b > product_prod_b_a,X1: a,X22: b] :
      ( ( produc6204743795672244857od_b_a @ F2 @ ( product_Pair_a_b @ X1 @ X22 ) )
      = ( F2 @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_169_old_Oprod_Ocase,axiom,
    ! [F2: a > b > $o,X1: a,X22: b] :
      ( ( produc3537405659489547051_a_b_o @ F2 @ ( product_Pair_a_b @ X1 @ X22 ) )
      = ( F2 @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_170_old_Oprod_Ocase,axiom,
    ! [F2: a > b > b,X1: a,X22: b] :
      ( ( produc6028431345588019474_a_b_b @ F2 @ ( product_Pair_a_b @ X1 @ X22 ) )
      = ( F2 @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_171_old_Oprod_Ocase,axiom,
    ! [F2: a > b > a,X1: a,X22: b] :
      ( ( produc6028431345588019473_a_b_a @ F2 @ ( product_Pair_a_b @ X1 @ X22 ) )
      = ( F2 @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_172_mem__Collect__eq,axiom,
    ! [A: produc5884233991663340231list_o,P2: produc5884233991663340231list_o > $o] :
      ( ( member7948383622993546480list_o @ A @ ( collec7805115074396547890list_o @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_173_mem__Collect__eq,axiom,
    ! [A: product_prod_a_a,P2: product_prod_a_a > $o] :
      ( ( member1426531477525435216od_a_a @ A @ ( collec3336397797384452498od_a_a @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_174_mem__Collect__eq,axiom,
    ! [A: set_a,P2: set_a > $o] :
      ( ( member_set_a @ A @ ( collect_set_a @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_175_mem__Collect__eq,axiom,
    ! [A: b,P2: b > $o] :
      ( ( member_b @ A @ ( collect_b @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_176_mem__Collect__eq,axiom,
    ! [A: product_prod_a_b,P2: product_prod_a_b > $o] :
      ( ( member1426531481828664017od_a_b @ A @ ( collec3336397801687681299od_a_b @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_177_mem__Collect__eq,axiom,
    ! [A: produc2233330766403592615od_a_b,P2: produc2233330766403592615od_a_b > $o] :
      ( ( member5065438833300252112od_a_b @ A @ ( collec7968472090841016338od_a_b @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_178_mem__Collect__eq,axiom,
    ! [A: a > option_list_o,P2: ( a > option_list_o ) > $o] :
      ( ( member1229624942697005410list_o @ A @ ( collec1572211083203048352list_o @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_179_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_180_mem__Collect__eq,axiom,
    ! [A: a,P2: a > $o] :
      ( ( member_a @ A @ ( collect_a @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_181_Collect__mem__eq,axiom,
    ! [A4: set_Pr3077528766752018087list_o] :
      ( ( collec7805115074396547890list_o
        @ ^ [X: produc5884233991663340231list_o] : ( member7948383622993546480list_o @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_182_Collect__mem__eq,axiom,
    ! [A4: set_Product_prod_a_a] :
      ( ( collec3336397797384452498od_a_a
        @ ^ [X: product_prod_a_a] : ( member1426531477525435216od_a_a @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_183_Collect__mem__eq,axiom,
    ! [A4: set_set_a] :
      ( ( collect_set_a
        @ ^ [X: set_a] : ( member_set_a @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_184_Collect__mem__eq,axiom,
    ! [A4: set_b] :
      ( ( collect_b
        @ ^ [X: b] : ( member_b @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_185_Collect__mem__eq,axiom,
    ! [A4: set_Product_prod_a_b] :
      ( ( collec3336397801687681299od_a_b
        @ ^ [X: product_prod_a_b] : ( member1426531481828664017od_a_b @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_186_Collect__mem__eq,axiom,
    ! [A4: set_Pr5493900063595384711od_a_b] :
      ( ( collec7968472090841016338od_a_b
        @ ^ [X: produc2233330766403592615od_a_b] : ( member5065438833300252112od_a_b @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_187_Collect__mem__eq,axiom,
    ! [A4: set_a_option_list_o] :
      ( ( collec1572211083203048352list_o
        @ ^ [X: a > option_list_o] : ( member1229624942697005410list_o @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_188_Collect__mem__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_189_Collect__mem__eq,axiom,
    ! [A4: set_a] :
      ( ( collect_a
        @ ^ [X: a] : ( member_a @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_190_Collect__cong,axiom,
    ! [P2: a > $o,Q2: a > $o] :
      ( ! [X2: a] :
          ( ( P2 @ X2 )
          = ( Q2 @ X2 ) )
     => ( ( collect_a @ P2 )
        = ( collect_a @ Q2 ) ) ) ).

% Collect_cong
thf(fact_191_Collect__cong,axiom,
    ! [P2: product_prod_a_b > $o,Q2: product_prod_a_b > $o] :
      ( ! [X2: product_prod_a_b] :
          ( ( P2 @ X2 )
          = ( Q2 @ X2 ) )
     => ( ( collec3336397801687681299od_a_b @ P2 )
        = ( collec3336397801687681299od_a_b @ Q2 ) ) ) ).

% Collect_cong
thf(fact_192_Collect__cong,axiom,
    ! [P2: produc2233330766403592615od_a_b > $o,Q2: produc2233330766403592615od_a_b > $o] :
      ( ! [X2: produc2233330766403592615od_a_b] :
          ( ( P2 @ X2 )
          = ( Q2 @ X2 ) )
     => ( ( collec7968472090841016338od_a_b @ P2 )
        = ( collec7968472090841016338od_a_b @ Q2 ) ) ) ).

% Collect_cong
thf(fact_193_Collect__cong,axiom,
    ! [P2: ( a > option_list_o ) > $o,Q2: ( a > option_list_o ) > $o] :
      ( ! [X2: a > option_list_o] :
          ( ( P2 @ X2 )
          = ( Q2 @ X2 ) )
     => ( ( collec1572211083203048352list_o @ P2 )
        = ( collec1572211083203048352list_o @ Q2 ) ) ) ).

% Collect_cong
thf(fact_194_Collect__cong,axiom,
    ! [P2: product_prod_nat_nat > $o,Q2: product_prod_nat_nat > $o] :
      ( ! [X2: product_prod_nat_nat] :
          ( ( P2 @ X2 )
          = ( Q2 @ X2 ) )
     => ( ( collec3392354462482085612at_nat @ P2 )
        = ( collec3392354462482085612at_nat @ Q2 ) ) ) ).

% Collect_cong
thf(fact_195_old_Oprod_Oexhaust,axiom,
    ! [Y4: product_prod_b_b] :
      ~ ! [A5: b,B3: b] :
          ( Y4
         != ( product_Pair_b_b @ A5 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_196_old_Oprod_Oexhaust,axiom,
    ! [Y4: product_prod_b_a] :
      ~ ! [A5: b,B3: a] :
          ( Y4
         != ( product_Pair_b_a @ A5 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_197_old_Oprod_Oexhaust,axiom,
    ! [Y4: produc5884233991663340231list_o] :
      ~ ! [A5: a,B3: list_o] :
          ( Y4
         != ( produc6899623729582506167list_o @ A5 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_198_old_Oprod_Oexhaust,axiom,
    ! [Y4: product_prod_a_b] :
      ~ ! [A5: a,B3: b] :
          ( Y4
         != ( product_Pair_a_b @ A5 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_199_old_Oprod_Oexhaust,axiom,
    ! [Y4: product_prod_a_a] :
      ~ ! [A5: a,B3: a] :
          ( Y4
         != ( product_Pair_a_a @ A5 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_200_surj__pair,axiom,
    ! [P: product_prod_b_b] :
    ? [X2: b,Y2: b] :
      ( P
      = ( product_Pair_b_b @ X2 @ Y2 ) ) ).

% surj_pair
thf(fact_201_surj__pair,axiom,
    ! [P: product_prod_b_a] :
    ? [X2: b,Y2: a] :
      ( P
      = ( product_Pair_b_a @ X2 @ Y2 ) ) ).

% surj_pair
thf(fact_202_surj__pair,axiom,
    ! [P: produc5884233991663340231list_o] :
    ? [X2: a,Y2: list_o] :
      ( P
      = ( produc6899623729582506167list_o @ X2 @ Y2 ) ) ).

% surj_pair
thf(fact_203_surj__pair,axiom,
    ! [P: product_prod_a_b] :
    ? [X2: a,Y2: b] :
      ( P
      = ( product_Pair_a_b @ X2 @ Y2 ) ) ).

% surj_pair
thf(fact_204_surj__pair,axiom,
    ! [P: product_prod_a_a] :
    ? [X2: a,Y2: a] :
      ( P
      = ( product_Pair_a_a @ X2 @ Y2 ) ) ).

% surj_pair
thf(fact_205_prod__cases,axiom,
    ! [P2: product_prod_b_b > $o,P: product_prod_b_b] :
      ( ! [A5: b,B3: b] : ( P2 @ ( product_Pair_b_b @ A5 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_206_prod__cases,axiom,
    ! [P2: product_prod_b_a > $o,P: product_prod_b_a] :
      ( ! [A5: b,B3: a] : ( P2 @ ( product_Pair_b_a @ A5 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_207_prod__cases,axiom,
    ! [P2: produc5884233991663340231list_o > $o,P: produc5884233991663340231list_o] :
      ( ! [A5: a,B3: list_o] : ( P2 @ ( produc6899623729582506167list_o @ A5 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_208_prod__cases,axiom,
    ! [P2: product_prod_a_b > $o,P: product_prod_a_b] :
      ( ! [A5: a,B3: b] : ( P2 @ ( product_Pair_a_b @ A5 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_209_prod__cases,axiom,
    ! [P2: product_prod_a_a > $o,P: product_prod_a_a] :
      ( ! [A5: a,B3: a] : ( P2 @ ( product_Pair_a_a @ A5 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_210_Pair__inject,axiom,
    ! [A: b,B: b,A3: b,B2: b] :
      ( ( ( product_Pair_b_b @ A @ B )
        = ( product_Pair_b_b @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_211_Pair__inject,axiom,
    ! [A: b,B: a,A3: b,B2: a] :
      ( ( ( product_Pair_b_a @ A @ B )
        = ( product_Pair_b_a @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_212_Pair__inject,axiom,
    ! [A: a,B: list_o,A3: a,B2: list_o] :
      ( ( ( produc6899623729582506167list_o @ A @ B )
        = ( produc6899623729582506167list_o @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_213_Pair__inject,axiom,
    ! [A: a,B: b,A3: a,B2: b] :
      ( ( ( product_Pair_a_b @ A @ B )
        = ( product_Pair_a_b @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_214_Pair__inject,axiom,
    ! [A: a,B: a,A3: a,B2: a] :
      ( ( ( product_Pair_a_a @ A @ B )
        = ( product_Pair_a_a @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_215_mem__case__prodE,axiom,
    ! [Z2: a,C: b > b > set_a,P: product_prod_b_b] :
      ( ( member_a @ Z2 @ ( produc2614443670078588402_set_a @ C @ P ) )
     => ~ ! [X2: b,Y2: b] :
            ( ( P
              = ( product_Pair_b_b @ X2 @ Y2 ) )
           => ~ ( member_a @ Z2 @ ( C @ X2 @ Y2 ) ) ) ) ).

% mem_case_prodE
thf(fact_216_mem__case__prodE,axiom,
    ! [Z2: b,C: b > b > set_b,P: product_prod_b_b] :
      ( ( member_b @ Z2 @ ( produc2614443674381817203_set_b @ C @ P ) )
     => ~ ! [X2: b,Y2: b] :
            ( ( P
              = ( product_Pair_b_b @ X2 @ Y2 ) )
           => ~ ( member_b @ Z2 @ ( C @ X2 @ Y2 ) ) ) ) ).

% mem_case_prodE
thf(fact_217_mem__case__prodE,axiom,
    ! [Z2: a,C: b > a > set_a,P: product_prod_b_a] :
      ( ( member_a @ Z2 @ ( produc1269978633269100467_set_a @ C @ P ) )
     => ~ ! [X2: b,Y2: a] :
            ( ( P
              = ( product_Pair_b_a @ X2 @ Y2 ) )
           => ~ ( member_a @ Z2 @ ( C @ X2 @ Y2 ) ) ) ) ).

% mem_case_prodE
thf(fact_218_mem__case__prodE,axiom,
    ! [Z2: b,C: b > a > set_b,P: product_prod_b_a] :
      ( ( member_b @ Z2 @ ( produc1269978637572329268_set_b @ C @ P ) )
     => ~ ! [X2: b,Y2: a] :
            ( ( P
              = ( product_Pair_b_a @ X2 @ Y2 ) )
           => ~ ( member_b @ Z2 @ ( C @ X2 @ Y2 ) ) ) ) ).

% mem_case_prodE
thf(fact_219_mem__case__prodE,axiom,
    ! [Z2: a,C: a > b > set_a,P: product_prod_a_b] :
      ( ( member_a @ Z2 @ ( produc1338550822707691121_set_a @ C @ P ) )
     => ~ ! [X2: a,Y2: b] :
            ( ( P
              = ( product_Pair_a_b @ X2 @ Y2 ) )
           => ~ ( member_a @ Z2 @ ( C @ X2 @ Y2 ) ) ) ) ).

% mem_case_prodE
thf(fact_220_mem__case__prodE,axiom,
    ! [Z2: b,C: a > b > set_b,P: product_prod_a_b] :
      ( ( member_b @ Z2 @ ( produc1338550827010919922_set_b @ C @ P ) )
     => ~ ! [X2: a,Y2: b] :
            ( ( P
              = ( product_Pair_a_b @ X2 @ Y2 ) )
           => ~ ( member_b @ Z2 @ ( C @ X2 @ Y2 ) ) ) ) ).

% mem_case_prodE
thf(fact_221_mem__case__prodE,axiom,
    ! [Z2: a,C: a > a > set_a,P: product_prod_a_a] :
      ( ( member_a @ Z2 @ ( produc9217457822752978994_set_a @ C @ P ) )
     => ~ ! [X2: a,Y2: a] :
            ( ( P
              = ( product_Pair_a_a @ X2 @ Y2 ) )
           => ~ ( member_a @ Z2 @ ( C @ X2 @ Y2 ) ) ) ) ).

% mem_case_prodE
thf(fact_222_mem__case__prodE,axiom,
    ! [Z2: b,C: a > a > set_b,P: product_prod_a_a] :
      ( ( member_b @ Z2 @ ( produc9217457827056207795_set_b @ C @ P ) )
     => ~ ! [X2: a,Y2: a] :
            ( ( P
              = ( product_Pair_a_a @ X2 @ Y2 ) )
           => ~ ( member_b @ Z2 @ ( C @ X2 @ Y2 ) ) ) ) ).

% mem_case_prodE
thf(fact_223_mem__case__prodE,axiom,
    ! [Z2: set_a,C: b > b > set_set_a,P: product_prod_b_b] :
      ( ( member_set_a @ Z2 @ ( produc5401209847792589650_set_a @ C @ P ) )
     => ~ ! [X2: b,Y2: b] :
            ( ( P
              = ( product_Pair_b_b @ X2 @ Y2 ) )
           => ~ ( member_set_a @ Z2 @ ( C @ X2 @ Y2 ) ) ) ) ).

% mem_case_prodE
thf(fact_224_mem__case__prodE,axiom,
    ! [Z2: set_a,C: b > a > set_set_a,P: product_prod_b_a] :
      ( ( member_set_a @ Z2 @ ( produc2483618069268599059_set_a @ C @ P ) )
     => ~ ! [X2: b,Y2: a] :
            ( ( P
              = ( product_Pair_b_a @ X2 @ Y2 ) )
           => ~ ( member_set_a @ Z2 @ ( C @ X2 @ Y2 ) ) ) ) ).

% mem_case_prodE
thf(fact_225_prod_Osplit__sel__asm,axiom,
    ! [P2: product_prod_a_b > $o,F2: b > a > product_prod_a_b,Prod: product_prod_b_a] :
      ( ( P2 @ ( produc4348216232050026237od_a_b @ F2 @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair_b_a @ ( product_fst_b_a @ Prod ) @ ( product_snd_b_a @ Prod ) ) )
            & ~ ( P2 @ ( F2 @ ( product_fst_b_a @ Prod ) @ ( product_snd_b_a @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_226_prod_Osplit__sel__asm,axiom,
    ! [P2: product_prod_b_a > $o,F2: a > b > product_prod_b_a,Prod: product_prod_a_b] :
      ( ( P2 @ ( produc6204743795672244857od_b_a @ F2 @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair_a_b @ ( product_fst_a_b @ Prod ) @ ( product_snd_a_b @ Prod ) ) )
            & ~ ( P2 @ ( F2 @ ( product_fst_a_b @ Prod ) @ ( product_snd_a_b @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_227_prod_Osplit__sel__asm,axiom,
    ! [P2: $o > $o,F2: a > b > $o,Prod: product_prod_a_b] :
      ( ( P2 @ ( produc3537405659489547051_a_b_o @ F2 @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair_a_b @ ( product_fst_a_b @ Prod ) @ ( product_snd_a_b @ Prod ) ) )
            & ~ ( P2 @ ( F2 @ ( product_fst_a_b @ Prod ) @ ( product_snd_a_b @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_228_prod_Osplit__sel__asm,axiom,
    ! [P2: b > $o,F2: a > b > b,Prod: product_prod_a_b] :
      ( ( P2 @ ( produc6028431345588019474_a_b_b @ F2 @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair_a_b @ ( product_fst_a_b @ Prod ) @ ( product_snd_a_b @ Prod ) ) )
            & ~ ( P2 @ ( F2 @ ( product_fst_a_b @ Prod ) @ ( product_snd_a_b @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_229_prod_Osplit__sel__asm,axiom,
    ! [P2: a > $o,F2: a > b > a,Prod: product_prod_a_b] :
      ( ( P2 @ ( produc6028431345588019473_a_b_a @ F2 @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair_a_b @ ( product_fst_a_b @ Prod ) @ ( product_snd_a_b @ Prod ) ) )
            & ~ ( P2 @ ( F2 @ ( product_fst_a_b @ Prod ) @ ( product_snd_a_b @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_230_prod_Osplit__sel,axiom,
    ! [P2: product_prod_a_b > $o,F2: b > a > product_prod_a_b,Prod: product_prod_b_a] :
      ( ( P2 @ ( produc4348216232050026237od_a_b @ F2 @ Prod ) )
      = ( ( Prod
          = ( product_Pair_b_a @ ( product_fst_b_a @ Prod ) @ ( product_snd_b_a @ Prod ) ) )
       => ( P2 @ ( F2 @ ( product_fst_b_a @ Prod ) @ ( product_snd_b_a @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_231_prod_Osplit__sel,axiom,
    ! [P2: product_prod_b_a > $o,F2: a > b > product_prod_b_a,Prod: product_prod_a_b] :
      ( ( P2 @ ( produc6204743795672244857od_b_a @ F2 @ Prod ) )
      = ( ( Prod
          = ( product_Pair_a_b @ ( product_fst_a_b @ Prod ) @ ( product_snd_a_b @ Prod ) ) )
       => ( P2 @ ( F2 @ ( product_fst_a_b @ Prod ) @ ( product_snd_a_b @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_232_prod_Osplit__sel,axiom,
    ! [P2: $o > $o,F2: a > b > $o,Prod: product_prod_a_b] :
      ( ( P2 @ ( produc3537405659489547051_a_b_o @ F2 @ Prod ) )
      = ( ( Prod
          = ( product_Pair_a_b @ ( product_fst_a_b @ Prod ) @ ( product_snd_a_b @ Prod ) ) )
       => ( P2 @ ( F2 @ ( product_fst_a_b @ Prod ) @ ( product_snd_a_b @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_233_prod_Osplit__sel,axiom,
    ! [P2: b > $o,F2: a > b > b,Prod: product_prod_a_b] :
      ( ( P2 @ ( produc6028431345588019474_a_b_b @ F2 @ Prod ) )
      = ( ( Prod
          = ( product_Pair_a_b @ ( product_fst_a_b @ Prod ) @ ( product_snd_a_b @ Prod ) ) )
       => ( P2 @ ( F2 @ ( product_fst_a_b @ Prod ) @ ( product_snd_a_b @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_234_prod_Osplit__sel,axiom,
    ! [P2: a > $o,F2: a > b > a,Prod: product_prod_a_b] :
      ( ( P2 @ ( produc6028431345588019473_a_b_a @ F2 @ Prod ) )
      = ( ( Prod
          = ( product_Pair_a_b @ ( product_fst_a_b @ Prod ) @ ( product_snd_a_b @ Prod ) ) )
       => ( P2 @ ( F2 @ ( product_fst_a_b @ Prod ) @ ( product_snd_a_b @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_235_fst__conv,axiom,
    ! [X1: b,X22: a] :
      ( ( product_fst_b_a @ ( product_Pair_b_a @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_236_fst__conv,axiom,
    ! [X1: a,X22: a] :
      ( ( product_fst_a_a @ ( product_Pair_a_a @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_237_fst__conv,axiom,
    ! [X1: b,X22: b] :
      ( ( product_fst_b_b @ ( product_Pair_b_b @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_238_fst__conv,axiom,
    ! [X1: a,X22: list_o] :
      ( ( product_fst_a_list_o @ ( produc6899623729582506167list_o @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_239_fst__conv,axiom,
    ! [X1: a,X22: b] :
      ( ( product_fst_a_b @ ( product_Pair_a_b @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_240_fst__eqD,axiom,
    ! [X3: b,Y4: a,A: b] :
      ( ( ( product_fst_b_a @ ( product_Pair_b_a @ X3 @ Y4 ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_241_fst__eqD,axiom,
    ! [X3: a,Y4: a,A: a] :
      ( ( ( product_fst_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_242_fst__eqD,axiom,
    ! [X3: b,Y4: b,A: b] :
      ( ( ( product_fst_b_b @ ( product_Pair_b_b @ X3 @ Y4 ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_243_fst__eqD,axiom,
    ! [X3: a,Y4: list_o,A: a] :
      ( ( ( product_fst_a_list_o @ ( produc6899623729582506167list_o @ X3 @ Y4 ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_244_fst__eqD,axiom,
    ! [X3: a,Y4: b,A: a] :
      ( ( ( product_fst_a_b @ ( product_Pair_a_b @ X3 @ Y4 ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_245_snd__eqD,axiom,
    ! [X3: a,Y4: list_o,A: list_o] :
      ( ( ( product_snd_a_list_o @ ( produc6899623729582506167list_o @ X3 @ Y4 ) )
        = A )
     => ( Y4 = A ) ) ).

% snd_eqD
thf(fact_246_snd__eqD,axiom,
    ! [X3: b,Y4: a,A: a] :
      ( ( ( product_snd_b_a @ ( product_Pair_b_a @ X3 @ Y4 ) )
        = A )
     => ( Y4 = A ) ) ).

% snd_eqD
thf(fact_247_snd__eqD,axiom,
    ! [X3: b,Y4: b,A: b] :
      ( ( ( product_snd_b_b @ ( product_Pair_b_b @ X3 @ Y4 ) )
        = A )
     => ( Y4 = A ) ) ).

% snd_eqD
thf(fact_248_snd__eqD,axiom,
    ! [X3: a,Y4: a,A: a] :
      ( ( ( product_snd_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) )
        = A )
     => ( Y4 = A ) ) ).

% snd_eqD
thf(fact_249_snd__eqD,axiom,
    ! [X3: a,Y4: b,A: b] :
      ( ( ( product_snd_a_b @ ( product_Pair_a_b @ X3 @ Y4 ) )
        = A )
     => ( Y4 = A ) ) ).

% snd_eqD
thf(fact_250_snd__conv,axiom,
    ! [X1: a,X22: list_o] :
      ( ( product_snd_a_list_o @ ( produc6899623729582506167list_o @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_251_snd__conv,axiom,
    ! [X1: b,X22: a] :
      ( ( product_snd_b_a @ ( product_Pair_b_a @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_252_snd__conv,axiom,
    ! [X1: b,X22: b] :
      ( ( product_snd_b_b @ ( product_Pair_b_b @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_253_snd__conv,axiom,
    ! [X1: a,X22: a] :
      ( ( product_snd_a_a @ ( product_Pair_a_a @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_254_snd__conv,axiom,
    ! [X1: a,X22: b] :
      ( ( product_snd_a_b @ ( product_Pair_a_b @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_255_prod_Oswap__def,axiom,
    ( produc84879911036722967st_o_a
    = ( ^ [P3: produc6380261841376349063st_o_a] : ( produc6899623729582506167list_o @ ( product_snd_list_o_a @ P3 ) @ ( product_fst_list_o_a @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_256_prod_Oswap__def,axiom,
    ( produc370354279364378199list_o
    = ( ^ [P3: produc5884233991663340231list_o] : ( produc6614149361254850935st_o_a @ ( product_snd_a_list_o @ P3 ) @ ( product_fst_a_list_o @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_257_prod_Oswap__def,axiom,
    ( product_swap_b_b
    = ( ^ [P3: product_prod_b_b] : ( product_Pair_b_b @ ( product_snd_b_b @ P3 ) @ ( product_fst_b_b @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_258_prod_Oswap__def,axiom,
    ( product_swap_a_a
    = ( ^ [P3: product_prod_a_a] : ( product_Pair_a_a @ ( product_snd_a_a @ P3 ) @ ( product_fst_a_a @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_259_prod_Oswap__def,axiom,
    ( product_swap_b_a
    = ( ^ [P3: product_prod_b_a] : ( product_Pair_a_b @ ( product_snd_b_a @ P3 ) @ ( product_fst_b_a @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_260_prod_Oswap__def,axiom,
    ( product_swap_a_b
    = ( ^ [P3: product_prod_a_b] : ( product_Pair_b_a @ ( product_snd_a_b @ P3 ) @ ( product_fst_a_b @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_261_apsnd__compose,axiom,
    ! [F2: b > b,G: product_prod_a_b > b,X3: produc4115131626046251985od_a_b] :
      ( ( product_apsnd_b_b_a @ F2 @ ( produc2551226510945298153_b_b_a @ G @ X3 ) )
      = ( produc2551226510945298153_b_b_a @ ( comp_b3886954628874447685od_a_b @ F2 @ G ) @ X3 ) ) ).

% apsnd_compose
thf(fact_262_apsnd__compose,axiom,
    ! [F2: b > b,G: b > b,X3: product_prod_a_b] :
      ( ( product_apsnd_b_b_a @ F2 @ ( product_apsnd_b_b_a @ G @ X3 ) )
      = ( product_apsnd_b_b_a @ ( comp_b_b_b @ F2 @ G ) @ X3 ) ) ).

% apsnd_compose
thf(fact_263_Product__Type_OCollect__case__prodD,axiom,
    ! [X3: produc2233330766403592615od_a_b,A4: product_prod_a_b > product_prod_a_b > $o] :
      ( ( member5065438833300252112od_a_b @ X3 @ ( collec7968472090841016338od_a_b @ ( produc5413562937804622316_a_b_o @ A4 ) ) )
     => ( A4 @ ( produc5013601669340818563od_a_b @ X3 ) @ ( produc8875788424405714885od_a_b @ X3 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_264_Product__Type_OCollect__case__prodD,axiom,
    ! [X3: product_prod_nat_nat,A4: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ X3 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ A4 ) ) )
     => ( A4 @ ( product_fst_nat_nat @ X3 ) @ ( product_snd_nat_nat @ X3 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_265_Product__Type_OCollect__case__prodD,axiom,
    ! [X3: produc5884233991663340231list_o,A4: a > list_o > $o] :
      ( ( member7948383622993546480list_o @ X3 @ ( collec7805115074396547890list_o @ ( produc8342644129638033356st_o_o @ A4 ) ) )
     => ( A4 @ ( product_fst_a_list_o @ X3 ) @ ( product_snd_a_list_o @ X3 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_266_Product__Type_OCollect__case__prodD,axiom,
    ! [X3: product_prod_b_a,A4: b > a > $o] :
      ( ( member7862447932407534991od_b_a @ X3 @ ( collec548942215411776465od_b_a @ ( produc7672072387910335853_b_a_o @ A4 ) ) )
     => ( A4 @ ( product_fst_b_a @ X3 ) @ ( product_snd_b_a @ X3 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_267_Product__Type_OCollect__case__prodD,axiom,
    ! [X3: product_prod_b_b,A4: b > b > $o] :
      ( ( member7862447936710763792od_b_b @ X3 @ ( collec548942219715005266od_b_b @ ( produc4772849988445941548_b_b_o @ A4 ) ) )
     => ( A4 @ ( product_fst_b_b @ X3 ) @ ( product_snd_b_b @ X3 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_268_Product__Type_OCollect__case__prodD,axiom,
    ! [X3: product_prod_a_a,A4: a > a > $o] :
      ( ( member1426531477525435216od_a_a @ X3 @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ A4 ) ) )
     => ( A4 @ ( product_fst_a_a @ X3 ) @ ( product_snd_a_a @ X3 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_269_Product__Type_OCollect__case__prodD,axiom,
    ! [X3: product_prod_a_b,A4: a > b > $o] :
      ( ( member1426531481828664017od_a_b @ X3 @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ A4 ) ) )
     => ( A4 @ ( product_fst_a_b @ X3 ) @ ( product_snd_a_b @ X3 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_270_prod_Oexhaust__sel,axiom,
    ! [Prod: produc5884233991663340231list_o] :
      ( Prod
      = ( produc6899623729582506167list_o @ ( product_fst_a_list_o @ Prod ) @ ( product_snd_a_list_o @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_271_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_b_a] :
      ( Prod
      = ( product_Pair_b_a @ ( product_fst_b_a @ Prod ) @ ( product_snd_b_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_272_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_b_b] :
      ( Prod
      = ( product_Pair_b_b @ ( product_fst_b_b @ Prod ) @ ( product_snd_b_b @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_273_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_a_a] :
      ( Prod
      = ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_274_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_a_b] :
      ( Prod
      = ( product_Pair_a_b @ ( product_fst_a_b @ Prod ) @ ( product_snd_a_b @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_275_exI__realizer,axiom,
    ! [P2: list_o > a > $o,Y4: list_o,X3: a] :
      ( ( P2 @ Y4 @ X3 )
     => ( P2 @ ( product_snd_a_list_o @ ( produc6899623729582506167list_o @ X3 @ Y4 ) ) @ ( product_fst_a_list_o @ ( produc6899623729582506167list_o @ X3 @ Y4 ) ) ) ) ).

% exI_realizer
thf(fact_276_exI__realizer,axiom,
    ! [P2: a > b > $o,Y4: a,X3: b] :
      ( ( P2 @ Y4 @ X3 )
     => ( P2 @ ( product_snd_b_a @ ( product_Pair_b_a @ X3 @ Y4 ) ) @ ( product_fst_b_a @ ( product_Pair_b_a @ X3 @ Y4 ) ) ) ) ).

% exI_realizer
thf(fact_277_exI__realizer,axiom,
    ! [P2: b > b > $o,Y4: b,X3: b] :
      ( ( P2 @ Y4 @ X3 )
     => ( P2 @ ( product_snd_b_b @ ( product_Pair_b_b @ X3 @ Y4 ) ) @ ( product_fst_b_b @ ( product_Pair_b_b @ X3 @ Y4 ) ) ) ) ).

% exI_realizer
thf(fact_278_exI__realizer,axiom,
    ! [P2: a > a > $o,Y4: a,X3: a] :
      ( ( P2 @ Y4 @ X3 )
     => ( P2 @ ( product_snd_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) ) @ ( product_fst_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) ) ) ) ).

% exI_realizer
thf(fact_279_exI__realizer,axiom,
    ! [P2: b > a > $o,Y4: b,X3: a] :
      ( ( P2 @ Y4 @ X3 )
     => ( P2 @ ( product_snd_a_b @ ( product_Pair_a_b @ X3 @ Y4 ) ) @ ( product_fst_a_b @ ( product_Pair_a_b @ X3 @ Y4 ) ) ) ) ).

% exI_realizer
thf(fact_280_conjI__realizer,axiom,
    ! [P2: a > $o,P: a,Q2: list_o > $o,Q: list_o] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( product_fst_a_list_o @ ( produc6899623729582506167list_o @ P @ Q ) ) )
          & ( Q2 @ ( product_snd_a_list_o @ ( produc6899623729582506167list_o @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_281_conjI__realizer,axiom,
    ! [P2: b > $o,P: b,Q2: a > $o,Q: a] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( product_fst_b_a @ ( product_Pair_b_a @ P @ Q ) ) )
          & ( Q2 @ ( product_snd_b_a @ ( product_Pair_b_a @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_282_conjI__realizer,axiom,
    ! [P2: b > $o,P: b,Q2: b > $o,Q: b] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( product_fst_b_b @ ( product_Pair_b_b @ P @ Q ) ) )
          & ( Q2 @ ( product_snd_b_b @ ( product_Pair_b_b @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_283_conjI__realizer,axiom,
    ! [P2: a > $o,P: a,Q2: a > $o,Q: a] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( product_fst_a_a @ ( product_Pair_a_a @ P @ Q ) ) )
          & ( Q2 @ ( product_snd_a_a @ ( product_Pair_a_a @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_284_conjI__realizer,axiom,
    ! [P2: a > $o,P: a,Q2: b > $o,Q: b] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( product_fst_a_b @ ( product_Pair_a_b @ P @ Q ) ) )
          & ( Q2 @ ( product_snd_a_b @ ( product_Pair_a_b @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_285_surjective__pairing,axiom,
    ! [T2: produc5884233991663340231list_o] :
      ( T2
      = ( produc6899623729582506167list_o @ ( product_fst_a_list_o @ T2 ) @ ( product_snd_a_list_o @ T2 ) ) ) ).

% surjective_pairing
thf(fact_286_surjective__pairing,axiom,
    ! [T2: product_prod_b_a] :
      ( T2
      = ( product_Pair_b_a @ ( product_fst_b_a @ T2 ) @ ( product_snd_b_a @ T2 ) ) ) ).

% surjective_pairing
thf(fact_287_surjective__pairing,axiom,
    ! [T2: product_prod_b_b] :
      ( T2
      = ( product_Pair_b_b @ ( product_fst_b_b @ T2 ) @ ( product_snd_b_b @ T2 ) ) ) ).

% surjective_pairing
thf(fact_288_surjective__pairing,axiom,
    ! [T2: product_prod_a_a] :
      ( T2
      = ( product_Pair_a_a @ ( product_fst_a_a @ T2 ) @ ( product_snd_a_a @ T2 ) ) ) ).

% surjective_pairing
thf(fact_289_surjective__pairing,axiom,
    ! [T2: product_prod_a_b] :
      ( T2
      = ( product_Pair_a_b @ ( product_fst_a_b @ T2 ) @ ( product_snd_a_b @ T2 ) ) ) ).

% surjective_pairing
thf(fact_290_split__beta,axiom,
    ( produc4348216232050026237od_a_b
    = ( ^ [F: b > a > product_prod_a_b,Prod3: product_prod_b_a] : ( F @ ( product_fst_b_a @ Prod3 ) @ ( product_snd_b_a @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_291_split__beta,axiom,
    ( produc6204743795672244857od_b_a
    = ( ^ [F: a > b > product_prod_b_a,Prod3: product_prod_a_b] : ( F @ ( product_fst_a_b @ Prod3 ) @ ( product_snd_a_b @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_292_split__beta,axiom,
    ( produc3537405659489547051_a_b_o
    = ( ^ [F: a > b > $o,Prod3: product_prod_a_b] : ( F @ ( product_fst_a_b @ Prod3 ) @ ( product_snd_a_b @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_293_split__beta,axiom,
    ( produc6028431345588019474_a_b_b
    = ( ^ [F: a > b > b,Prod3: product_prod_a_b] : ( F @ ( product_fst_a_b @ Prod3 ) @ ( product_snd_a_b @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_294_split__beta,axiom,
    ( produc6028431345588019473_a_b_a
    = ( ^ [F: a > b > a,Prod3: product_prod_a_b] : ( F @ ( product_fst_a_b @ Prod3 ) @ ( product_snd_a_b @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_295_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: a > list_o > $o,X3: a,Y4: list_o,A: produc5884233991663340231list_o] :
      ( ( P2 @ X3 @ Y4 )
     => ( ( A
          = ( produc6899623729582506167list_o @ X3 @ Y4 ) )
       => ( P2 @ ( product_fst_a_list_o @ A ) @ ( product_snd_a_list_o @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_296_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: b > a > $o,X3: b,Y4: a,A: product_prod_b_a] :
      ( ( P2 @ X3 @ Y4 )
     => ( ( A
          = ( product_Pair_b_a @ X3 @ Y4 ) )
       => ( P2 @ ( product_fst_b_a @ A ) @ ( product_snd_b_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_297_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: b > b > $o,X3: b,Y4: b,A: product_prod_b_b] :
      ( ( P2 @ X3 @ Y4 )
     => ( ( A
          = ( product_Pair_b_b @ X3 @ Y4 ) )
       => ( P2 @ ( product_fst_b_b @ A ) @ ( product_snd_b_b @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_298_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: a > a > $o,X3: a,Y4: a,A: product_prod_a_a] :
      ( ( P2 @ X3 @ Y4 )
     => ( ( A
          = ( product_Pair_a_a @ X3 @ Y4 ) )
       => ( P2 @ ( product_fst_a_a @ A ) @ ( product_snd_a_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_299_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: a > b > $o,X3: a,Y4: b,A: product_prod_a_b] :
      ( ( P2 @ X3 @ Y4 )
     => ( ( A
          = ( product_Pair_a_b @ X3 @ Y4 ) )
       => ( P2 @ ( product_fst_a_b @ A ) @ ( product_snd_a_b @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_300_comp__apply,axiom,
    ( comp_b3886954628874447685od_a_b
    = ( ^ [F: b > b,G2: product_prod_a_b > b,X: product_prod_a_b] : ( F @ ( G2 @ X ) ) ) ) ).

% comp_apply
thf(fact_301_comp__apply,axiom,
    ( comp_a1036870397537576092od_a_b
    = ( ^ [F: a > product_prod_a_a,G2: product_prod_a_b > a,X: product_prod_a_b] : ( F @ ( G2 @ X ) ) ) ) ).

% comp_apply
thf(fact_302_comp__apply,axiom,
    ( comp_a9170378079104387268od_a_b
    = ( ^ [F: a > b,G2: product_prod_a_b > a,X: product_prod_a_b] : ( F @ ( G2 @ X ) ) ) ) ).

% comp_apply
thf(fact_303_comp__apply,axiom,
    ( comp_a586446342581427589od_a_b
    = ( ^ [F: a > a,G2: product_prod_a_b > a,X: product_prod_a_b] : ( F @ ( G2 @ X ) ) ) ) ).

% comp_apply
thf(fact_304_comp__apply,axiom,
    ( comp_a_a_a
    = ( ^ [F: a > a,G2: a > a,X: a] : ( F @ ( G2 @ X ) ) ) ) ).

% comp_apply
thf(fact_305_apsnd__apfst,axiom,
    ! [F2: b > b,G: a > b,X3: product_prod_a_b] :
      ( ( product_apsnd_b_b_b @ F2 @ ( product_apfst_a_b_b @ G @ X3 ) )
      = ( product_Pair_b_b @ ( G @ ( product_fst_a_b @ X3 ) ) @ ( F2 @ ( product_snd_a_b @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_306_apsnd__apfst,axiom,
    ! [F2: b > a,G: a > b,X3: product_prod_a_b] :
      ( ( product_apsnd_b_a_b @ F2 @ ( product_apfst_a_b_b @ G @ X3 ) )
      = ( product_Pair_b_a @ ( G @ ( product_fst_a_b @ X3 ) ) @ ( F2 @ ( product_snd_a_b @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_307_apsnd__apfst,axiom,
    ! [F2: b > a,G: a > a,X3: product_prod_a_b] :
      ( ( product_apsnd_b_a_a @ F2 @ ( product_apfst_a_a_b @ G @ X3 ) )
      = ( product_Pair_a_a @ ( G @ ( product_fst_a_b @ X3 ) ) @ ( F2 @ ( product_snd_a_b @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_308_apsnd__apfst,axiom,
    ! [F2: b > b,G: a > a,X3: product_prod_a_b] :
      ( ( product_apsnd_b_b_a @ F2 @ ( product_apfst_a_a_b @ G @ X3 ) )
      = ( product_Pair_a_b @ ( G @ ( product_fst_a_b @ X3 ) ) @ ( F2 @ ( product_snd_a_b @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_309_apsnd__apfst,axiom,
    ! [F2: a > b,G: b > b,X3: product_prod_b_a] :
      ( ( product_apsnd_a_b_b @ F2 @ ( product_apfst_b_b_a @ G @ X3 ) )
      = ( product_Pair_b_b @ ( G @ ( product_fst_b_a @ X3 ) ) @ ( F2 @ ( product_snd_b_a @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_310_apsnd__apfst,axiom,
    ! [F2: a > a,G: b > b,X3: product_prod_b_a] :
      ( ( product_apsnd_a_a_b @ F2 @ ( product_apfst_b_b_a @ G @ X3 ) )
      = ( product_Pair_b_a @ ( G @ ( product_fst_b_a @ X3 ) ) @ ( F2 @ ( product_snd_b_a @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_311_apsnd__apfst,axiom,
    ! [F2: a > b,G: b > a,X3: product_prod_b_a] :
      ( ( product_apsnd_a_b_a @ F2 @ ( product_apfst_b_a_a @ G @ X3 ) )
      = ( product_Pair_a_b @ ( G @ ( product_fst_b_a @ X3 ) ) @ ( F2 @ ( product_snd_b_a @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_312_apsnd__apfst,axiom,
    ! [F2: a > a,G: b > a,X3: product_prod_b_a] :
      ( ( product_apsnd_a_a_a @ F2 @ ( product_apfst_b_a_a @ G @ X3 ) )
      = ( product_Pair_a_a @ ( G @ ( product_fst_b_a @ X3 ) ) @ ( F2 @ ( product_snd_b_a @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_313_apsnd__apfst,axiom,
    ! [F2: b > b,G: b > b,X3: product_prod_b_b] :
      ( ( product_apsnd_b_b_b @ F2 @ ( product_apfst_b_b_b @ G @ X3 ) )
      = ( product_Pair_b_b @ ( G @ ( product_fst_b_b @ X3 ) ) @ ( F2 @ ( product_snd_b_b @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_314_apsnd__apfst,axiom,
    ! [F2: b > a,G: b > b,X3: product_prod_b_b] :
      ( ( product_apsnd_b_a_b @ F2 @ ( product_apfst_b_b_b @ G @ X3 ) )
      = ( product_Pair_b_a @ ( G @ ( product_fst_b_b @ X3 ) ) @ ( F2 @ ( product_snd_b_b @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_315_apfst__apsnd,axiom,
    ! [F2: a > b,G: b > a,X3: product_prod_a_b] :
      ( ( product_apfst_a_b_a @ F2 @ ( product_apsnd_b_a_a @ G @ X3 ) )
      = ( product_Pair_b_a @ ( F2 @ ( product_fst_a_b @ X3 ) ) @ ( G @ ( product_snd_a_b @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_316_apfst__apsnd,axiom,
    ! [F2: a > a,G: b > a,X3: product_prod_a_b] :
      ( ( product_apfst_a_a_a @ F2 @ ( product_apsnd_b_a_a @ G @ X3 ) )
      = ( product_Pair_a_a @ ( F2 @ ( product_fst_a_b @ X3 ) ) @ ( G @ ( product_snd_a_b @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_317_apfst__apsnd,axiom,
    ! [F2: a > b,G: b > b,X3: product_prod_a_b] :
      ( ( product_apfst_a_b_b @ F2 @ ( product_apsnd_b_b_a @ G @ X3 ) )
      = ( product_Pair_b_b @ ( F2 @ ( product_fst_a_b @ X3 ) ) @ ( G @ ( product_snd_a_b @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_318_apfst__apsnd,axiom,
    ! [F2: a > a,G: b > b,X3: product_prod_a_b] :
      ( ( product_apfst_a_a_b @ F2 @ ( product_apsnd_b_b_a @ G @ X3 ) )
      = ( product_Pair_a_b @ ( F2 @ ( product_fst_a_b @ X3 ) ) @ ( G @ ( product_snd_a_b @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_319_apfst__apsnd,axiom,
    ! [F2: b > b,G: a > b,X3: product_prod_b_a] :
      ( ( product_apfst_b_b_b @ F2 @ ( product_apsnd_a_b_b @ G @ X3 ) )
      = ( product_Pair_b_b @ ( F2 @ ( product_fst_b_a @ X3 ) ) @ ( G @ ( product_snd_b_a @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_320_apfst__apsnd,axiom,
    ! [F2: b > b,G: a > a,X3: product_prod_b_a] :
      ( ( product_apfst_b_b_a @ F2 @ ( product_apsnd_a_a_b @ G @ X3 ) )
      = ( product_Pair_b_a @ ( F2 @ ( product_fst_b_a @ X3 ) ) @ ( G @ ( product_snd_b_a @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_321_apfst__apsnd,axiom,
    ! [F2: b > a,G: a > b,X3: product_prod_b_a] :
      ( ( product_apfst_b_a_b @ F2 @ ( product_apsnd_a_b_b @ G @ X3 ) )
      = ( product_Pair_a_b @ ( F2 @ ( product_fst_b_a @ X3 ) ) @ ( G @ ( product_snd_b_a @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_322_apfst__apsnd,axiom,
    ! [F2: b > a,G: a > a,X3: product_prod_b_a] :
      ( ( product_apfst_b_a_a @ F2 @ ( product_apsnd_a_a_b @ G @ X3 ) )
      = ( product_Pair_a_a @ ( F2 @ ( product_fst_b_a @ X3 ) ) @ ( G @ ( product_snd_b_a @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_323_apfst__apsnd,axiom,
    ! [F2: b > b,G: b > b,X3: product_prod_b_b] :
      ( ( product_apfst_b_b_b @ F2 @ ( product_apsnd_b_b_b @ G @ X3 ) )
      = ( product_Pair_b_b @ ( F2 @ ( product_fst_b_b @ X3 ) ) @ ( G @ ( product_snd_b_b @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_324_apfst__apsnd,axiom,
    ! [F2: b > b,G: b > a,X3: product_prod_b_b] :
      ( ( product_apfst_b_b_a @ F2 @ ( product_apsnd_b_a_b @ G @ X3 ) )
      = ( product_Pair_b_a @ ( F2 @ ( product_fst_b_b @ X3 ) ) @ ( G @ ( product_snd_b_b @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_325_inj__on__imageI2,axiom,
    ! [F3: b > b,F2: product_prod_a_b > b,A4: set_Product_prod_a_b] :
      ( ( inj_on4343330552946057672_a_b_b @ ( comp_b3886954628874447685od_a_b @ F3 @ F2 ) @ A4 )
     => ( inj_on4343330552946057672_a_b_b @ F2 @ A4 ) ) ).

% inj_on_imageI2
thf(fact_326_inj__on__imageI2,axiom,
    ! [F3: a > product_prod_a_a,F2: product_prod_a_b > a,A4: set_Product_prod_a_b] :
      ( ( inj_on1230094050847898352od_a_a @ ( comp_a1036870397537576092od_a_b @ F3 @ F2 ) @ A4 )
     => ( inj_on4343330552946057671_a_b_a @ F2 @ A4 ) ) ).

% inj_on_imageI2
thf(fact_327_inj__on__imageI2,axiom,
    ! [F3: a > b,F2: product_prod_a_b > a,A4: set_Product_prod_a_b] :
      ( ( inj_on4343330552946057672_a_b_b @ ( comp_a9170378079104387268od_a_b @ F3 @ F2 ) @ A4 )
     => ( inj_on4343330552946057671_a_b_a @ F2 @ A4 ) ) ).

% inj_on_imageI2
thf(fact_328_inj__on__imageI2,axiom,
    ! [F3: option_list_o > option_list_o,F2: b > option_list_o,A4: set_b] :
      ( ( inj_on8261448415883032086list_o @ ( comp_o7170176683143658716st_o_b @ F3 @ F2 ) @ A4 )
     => ( inj_on8261448415883032086list_o @ F2 @ A4 ) ) ).

% inj_on_imageI2
thf(fact_329_inj__on__imageI2,axiom,
    ! [F3: option_list_o > option_list_o,F2: a > option_list_o,A4: set_a] :
      ( ( inj_on374126998980950615list_o @ ( comp_o7170176683143658715st_o_a @ F3 @ F2 ) @ A4 )
     => ( inj_on374126998980950615list_o @ F2 @ A4 ) ) ).

% inj_on_imageI2
thf(fact_330_inj__on__imageI2,axiom,
    ! [F3: a > option_list_o,F2: a > a,A4: set_a] :
      ( ( inj_on374126998980950615list_o @ ( comp_a6425641528110818421st_o_a @ F3 @ F2 ) @ A4 )
     => ( inj_on_a_a @ F2 @ A4 ) ) ).

% inj_on_imageI2
thf(fact_331_inj__on__imageI2,axiom,
    ! [F3: option_list_o > a,F2: a > option_list_o,A4: set_a] :
      ( ( inj_on_a_a @ ( comp_o695885166685416001_o_a_a @ F3 @ F2 ) @ A4 )
     => ( inj_on374126998980950615list_o @ F2 @ A4 ) ) ).

% inj_on_imageI2
thf(fact_332_inj__on__imageI2,axiom,
    ! [F3: a > a,F2: a > a,A4: set_a] :
      ( ( inj_on_a_a @ ( comp_a_a_a @ F3 @ F2 ) @ A4 )
     => ( inj_on_a_a @ F2 @ A4 ) ) ).

% inj_on_imageI2
thf(fact_333_inj__on__imageI2,axiom,
    ! [F3: a > a,F2: product_prod_a_b > a,A4: set_Product_prod_a_b] :
      ( ( inj_on4343330552946057671_a_b_a @ ( comp_a586446342581427589od_a_b @ F3 @ F2 ) @ A4 )
     => ( inj_on4343330552946057671_a_b_a @ F2 @ A4 ) ) ).

% inj_on_imageI2
thf(fact_334_sndI,axiom,
    ! [X3: produc5884233991663340231list_o,Y4: a,Z2: list_o] :
      ( ( X3
        = ( produc6899623729582506167list_o @ Y4 @ Z2 ) )
     => ( ( product_snd_a_list_o @ X3 )
        = Z2 ) ) ).

% sndI
thf(fact_335_sndI,axiom,
    ! [X3: product_prod_b_a,Y4: b,Z2: a] :
      ( ( X3
        = ( product_Pair_b_a @ Y4 @ Z2 ) )
     => ( ( product_snd_b_a @ X3 )
        = Z2 ) ) ).

% sndI
thf(fact_336_sndI,axiom,
    ! [X3: product_prod_b_b,Y4: b,Z2: b] :
      ( ( X3
        = ( product_Pair_b_b @ Y4 @ Z2 ) )
     => ( ( product_snd_b_b @ X3 )
        = Z2 ) ) ).

% sndI
thf(fact_337_sndI,axiom,
    ! [X3: product_prod_a_a,Y4: a,Z2: a] :
      ( ( X3
        = ( product_Pair_a_a @ Y4 @ Z2 ) )
     => ( ( product_snd_a_a @ X3 )
        = Z2 ) ) ).

% sndI
thf(fact_338_sndI,axiom,
    ! [X3: product_prod_a_b,Y4: a,Z2: b] :
      ( ( X3
        = ( product_Pair_a_b @ Y4 @ Z2 ) )
     => ( ( product_snd_a_b @ X3 )
        = Z2 ) ) ).

% sndI
thf(fact_339_eq__snd__iff,axiom,
    ! [B: list_o,P: produc5884233991663340231list_o] :
      ( ( B
        = ( product_snd_a_list_o @ P ) )
      = ( ? [A2: a] :
            ( P
            = ( produc6899623729582506167list_o @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_340_eq__snd__iff,axiom,
    ! [B: a,P: product_prod_b_a] :
      ( ( B
        = ( product_snd_b_a @ P ) )
      = ( ? [A2: b] :
            ( P
            = ( product_Pair_b_a @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_341_eq__snd__iff,axiom,
    ! [B: b,P: product_prod_b_b] :
      ( ( B
        = ( product_snd_b_b @ P ) )
      = ( ? [A2: b] :
            ( P
            = ( product_Pair_b_b @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_342_eq__snd__iff,axiom,
    ! [B: a,P: product_prod_a_a] :
      ( ( B
        = ( product_snd_a_a @ P ) )
      = ( ? [A2: a] :
            ( P
            = ( product_Pair_a_a @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_343_eq__snd__iff,axiom,
    ! [B: b,P: product_prod_a_b] :
      ( ( B
        = ( product_snd_a_b @ P ) )
      = ( ? [A2: a] :
            ( P
            = ( product_Pair_a_b @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_344_eq__fst__iff,axiom,
    ! [A: b,P: product_prod_b_a] :
      ( ( A
        = ( product_fst_b_a @ P ) )
      = ( ? [B4: a] :
            ( P
            = ( product_Pair_b_a @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_345_eq__fst__iff,axiom,
    ! [A: a,P: product_prod_a_a] :
      ( ( A
        = ( product_fst_a_a @ P ) )
      = ( ? [B4: a] :
            ( P
            = ( product_Pair_a_a @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_346_eq__fst__iff,axiom,
    ! [A: b,P: product_prod_b_b] :
      ( ( A
        = ( product_fst_b_b @ P ) )
      = ( ? [B4: b] :
            ( P
            = ( product_Pair_b_b @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_347_eq__fst__iff,axiom,
    ! [A: a,P: produc5884233991663340231list_o] :
      ( ( A
        = ( product_fst_a_list_o @ P ) )
      = ( ? [B4: list_o] :
            ( P
            = ( produc6899623729582506167list_o @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_348_eq__fst__iff,axiom,
    ! [A: a,P: product_prod_a_b] :
      ( ( A
        = ( product_fst_a_b @ P ) )
      = ( ? [B4: b] :
            ( P
            = ( product_Pair_a_b @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_349_fstI,axiom,
    ! [X3: product_prod_b_a,Y4: b,Z2: a] :
      ( ( X3
        = ( product_Pair_b_a @ Y4 @ Z2 ) )
     => ( ( product_fst_b_a @ X3 )
        = Y4 ) ) ).

% fstI
thf(fact_350_fstI,axiom,
    ! [X3: product_prod_a_a,Y4: a,Z2: a] :
      ( ( X3
        = ( product_Pair_a_a @ Y4 @ Z2 ) )
     => ( ( product_fst_a_a @ X3 )
        = Y4 ) ) ).

% fstI
thf(fact_351_fstI,axiom,
    ! [X3: product_prod_b_b,Y4: b,Z2: b] :
      ( ( X3
        = ( product_Pair_b_b @ Y4 @ Z2 ) )
     => ( ( product_fst_b_b @ X3 )
        = Y4 ) ) ).

% fstI
thf(fact_352_fstI,axiom,
    ! [X3: produc5884233991663340231list_o,Y4: a,Z2: list_o] :
      ( ( X3
        = ( produc6899623729582506167list_o @ Y4 @ Z2 ) )
     => ( ( product_fst_a_list_o @ X3 )
        = Y4 ) ) ).

% fstI
thf(fact_353_fstI,axiom,
    ! [X3: product_prod_a_b,Y4: a,Z2: b] :
      ( ( X3
        = ( product_Pair_a_b @ Y4 @ Z2 ) )
     => ( ( product_fst_a_b @ X3 )
        = Y4 ) ) ).

% fstI
thf(fact_354_split__cong,axiom,
    ! [Q: product_prod_b_a,F2: b > a > product_prod_a_b,G: b > a > product_prod_a_b,P: product_prod_b_a] :
      ( ! [X2: b,Y2: a] :
          ( ( ( product_Pair_b_a @ X2 @ Y2 )
            = Q )
         => ( ( F2 @ X2 @ Y2 )
            = ( G @ X2 @ Y2 ) ) )
     => ( ( P = Q )
       => ( ( produc4348216232050026237od_a_b @ F2 @ P )
          = ( produc4348216232050026237od_a_b @ G @ Q ) ) ) ) ).

% split_cong
thf(fact_355_split__cong,axiom,
    ! [Q: product_prod_a_b,F2: a > b > product_prod_b_a,G: a > b > product_prod_b_a,P: product_prod_a_b] :
      ( ! [X2: a,Y2: b] :
          ( ( ( product_Pair_a_b @ X2 @ Y2 )
            = Q )
         => ( ( F2 @ X2 @ Y2 )
            = ( G @ X2 @ Y2 ) ) )
     => ( ( P = Q )
       => ( ( produc6204743795672244857od_b_a @ F2 @ P )
          = ( produc6204743795672244857od_b_a @ G @ Q ) ) ) ) ).

% split_cong
thf(fact_356_split__cong,axiom,
    ! [Q: product_prod_a_b,F2: a > b > $o,G: a > b > $o,P: product_prod_a_b] :
      ( ! [X2: a,Y2: b] :
          ( ( ( product_Pair_a_b @ X2 @ Y2 )
            = Q )
         => ( ( F2 @ X2 @ Y2 )
            = ( G @ X2 @ Y2 ) ) )
     => ( ( P = Q )
       => ( ( produc3537405659489547051_a_b_o @ F2 @ P )
          = ( produc3537405659489547051_a_b_o @ G @ Q ) ) ) ) ).

% split_cong
thf(fact_357_split__cong,axiom,
    ! [Q: product_prod_a_b,F2: a > b > b,G: a > b > b,P: product_prod_a_b] :
      ( ! [X2: a,Y2: b] :
          ( ( ( product_Pair_a_b @ X2 @ Y2 )
            = Q )
         => ( ( F2 @ X2 @ Y2 )
            = ( G @ X2 @ Y2 ) ) )
     => ( ( P = Q )
       => ( ( produc6028431345588019474_a_b_b @ F2 @ P )
          = ( produc6028431345588019474_a_b_b @ G @ Q ) ) ) ) ).

% split_cong
thf(fact_358_split__cong,axiom,
    ! [Q: product_prod_a_b,F2: a > b > a,G: a > b > a,P: product_prod_a_b] :
      ( ! [X2: a,Y2: b] :
          ( ( ( product_Pair_a_b @ X2 @ Y2 )
            = Q )
         => ( ( F2 @ X2 @ Y2 )
            = ( G @ X2 @ Y2 ) ) )
     => ( ( P = Q )
       => ( ( produc6028431345588019473_a_b_a @ F2 @ P )
          = ( produc6028431345588019473_a_b_a @ G @ Q ) ) ) ) ).

% split_cong
thf(fact_359_apfst__conv,axiom,
    ! [F2: b > b,X3: b,Y4: b] :
      ( ( product_apfst_b_b_b @ F2 @ ( product_Pair_b_b @ X3 @ Y4 ) )
      = ( product_Pair_b_b @ ( F2 @ X3 ) @ Y4 ) ) ).

% apfst_conv
thf(fact_360_apfst__conv,axiom,
    ! [F2: b > a,X3: b,Y4: b] :
      ( ( product_apfst_b_a_b @ F2 @ ( product_Pair_b_b @ X3 @ Y4 ) )
      = ( product_Pair_a_b @ ( F2 @ X3 ) @ Y4 ) ) ).

% apfst_conv
thf(fact_361_apfst__conv,axiom,
    ! [F2: b > b,X3: b,Y4: a] :
      ( ( product_apfst_b_b_a @ F2 @ ( product_Pair_b_a @ X3 @ Y4 ) )
      = ( product_Pair_b_a @ ( F2 @ X3 ) @ Y4 ) ) ).

% apfst_conv
thf(fact_362_apfst__conv,axiom,
    ! [F2: b > a,X3: b,Y4: a] :
      ( ( product_apfst_b_a_a @ F2 @ ( product_Pair_b_a @ X3 @ Y4 ) )
      = ( product_Pair_a_a @ ( F2 @ X3 ) @ Y4 ) ) ).

% apfst_conv
thf(fact_363_apfst__conv,axiom,
    ! [F2: a > a,X3: a,Y4: list_o] :
      ( ( produc948369090323839586list_o @ F2 @ ( produc6899623729582506167list_o @ X3 @ Y4 ) )
      = ( produc6899623729582506167list_o @ ( F2 @ X3 ) @ Y4 ) ) ).

% apfst_conv
thf(fact_364_apfst__conv,axiom,
    ! [F2: a > b,X3: a,Y4: b] :
      ( ( product_apfst_a_b_b @ F2 @ ( product_Pair_a_b @ X3 @ Y4 ) )
      = ( product_Pair_b_b @ ( F2 @ X3 ) @ Y4 ) ) ).

% apfst_conv
thf(fact_365_apfst__conv,axiom,
    ! [F2: a > b,X3: a,Y4: a] :
      ( ( product_apfst_a_b_a @ F2 @ ( product_Pair_a_a @ X3 @ Y4 ) )
      = ( product_Pair_b_a @ ( F2 @ X3 ) @ Y4 ) ) ).

% apfst_conv
thf(fact_366_apfst__conv,axiom,
    ! [F2: a > a,X3: a,Y4: a] :
      ( ( product_apfst_a_a_a @ F2 @ ( product_Pair_a_a @ X3 @ Y4 ) )
      = ( product_Pair_a_a @ ( F2 @ X3 ) @ Y4 ) ) ).

% apfst_conv
thf(fact_367_apfst__conv,axiom,
    ! [F2: a > a,X3: a,Y4: b] :
      ( ( product_apfst_a_a_b @ F2 @ ( product_Pair_a_b @ X3 @ Y4 ) )
      = ( product_Pair_a_b @ ( F2 @ X3 ) @ Y4 ) ) ).

% apfst_conv
thf(fact_368_fst__apfst,axiom,
    ! [F2: b > a,X3: product_prod_b_b] :
      ( ( product_fst_a_b @ ( product_apfst_b_a_b @ F2 @ X3 ) )
      = ( F2 @ ( product_fst_b_b @ X3 ) ) ) ).

% fst_apfst
thf(fact_369_fst__apfst,axiom,
    ! [F2: b > b,X3: product_prod_b_a] :
      ( ( product_fst_b_a @ ( product_apfst_b_b_a @ F2 @ X3 ) )
      = ( F2 @ ( product_fst_b_a @ X3 ) ) ) ).

% fst_apfst
thf(fact_370_fst__apfst,axiom,
    ! [F2: a > b,X3: product_prod_a_a] :
      ( ( product_fst_b_a @ ( product_apfst_a_b_a @ F2 @ X3 ) )
      = ( F2 @ ( product_fst_a_a @ X3 ) ) ) ).

% fst_apfst
thf(fact_371_fst__apfst,axiom,
    ! [F2: b > a,X3: product_prod_b_a] :
      ( ( product_fst_a_a @ ( product_apfst_b_a_a @ F2 @ X3 ) )
      = ( F2 @ ( product_fst_b_a @ X3 ) ) ) ).

% fst_apfst
thf(fact_372_fst__apfst,axiom,
    ! [F2: a > a,X3: product_prod_a_a] :
      ( ( product_fst_a_a @ ( product_apfst_a_a_a @ F2 @ X3 ) )
      = ( F2 @ ( product_fst_a_a @ X3 ) ) ) ).

% fst_apfst
thf(fact_373_fst__apfst,axiom,
    ! [F2: a > b,X3: product_prod_a_b] :
      ( ( product_fst_b_b @ ( product_apfst_a_b_b @ F2 @ X3 ) )
      = ( F2 @ ( product_fst_a_b @ X3 ) ) ) ).

% fst_apfst
thf(fact_374_fst__apfst,axiom,
    ! [F2: b > b,X3: product_prod_b_b] :
      ( ( product_fst_b_b @ ( product_apfst_b_b_b @ F2 @ X3 ) )
      = ( F2 @ ( product_fst_b_b @ X3 ) ) ) ).

% fst_apfst
thf(fact_375_fst__apfst,axiom,
    ! [F2: a > a,X3: produc5884233991663340231list_o] :
      ( ( product_fst_a_list_o @ ( produc948369090323839586list_o @ F2 @ X3 ) )
      = ( F2 @ ( product_fst_a_list_o @ X3 ) ) ) ).

% fst_apfst
thf(fact_376_fst__apfst,axiom,
    ! [F2: a > a,X3: product_prod_a_b] :
      ( ( product_fst_a_b @ ( product_apfst_a_a_b @ F2 @ X3 ) )
      = ( F2 @ ( product_fst_a_b @ X3 ) ) ) ).

% fst_apfst
thf(fact_377_apfst__eq__conv,axiom,
    ! [F2: a > a,X3: product_prod_a_b,G: a > a] :
      ( ( ( product_apfst_a_a_b @ F2 @ X3 )
        = ( product_apfst_a_a_b @ G @ X3 ) )
      = ( ( F2 @ ( product_fst_a_b @ X3 ) )
        = ( G @ ( product_fst_a_b @ X3 ) ) ) ) ).

% apfst_eq_conv
thf(fact_378_snd__apfst,axiom,
    ! [F2: b > a,X3: product_prod_b_b] :
      ( ( product_snd_a_b @ ( product_apfst_b_a_b @ F2 @ X3 ) )
      = ( product_snd_b_b @ X3 ) ) ).

% snd_apfst
thf(fact_379_snd__apfst,axiom,
    ! [F2: b > b,X3: product_prod_b_a] :
      ( ( product_snd_b_a @ ( product_apfst_b_b_a @ F2 @ X3 ) )
      = ( product_snd_b_a @ X3 ) ) ).

% snd_apfst
thf(fact_380_snd__apfst,axiom,
    ! [F2: a > b,X3: product_prod_a_a] :
      ( ( product_snd_b_a @ ( product_apfst_a_b_a @ F2 @ X3 ) )
      = ( product_snd_a_a @ X3 ) ) ).

% snd_apfst
thf(fact_381_snd__apfst,axiom,
    ! [F2: a > b,X3: product_prod_a_b] :
      ( ( product_snd_b_b @ ( product_apfst_a_b_b @ F2 @ X3 ) )
      = ( product_snd_a_b @ X3 ) ) ).

% snd_apfst
thf(fact_382_snd__apfst,axiom,
    ! [F2: b > b,X3: product_prod_b_b] :
      ( ( product_snd_b_b @ ( product_apfst_b_b_b @ F2 @ X3 ) )
      = ( product_snd_b_b @ X3 ) ) ).

% snd_apfst
thf(fact_383_snd__apfst,axiom,
    ! [F2: b > a,X3: product_prod_b_a] :
      ( ( product_snd_a_a @ ( product_apfst_b_a_a @ F2 @ X3 ) )
      = ( product_snd_b_a @ X3 ) ) ).

% snd_apfst
thf(fact_384_snd__apfst,axiom,
    ! [F2: a > a,X3: product_prod_a_a] :
      ( ( product_snd_a_a @ ( product_apfst_a_a_a @ F2 @ X3 ) )
      = ( product_snd_a_a @ X3 ) ) ).

% snd_apfst
thf(fact_385_snd__apfst,axiom,
    ! [F2: a > a,X3: product_prod_a_b] :
      ( ( product_snd_a_b @ ( product_apfst_a_a_b @ F2 @ X3 ) )
      = ( product_snd_a_b @ X3 ) ) ).

% snd_apfst
thf(fact_386_snd__comp__apfst,axiom,
    ! [F2: b > a] :
      ( ( comp_P8445432447316551853od_b_b @ product_snd_a_b @ ( product_apfst_b_a_b @ F2 ) )
      = product_snd_b_b ) ).

% snd_comp_apfst
thf(fact_387_snd__comp__apfst,axiom,
    ! [F2: b > b] :
      ( ( comp_P7811403737996458923od_b_a @ product_snd_b_a @ ( product_apfst_b_b_a @ F2 ) )
      = product_snd_b_a ) ).

% snd_comp_apfst
thf(fact_388_snd__comp__apfst,axiom,
    ! [F2: a > b] :
      ( ( comp_P1375487283114359148od_a_a @ product_snd_b_a @ ( product_apfst_a_b_a @ F2 ) )
      = product_snd_a_a ) ).

% snd_comp_apfst
thf(fact_389_snd__comp__apfst,axiom,
    ! [F2: a > b] :
      ( ( comp_P5715990135234274861od_a_b @ product_snd_b_b @ ( product_apfst_a_b_b @ F2 ) )
      = product_snd_a_b ) ).

% snd_comp_apfst
thf(fact_390_snd__comp__apfst,axiom,
    ! [F2: b > b] :
      ( ( comp_P2928534553261598828od_b_b @ product_snd_b_b @ ( product_apfst_b_b_b @ F2 ) )
      = product_snd_b_b ) ).

% snd_comp_apfst
thf(fact_391_snd__comp__apfst,axiom,
    ! [F2: b > a] :
      ( ( comp_P4104929595196636140od_b_a @ product_snd_a_a @ ( product_apfst_b_a_a @ F2 ) )
      = product_snd_b_a ) ).

% snd_comp_apfst
thf(fact_392_snd__comp__apfst,axiom,
    ! [F2: a > a] :
      ( ( comp_P6892385177169312173od_a_a @ product_snd_a_a @ ( product_apfst_a_a_a @ F2 ) )
      = product_snd_a_a ) ).

% snd_comp_apfst
thf(fact_393_snd__comp__apfst,axiom,
    ! [F2: a > a] :
      ( ( comp_P2009515992434452078od_a_b @ product_snd_a_b @ ( product_apfst_a_a_b @ F2 ) )
      = product_snd_a_b ) ).

% snd_comp_apfst
thf(fact_394_fst__comp__apfst,axiom,
    ! [F2: a > product_prod_a_a] :
      ( ( comp_P8899913933205946077od_a_b @ produc2553620009333216027_a_a_b @ ( produc8216020251299224538_a_a_b @ F2 ) )
      = ( comp_a1036870397537576092od_a_b @ F2 @ product_fst_a_b ) ) ).

% fst_comp_apfst
thf(fact_395_fst__comp__apfst,axiom,
    ! [F2: b > a] :
      ( ( comp_P9084872747648367982od_b_b @ product_fst_a_b @ ( product_apfst_b_a_b @ F2 ) )
      = ( comp_b1738939347233587781od_b_b @ F2 @ product_fst_b_b ) ) ).

% fst_comp_apfst
thf(fact_396_fst__comp__apfst,axiom,
    ! [F2: b > b] :
      ( ( comp_P7171963437664642794od_b_a @ product_fst_b_a @ ( product_apfst_b_b_a @ F2 ) )
      = ( comp_b1099499042598542851od_b_a @ F2 @ product_fst_b_a ) ) ).

% fst_comp_apfst
thf(fact_397_fst__comp__apfst,axiom,
    ! [F2: a > b] :
      ( ( comp_P736046982782543019od_a_a @ product_fst_b_a @ ( product_apfst_a_b_a @ F2 ) )
      = ( comp_a9170378074801158467od_a_a @ F2 @ product_fst_a_a ) ) ).

% fst_comp_apfst
thf(fact_398_fst__comp__apfst,axiom,
    ! [F2: b > a] :
      ( ( comp_P4104929595196636140od_b_a @ product_fst_a_a @ ( product_apfst_b_a_a @ F2 ) )
      = ( comp_b1738939342930358980od_b_a @ F2 @ product_fst_b_a ) ) ).

% fst_comp_apfst
thf(fact_399_fst__comp__apfst,axiom,
    ! [F2: a > a] :
      ( ( comp_P6892385177169312173od_a_a @ product_fst_a_a @ ( product_apfst_a_a_a @ F2 ) )
      = ( comp_a586446338278198788od_a_a @ F2 @ product_fst_a_a ) ) ).

% fst_comp_apfst
thf(fact_400_fst__comp__apfst,axiom,
    ! [F2: a > b] :
      ( ( comp_P5715990135234274861od_a_b @ product_fst_b_b @ ( product_apfst_a_b_b @ F2 ) )
      = ( comp_a9170378079104387268od_a_b @ F2 @ product_fst_a_b ) ) ).

% fst_comp_apfst
thf(fact_401_fst__comp__apfst,axiom,
    ! [F2: b > b] :
      ( ( comp_P2928534553261598828od_b_b @ product_fst_b_b @ ( product_apfst_b_b_b @ F2 ) )
      = ( comp_b1099499046901771652od_b_b @ F2 @ product_fst_b_b ) ) ).

% fst_comp_apfst
thf(fact_402_fst__comp__apfst,axiom,
    ! [F2: a > a] :
      ( ( comp_P9144535235892738285list_o @ product_fst_a_list_o @ ( produc948369090323839586list_o @ F2 ) )
      = ( comp_a1590955686735427492list_o @ F2 @ product_fst_a_list_o ) ) ).

% fst_comp_apfst
thf(fact_403_fst__comp__apfst,axiom,
    ! [F2: a > a] :
      ( ( comp_P2648956292766268207od_a_b @ product_fst_a_b @ ( product_apfst_a_a_b @ F2 ) )
      = ( comp_a586446342581427589od_a_b @ F2 @ product_fst_a_b ) ) ).

% fst_comp_apfst
thf(fact_404_inj__swap,axiom,
    ! [A4: set_Product_prod_b_a] : ( inj_on6506093261728657715od_a_b @ product_swap_b_a @ A4 ) ).

% inj_swap
thf(fact_405_inj__swap,axiom,
    ! [A4: set_Product_prod_a_b] : ( inj_on7666010505729998127od_b_a @ product_swap_a_b @ A4 ) ).

% inj_swap
thf(fact_406_apfst__compose,axiom,
    ! [F2: a > a,G: product_prod_a_b > a,X3: produc3167243053651132542_a_b_b] :
      ( ( product_apfst_a_a_b @ F2 @ ( produc5567899478585720173_b_a_b @ G @ X3 ) )
      = ( produc5567899478585720173_b_a_b @ ( comp_a586446342581427589od_a_b @ F2 @ G ) @ X3 ) ) ).

% apfst_compose
thf(fact_407_apfst__compose,axiom,
    ! [F2: a > a,G: a > a,X3: product_prod_a_b] :
      ( ( product_apfst_a_a_b @ F2 @ ( product_apfst_a_a_b @ G @ X3 ) )
      = ( product_apfst_a_a_b @ ( comp_a_a_a @ F2 @ G ) @ X3 ) ) ).

% apfst_compose
thf(fact_408_apsnd__apfst__commute,axiom,
    ! [F2: b > b,G: a > a,P: product_prod_a_b] :
      ( ( product_apsnd_b_b_a @ F2 @ ( product_apfst_a_a_b @ G @ P ) )
      = ( product_apfst_a_a_b @ G @ ( product_apsnd_b_b_a @ F2 @ P ) ) ) ).

% apsnd_apfst_commute
thf(fact_409_comp__def,axiom,
    ( comp_b3886954628874447685od_a_b
    = ( ^ [F: b > b,G2: product_prod_a_b > b,X: product_prod_a_b] : ( F @ ( G2 @ X ) ) ) ) ).

% comp_def
thf(fact_410_comp__def,axiom,
    ( comp_a1036870397537576092od_a_b
    = ( ^ [F: a > product_prod_a_a,G2: product_prod_a_b > a,X: product_prod_a_b] : ( F @ ( G2 @ X ) ) ) ) ).

% comp_def
thf(fact_411_comp__def,axiom,
    ( comp_a9170378079104387268od_a_b
    = ( ^ [F: a > b,G2: product_prod_a_b > a,X: product_prod_a_b] : ( F @ ( G2 @ X ) ) ) ) ).

% comp_def
thf(fact_412_comp__def,axiom,
    ( comp_a586446342581427589od_a_b
    = ( ^ [F: a > a,G2: product_prod_a_b > a,X: product_prod_a_b] : ( F @ ( G2 @ X ) ) ) ) ).

% comp_def
thf(fact_413_comp__def,axiom,
    ( comp_a_a_a
    = ( ^ [F: a > a,G2: a > a,X: a] : ( F @ ( G2 @ X ) ) ) ) ).

% comp_def
thf(fact_414_comp__assoc,axiom,
    ! [F2: a > a,G: a > a,H: a > a] :
      ( ( comp_a_a_a @ ( comp_a_a_a @ F2 @ G ) @ H )
      = ( comp_a_a_a @ F2 @ ( comp_a_a_a @ G @ H ) ) ) ).

% comp_assoc
thf(fact_415_comp__assoc,axiom,
    ! [F2: a > a,G: product_prod_a_b > a,H: a > product_prod_a_b] :
      ( ( comp_P3199017072728932485_b_a_a @ ( comp_a586446342581427589od_a_b @ F2 @ G ) @ H )
      = ( comp_a_a_a @ F2 @ ( comp_P3199017072728932485_b_a_a @ G @ H ) ) ) ).

% comp_assoc
thf(fact_416_comp__assoc,axiom,
    ! [F2: b > b,G: b > b,H: product_prod_a_b > b] :
      ( ( comp_b3886954628874447685od_a_b @ ( comp_b_b_b @ F2 @ G ) @ H )
      = ( comp_b3886954628874447685od_a_b @ F2 @ ( comp_b3886954628874447685od_a_b @ G @ H ) ) ) ).

% comp_assoc
thf(fact_417_comp__assoc,axiom,
    ! [F2: a > b,G: b > a,H: product_prod_a_b > b] :
      ( ( comp_b3886954628874447685od_a_b @ ( comp_a_b_b @ F2 @ G ) @ H )
      = ( comp_a9170378079104387268od_a_b @ F2 @ ( comp_b4526394929206263814od_a_b @ G @ H ) ) ) ).

% comp_assoc
thf(fact_418_comp__assoc,axiom,
    ! [F2: b > b,G: a > b,H: product_prod_a_b > a] :
      ( ( comp_a9170378079104387268od_a_b @ ( comp_b_b_a @ F2 @ G ) @ H )
      = ( comp_b3886954628874447685od_a_b @ F2 @ ( comp_a9170378079104387268od_a_b @ G @ H ) ) ) ).

% comp_assoc
thf(fact_419_comp__assoc,axiom,
    ! [F2: a > b,G: a > a,H: product_prod_a_b > a] :
      ( ( comp_a9170378079104387268od_a_b @ ( comp_a_b_a @ F2 @ G ) @ H )
      = ( comp_a9170378079104387268od_a_b @ F2 @ ( comp_a586446342581427589od_a_b @ G @ H ) ) ) ).

% comp_assoc
thf(fact_420_comp__assoc,axiom,
    ! [F2: b > a,G: a > b,H: product_prod_a_b > a] :
      ( ( comp_a586446342581427589od_a_b @ ( comp_b_a_a @ F2 @ G ) @ H )
      = ( comp_b4526394929206263814od_a_b @ F2 @ ( comp_a9170378079104387268od_a_b @ G @ H ) ) ) ).

% comp_assoc
thf(fact_421_comp__assoc,axiom,
    ! [F2: a > a,G: a > a,H: product_prod_a_b > a] :
      ( ( comp_a586446342581427589od_a_b @ ( comp_a_a_a @ F2 @ G ) @ H )
      = ( comp_a586446342581427589od_a_b @ F2 @ ( comp_a586446342581427589od_a_b @ G @ H ) ) ) ).

% comp_assoc
thf(fact_422_comp__assoc,axiom,
    ! [F2: b > b,G: product_prod_a_b > b,H: product_prod_a_b > product_prod_a_b] :
      ( ( comp_P2009515992434452078od_a_b @ ( comp_b3886954628874447685od_a_b @ F2 @ G ) @ H )
      = ( comp_b3886954628874447685od_a_b @ F2 @ ( comp_P2009515992434452078od_a_b @ G @ H ) ) ) ).

% comp_assoc
thf(fact_423_comp__assoc,axiom,
    ! [F2: a > b,G: product_prod_a_b > a,H: product_prod_a_b > product_prod_a_b] :
      ( ( comp_P2009515992434452078od_a_b @ ( comp_a9170378079104387268od_a_b @ F2 @ G ) @ H )
      = ( comp_a9170378079104387268od_a_b @ F2 @ ( comp_P2648956292766268207od_a_b @ G @ H ) ) ) ).

% comp_assoc
thf(fact_424_comp__eq__dest,axiom,
    ! [A: b > b,B: product_prod_a_b > b,C: b > b,D: product_prod_a_b > b,V: product_prod_a_b] :
      ( ( ( comp_b3886954628874447685od_a_b @ A @ B )
        = ( comp_b3886954628874447685od_a_b @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_425_comp__eq__dest,axiom,
    ! [A: b > b,B: product_prod_a_b > b,C: a > b,D: product_prod_a_b > a,V: product_prod_a_b] :
      ( ( ( comp_b3886954628874447685od_a_b @ A @ B )
        = ( comp_a9170378079104387268od_a_b @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_426_comp__eq__dest,axiom,
    ! [A: a > product_prod_a_a,B: product_prod_a_b > a,C: a > product_prod_a_a,D: product_prod_a_b > a,V: product_prod_a_b] :
      ( ( ( comp_a1036870397537576092od_a_b @ A @ B )
        = ( comp_a1036870397537576092od_a_b @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_427_comp__eq__dest,axiom,
    ! [A: a > b,B: product_prod_a_b > a,C: b > b,D: product_prod_a_b > b,V: product_prod_a_b] :
      ( ( ( comp_a9170378079104387268od_a_b @ A @ B )
        = ( comp_b3886954628874447685od_a_b @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_428_comp__eq__dest,axiom,
    ! [A: a > b,B: product_prod_a_b > a,C: a > b,D: product_prod_a_b > a,V: product_prod_a_b] :
      ( ( ( comp_a9170378079104387268od_a_b @ A @ B )
        = ( comp_a9170378079104387268od_a_b @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_429_comp__eq__dest,axiom,
    ! [A: a > a,B: product_prod_a_b > a,C: a > a,D: product_prod_a_b > a,V: product_prod_a_b] :
      ( ( ( comp_a586446342581427589od_a_b @ A @ B )
        = ( comp_a586446342581427589od_a_b @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_430_comp__eq__dest,axiom,
    ! [A: a > a,B: a > a,C: a > a,D: a > a,V: a] :
      ( ( ( comp_a_a_a @ A @ B )
        = ( comp_a_a_a @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_431_comp__eq__elim,axiom,
    ! [A: b > b,B: product_prod_a_b > b,C: b > b,D: product_prod_a_b > b] :
      ( ( ( comp_b3886954628874447685od_a_b @ A @ B )
        = ( comp_b3886954628874447685od_a_b @ C @ D ) )
     => ! [V2: product_prod_a_b] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_432_comp__eq__elim,axiom,
    ! [A: b > b,B: product_prod_a_b > b,C: a > b,D: product_prod_a_b > a] :
      ( ( ( comp_b3886954628874447685od_a_b @ A @ B )
        = ( comp_a9170378079104387268od_a_b @ C @ D ) )
     => ! [V2: product_prod_a_b] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_433_comp__eq__elim,axiom,
    ! [A: a > product_prod_a_a,B: product_prod_a_b > a,C: a > product_prod_a_a,D: product_prod_a_b > a] :
      ( ( ( comp_a1036870397537576092od_a_b @ A @ B )
        = ( comp_a1036870397537576092od_a_b @ C @ D ) )
     => ! [V2: product_prod_a_b] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_434_comp__eq__elim,axiom,
    ! [A: a > b,B: product_prod_a_b > a,C: b > b,D: product_prod_a_b > b] :
      ( ( ( comp_a9170378079104387268od_a_b @ A @ B )
        = ( comp_b3886954628874447685od_a_b @ C @ D ) )
     => ! [V2: product_prod_a_b] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_435_comp__eq__elim,axiom,
    ! [A: a > b,B: product_prod_a_b > a,C: a > b,D: product_prod_a_b > a] :
      ( ( ( comp_a9170378079104387268od_a_b @ A @ B )
        = ( comp_a9170378079104387268od_a_b @ C @ D ) )
     => ! [V2: product_prod_a_b] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_436_comp__eq__elim,axiom,
    ! [A: a > a,B: product_prod_a_b > a,C: a > a,D: product_prod_a_b > a] :
      ( ( ( comp_a586446342581427589od_a_b @ A @ B )
        = ( comp_a586446342581427589od_a_b @ C @ D ) )
     => ! [V2: product_prod_a_b] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_437_comp__eq__elim,axiom,
    ! [A: a > a,B: a > a,C: a > a,D: a > a] :
      ( ( ( comp_a_a_a @ A @ B )
        = ( comp_a_a_a @ C @ D ) )
     => ! [V2: a] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_438_comp__cong,axiom,
    ! [F2: b > b,G: product_prod_a_b > b,X3: product_prod_a_b,F3: b > b,G3: product_prod_a_b > b,X4: product_prod_a_b] :
      ( ( ( F2 @ ( G @ X3 ) )
        = ( F3 @ ( G3 @ X4 ) ) )
     => ( ( comp_b3886954628874447685od_a_b @ F2 @ G @ X3 )
        = ( comp_b3886954628874447685od_a_b @ F3 @ G3 @ X4 ) ) ) ).

% comp_cong
thf(fact_439_comp__cong,axiom,
    ! [F2: b > b,G: product_prod_a_b > b,X3: product_prod_a_b,F3: a > b,G3: product_prod_a_b > a,X4: product_prod_a_b] :
      ( ( ( F2 @ ( G @ X3 ) )
        = ( F3 @ ( G3 @ X4 ) ) )
     => ( ( comp_b3886954628874447685od_a_b @ F2 @ G @ X3 )
        = ( comp_a9170378079104387268od_a_b @ F3 @ G3 @ X4 ) ) ) ).

% comp_cong
thf(fact_440_comp__cong,axiom,
    ! [F2: a > product_prod_a_a,G: product_prod_a_b > a,X3: product_prod_a_b,F3: a > product_prod_a_a,G3: product_prod_a_b > a,X4: product_prod_a_b] :
      ( ( ( F2 @ ( G @ X3 ) )
        = ( F3 @ ( G3 @ X4 ) ) )
     => ( ( comp_a1036870397537576092od_a_b @ F2 @ G @ X3 )
        = ( comp_a1036870397537576092od_a_b @ F3 @ G3 @ X4 ) ) ) ).

% comp_cong
thf(fact_441_comp__cong,axiom,
    ! [F2: a > b,G: product_prod_a_b > a,X3: product_prod_a_b,F3: b > b,G3: product_prod_a_b > b,X4: product_prod_a_b] :
      ( ( ( F2 @ ( G @ X3 ) )
        = ( F3 @ ( G3 @ X4 ) ) )
     => ( ( comp_a9170378079104387268od_a_b @ F2 @ G @ X3 )
        = ( comp_b3886954628874447685od_a_b @ F3 @ G3 @ X4 ) ) ) ).

% comp_cong
thf(fact_442_comp__cong,axiom,
    ! [F2: a > b,G: product_prod_a_b > a,X3: product_prod_a_b,F3: a > b,G3: product_prod_a_b > a,X4: product_prod_a_b] :
      ( ( ( F2 @ ( G @ X3 ) )
        = ( F3 @ ( G3 @ X4 ) ) )
     => ( ( comp_a9170378079104387268od_a_b @ F2 @ G @ X3 )
        = ( comp_a9170378079104387268od_a_b @ F3 @ G3 @ X4 ) ) ) ).

% comp_cong
thf(fact_443_comp__cong,axiom,
    ! [F2: a > a,G: product_prod_a_b > a,X3: product_prod_a_b,F3: a > a,G3: product_prod_a_b > a,X4: product_prod_a_b] :
      ( ( ( F2 @ ( G @ X3 ) )
        = ( F3 @ ( G3 @ X4 ) ) )
     => ( ( comp_a586446342581427589od_a_b @ F2 @ G @ X3 )
        = ( comp_a586446342581427589od_a_b @ F3 @ G3 @ X4 ) ) ) ).

% comp_cong
thf(fact_444_comp__cong,axiom,
    ! [F2: a > a,G: product_prod_a_b > a,X3: product_prod_a_b,F3: a > a,G3: a > a,X4: a] :
      ( ( ( F2 @ ( G @ X3 ) )
        = ( F3 @ ( G3 @ X4 ) ) )
     => ( ( comp_a586446342581427589od_a_b @ F2 @ G @ X3 )
        = ( comp_a_a_a @ F3 @ G3 @ X4 ) ) ) ).

% comp_cong
thf(fact_445_comp__cong,axiom,
    ! [F2: a > a,G: a > a,X3: a,F3: a > a,G3: product_prod_a_b > a,X4: product_prod_a_b] :
      ( ( ( F2 @ ( G @ X3 ) )
        = ( F3 @ ( G3 @ X4 ) ) )
     => ( ( comp_a_a_a @ F2 @ G @ X3 )
        = ( comp_a586446342581427589od_a_b @ F3 @ G3 @ X4 ) ) ) ).

% comp_cong
thf(fact_446_comp__cong,axiom,
    ! [F2: a > a,G: a > a,X3: a,F3: a > a,G3: a > a,X4: a] :
      ( ( ( F2 @ ( G @ X3 ) )
        = ( F3 @ ( G3 @ X4 ) ) )
     => ( ( comp_a_a_a @ F2 @ G @ X3 )
        = ( comp_a_a_a @ F3 @ G3 @ X4 ) ) ) ).

% comp_cong
thf(fact_447_comp__eq__dest__lhs,axiom,
    ! [A: b > b,B: product_prod_a_b > b,C: product_prod_a_b > b,V: product_prod_a_b] :
      ( ( ( comp_b3886954628874447685od_a_b @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_448_comp__eq__dest__lhs,axiom,
    ! [A: a > product_prod_a_a,B: product_prod_a_b > a,C: product_prod_a_b > product_prod_a_a,V: product_prod_a_b] :
      ( ( ( comp_a1036870397537576092od_a_b @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_449_comp__eq__dest__lhs,axiom,
    ! [A: a > b,B: product_prod_a_b > a,C: product_prod_a_b > b,V: product_prod_a_b] :
      ( ( ( comp_a9170378079104387268od_a_b @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_450_comp__eq__dest__lhs,axiom,
    ! [A: a > a,B: product_prod_a_b > a,C: product_prod_a_b > a,V: product_prod_a_b] :
      ( ( ( comp_a586446342581427589od_a_b @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_451_comp__eq__dest__lhs,axiom,
    ! [A: a > a,B: a > a,C: a > a,V: a] :
      ( ( ( comp_a_a_a @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_452_comp__apply__eq,axiom,
    ! [F2: b > b,G: product_prod_a_b > b,X3: product_prod_a_b,H: b > b,K: product_prod_a_b > b] :
      ( ( ( F2 @ ( G @ X3 ) )
        = ( H @ ( K @ X3 ) ) )
     => ( ( comp_b3886954628874447685od_a_b @ F2 @ G @ X3 )
        = ( comp_b3886954628874447685od_a_b @ H @ K @ X3 ) ) ) ).

% comp_apply_eq
thf(fact_453_comp__apply__eq,axiom,
    ! [F2: b > b,G: product_prod_a_b > b,X3: product_prod_a_b,H: a > b,K: product_prod_a_b > a] :
      ( ( ( F2 @ ( G @ X3 ) )
        = ( H @ ( K @ X3 ) ) )
     => ( ( comp_b3886954628874447685od_a_b @ F2 @ G @ X3 )
        = ( comp_a9170378079104387268od_a_b @ H @ K @ X3 ) ) ) ).

% comp_apply_eq
thf(fact_454_comp__apply__eq,axiom,
    ! [F2: a > product_prod_a_a,G: product_prod_a_b > a,X3: product_prod_a_b,H: a > product_prod_a_a,K: product_prod_a_b > a] :
      ( ( ( F2 @ ( G @ X3 ) )
        = ( H @ ( K @ X3 ) ) )
     => ( ( comp_a1036870397537576092od_a_b @ F2 @ G @ X3 )
        = ( comp_a1036870397537576092od_a_b @ H @ K @ X3 ) ) ) ).

% comp_apply_eq
thf(fact_455_comp__apply__eq,axiom,
    ! [F2: a > b,G: product_prod_a_b > a,X3: product_prod_a_b,H: b > b,K: product_prod_a_b > b] :
      ( ( ( F2 @ ( G @ X3 ) )
        = ( H @ ( K @ X3 ) ) )
     => ( ( comp_a9170378079104387268od_a_b @ F2 @ G @ X3 )
        = ( comp_b3886954628874447685od_a_b @ H @ K @ X3 ) ) ) ).

% comp_apply_eq
thf(fact_456_comp__apply__eq,axiom,
    ! [F2: a > b,G: product_prod_a_b > a,X3: product_prod_a_b,H: a > b,K: product_prod_a_b > a] :
      ( ( ( F2 @ ( G @ X3 ) )
        = ( H @ ( K @ X3 ) ) )
     => ( ( comp_a9170378079104387268od_a_b @ F2 @ G @ X3 )
        = ( comp_a9170378079104387268od_a_b @ H @ K @ X3 ) ) ) ).

% comp_apply_eq
thf(fact_457_comp__apply__eq,axiom,
    ! [F2: a > a,G: product_prod_a_b > a,X3: product_prod_a_b,H: a > a,K: product_prod_a_b > a] :
      ( ( ( F2 @ ( G @ X3 ) )
        = ( H @ ( K @ X3 ) ) )
     => ( ( comp_a586446342581427589od_a_b @ F2 @ G @ X3 )
        = ( comp_a586446342581427589od_a_b @ H @ K @ X3 ) ) ) ).

% comp_apply_eq
thf(fact_458_comp__apply__eq,axiom,
    ! [F2: a > a,G: a > a,X3: a,H: a > a,K: a > a] :
      ( ( ( F2 @ ( G @ X3 ) )
        = ( H @ ( K @ X3 ) ) )
     => ( ( comp_a_a_a @ F2 @ G @ X3 )
        = ( comp_a_a_a @ H @ K @ X3 ) ) ) ).

% comp_apply_eq
thf(fact_459_inj__onD,axiom,
    ! [F2: a > option_list_o,A4: set_a,X3: a,Y4: a] :
      ( ( inj_on374126998980950615list_o @ F2 @ A4 )
     => ( ( ( F2 @ X3 )
          = ( F2 @ Y4 ) )
       => ( ( member_a @ X3 @ A4 )
         => ( ( member_a @ Y4 @ A4 )
           => ( X3 = Y4 ) ) ) ) ) ).

% inj_onD
thf(fact_460_inj__onD,axiom,
    ! [F2: a > a,A4: set_a,X3: a,Y4: a] :
      ( ( inj_on_a_a @ F2 @ A4 )
     => ( ( ( F2 @ X3 )
          = ( F2 @ Y4 ) )
       => ( ( member_a @ X3 @ A4 )
         => ( ( member_a @ Y4 @ A4 )
           => ( X3 = Y4 ) ) ) ) ) ).

% inj_onD
thf(fact_461_inj__onD,axiom,
    ! [F2: product_prod_a_b > a,A4: set_Product_prod_a_b,X3: product_prod_a_b,Y4: product_prod_a_b] :
      ( ( inj_on4343330552946057671_a_b_a @ F2 @ A4 )
     => ( ( ( F2 @ X3 )
          = ( F2 @ Y4 ) )
       => ( ( member1426531481828664017od_a_b @ X3 @ A4 )
         => ( ( member1426531481828664017od_a_b @ Y4 @ A4 )
           => ( X3 = Y4 ) ) ) ) ) ).

% inj_onD
thf(fact_462_inj__onD,axiom,
    ! [F2: b > option_list_o,A4: set_b,X3: b,Y4: b] :
      ( ( inj_on8261448415883032086list_o @ F2 @ A4 )
     => ( ( ( F2 @ X3 )
          = ( F2 @ Y4 ) )
       => ( ( member_b @ X3 @ A4 )
         => ( ( member_b @ Y4 @ A4 )
           => ( X3 = Y4 ) ) ) ) ) ).

% inj_onD
thf(fact_463_inj__onI,axiom,
    ! [A4: set_a,F2: a > option_list_o] :
      ( ! [X2: a,Y2: a] :
          ( ( member_a @ X2 @ A4 )
         => ( ( member_a @ Y2 @ A4 )
           => ( ( ( F2 @ X2 )
                = ( F2 @ Y2 ) )
             => ( X2 = Y2 ) ) ) )
     => ( inj_on374126998980950615list_o @ F2 @ A4 ) ) ).

% inj_onI
thf(fact_464_inj__onI,axiom,
    ! [A4: set_a,F2: a > a] :
      ( ! [X2: a,Y2: a] :
          ( ( member_a @ X2 @ A4 )
         => ( ( member_a @ Y2 @ A4 )
           => ( ( ( F2 @ X2 )
                = ( F2 @ Y2 ) )
             => ( X2 = Y2 ) ) ) )
     => ( inj_on_a_a @ F2 @ A4 ) ) ).

% inj_onI
thf(fact_465_inj__onI,axiom,
    ! [A4: set_Product_prod_a_b,F2: product_prod_a_b > a] :
      ( ! [X2: product_prod_a_b,Y2: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X2 @ A4 )
         => ( ( member1426531481828664017od_a_b @ Y2 @ A4 )
           => ( ( ( F2 @ X2 )
                = ( F2 @ Y2 ) )
             => ( X2 = Y2 ) ) ) )
     => ( inj_on4343330552946057671_a_b_a @ F2 @ A4 ) ) ).

% inj_onI
thf(fact_466_inj__onI,axiom,
    ! [A4: set_b,F2: b > option_list_o] :
      ( ! [X2: b,Y2: b] :
          ( ( member_b @ X2 @ A4 )
         => ( ( member_b @ Y2 @ A4 )
           => ( ( ( F2 @ X2 )
                = ( F2 @ Y2 ) )
             => ( X2 = Y2 ) ) ) )
     => ( inj_on8261448415883032086list_o @ F2 @ A4 ) ) ).

% inj_onI
thf(fact_467_inj__on__def,axiom,
    ( inj_on8261448415883032086list_o
    = ( ^ [F: b > option_list_o,A6: set_b] :
        ! [X: b] :
          ( ( member_b @ X @ A6 )
         => ! [Y: b] :
              ( ( member_b @ Y @ A6 )
             => ( ( ( F @ X )
                  = ( F @ Y ) )
               => ( X = Y ) ) ) ) ) ) ).

% inj_on_def
thf(fact_468_inj__on__def,axiom,
    ( inj_on374126998980950615list_o
    = ( ^ [F: a > option_list_o,A6: set_a] :
        ! [X: a] :
          ( ( member_a @ X @ A6 )
         => ! [Y: a] :
              ( ( member_a @ Y @ A6 )
             => ( ( ( F @ X )
                  = ( F @ Y ) )
               => ( X = Y ) ) ) ) ) ) ).

% inj_on_def
thf(fact_469_inj__on__def,axiom,
    ( inj_on_a_a
    = ( ^ [F: a > a,A6: set_a] :
        ! [X: a] :
          ( ( member_a @ X @ A6 )
         => ! [Y: a] :
              ( ( member_a @ Y @ A6 )
             => ( ( ( F @ X )
                  = ( F @ Y ) )
               => ( X = Y ) ) ) ) ) ) ).

% inj_on_def
thf(fact_470_inj__on__def,axiom,
    ( inj_on4343330552946057671_a_b_a
    = ( ^ [F: product_prod_a_b > a,A6: set_Product_prod_a_b] :
        ! [X: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X @ A6 )
         => ! [Y: product_prod_a_b] :
              ( ( member1426531481828664017od_a_b @ Y @ A6 )
             => ( ( ( F @ X )
                  = ( F @ Y ) )
               => ( X = Y ) ) ) ) ) ) ).

% inj_on_def
thf(fact_471_inj__on__cong,axiom,
    ! [A4: set_a,F2: a > option_list_o,G: a > option_list_o] :
      ( ! [A5: a] :
          ( ( member_a @ A5 @ A4 )
         => ( ( F2 @ A5 )
            = ( G @ A5 ) ) )
     => ( ( inj_on374126998980950615list_o @ F2 @ A4 )
        = ( inj_on374126998980950615list_o @ G @ A4 ) ) ) ).

% inj_on_cong
thf(fact_472_inj__on__cong,axiom,
    ! [A4: set_a,F2: a > a,G: a > a] :
      ( ! [A5: a] :
          ( ( member_a @ A5 @ A4 )
         => ( ( F2 @ A5 )
            = ( G @ A5 ) ) )
     => ( ( inj_on_a_a @ F2 @ A4 )
        = ( inj_on_a_a @ G @ A4 ) ) ) ).

% inj_on_cong
thf(fact_473_inj__on__cong,axiom,
    ! [A4: set_Product_prod_a_b,F2: product_prod_a_b > a,G: product_prod_a_b > a] :
      ( ! [A5: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ A5 @ A4 )
         => ( ( F2 @ A5 )
            = ( G @ A5 ) ) )
     => ( ( inj_on4343330552946057671_a_b_a @ F2 @ A4 )
        = ( inj_on4343330552946057671_a_b_a @ G @ A4 ) ) ) ).

% inj_on_cong
thf(fact_474_inj__on__cong,axiom,
    ! [A4: set_b,F2: b > option_list_o,G: b > option_list_o] :
      ( ! [A5: b] :
          ( ( member_b @ A5 @ A4 )
         => ( ( F2 @ A5 )
            = ( G @ A5 ) ) )
     => ( ( inj_on8261448415883032086list_o @ F2 @ A4 )
        = ( inj_on8261448415883032086list_o @ G @ A4 ) ) ) ).

% inj_on_cong
thf(fact_475_inj__on__eq__iff,axiom,
    ! [F2: a > option_list_o,A4: set_a,X3: a,Y4: a] :
      ( ( inj_on374126998980950615list_o @ F2 @ A4 )
     => ( ( member_a @ X3 @ A4 )
       => ( ( member_a @ Y4 @ A4 )
         => ( ( ( F2 @ X3 )
              = ( F2 @ Y4 ) )
            = ( X3 = Y4 ) ) ) ) ) ).

% inj_on_eq_iff
thf(fact_476_inj__on__eq__iff,axiom,
    ! [F2: a > a,A4: set_a,X3: a,Y4: a] :
      ( ( inj_on_a_a @ F2 @ A4 )
     => ( ( member_a @ X3 @ A4 )
       => ( ( member_a @ Y4 @ A4 )
         => ( ( ( F2 @ X3 )
              = ( F2 @ Y4 ) )
            = ( X3 = Y4 ) ) ) ) ) ).

% inj_on_eq_iff
thf(fact_477_inj__on__eq__iff,axiom,
    ! [F2: product_prod_a_b > a,A4: set_Product_prod_a_b,X3: product_prod_a_b,Y4: product_prod_a_b] :
      ( ( inj_on4343330552946057671_a_b_a @ F2 @ A4 )
     => ( ( member1426531481828664017od_a_b @ X3 @ A4 )
       => ( ( member1426531481828664017od_a_b @ Y4 @ A4 )
         => ( ( ( F2 @ X3 )
              = ( F2 @ Y4 ) )
            = ( X3 = Y4 ) ) ) ) ) ).

% inj_on_eq_iff
thf(fact_478_inj__on__eq__iff,axiom,
    ! [F2: b > option_list_o,A4: set_b,X3: b,Y4: b] :
      ( ( inj_on8261448415883032086list_o @ F2 @ A4 )
     => ( ( member_b @ X3 @ A4 )
       => ( ( member_b @ Y4 @ A4 )
         => ( ( ( F2 @ X3 )
              = ( F2 @ Y4 ) )
            = ( X3 = Y4 ) ) ) ) ) ).

% inj_on_eq_iff
thf(fact_479_inj__on__contraD,axiom,
    ! [F2: a > option_list_o,A4: set_a,X3: a,Y4: a] :
      ( ( inj_on374126998980950615list_o @ F2 @ A4 )
     => ( ( X3 != Y4 )
       => ( ( member_a @ X3 @ A4 )
         => ( ( member_a @ Y4 @ A4 )
           => ( ( F2 @ X3 )
             != ( F2 @ Y4 ) ) ) ) ) ) ).

% inj_on_contraD
thf(fact_480_inj__on__contraD,axiom,
    ! [F2: a > a,A4: set_a,X3: a,Y4: a] :
      ( ( inj_on_a_a @ F2 @ A4 )
     => ( ( X3 != Y4 )
       => ( ( member_a @ X3 @ A4 )
         => ( ( member_a @ Y4 @ A4 )
           => ( ( F2 @ X3 )
             != ( F2 @ Y4 ) ) ) ) ) ) ).

% inj_on_contraD
thf(fact_481_inj__on__contraD,axiom,
    ! [F2: product_prod_a_b > a,A4: set_Product_prod_a_b,X3: product_prod_a_b,Y4: product_prod_a_b] :
      ( ( inj_on4343330552946057671_a_b_a @ F2 @ A4 )
     => ( ( X3 != Y4 )
       => ( ( member1426531481828664017od_a_b @ X3 @ A4 )
         => ( ( member1426531481828664017od_a_b @ Y4 @ A4 )
           => ( ( F2 @ X3 )
             != ( F2 @ Y4 ) ) ) ) ) ) ).

% inj_on_contraD
thf(fact_482_inj__on__contraD,axiom,
    ! [F2: b > option_list_o,A4: set_b,X3: b,Y4: b] :
      ( ( inj_on8261448415883032086list_o @ F2 @ A4 )
     => ( ( X3 != Y4 )
       => ( ( member_b @ X3 @ A4 )
         => ( ( member_b @ Y4 @ A4 )
           => ( ( F2 @ X3 )
             != ( F2 @ Y4 ) ) ) ) ) ) ).

% inj_on_contraD
thf(fact_483_inj__on__inverseI,axiom,
    ! [A4: set_a,G: option_list_o > a,F2: a > option_list_o] :
      ( ! [X2: a] :
          ( ( member_a @ X2 @ A4 )
         => ( ( G @ ( F2 @ X2 ) )
            = X2 ) )
     => ( inj_on374126998980950615list_o @ F2 @ A4 ) ) ).

% inj_on_inverseI
thf(fact_484_inj__on__inverseI,axiom,
    ! [A4: set_a,G: a > a,F2: a > a] :
      ( ! [X2: a] :
          ( ( member_a @ X2 @ A4 )
         => ( ( G @ ( F2 @ X2 ) )
            = X2 ) )
     => ( inj_on_a_a @ F2 @ A4 ) ) ).

% inj_on_inverseI
thf(fact_485_inj__on__inverseI,axiom,
    ! [A4: set_Product_prod_a_b,G: a > product_prod_a_b,F2: product_prod_a_b > a] :
      ( ! [X2: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X2 @ A4 )
         => ( ( G @ ( F2 @ X2 ) )
            = X2 ) )
     => ( inj_on4343330552946057671_a_b_a @ F2 @ A4 ) ) ).

% inj_on_inverseI
thf(fact_486_inj__on__inverseI,axiom,
    ! [A4: set_b,G: option_list_o > b,F2: b > option_list_o] :
      ( ! [X2: b] :
          ( ( member_b @ X2 @ A4 )
         => ( ( G @ ( F2 @ X2 ) )
            = X2 ) )
     => ( inj_on8261448415883032086list_o @ F2 @ A4 ) ) ).

% inj_on_inverseI
thf(fact_487_apfst__convE,axiom,
    ! [Q: product_prod_b_b,F2: b > b,P: product_prod_b_b] :
      ( ( Q
        = ( product_apfst_b_b_b @ F2 @ P ) )
     => ~ ! [X2: b,Y2: b] :
            ( ( P
              = ( product_Pair_b_b @ X2 @ Y2 ) )
           => ( Q
             != ( product_Pair_b_b @ ( F2 @ X2 ) @ Y2 ) ) ) ) ).

% apfst_convE
thf(fact_488_apfst__convE,axiom,
    ! [Q: product_prod_a_b,F2: b > a,P: product_prod_b_b] :
      ( ( Q
        = ( product_apfst_b_a_b @ F2 @ P ) )
     => ~ ! [X2: b,Y2: b] :
            ( ( P
              = ( product_Pair_b_b @ X2 @ Y2 ) )
           => ( Q
             != ( product_Pair_a_b @ ( F2 @ X2 ) @ Y2 ) ) ) ) ).

% apfst_convE
thf(fact_489_apfst__convE,axiom,
    ! [Q: product_prod_b_a,F2: b > b,P: product_prod_b_a] :
      ( ( Q
        = ( product_apfst_b_b_a @ F2 @ P ) )
     => ~ ! [X2: b,Y2: a] :
            ( ( P
              = ( product_Pair_b_a @ X2 @ Y2 ) )
           => ( Q
             != ( product_Pair_b_a @ ( F2 @ X2 ) @ Y2 ) ) ) ) ).

% apfst_convE
thf(fact_490_apfst__convE,axiom,
    ! [Q: product_prod_a_a,F2: b > a,P: product_prod_b_a] :
      ( ( Q
        = ( product_apfst_b_a_a @ F2 @ P ) )
     => ~ ! [X2: b,Y2: a] :
            ( ( P
              = ( product_Pair_b_a @ X2 @ Y2 ) )
           => ( Q
             != ( product_Pair_a_a @ ( F2 @ X2 ) @ Y2 ) ) ) ) ).

% apfst_convE
thf(fact_491_apfst__convE,axiom,
    ! [Q: produc5884233991663340231list_o,F2: a > a,P: produc5884233991663340231list_o] :
      ( ( Q
        = ( produc948369090323839586list_o @ F2 @ P ) )
     => ~ ! [X2: a,Y2: list_o] :
            ( ( P
              = ( produc6899623729582506167list_o @ X2 @ Y2 ) )
           => ( Q
             != ( produc6899623729582506167list_o @ ( F2 @ X2 ) @ Y2 ) ) ) ) ).

% apfst_convE
thf(fact_492_apfst__convE,axiom,
    ! [Q: product_prod_b_b,F2: a > b,P: product_prod_a_b] :
      ( ( Q
        = ( product_apfst_a_b_b @ F2 @ P ) )
     => ~ ! [X2: a,Y2: b] :
            ( ( P
              = ( product_Pair_a_b @ X2 @ Y2 ) )
           => ( Q
             != ( product_Pair_b_b @ ( F2 @ X2 ) @ Y2 ) ) ) ) ).

% apfst_convE
thf(fact_493_apfst__convE,axiom,
    ! [Q: product_prod_b_a,F2: a > b,P: product_prod_a_a] :
      ( ( Q
        = ( product_apfst_a_b_a @ F2 @ P ) )
     => ~ ! [X2: a,Y2: a] :
            ( ( P
              = ( product_Pair_a_a @ X2 @ Y2 ) )
           => ( Q
             != ( product_Pair_b_a @ ( F2 @ X2 ) @ Y2 ) ) ) ) ).

% apfst_convE
thf(fact_494_apfst__convE,axiom,
    ! [Q: product_prod_a_a,F2: a > a,P: product_prod_a_a] :
      ( ( Q
        = ( product_apfst_a_a_a @ F2 @ P ) )
     => ~ ! [X2: a,Y2: a] :
            ( ( P
              = ( product_Pair_a_a @ X2 @ Y2 ) )
           => ( Q
             != ( product_Pair_a_a @ ( F2 @ X2 ) @ Y2 ) ) ) ) ).

% apfst_convE
thf(fact_495_apfst__convE,axiom,
    ! [Q: product_prod_a_b,F2: a > a,P: product_prod_a_b] :
      ( ( Q
        = ( product_apfst_a_a_b @ F2 @ P ) )
     => ~ ! [X2: a,Y2: b] :
            ( ( P
              = ( product_Pair_a_b @ X2 @ Y2 ) )
           => ( Q
             != ( product_Pair_a_b @ ( F2 @ X2 ) @ Y2 ) ) ) ) ).

% apfst_convE
thf(fact_496_snd__sndOp,axiom,
    ! [P2: a > b > $o,Q2: b > b > $o] :
      ( product_snd_a_b
      = ( comp_P5715990135234274861od_a_b @ product_snd_b_b @ ( bNF_sndOp_a_b_b @ P2 @ Q2 ) ) ) ).

% snd_sndOp
thf(fact_497_snd__sndOp,axiom,
    ! [P2: b > b > $o,Q2: b > a > $o] :
      ( product_snd_b_a
      = ( comp_P7811403737996458923od_b_a @ product_snd_b_a @ ( bNF_sndOp_b_b_a @ P2 @ Q2 ) ) ) ).

% snd_sndOp
thf(fact_498_snd__sndOp,axiom,
    ! [P2: b > a > $o,Q2: a > a > $o] :
      ( product_snd_b_a
      = ( comp_P4104929595196636140od_b_a @ product_snd_a_a @ ( bNF_sndOp_b_a_a @ P2 @ Q2 ) ) ) ).

% snd_sndOp
thf(fact_499_snd__sndOp,axiom,
    ! [P2: b > a > $o,Q2: a > b > $o] :
      ( product_snd_b_b
      = ( comp_P8445432447316551853od_b_b @ product_snd_a_b @ ( bNF_sndOp_b_a_b @ P2 @ Q2 ) ) ) ).

% snd_sndOp
thf(fact_500_snd__sndOp,axiom,
    ! [P2: b > b > $o,Q2: b > b > $o] :
      ( product_snd_b_b
      = ( comp_P2928534553261598828od_b_b @ product_snd_b_b @ ( bNF_sndOp_b_b_b @ P2 @ Q2 ) ) ) ).

% snd_sndOp
thf(fact_501_snd__sndOp,axiom,
    ! [P2: a > b > $o,Q2: b > a > $o] :
      ( product_snd_a_a
      = ( comp_P1375487283114359148od_a_a @ product_snd_b_a @ ( bNF_sndOp_a_b_a @ P2 @ Q2 ) ) ) ).

% snd_sndOp
thf(fact_502_snd__sndOp,axiom,
    ! [P2: a > a > $o,Q2: a > a > $o] :
      ( product_snd_a_a
      = ( comp_P6892385177169312173od_a_a @ product_snd_a_a @ ( bNF_sndOp_a_a_a @ P2 @ Q2 ) ) ) ).

% snd_sndOp
thf(fact_503_snd__sndOp,axiom,
    ! [P2: a > a > $o,Q2: a > b > $o] :
      ( product_snd_a_b
      = ( comp_P2009515992434452078od_a_b @ product_snd_a_b @ ( bNF_sndOp_a_a_b @ P2 @ Q2 ) ) ) ).

% snd_sndOp
thf(fact_504_fst__fstOp,axiom,
    ! [P2: a > b > $o,Q2: b > b > $o] :
      ( product_fst_a_b
      = ( comp_P2648956292766268207od_a_b @ product_fst_a_b @ ( bNF_fstOp_a_b_b @ P2 @ Q2 ) ) ) ).

% fst_fstOp
thf(fact_505_fst__fstOp,axiom,
    ! [P2: a > a > $o,Q2: a > b > $o] :
      ( product_fst_a_b
      = ( comp_P6892385181472540974od_a_b @ product_fst_a_a @ ( bNF_fstOp_a_a_b @ P2 @ Q2 ) ) ) ).

% fst_fstOp
thf(fact_506_fst__fstOp,axiom,
    ! [P2: b > a > $o,Q2: a > a > $o] :
      ( product_fst_b_a
      = ( comp_P7171963437664642794od_b_a @ product_fst_b_a @ ( bNF_fstOp_b_a_a @ P2 @ Q2 ) ) ) ).

% fst_fstOp
thf(fact_507_fst__fstOp,axiom,
    ! [P2: b > b > $o,Q2: b > a > $o] :
      ( product_fst_b_a
      = ( comp_P2928534548958370027od_b_a @ product_fst_b_b @ ( bNF_fstOp_b_b_a @ P2 @ Q2 ) ) ) ).

% fst_fstOp
thf(fact_508_fst__fstOp,axiom,
    ! [P2: a > b > $o,Q2: b > a > $o] :
      ( product_fst_a_a
      = ( comp_P2648956288463039406od_a_a @ product_fst_a_b @ ( bNF_fstOp_a_b_a @ P2 @ Q2 ) ) ) ).

% fst_fstOp
thf(fact_509_fst__fstOp,axiom,
    ! [P2: a > a > $o,Q2: a > a > $o] :
      ( product_fst_a_a
      = ( comp_P6892385177169312173od_a_a @ product_fst_a_a @ ( bNF_fstOp_a_a_a @ P2 @ Q2 ) ) ) ).

% fst_fstOp
thf(fact_510_fst__fstOp,axiom,
    ! [P2: b > a > $o,Q2: a > b > $o] :
      ( product_fst_b_b
      = ( comp_P7171963441967871595od_b_b @ product_fst_b_a @ ( bNF_fstOp_b_a_b @ P2 @ Q2 ) ) ) ).

% fst_fstOp
thf(fact_511_fst__fstOp,axiom,
    ! [P2: b > b > $o,Q2: b > b > $o] :
      ( product_fst_b_b
      = ( comp_P2928534553261598828od_b_b @ product_fst_b_b @ ( bNF_fstOp_b_b_b @ P2 @ Q2 ) ) ) ).

% fst_fstOp
thf(fact_512_fst__fstOp,axiom,
    ! [P2: a > list_o > $o,Q2: list_o > b > $o] :
      ( product_fst_a_b
      = ( comp_P2559030039514677966od_a_b @ product_fst_a_list_o @ ( bNF_fstOp_a_list_o_b @ P2 @ Q2 ) ) ) ).

% fst_fstOp
thf(fact_513_fst__fstOp,axiom,
    ! [P2: a > list_o > $o,Q2: list_o > a > $o] :
      ( product_fst_a_a
      = ( comp_P2559030035211449165od_a_a @ product_fst_a_list_o @ ( bNF_fstOp_a_list_o_a @ P2 @ Q2 ) ) ) ).

% fst_fstOp
thf(fact_514_Collect__case__prod__Grp__eqD,axiom,
    ! [Z2: product_prod_a_b,A4: set_a,F2: a > b] :
      ( ( member1426531481828664017od_a_b @ Z2 @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ ( bNF_Grp_a_b @ A4 @ F2 ) ) ) )
     => ( ( comp_a9170378079104387268od_a_b @ F2 @ product_fst_a_b @ Z2 )
        = ( product_snd_a_b @ Z2 ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_515_Collect__case__prod__Grp__eqD,axiom,
    ! [Z2: product_prod_nat_nat,A4: set_nat,F2: nat > nat] :
      ( ( member8440522571783428010at_nat @ Z2 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( bNF_Grp_nat_nat @ A4 @ F2 ) ) ) )
     => ( ( comp_n2711240574518091068at_nat @ F2 @ product_fst_nat_nat @ Z2 )
        = ( product_snd_nat_nat @ Z2 ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_516_Collect__case__prod__Grp__eqD,axiom,
    ! [Z2: product_prod_b_a,A4: set_b,F2: b > a] :
      ( ( member7862447932407534991od_b_a @ Z2 @ ( collec548942215411776465od_b_a @ ( produc7672072387910335853_b_a_o @ ( bNF_Grp_b_a @ A4 @ F2 ) ) ) )
     => ( ( comp_b1738939342930358980od_b_a @ F2 @ product_fst_b_a @ Z2 )
        = ( product_snd_b_a @ Z2 ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_517_Collect__case__prod__Grp__eqD,axiom,
    ! [Z2: product_prod_b_b,A4: set_b,F2: b > b] :
      ( ( member7862447936710763792od_b_b @ Z2 @ ( collec548942219715005266od_b_b @ ( produc4772849988445941548_b_b_o @ ( bNF_Grp_b_b @ A4 @ F2 ) ) ) )
     => ( ( comp_b1099499046901771652od_b_b @ F2 @ product_fst_b_b @ Z2 )
        = ( product_snd_b_b @ Z2 ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_518_Collect__case__prod__Grp__eqD,axiom,
    ! [Z2: product_prod_a_a,A4: set_a,F2: a > a] :
      ( ( member1426531477525435216od_a_a @ Z2 @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ ( bNF_Grp_a_a @ A4 @ F2 ) ) ) )
     => ( ( comp_a586446338278198788od_a_a @ F2 @ product_fst_a_a @ Z2 )
        = ( product_snd_a_a @ Z2 ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_519_Collect__case__prod__Grp__eqD,axiom,
    ! [Z2: produc5884233991663340231list_o,A4: set_a,F2: a > list_o] :
      ( ( member7948383622993546480list_o @ Z2 @ ( collec7805115074396547890list_o @ ( produc8342644129638033356st_o_o @ ( bNF_Grp_a_list_o @ A4 @ F2 ) ) ) )
     => ( ( comp_a7612839990368905732list_o @ F2 @ product_fst_a_list_o @ Z2 )
        = ( product_snd_a_list_o @ Z2 ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_520_Collect__case__prod__Grp__eqD,axiom,
    ! [Z2: produc3167243049347903741_a_b_a,A4: set_Product_prod_a_b,F2: product_prod_a_b > a] :
      ( ( member2194267517197600294_a_b_a @ Z2 @ ( collec4168926458763418216_a_b_a @ ( produc1879355882332503830_b_a_o @ ( bNF_Gr7842136747927947300_a_b_a @ A4 @ F2 ) ) ) )
     => ( ( comp_P2639128986261590148_a_b_a @ F2 @ produc1917971008728229337_a_b_a @ Z2 )
        = ( produc6837468258142229275_a_b_a @ Z2 ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_521_Collect__case__prod__Grp__eqD,axiom,
    ! [Z2: produc3167243053651132542_a_b_b,A4: set_Product_prod_a_b,F2: product_prod_a_b > b] :
      ( ( member2194267521500829095_a_b_b @ Z2 @ ( collec4168926463066647017_a_b_b @ ( produc8203505519722885333_b_b_o @ ( bNF_Gr7842136747927947301_a_b_b @ A4 @ F2 ) ) ) )
     => ( ( comp_P7811183620037784644_a_b_b @ F2 @ produc1917971008728229338_a_b_b @ Z2 )
        = ( produc6837468258142229276_a_b_b @ Z2 ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_522_Collect__case__prod__Grp__eqD,axiom,
    ! [Z2: produc2233330766403592615od_a_b,A4: set_Product_prod_a_b,F2: product_prod_a_b > product_prod_a_b] :
      ( ( member5065438833300252112od_a_b @ Z2 @ ( collec7968472090841016338od_a_b @ ( produc5413562937804622316_a_b_o @ ( bNF_Gr7211640137128594382od_a_b @ A4 @ F2 ) ) ) )
     => ( ( comp_P6366629489221263108od_a_b @ F2 @ produc5013601669340818563od_a_b @ Z2 )
        = ( produc8875788424405714885od_a_b @ Z2 ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_523_Collect__case__prod__Grp__eqD,axiom,
    ! [Z2: produc1065045136403542119od_a_a,A4: set_Pr5493900063595384711od_a_b,F2: produc2233330766403592615od_a_b > product_prod_a_a] :
      ( ( member4368417410682638992od_a_a @ Z2 @ ( collec5379532638810960082od_a_a @ ( produc5892353234140082604_a_a_o @ ( bNF_Gr248057877874286094od_a_a @ A4 @ F2 ) ) ) )
     => ( ( comp_P3398507451812940164od_a_a @ F2 @ produc7653073535890015427od_a_a @ Z2 )
        = ( produc4124993899221236741od_a_a @ Z2 ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_524_internal__case__prod__def,axiom,
    produc8376542487572969895od_a_b = produc4348216232050026237od_a_b ).

% internal_case_prod_def
thf(fact_525_internal__case__prod__def,axiom,
    produc1009698014340412707od_b_a = produc6204743795672244857od_b_a ).

% internal_case_prod_def
thf(fact_526_internal__case__prod__def,axiom,
    produc4617426408912073941_a_b_o = produc3537405659489547051_a_b_o ).

% internal_case_prod_def
thf(fact_527_internal__case__prod__def,axiom,
    produc318858384453529020_a_b_b = produc6028431345588019474_a_b_b ).

% internal_case_prod_def
thf(fact_528_internal__case__prod__def,axiom,
    produc318858384453529019_a_b_a = produc6028431345588019473_a_b_a ).

% internal_case_prod_def
thf(fact_529_snd__comp__map__prod,axiom,
    ! [F2: a > a,G: b > b] :
      ( ( comp_P2009515992434452078od_a_b @ product_snd_a_b @ ( produc1231242867592151606_a_b_b @ F2 @ G ) )
      = ( comp_b3886954628874447685od_a_b @ G @ product_snd_a_b ) ) ).

% snd_comp_map_prod
thf(fact_530_snd__comp__map__prod,axiom,
    ! [F2: b > a,G: a > b] :
      ( ( comp_P8445432443013323052od_b_a @ product_snd_a_b @ ( produc5506660486859626166_a_a_b @ F2 @ G ) )
      = ( comp_a6382922492828482434od_b_a @ G @ product_snd_b_a ) ) ).

% snd_comp_map_prod
thf(fact_531_snd__comp__map__prod,axiom,
    ! [F2: b > a,G: b > b] :
      ( ( comp_P8445432447316551853od_b_b @ product_snd_a_b @ ( produc2719204904886950133_a_b_b @ F2 @ G ) )
      = ( comp_b1099499046901771652od_b_b @ G @ product_snd_b_b ) ) ).

% snd_comp_map_prod
thf(fact_532_snd__comp__map__prod,axiom,
    ! [F2: a > a,G: a > b] :
      ( ( comp_P2009515988131223277od_a_a @ product_snd_a_b @ ( produc4018698449564827639_a_a_b @ F2 @ G ) )
      = ( comp_a9170378074801158467od_a_a @ G @ product_snd_a_a ) ) ).

% snd_comp_map_prod
thf(fact_533_snd__comp__map__prod,axiom,
    ! [F2: a > b,G: b > a] :
      ( ( comp_P1375487287417587949od_a_b @ product_snd_b_a @ ( produc6827461482780282294_b_b_a @ F2 @ G ) )
      = ( comp_b4526394929206263814od_a_b @ G @ product_snd_a_b ) ) ).

% snd_comp_map_prod
thf(fact_534_snd__comp__map__prod,axiom,
    ! [F2: b > b,G: a > a] :
      ( ( comp_P7811403737996458923od_b_a @ product_snd_b_a @ ( produc1879507065192981046_b_a_a @ F2 @ G ) )
      = ( comp_a7022362793160298563od_b_a @ G @ product_snd_b_a ) ) ).

% snd_comp_map_prod
thf(fact_535_snd__comp__map__prod,axiom,
    ! [F2: b > b,G: b > a] :
      ( ( comp_P7811403742299687724od_b_b @ product_snd_b_a @ ( produc8315423520075080821_b_b_a @ F2 @ G ) )
      = ( comp_b1738939347233587781od_b_b @ G @ product_snd_b_b ) ) ).

% snd_comp_map_prod
thf(fact_536_snd__comp__map__prod,axiom,
    ! [F2: a > b,G: a > a] :
      ( ( comp_P1375487283114359148od_a_a @ product_snd_b_a @ ( produc391545027898182519_b_a_a @ F2 @ G ) )
      = ( comp_a586446338278198788od_a_a @ G @ product_snd_a_a ) ) ).

% snd_comp_map_prod
thf(fact_537_snd__comp__map__prod,axiom,
    ! [F2: a > b,G: b > b] :
      ( ( comp_P5715990135234274861od_a_b @ product_snd_b_b @ ( produc6827461482780282295_b_b_b @ F2 @ G ) )
      = ( comp_b3886954628874447685od_a_b @ G @ product_snd_a_b ) ) ).

% snd_comp_map_prod
thf(fact_538_snd__comp__map__prod,axiom,
    ! [F2: b > b,G: a > b] :
      ( ( comp_P2928534548958370027od_b_a @ product_snd_b_b @ ( produc1879507065192981047_b_a_b @ F2 @ G ) )
      = ( comp_a6382922492828482434od_b_a @ G @ product_snd_b_a ) ) ).

% snd_comp_map_prod
thf(fact_539_fst__comp__map__prod,axiom,
    ! [F2: a > a,G: b > b] :
      ( ( comp_P2648956292766268207od_a_b @ product_fst_a_b @ ( produc1231242867592151606_a_b_b @ F2 @ G ) )
      = ( comp_a586446342581427589od_a_b @ F2 @ product_fst_a_b ) ) ).

% fst_comp_map_prod
thf(fact_540_fst__comp__map__prod,axiom,
    ! [F2: b > a,G: a > b] :
      ( ( comp_P9084872743345139181od_b_a @ product_fst_a_b @ ( produc5506660486859626166_a_a_b @ F2 @ G ) )
      = ( comp_b1738939342930358980od_b_a @ F2 @ product_fst_b_a ) ) ).

% fst_comp_map_prod
thf(fact_541_fst__comp__map__prod,axiom,
    ! [F2: a > a,G: a > b] :
      ( ( comp_P2648956288463039406od_a_a @ product_fst_a_b @ ( produc4018698449564827639_a_a_b @ F2 @ G ) )
      = ( comp_a586446338278198788od_a_a @ F2 @ product_fst_a_a ) ) ).

% fst_comp_map_prod
thf(fact_542_fst__comp__map__prod,axiom,
    ! [F2: b > a,G: b > b] :
      ( ( comp_P9084872747648367982od_b_b @ product_fst_a_b @ ( produc2719204904886950133_a_b_b @ F2 @ G ) )
      = ( comp_b1738939347233587781od_b_b @ F2 @ product_fst_b_b ) ) ).

% fst_comp_map_prod
thf(fact_543_fst__comp__map__prod,axiom,
    ! [F2: a > b,G: b > a] :
      ( ( comp_P736046987085771820od_a_b @ product_fst_b_a @ ( produc6827461482780282294_b_b_a @ F2 @ G ) )
      = ( comp_a9170378079104387268od_a_b @ F2 @ product_fst_a_b ) ) ).

% fst_comp_map_prod
thf(fact_544_fst__comp__map__prod,axiom,
    ! [F2: b > b,G: a > a] :
      ( ( comp_P7171963437664642794od_b_a @ product_fst_b_a @ ( produc1879507065192981046_b_a_a @ F2 @ G ) )
      = ( comp_b1099499042598542851od_b_a @ F2 @ product_fst_b_a ) ) ).

% fst_comp_map_prod
thf(fact_545_fst__comp__map__prod,axiom,
    ! [F2: a > b,G: a > a] :
      ( ( comp_P736046982782543019od_a_a @ product_fst_b_a @ ( produc391545027898182519_b_a_a @ F2 @ G ) )
      = ( comp_a9170378074801158467od_a_a @ F2 @ product_fst_a_a ) ) ).

% fst_comp_map_prod
thf(fact_546_fst__comp__map__prod,axiom,
    ! [F2: b > b,G: b > a] :
      ( ( comp_P7171963441967871595od_b_b @ product_fst_b_a @ ( produc8315423520075080821_b_b_a @ F2 @ G ) )
      = ( comp_b1099499046901771652od_b_b @ F2 @ product_fst_b_b ) ) ).

% fst_comp_map_prod
thf(fact_547_fst__comp__map__prod,axiom,
    ! [F2: a > a,G: b > a] :
      ( ( comp_P6892385181472540974od_a_b @ product_fst_a_a @ ( produc1231242867592151605_a_b_a @ F2 @ G ) )
      = ( comp_a586446342581427589od_a_b @ F2 @ product_fst_a_b ) ) ).

% fst_comp_map_prod
thf(fact_548_fst__comp__map__prod,axiom,
    ! [F2: b > a,G: a > a] :
      ( ( comp_P4104929595196636140od_b_a @ product_fst_a_a @ ( produc5506660486859626165_a_a_a @ F2 @ G ) )
      = ( comp_b1738939342930358980od_b_a @ F2 @ product_fst_b_a ) ) ).

% fst_comp_map_prod
thf(fact_549_fstOp__def,axiom,
    ( bNF_fstOp_a_b_b
    = ( ^ [P4: a > b > $o,Q3: b > b > $o,Ac: product_prod_a_b] : ( product_Pair_a_b @ ( product_fst_a_b @ Ac ) @ ( bNF_pi7997277732171029722_a_b_b @ P4 @ Q3 @ ( product_fst_a_b @ Ac ) @ ( product_snd_a_b @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_550_fstOp__def,axiom,
    ( bNF_fstOp_a_a_b
    = ( ^ [P4: a > a > $o,Q3: a > b > $o,Ac: product_prod_a_b] : ( product_Pair_a_a @ ( product_fst_a_b @ Ac ) @ ( bNF_pi1561361277288929947_a_a_b @ P4 @ Q3 @ ( product_fst_a_b @ Ac ) @ ( product_snd_a_b @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_551_fstOp__def,axiom,
    ( bNF_fstOp_b_b_a
    = ( ^ [P4: b > b > $o,Q3: b > a > $o,Ac: product_prod_b_a] : ( product_Pair_b_b @ ( product_fst_b_a @ Ac ) @ ( bNF_pi4370124310504384602_b_b_a @ P4 @ Q3 @ ( product_fst_b_a @ Ac ) @ ( product_snd_b_a @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_552_fstOp__def,axiom,
    ( bNF_fstOp_b_a_a
    = ( ^ [P4: b > a > $o,Q3: a > a > $o,Ac: product_prod_b_a] : ( product_Pair_b_a @ ( product_fst_b_a @ Ac ) @ ( bNF_pi7157579892477060635_b_a_a @ P4 @ Q3 @ ( product_fst_b_a @ Ac ) @ ( product_snd_b_a @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_553_fstOp__def,axiom,
    ( bNF_fstOp_b_b_b
    = ( ^ [P4: b > b > $o,Q3: b > b > $o,Ac: product_prod_b_b] : ( product_Pair_b_b @ ( product_fst_b_b @ Ac ) @ ( bNF_pi4370124310504384603_b_b_b @ P4 @ Q3 @ ( product_fst_b_b @ Ac ) @ ( product_snd_b_b @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_554_fstOp__def,axiom,
    ( bNF_fstOp_b_a_b
    = ( ^ [P4: b > a > $o,Q3: a > b > $o,Ac: product_prod_b_b] : ( product_Pair_b_a @ ( product_fst_b_b @ Ac ) @ ( bNF_pi7157579892477060636_b_a_b @ P4 @ Q3 @ ( product_fst_b_b @ Ac ) @ ( product_snd_b_b @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_555_fstOp__def,axiom,
    ( bNF_fstOp_a_b_a
    = ( ^ [P4: a > b > $o,Q3: b > a > $o,Ac: product_prod_a_a] : ( product_Pair_a_b @ ( product_fst_a_a @ Ac ) @ ( bNF_pi7997277732171029721_a_b_a @ P4 @ Q3 @ ( product_fst_a_a @ Ac ) @ ( product_snd_a_a @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_556_fstOp__def,axiom,
    ( bNF_fstOp_a_a_a
    = ( ^ [P4: a > a > $o,Q3: a > a > $o,Ac: product_prod_a_a] : ( product_Pair_a_a @ ( product_fst_a_a @ Ac ) @ ( bNF_pi1561361277288929946_a_a_a @ P4 @ Q3 @ ( product_fst_a_a @ Ac ) @ ( product_snd_a_a @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_557_fstOp__def,axiom,
    ( bNF_fstOp_a_b_list_o
    = ( ^ [P4: a > b > $o,Q3: b > list_o > $o,Ac: produc5884233991663340231list_o] : ( product_Pair_a_b @ ( product_fst_a_list_o @ Ac ) @ ( bNF_pi5777870426848302713list_o @ P4 @ Q3 @ ( product_fst_a_list_o @ Ac ) @ ( product_snd_a_list_o @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_558_fstOp__def,axiom,
    ( bNF_fstOp_a_a_list_o
    = ( ^ [P4: a > a > $o,Q3: a > list_o > $o,Ac: produc5884233991663340231list_o] : ( product_Pair_a_a @ ( product_fst_a_list_o @ Ac ) @ ( bNF_pi4433405390038814778list_o @ P4 @ Q3 @ ( product_fst_a_list_o @ Ac ) @ ( product_snd_a_list_o @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_559_sndOp__def,axiom,
    ( bNF_sndOp_a_a_list_o
    = ( ^ [P4: a > a > $o,Q3: a > list_o > $o,Ac: produc5884233991663340231list_o] : ( produc6899623729582506167list_o @ ( bNF_pi4433405390038814778list_o @ P4 @ Q3 @ ( product_fst_a_list_o @ Ac ) @ ( product_snd_a_list_o @ Ac ) ) @ ( product_snd_a_list_o @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_560_sndOp__def,axiom,
    ( bNF_sndOp_a_b_b
    = ( ^ [P4: a > b > $o,Q3: b > b > $o,Ac: product_prod_a_b] : ( product_Pair_b_b @ ( bNF_pi7997277732171029722_a_b_b @ P4 @ Q3 @ ( product_fst_a_b @ Ac ) @ ( product_snd_a_b @ Ac ) ) @ ( product_snd_a_b @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_561_sndOp__def,axiom,
    ( bNF_sndOp_a_a_b
    = ( ^ [P4: a > a > $o,Q3: a > b > $o,Ac: product_prod_a_b] : ( product_Pair_a_b @ ( bNF_pi1561361277288929947_a_a_b @ P4 @ Q3 @ ( product_fst_a_b @ Ac ) @ ( product_snd_a_b @ Ac ) ) @ ( product_snd_a_b @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_562_sndOp__def,axiom,
    ( bNF_sndOp_b_b_a
    = ( ^ [P4: b > b > $o,Q3: b > a > $o,Ac: product_prod_b_a] : ( product_Pair_b_a @ ( bNF_pi4370124310504384602_b_b_a @ P4 @ Q3 @ ( product_fst_b_a @ Ac ) @ ( product_snd_b_a @ Ac ) ) @ ( product_snd_b_a @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_563_sndOp__def,axiom,
    ( bNF_sndOp_b_a_a
    = ( ^ [P4: b > a > $o,Q3: a > a > $o,Ac: product_prod_b_a] : ( product_Pair_a_a @ ( bNF_pi7157579892477060635_b_a_a @ P4 @ Q3 @ ( product_fst_b_a @ Ac ) @ ( product_snd_b_a @ Ac ) ) @ ( product_snd_b_a @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_564_sndOp__def,axiom,
    ( bNF_sndOp_b_b_b
    = ( ^ [P4: b > b > $o,Q3: b > b > $o,Ac: product_prod_b_b] : ( product_Pair_b_b @ ( bNF_pi4370124310504384603_b_b_b @ P4 @ Q3 @ ( product_fst_b_b @ Ac ) @ ( product_snd_b_b @ Ac ) ) @ ( product_snd_b_b @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_565_sndOp__def,axiom,
    ( bNF_sndOp_b_a_b
    = ( ^ [P4: b > a > $o,Q3: a > b > $o,Ac: product_prod_b_b] : ( product_Pair_a_b @ ( bNF_pi7157579892477060636_b_a_b @ P4 @ Q3 @ ( product_fst_b_b @ Ac ) @ ( product_snd_b_b @ Ac ) ) @ ( product_snd_b_b @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_566_sndOp__def,axiom,
    ( bNF_sndOp_a_b_a
    = ( ^ [P4: a > b > $o,Q3: b > a > $o,Ac: product_prod_a_a] : ( product_Pair_b_a @ ( bNF_pi7997277732171029721_a_b_a @ P4 @ Q3 @ ( product_fst_a_a @ Ac ) @ ( product_snd_a_a @ Ac ) ) @ ( product_snd_a_a @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_567_sndOp__def,axiom,
    ( bNF_sndOp_a_a_a
    = ( ^ [P4: a > a > $o,Q3: a > a > $o,Ac: product_prod_a_a] : ( product_Pair_a_a @ ( bNF_pi1561361277288929946_a_a_a @ P4 @ Q3 @ ( product_fst_a_a @ Ac ) @ ( product_snd_a_a @ Ac ) ) @ ( product_snd_a_a @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_568_map__prod__simp,axiom,
    ! [F2: b > b,G: b > b,A: b,B: b] :
      ( ( produc8315423520075080822_b_b_b @ F2 @ G @ ( product_Pair_b_b @ A @ B ) )
      = ( product_Pair_b_b @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_569_map__prod__simp,axiom,
    ! [F2: b > b,G: b > a,A: b,B: b] :
      ( ( produc8315423520075080821_b_b_a @ F2 @ G @ ( product_Pair_b_b @ A @ B ) )
      = ( product_Pair_b_a @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_570_map__prod__simp,axiom,
    ! [F2: b > a,G: b > b,A: b,B: b] :
      ( ( produc2719204904886950133_a_b_b @ F2 @ G @ ( product_Pair_b_b @ A @ B ) )
      = ( product_Pair_a_b @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_571_map__prod__simp,axiom,
    ! [F2: b > a,G: b > a,A: b,B: b] :
      ( ( produc2719204904886950132_a_b_a @ F2 @ G @ ( product_Pair_b_b @ A @ B ) )
      = ( product_Pair_a_a @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_572_map__prod__simp,axiom,
    ! [F2: b > b,G: a > b,A: b,B: a] :
      ( ( produc1879507065192981047_b_a_b @ F2 @ G @ ( product_Pair_b_a @ A @ B ) )
      = ( product_Pair_b_b @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_573_map__prod__simp,axiom,
    ! [F2: b > b,G: a > a,A: b,B: a] :
      ( ( produc1879507065192981046_b_a_a @ F2 @ G @ ( product_Pair_b_a @ A @ B ) )
      = ( product_Pair_b_a @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_574_map__prod__simp,axiom,
    ! [F2: b > a,G: a > b,A: b,B: a] :
      ( ( produc5506660486859626166_a_a_b @ F2 @ G @ ( product_Pair_b_a @ A @ B ) )
      = ( product_Pair_a_b @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_575_map__prod__simp,axiom,
    ! [F2: b > a,G: a > a,A: b,B: a] :
      ( ( produc5506660486859626165_a_a_a @ F2 @ G @ ( product_Pair_b_a @ A @ B ) )
      = ( product_Pair_a_a @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_576_map__prod__simp,axiom,
    ! [F2: a > b,G: b > b,A: a,B: b] :
      ( ( produc6827461482780282295_b_b_b @ F2 @ G @ ( product_Pair_a_b @ A @ B ) )
      = ( product_Pair_b_b @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_577_map__prod__simp,axiom,
    ! [F2: a > b,G: b > a,A: a,B: b] :
      ( ( produc6827461482780282294_b_b_a @ F2 @ G @ ( product_Pair_a_b @ A @ B ) )
      = ( product_Pair_b_a @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_578_fst__map__prod,axiom,
    ! [F2: a > a,G: b > b,X3: product_prod_a_b] :
      ( ( product_fst_a_b @ ( produc1231242867592151606_a_b_b @ F2 @ G @ X3 ) )
      = ( F2 @ ( product_fst_a_b @ X3 ) ) ) ).

% fst_map_prod
thf(fact_579_fst__map__prod,axiom,
    ! [F2: b > a,G: a > b,X3: product_prod_b_a] :
      ( ( product_fst_a_b @ ( produc5506660486859626166_a_a_b @ F2 @ G @ X3 ) )
      = ( F2 @ ( product_fst_b_a @ X3 ) ) ) ).

% fst_map_prod
thf(fact_580_fst__map__prod,axiom,
    ! [F2: a > a,G: a > b,X3: product_prod_a_a] :
      ( ( product_fst_a_b @ ( produc4018698449564827639_a_a_b @ F2 @ G @ X3 ) )
      = ( F2 @ ( product_fst_a_a @ X3 ) ) ) ).

% fst_map_prod
thf(fact_581_fst__map__prod,axiom,
    ! [F2: b > a,G: b > b,X3: product_prod_b_b] :
      ( ( product_fst_a_b @ ( produc2719204904886950133_a_b_b @ F2 @ G @ X3 ) )
      = ( F2 @ ( product_fst_b_b @ X3 ) ) ) ).

% fst_map_prod
thf(fact_582_fst__map__prod,axiom,
    ! [F2: a > b,G: b > a,X3: product_prod_a_b] :
      ( ( product_fst_b_a @ ( produc6827461482780282294_b_b_a @ F2 @ G @ X3 ) )
      = ( F2 @ ( product_fst_a_b @ X3 ) ) ) ).

% fst_map_prod
thf(fact_583_fst__map__prod,axiom,
    ! [F2: b > b,G: a > a,X3: product_prod_b_a] :
      ( ( product_fst_b_a @ ( produc1879507065192981046_b_a_a @ F2 @ G @ X3 ) )
      = ( F2 @ ( product_fst_b_a @ X3 ) ) ) ).

% fst_map_prod
thf(fact_584_fst__map__prod,axiom,
    ! [F2: a > b,G: a > a,X3: product_prod_a_a] :
      ( ( product_fst_b_a @ ( produc391545027898182519_b_a_a @ F2 @ G @ X3 ) )
      = ( F2 @ ( product_fst_a_a @ X3 ) ) ) ).

% fst_map_prod
thf(fact_585_fst__map__prod,axiom,
    ! [F2: b > b,G: b > a,X3: product_prod_b_b] :
      ( ( product_fst_b_a @ ( produc8315423520075080821_b_b_a @ F2 @ G @ X3 ) )
      = ( F2 @ ( product_fst_b_b @ X3 ) ) ) ).

% fst_map_prod
thf(fact_586_fst__map__prod,axiom,
    ! [F2: a > a,G: b > a,X3: product_prod_a_b] :
      ( ( product_fst_a_a @ ( produc1231242867592151605_a_b_a @ F2 @ G @ X3 ) )
      = ( F2 @ ( product_fst_a_b @ X3 ) ) ) ).

% fst_map_prod
thf(fact_587_fst__map__prod,axiom,
    ! [F2: b > a,G: a > a,X3: product_prod_b_a] :
      ( ( product_fst_a_a @ ( produc5506660486859626165_a_a_a @ F2 @ G @ X3 ) )
      = ( F2 @ ( product_fst_b_a @ X3 ) ) ) ).

% fst_map_prod
thf(fact_588_snd__map__prod,axiom,
    ! [F2: a > a,G: b > b,X3: product_prod_a_b] :
      ( ( product_snd_a_b @ ( produc1231242867592151606_a_b_b @ F2 @ G @ X3 ) )
      = ( G @ ( product_snd_a_b @ X3 ) ) ) ).

% snd_map_prod
thf(fact_589_snd__map__prod,axiom,
    ! [F2: b > a,G: a > b,X3: product_prod_b_a] :
      ( ( product_snd_a_b @ ( produc5506660486859626166_a_a_b @ F2 @ G @ X3 ) )
      = ( G @ ( product_snd_b_a @ X3 ) ) ) ).

% snd_map_prod
thf(fact_590_snd__map__prod,axiom,
    ! [F2: b > a,G: b > b,X3: product_prod_b_b] :
      ( ( product_snd_a_b @ ( produc2719204904886950133_a_b_b @ F2 @ G @ X3 ) )
      = ( G @ ( product_snd_b_b @ X3 ) ) ) ).

% snd_map_prod
thf(fact_591_snd__map__prod,axiom,
    ! [F2: a > a,G: a > b,X3: product_prod_a_a] :
      ( ( product_snd_a_b @ ( produc4018698449564827639_a_a_b @ F2 @ G @ X3 ) )
      = ( G @ ( product_snd_a_a @ X3 ) ) ) ).

% snd_map_prod
thf(fact_592_snd__map__prod,axiom,
    ! [F2: a > b,G: b > a,X3: product_prod_a_b] :
      ( ( product_snd_b_a @ ( produc6827461482780282294_b_b_a @ F2 @ G @ X3 ) )
      = ( G @ ( product_snd_a_b @ X3 ) ) ) ).

% snd_map_prod
thf(fact_593_snd__map__prod,axiom,
    ! [F2: b > b,G: a > a,X3: product_prod_b_a] :
      ( ( product_snd_b_a @ ( produc1879507065192981046_b_a_a @ F2 @ G @ X3 ) )
      = ( G @ ( product_snd_b_a @ X3 ) ) ) ).

% snd_map_prod
thf(fact_594_snd__map__prod,axiom,
    ! [F2: b > b,G: b > a,X3: product_prod_b_b] :
      ( ( product_snd_b_a @ ( produc8315423520075080821_b_b_a @ F2 @ G @ X3 ) )
      = ( G @ ( product_snd_b_b @ X3 ) ) ) ).

% snd_map_prod
thf(fact_595_snd__map__prod,axiom,
    ! [F2: a > b,G: a > a,X3: product_prod_a_a] :
      ( ( product_snd_b_a @ ( produc391545027898182519_b_a_a @ F2 @ G @ X3 ) )
      = ( G @ ( product_snd_a_a @ X3 ) ) ) ).

% snd_map_prod
thf(fact_596_snd__map__prod,axiom,
    ! [F2: a > b,G: b > b,X3: product_prod_a_b] :
      ( ( product_snd_b_b @ ( produc6827461482780282295_b_b_b @ F2 @ G @ X3 ) )
      = ( G @ ( product_snd_a_b @ X3 ) ) ) ).

% snd_map_prod
thf(fact_597_snd__map__prod,axiom,
    ! [F2: b > b,G: a > b,X3: product_prod_b_a] :
      ( ( product_snd_b_b @ ( produc1879507065192981047_b_a_b @ F2 @ G @ X3 ) )
      = ( G @ ( product_snd_b_a @ X3 ) ) ) ).

% snd_map_prod
thf(fact_598_GrpE,axiom,
    ! [A4: set_a,F2: a > b,X3: a,Y4: b] :
      ( ( bNF_Grp_a_b @ A4 @ F2 @ X3 @ Y4 )
     => ~ ( ( ( F2 @ X3 )
            = Y4 )
         => ~ ( member_a @ X3 @ A4 ) ) ) ).

% GrpE
thf(fact_599_GrpE,axiom,
    ! [A4: set_Product_prod_a_b,F2: product_prod_a_b > a,X3: product_prod_a_b,Y4: a] :
      ( ( bNF_Gr7842136747927947300_a_b_a @ A4 @ F2 @ X3 @ Y4 )
     => ~ ( ( ( F2 @ X3 )
            = Y4 )
         => ~ ( member1426531481828664017od_a_b @ X3 @ A4 ) ) ) ).

% GrpE
thf(fact_600_GrpE,axiom,
    ! [A4: set_Product_prod_a_b,F2: product_prod_a_b > b,X3: product_prod_a_b,Y4: b] :
      ( ( bNF_Gr7842136747927947301_a_b_b @ A4 @ F2 @ X3 @ Y4 )
     => ~ ( ( ( F2 @ X3 )
            = Y4 )
         => ~ ( member1426531481828664017od_a_b @ X3 @ A4 ) ) ) ).

% GrpE
thf(fact_601_GrpE,axiom,
    ! [A4: set_Pr5493900063595384711od_a_b,F2: produc2233330766403592615od_a_b > product_prod_a_a,X3: produc2233330766403592615od_a_b,Y4: product_prod_a_a] :
      ( ( bNF_Gr248057877874286094od_a_a @ A4 @ F2 @ X3 @ Y4 )
     => ~ ( ( ( F2 @ X3 )
            = Y4 )
         => ~ ( member5065438833300252112od_a_b @ X3 @ A4 ) ) ) ).

% GrpE
thf(fact_602_GrpE,axiom,
    ! [A4: set_Pr5493900063595384711od_a_b,F2: produc2233330766403592615od_a_b > product_prod_b_b,X3: produc2233330766403592615od_a_b,Y4: product_prod_b_b] :
      ( ( bNF_Gr6683974337059614670od_b_b @ A4 @ F2 @ X3 @ Y4 )
     => ~ ( ( ( F2 @ X3 )
            = Y4 )
         => ~ ( member5065438833300252112od_a_b @ X3 @ A4 ) ) ) ).

% GrpE
thf(fact_603_GrpI,axiom,
    ! [F2: a > b,X3: a,Y4: b,A4: set_a] :
      ( ( ( F2 @ X3 )
        = Y4 )
     => ( ( member_a @ X3 @ A4 )
       => ( bNF_Grp_a_b @ A4 @ F2 @ X3 @ Y4 ) ) ) ).

% GrpI
thf(fact_604_GrpI,axiom,
    ! [F2: product_prod_a_b > a,X3: product_prod_a_b,Y4: a,A4: set_Product_prod_a_b] :
      ( ( ( F2 @ X3 )
        = Y4 )
     => ( ( member1426531481828664017od_a_b @ X3 @ A4 )
       => ( bNF_Gr7842136747927947300_a_b_a @ A4 @ F2 @ X3 @ Y4 ) ) ) ).

% GrpI
thf(fact_605_GrpI,axiom,
    ! [F2: product_prod_a_b > b,X3: product_prod_a_b,Y4: b,A4: set_Product_prod_a_b] :
      ( ( ( F2 @ X3 )
        = Y4 )
     => ( ( member1426531481828664017od_a_b @ X3 @ A4 )
       => ( bNF_Gr7842136747927947301_a_b_b @ A4 @ F2 @ X3 @ Y4 ) ) ) ).

% GrpI
thf(fact_606_GrpI,axiom,
    ! [F2: produc2233330766403592615od_a_b > product_prod_a_a,X3: produc2233330766403592615od_a_b,Y4: product_prod_a_a,A4: set_Pr5493900063595384711od_a_b] :
      ( ( ( F2 @ X3 )
        = Y4 )
     => ( ( member5065438833300252112od_a_b @ X3 @ A4 )
       => ( bNF_Gr248057877874286094od_a_a @ A4 @ F2 @ X3 @ Y4 ) ) ) ).

% GrpI
thf(fact_607_GrpI,axiom,
    ! [F2: produc2233330766403592615od_a_b > product_prod_b_b,X3: produc2233330766403592615od_a_b,Y4: product_prod_b_b,A4: set_Pr5493900063595384711od_a_b] :
      ( ( ( F2 @ X3 )
        = Y4 )
     => ( ( member5065438833300252112od_a_b @ X3 @ A4 )
       => ( bNF_Gr6683974337059614670od_b_b @ A4 @ F2 @ X3 @ Y4 ) ) ) ).

% GrpI
thf(fact_608_map__prod_Ocomp,axiom,
    ! [F2: a > a,G: b > b,H: a > a,I: b > b] :
      ( ( comp_P2392903256394990597od_a_b @ ( produc1231242867592151606_a_b_b @ F2 @ G ) @ ( produc1231242867592151606_a_b_b @ H @ I ) )
      = ( produc1231242867592151606_a_b_b @ ( comp_a_a_a @ F2 @ H ) @ ( comp_b_b_b @ G @ I ) ) ) ).

% map_prod.comp
thf(fact_609_map__prod_Ocomp,axiom,
    ! [F2: a > a,G: a > a,H: a > a,I: a > a] :
      ( ( comp_P1279500999426142148od_a_a @ ( produc4018698449564827638_a_a_a @ F2 @ G ) @ ( produc4018698449564827638_a_a_a @ H @ I ) )
      = ( produc4018698449564827638_a_a_a @ ( comp_a_a_a @ F2 @ H ) @ ( comp_a_a_a @ G @ I ) ) ) ).

% map_prod.comp
thf(fact_610_map__prod_Ocomp,axiom,
    ! [F2: b > b,G: a > a,H: product_prod_a_b > b,I: a > a] :
      ( ( comp_P256948100396980698_a_b_a @ ( produc1879507065192981046_b_a_a @ F2 @ G ) @ ( produc4881195996318342733_b_a_a @ H @ I ) )
      = ( produc4881195996318342733_b_a_a @ ( comp_b3886954628874447685od_a_b @ F2 @ H ) @ ( comp_a_a_a @ G @ I ) ) ) ).

% map_prod.comp
thf(fact_611_map__prod_Ocomp,axiom,
    ! [F2: a > b,G: a > a,H: product_prod_a_b > a,I: a > a] :
      ( ( comp_P6517114761765832731_a_b_a @ ( produc391545027898182519_b_a_a @ F2 @ G ) @ ( produc8508349417984987852_a_a_a @ H @ I ) )
      = ( produc4881195996318342733_b_a_a @ ( comp_a9170378079104387268od_a_b @ F2 @ H ) @ ( comp_a_a_a @ G @ I ) ) ) ).

% map_prod.comp
thf(fact_612_map__prod_Ocomp,axiom,
    ! [F2: a > a,G: a > b,H: a > a,I: product_prod_a_b > a] :
      ( ( comp_P8128989039197251117od_a_b @ ( produc4018698449564827639_a_a_b @ F2 @ G ) @ ( produc5685587858988807628_a_b_a @ H @ I ) )
      = ( produc5685587858988807629_a_b_b @ ( comp_a_a_a @ F2 @ H ) @ ( comp_a9170378079104387268od_a_b @ G @ I ) ) ) ).

% map_prod.comp
thf(fact_613_map__prod_Ocomp,axiom,
    ! [F2: a > a,G: b > b,H: a > a,I: product_prod_a_b > b] :
      ( ( comp_P3550021029019600430od_a_b @ ( produc1231242867592151606_a_b_b @ F2 @ G ) @ ( produc5685587858988807629_a_b_b @ H @ I ) )
      = ( produc5685587858988807629_a_b_b @ ( comp_a_a_a @ F2 @ H ) @ ( comp_b3886954628874447685od_a_b @ G @ I ) ) ) ).

% map_prod.comp
thf(fact_614_map__prod_Ocomp,axiom,
    ! [F2: a > a,G: a > a,H: product_prod_a_b > a,I: a > a] :
      ( ( comp_P3584601654168751770_a_b_a @ ( produc4018698449564827638_a_a_a @ F2 @ G ) @ ( produc8508349417984987852_a_a_a @ H @ I ) )
      = ( produc8508349417984987852_a_a_a @ ( comp_a586446342581427589od_a_b @ F2 @ H ) @ ( comp_a_a_a @ G @ I ) ) ) ).

% map_prod.comp
thf(fact_615_map__prod_Ocomp,axiom,
    ! [F2: a > a,G: a > a,H: a > a,I: product_prod_a_b > a] :
      ( ( comp_P4532490230867100014od_a_b @ ( produc4018698449564827638_a_a_a @ F2 @ G ) @ ( produc5685587858988807628_a_b_a @ H @ I ) )
      = ( produc5685587858988807628_a_b_a @ ( comp_a_a_a @ F2 @ H ) @ ( comp_a586446342581427589od_a_b @ G @ I ) ) ) ).

% map_prod.comp
thf(fact_616_map__prod_Ocomp,axiom,
    ! [F2: b > b,G: a > a,H: product_prod_a_b > b,I: product_prod_a_b > a] :
      ( ( comp_P1864535819969406340od_a_b @ ( produc1879507065192981046_b_a_a @ F2 @ G ) @ ( produc7998683168210511139_a_b_a @ H @ I ) )
      = ( produc7998683168210511139_a_b_a @ ( comp_b3886954628874447685od_a_b @ F2 @ H ) @ ( comp_a586446342581427589od_a_b @ G @ I ) ) ) ).

% map_prod.comp
thf(fact_617_map__prod_Ocomp,axiom,
    ! [F2: a > product_prod_a_a,G: a > a,H: product_prod_a_b > a,I: a > a] :
      ( ( comp_P7205593717040431683_a_b_a @ ( produc6451820394740312479_a_a_a @ F2 @ G ) @ ( produc8508349417984987852_a_a_a @ H @ I ) )
      = ( produc4824967117337179765_a_a_a @ ( comp_a1036870397537576092od_a_b @ F2 @ H ) @ ( comp_a_a_a @ G @ I ) ) ) ).

% map_prod.comp
thf(fact_618_prod_Omap__comp,axiom,
    ! [G1: a > a,G22: b > b,F1: a > a,F22: b > b,V: product_prod_a_b] :
      ( ( produc1231242867592151606_a_b_b @ G1 @ G22 @ ( produc1231242867592151606_a_b_b @ F1 @ F22 @ V ) )
      = ( produc1231242867592151606_a_b_b @ ( comp_a_a_a @ G1 @ F1 ) @ ( comp_b_b_b @ G22 @ F22 ) @ V ) ) ).

% prod.map_comp
thf(fact_619_prod_Omap__comp,axiom,
    ! [G1: a > a,G22: a > a,F1: a > a,F22: a > a,V: product_prod_a_a] :
      ( ( produc4018698449564827638_a_a_a @ G1 @ G22 @ ( produc4018698449564827638_a_a_a @ F1 @ F22 @ V ) )
      = ( produc4018698449564827638_a_a_a @ ( comp_a_a_a @ G1 @ F1 ) @ ( comp_a_a_a @ G22 @ F22 ) @ V ) ) ).

% prod.map_comp
thf(fact_620_prod_Omap__comp,axiom,
    ! [G1: b > b,G22: a > a,F1: product_prod_a_b > b,F22: a > a,V: produc3167243049347903741_a_b_a] :
      ( ( produc1879507065192981046_b_a_a @ G1 @ G22 @ ( produc4881195996318342733_b_a_a @ F1 @ F22 @ V ) )
      = ( produc4881195996318342733_b_a_a @ ( comp_b3886954628874447685od_a_b @ G1 @ F1 ) @ ( comp_a_a_a @ G22 @ F22 ) @ V ) ) ).

% prod.map_comp
thf(fact_621_prod_Omap__comp,axiom,
    ! [G1: a > b,G22: a > a,F1: product_prod_a_b > a,F22: a > a,V: produc3167243049347903741_a_b_a] :
      ( ( produc391545027898182519_b_a_a @ G1 @ G22 @ ( produc8508349417984987852_a_a_a @ F1 @ F22 @ V ) )
      = ( produc4881195996318342733_b_a_a @ ( comp_a9170378079104387268od_a_b @ G1 @ F1 ) @ ( comp_a_a_a @ G22 @ F22 ) @ V ) ) ).

% prod.map_comp
thf(fact_622_prod_Omap__comp,axiom,
    ! [G1: a > a,G22: a > b,F1: a > a,F22: product_prod_a_b > a,V: produc4115131626046251985od_a_b] :
      ( ( produc4018698449564827639_a_a_b @ G1 @ G22 @ ( produc5685587858988807628_a_b_a @ F1 @ F22 @ V ) )
      = ( produc5685587858988807629_a_b_b @ ( comp_a_a_a @ G1 @ F1 ) @ ( comp_a9170378079104387268od_a_b @ G22 @ F22 ) @ V ) ) ).

% prod.map_comp
thf(fact_623_prod_Omap__comp,axiom,
    ! [G1: a > a,G22: b > b,F1: a > a,F22: product_prod_a_b > b,V: produc4115131626046251985od_a_b] :
      ( ( produc1231242867592151606_a_b_b @ G1 @ G22 @ ( produc5685587858988807629_a_b_b @ F1 @ F22 @ V ) )
      = ( produc5685587858988807629_a_b_b @ ( comp_a_a_a @ G1 @ F1 ) @ ( comp_b3886954628874447685od_a_b @ G22 @ F22 ) @ V ) ) ).

% prod.map_comp
thf(fact_624_prod_Omap__comp,axiom,
    ! [G1: a > a,G22: a > a,F1: product_prod_a_b > a,F22: a > a,V: produc3167243049347903741_a_b_a] :
      ( ( produc4018698449564827638_a_a_a @ G1 @ G22 @ ( produc8508349417984987852_a_a_a @ F1 @ F22 @ V ) )
      = ( produc8508349417984987852_a_a_a @ ( comp_a586446342581427589od_a_b @ G1 @ F1 ) @ ( comp_a_a_a @ G22 @ F22 ) @ V ) ) ).

% prod.map_comp
thf(fact_625_prod_Omap__comp,axiom,
    ! [G1: a > a,G22: a > a,F1: a > a,F22: product_prod_a_b > a,V: produc4115131626046251985od_a_b] :
      ( ( produc4018698449564827638_a_a_a @ G1 @ G22 @ ( produc5685587858988807628_a_b_a @ F1 @ F22 @ V ) )
      = ( produc5685587858988807628_a_b_a @ ( comp_a_a_a @ G1 @ F1 ) @ ( comp_a586446342581427589od_a_b @ G22 @ F22 ) @ V ) ) ).

% prod.map_comp
thf(fact_626_prod_Omap__comp,axiom,
    ! [G1: b > b,G22: a > a,F1: product_prod_a_b > b,F22: product_prod_a_b > a,V: produc2233330766403592615od_a_b] :
      ( ( produc1879507065192981046_b_a_a @ G1 @ G22 @ ( produc7998683168210511139_a_b_a @ F1 @ F22 @ V ) )
      = ( produc7998683168210511139_a_b_a @ ( comp_b3886954628874447685od_a_b @ G1 @ F1 ) @ ( comp_a586446342581427589od_a_b @ G22 @ F22 ) @ V ) ) ).

% prod.map_comp
thf(fact_627_prod_Omap__comp,axiom,
    ! [G1: a > product_prod_a_a,G22: a > a,F1: product_prod_a_b > a,F22: a > a,V: produc3167243049347903741_a_b_a] :
      ( ( produc6451820394740312479_a_a_a @ G1 @ G22 @ ( produc8508349417984987852_a_a_a @ F1 @ F22 @ V ) )
      = ( produc4824967117337179765_a_a_a @ ( comp_a1036870397537576092od_a_b @ G1 @ F1 ) @ ( comp_a_a_a @ G22 @ F22 ) @ V ) ) ).

% prod.map_comp
thf(fact_628_map__prod_Ocompositionality,axiom,
    ! [F2: a > a,G: b > b,H: a > a,I: b > b,Prod: product_prod_a_b] :
      ( ( produc1231242867592151606_a_b_b @ F2 @ G @ ( produc1231242867592151606_a_b_b @ H @ I @ Prod ) )
      = ( produc1231242867592151606_a_b_b @ ( comp_a_a_a @ F2 @ H ) @ ( comp_b_b_b @ G @ I ) @ Prod ) ) ).

% map_prod.compositionality
thf(fact_629_map__prod_Ocompositionality,axiom,
    ! [F2: a > a,G: a > a,H: a > a,I: a > a,Prod: product_prod_a_a] :
      ( ( produc4018698449564827638_a_a_a @ F2 @ G @ ( produc4018698449564827638_a_a_a @ H @ I @ Prod ) )
      = ( produc4018698449564827638_a_a_a @ ( comp_a_a_a @ F2 @ H ) @ ( comp_a_a_a @ G @ I ) @ Prod ) ) ).

% map_prod.compositionality
thf(fact_630_map__prod_Ocompositionality,axiom,
    ! [F2: b > b,G: a > a,H: product_prod_a_b > b,I: a > a,Prod: produc3167243049347903741_a_b_a] :
      ( ( produc1879507065192981046_b_a_a @ F2 @ G @ ( produc4881195996318342733_b_a_a @ H @ I @ Prod ) )
      = ( produc4881195996318342733_b_a_a @ ( comp_b3886954628874447685od_a_b @ F2 @ H ) @ ( comp_a_a_a @ G @ I ) @ Prod ) ) ).

% map_prod.compositionality
thf(fact_631_map__prod_Ocompositionality,axiom,
    ! [F2: a > b,G: a > a,H: product_prod_a_b > a,I: a > a,Prod: produc3167243049347903741_a_b_a] :
      ( ( produc391545027898182519_b_a_a @ F2 @ G @ ( produc8508349417984987852_a_a_a @ H @ I @ Prod ) )
      = ( produc4881195996318342733_b_a_a @ ( comp_a9170378079104387268od_a_b @ F2 @ H ) @ ( comp_a_a_a @ G @ I ) @ Prod ) ) ).

% map_prod.compositionality
thf(fact_632_map__prod_Ocompositionality,axiom,
    ! [F2: a > a,G: a > b,H: a > a,I: product_prod_a_b > a,Prod: produc4115131626046251985od_a_b] :
      ( ( produc4018698449564827639_a_a_b @ F2 @ G @ ( produc5685587858988807628_a_b_a @ H @ I @ Prod ) )
      = ( produc5685587858988807629_a_b_b @ ( comp_a_a_a @ F2 @ H ) @ ( comp_a9170378079104387268od_a_b @ G @ I ) @ Prod ) ) ).

% map_prod.compositionality
thf(fact_633_map__prod_Ocompositionality,axiom,
    ! [F2: a > a,G: b > b,H: a > a,I: product_prod_a_b > b,Prod: produc4115131626046251985od_a_b] :
      ( ( produc1231242867592151606_a_b_b @ F2 @ G @ ( produc5685587858988807629_a_b_b @ H @ I @ Prod ) )
      = ( produc5685587858988807629_a_b_b @ ( comp_a_a_a @ F2 @ H ) @ ( comp_b3886954628874447685od_a_b @ G @ I ) @ Prod ) ) ).

% map_prod.compositionality
thf(fact_634_map__prod_Ocompositionality,axiom,
    ! [F2: a > a,G: a > a,H: product_prod_a_b > a,I: a > a,Prod: produc3167243049347903741_a_b_a] :
      ( ( produc4018698449564827638_a_a_a @ F2 @ G @ ( produc8508349417984987852_a_a_a @ H @ I @ Prod ) )
      = ( produc8508349417984987852_a_a_a @ ( comp_a586446342581427589od_a_b @ F2 @ H ) @ ( comp_a_a_a @ G @ I ) @ Prod ) ) ).

% map_prod.compositionality
thf(fact_635_map__prod_Ocompositionality,axiom,
    ! [F2: a > a,G: a > a,H: a > a,I: product_prod_a_b > a,Prod: produc4115131626046251985od_a_b] :
      ( ( produc4018698449564827638_a_a_a @ F2 @ G @ ( produc5685587858988807628_a_b_a @ H @ I @ Prod ) )
      = ( produc5685587858988807628_a_b_a @ ( comp_a_a_a @ F2 @ H ) @ ( comp_a586446342581427589od_a_b @ G @ I ) @ Prod ) ) ).

% map_prod.compositionality
thf(fact_636_map__prod_Ocompositionality,axiom,
    ! [F2: b > b,G: a > a,H: product_prod_a_b > b,I: product_prod_a_b > a,Prod: produc2233330766403592615od_a_b] :
      ( ( produc1879507065192981046_b_a_a @ F2 @ G @ ( produc7998683168210511139_a_b_a @ H @ I @ Prod ) )
      = ( produc7998683168210511139_a_b_a @ ( comp_b3886954628874447685od_a_b @ F2 @ H ) @ ( comp_a586446342581427589od_a_b @ G @ I ) @ Prod ) ) ).

% map_prod.compositionality
thf(fact_637_map__prod_Ocompositionality,axiom,
    ! [F2: a > product_prod_a_a,G: a > a,H: product_prod_a_b > a,I: a > a,Prod: produc3167243049347903741_a_b_a] :
      ( ( produc6451820394740312479_a_a_a @ F2 @ G @ ( produc8508349417984987852_a_a_a @ H @ I @ Prod ) )
      = ( produc4824967117337179765_a_a_a @ ( comp_a1036870397537576092od_a_b @ F2 @ H ) @ ( comp_a_a_a @ G @ I ) @ Prod ) ) ).

% map_prod.compositionality
thf(fact_638_map__prod__compose,axiom,
    ! [F1: a > a,F22: a > a,G1: b > b,G22: b > b] :
      ( ( produc1231242867592151606_a_b_b @ ( comp_a_a_a @ F1 @ F22 ) @ ( comp_b_b_b @ G1 @ G22 ) )
      = ( comp_P2392903256394990597od_a_b @ ( produc1231242867592151606_a_b_b @ F1 @ G1 ) @ ( produc1231242867592151606_a_b_b @ F22 @ G22 ) ) ) ).

% map_prod_compose
thf(fact_639_map__prod__compose,axiom,
    ! [F1: a > a,F22: a > a,G1: a > a,G22: a > a] :
      ( ( produc4018698449564827638_a_a_a @ ( comp_a_a_a @ F1 @ F22 ) @ ( comp_a_a_a @ G1 @ G22 ) )
      = ( comp_P1279500999426142148od_a_a @ ( produc4018698449564827638_a_a_a @ F1 @ G1 ) @ ( produc4018698449564827638_a_a_a @ F22 @ G22 ) ) ) ).

% map_prod_compose
thf(fact_640_map__prod__compose,axiom,
    ! [F1: b > b,F22: product_prod_a_b > b,G1: a > a,G22: a > a] :
      ( ( produc4881195996318342733_b_a_a @ ( comp_b3886954628874447685od_a_b @ F1 @ F22 ) @ ( comp_a_a_a @ G1 @ G22 ) )
      = ( comp_P256948100396980698_a_b_a @ ( produc1879507065192981046_b_a_a @ F1 @ G1 ) @ ( produc4881195996318342733_b_a_a @ F22 @ G22 ) ) ) ).

% map_prod_compose
thf(fact_641_map__prod__compose,axiom,
    ! [F1: a > b,F22: product_prod_a_b > a,G1: a > a,G22: a > a] :
      ( ( produc4881195996318342733_b_a_a @ ( comp_a9170378079104387268od_a_b @ F1 @ F22 ) @ ( comp_a_a_a @ G1 @ G22 ) )
      = ( comp_P6517114761765832731_a_b_a @ ( produc391545027898182519_b_a_a @ F1 @ G1 ) @ ( produc8508349417984987852_a_a_a @ F22 @ G22 ) ) ) ).

% map_prod_compose
thf(fact_642_map__prod__compose,axiom,
    ! [F1: a > a,F22: a > a,G1: a > b,G22: product_prod_a_b > a] :
      ( ( produc5685587858988807629_a_b_b @ ( comp_a_a_a @ F1 @ F22 ) @ ( comp_a9170378079104387268od_a_b @ G1 @ G22 ) )
      = ( comp_P8128989039197251117od_a_b @ ( produc4018698449564827639_a_a_b @ F1 @ G1 ) @ ( produc5685587858988807628_a_b_a @ F22 @ G22 ) ) ) ).

% map_prod_compose
thf(fact_643_map__prod__compose,axiom,
    ! [F1: a > a,F22: a > a,G1: b > b,G22: product_prod_a_b > b] :
      ( ( produc5685587858988807629_a_b_b @ ( comp_a_a_a @ F1 @ F22 ) @ ( comp_b3886954628874447685od_a_b @ G1 @ G22 ) )
      = ( comp_P3550021029019600430od_a_b @ ( produc1231242867592151606_a_b_b @ F1 @ G1 ) @ ( produc5685587858988807629_a_b_b @ F22 @ G22 ) ) ) ).

% map_prod_compose
thf(fact_644_map__prod__compose,axiom,
    ! [F1: a > a,F22: product_prod_a_b > a,G1: a > a,G22: a > a] :
      ( ( produc8508349417984987852_a_a_a @ ( comp_a586446342581427589od_a_b @ F1 @ F22 ) @ ( comp_a_a_a @ G1 @ G22 ) )
      = ( comp_P3584601654168751770_a_b_a @ ( produc4018698449564827638_a_a_a @ F1 @ G1 ) @ ( produc8508349417984987852_a_a_a @ F22 @ G22 ) ) ) ).

% map_prod_compose
thf(fact_645_map__prod__compose,axiom,
    ! [F1: a > a,F22: a > a,G1: a > a,G22: product_prod_a_b > a] :
      ( ( produc5685587858988807628_a_b_a @ ( comp_a_a_a @ F1 @ F22 ) @ ( comp_a586446342581427589od_a_b @ G1 @ G22 ) )
      = ( comp_P4532490230867100014od_a_b @ ( produc4018698449564827638_a_a_a @ F1 @ G1 ) @ ( produc5685587858988807628_a_b_a @ F22 @ G22 ) ) ) ).

% map_prod_compose
thf(fact_646_map__prod__compose,axiom,
    ! [F1: b > b,F22: product_prod_a_b > b,G1: a > a,G22: product_prod_a_b > a] :
      ( ( produc7998683168210511139_a_b_a @ ( comp_b3886954628874447685od_a_b @ F1 @ F22 ) @ ( comp_a586446342581427589od_a_b @ G1 @ G22 ) )
      = ( comp_P1864535819969406340od_a_b @ ( produc1879507065192981046_b_a_a @ F1 @ G1 ) @ ( produc7998683168210511139_a_b_a @ F22 @ G22 ) ) ) ).

% map_prod_compose
thf(fact_647_map__prod__compose,axiom,
    ! [F1: a > product_prod_a_a,F22: product_prod_a_b > a,G1: a > a,G22: a > a] :
      ( ( produc4824967117337179765_a_a_a @ ( comp_a1036870397537576092od_a_b @ F1 @ F22 ) @ ( comp_a_a_a @ G1 @ G22 ) )
      = ( comp_P7205593717040431683_a_b_a @ ( produc6451820394740312479_a_a_a @ F1 @ G1 ) @ ( produc8508349417984987852_a_a_a @ F22 @ G22 ) ) ) ).

% map_prod_compose
thf(fact_648_Collect__case__prod__Grp__in,axiom,
    ! [Z2: product_prod_a_b,A4: set_a,F2: a > b] :
      ( ( member1426531481828664017od_a_b @ Z2 @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ ( bNF_Grp_a_b @ A4 @ F2 ) ) ) )
     => ( member_a @ ( product_fst_a_b @ Z2 ) @ A4 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_649_Collect__case__prod__Grp__in,axiom,
    ! [Z2: product_prod_nat_nat,A4: set_nat,F2: nat > nat] :
      ( ( member8440522571783428010at_nat @ Z2 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( bNF_Grp_nat_nat @ A4 @ F2 ) ) ) )
     => ( member_nat @ ( product_fst_nat_nat @ Z2 ) @ A4 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_650_Collect__case__prod__Grp__in,axiom,
    ! [Z2: product_prod_b_a,A4: set_b,F2: b > a] :
      ( ( member7862447932407534991od_b_a @ Z2 @ ( collec548942215411776465od_b_a @ ( produc7672072387910335853_b_a_o @ ( bNF_Grp_b_a @ A4 @ F2 ) ) ) )
     => ( member_b @ ( product_fst_b_a @ Z2 ) @ A4 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_651_Collect__case__prod__Grp__in,axiom,
    ! [Z2: product_prod_a_a,A4: set_a,F2: a > a] :
      ( ( member1426531477525435216od_a_a @ Z2 @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ ( bNF_Grp_a_a @ A4 @ F2 ) ) ) )
     => ( member_a @ ( product_fst_a_a @ Z2 ) @ A4 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_652_Collect__case__prod__Grp__in,axiom,
    ! [Z2: product_prod_b_b,A4: set_b,F2: b > b] :
      ( ( member7862447936710763792od_b_b @ Z2 @ ( collec548942219715005266od_b_b @ ( produc4772849988445941548_b_b_o @ ( bNF_Grp_b_b @ A4 @ F2 ) ) ) )
     => ( member_b @ ( product_fst_b_b @ Z2 ) @ A4 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_653_Collect__case__prod__Grp__in,axiom,
    ! [Z2: produc5884233991663340231list_o,A4: set_a,F2: a > list_o] :
      ( ( member7948383622993546480list_o @ Z2 @ ( collec7805115074396547890list_o @ ( produc8342644129638033356st_o_o @ ( bNF_Grp_a_list_o @ A4 @ F2 ) ) ) )
     => ( member_a @ ( product_fst_a_list_o @ Z2 ) @ A4 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_654_Collect__case__prod__Grp__in,axiom,
    ! [Z2: produc3167243049347903741_a_b_a,A4: set_Product_prod_a_b,F2: product_prod_a_b > a] :
      ( ( member2194267517197600294_a_b_a @ Z2 @ ( collec4168926458763418216_a_b_a @ ( produc1879355882332503830_b_a_o @ ( bNF_Gr7842136747927947300_a_b_a @ A4 @ F2 ) ) ) )
     => ( member1426531481828664017od_a_b @ ( produc1917971008728229337_a_b_a @ Z2 ) @ A4 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_655_Collect__case__prod__Grp__in,axiom,
    ! [Z2: produc3167243053651132542_a_b_b,A4: set_Product_prod_a_b,F2: product_prod_a_b > b] :
      ( ( member2194267521500829095_a_b_b @ Z2 @ ( collec4168926463066647017_a_b_b @ ( produc8203505519722885333_b_b_o @ ( bNF_Gr7842136747927947301_a_b_b @ A4 @ F2 ) ) ) )
     => ( member1426531481828664017od_a_b @ ( produc1917971008728229338_a_b_b @ Z2 ) @ A4 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_656_Collect__case__prod__Grp__in,axiom,
    ! [Z2: produc2233330766403592615od_a_b,A4: set_Product_prod_a_b,F2: product_prod_a_b > product_prod_a_b] :
      ( ( member5065438833300252112od_a_b @ Z2 @ ( collec7968472090841016338od_a_b @ ( produc5413562937804622316_a_b_o @ ( bNF_Gr7211640137128594382od_a_b @ A4 @ F2 ) ) ) )
     => ( member1426531481828664017od_a_b @ ( produc5013601669340818563od_a_b @ Z2 ) @ A4 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_657_Collect__case__prod__Grp__in,axiom,
    ! [Z2: produc1065045136403542119od_a_a,A4: set_Pr5493900063595384711od_a_b,F2: produc2233330766403592615od_a_b > product_prod_a_a] :
      ( ( member4368417410682638992od_a_a @ Z2 @ ( collec5379532638810960082od_a_a @ ( produc5892353234140082604_a_a_o @ ( bNF_Gr248057877874286094od_a_a @ A4 @ F2 ) ) ) )
     => ( member5065438833300252112od_a_b @ ( produc7653073535890015427od_a_a @ Z2 ) @ A4 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_658_prod_Osize__gen__o__map,axiom,
    ! [F2: a > nat,Fa: b > nat,G: a > a,Ga: b > b] :
      ( ( comp_P6232115124411100305od_a_b @ ( basic_3571716426759296053od_a_b @ F2 @ Fa ) @ ( produc1231242867592151606_a_b_b @ G @ Ga ) )
      = ( basic_3571716426759296053od_a_b @ ( comp_a_nat_a @ F2 @ G ) @ ( comp_b_nat_b @ Fa @ Ga ) ) ) ).

% prod.size_gen_o_map
thf(fact_659_prod_Osize__gen__o__map,axiom,
    ! [F2: a > nat,Fa: a > nat,G: a > a,Ga: a > a] :
      ( ( comp_P2525640977308048721od_a_a @ ( basic_3571716426759296052od_a_a @ F2 @ Fa ) @ ( produc4018698449564827638_a_a_a @ G @ Ga ) )
      = ( basic_3571716426759296052od_a_a @ ( comp_a_nat_a @ F2 @ G ) @ ( comp_a_nat_a @ Fa @ Ga ) ) ) ).

% prod.size_gen_o_map
thf(fact_660_prod_Osize__gen__o__map,axiom,
    ! [F2: a > nat,Fa: a > nat,G: product_prod_a_b > a,Ga: product_prod_a_b > a] :
      ( ( comp_P2842972232926698961od_a_b @ ( basic_3571716426759296052od_a_a @ F2 @ Fa ) @ ( produc4058734581585674914_a_b_a @ G @ Ga ) )
      = ( basic_1112880799063329204od_a_b @ ( comp_a3081486988346654907od_a_b @ F2 @ G ) @ ( comp_a3081486988346654907od_a_b @ Fa @ Ga ) ) ) ).

% prod.size_gen_o_map
thf(fact_661_prod_Osize__gen__o__map,axiom,
    ! [F2: b > nat,Fa: b > nat,G: product_prod_a_b > b,Ga: product_prod_a_b > b] :
      ( ( comp_P1853936029605985297od_a_b @ ( basic_784260844786620020od_b_b @ F2 @ Fa ) @ ( produc7998683168210511140_a_b_b @ G @ Ga ) )
      = ( basic_1112880799063329204od_a_b @ ( comp_b4322612161844523002od_a_b @ F2 @ G ) @ ( comp_b4322612161844523002od_a_b @ Fa @ Ga ) ) ) ).

% prod.size_gen_o_map
thf(fact_662_csquare__fstOp__sndOp,axiom,
    ! [F2: ( a > a > $o ) > product_prod_a_a > $o,P2: a > b > $o,Q2: b > a > $o] : ( bNF_cs6251773214417337457od_b_a @ ( collec3336397797384452498od_a_a @ ( F2 @ ( relcompp_a_b_a @ P2 @ Q2 ) ) ) @ product_snd_a_b @ product_fst_b_a @ ( bNF_fstOp_a_b_a @ P2 @ Q2 ) @ ( bNF_sndOp_a_b_a @ P2 @ Q2 ) ) ).

% csquare_fstOp_sndOp
thf(fact_663_csquare__fstOp__sndOp,axiom,
    ! [F2: ( a > b > $o ) > product_prod_a_b > $o,P2: a > b > $o,Q2: b > b > $o] : ( bNF_cs7476683058888131889od_b_b @ ( collec3336397801687681299od_a_b @ ( F2 @ ( relcompp_a_b_b @ P2 @ Q2 ) ) ) @ product_snd_a_b @ product_fst_b_b @ ( bNF_fstOp_a_b_b @ P2 @ Q2 ) @ ( bNF_sndOp_a_b_b @ P2 @ Q2 ) ) ).

% csquare_fstOp_sndOp
thf(fact_664_csquare__fstOp__sndOp,axiom,
    ! [F2: ( b > b > $o ) > product_prod_b_b > $o,P2: b > a > $o,Q2: a > b > $o] : ( bNF_cs7383540917916740210od_a_b @ ( collec548942219715005266od_b_b @ ( F2 @ ( relcompp_b_a_b @ P2 @ Q2 ) ) ) @ product_snd_b_a @ product_fst_a_b @ ( bNF_fstOp_b_a_b @ P2 @ Q2 ) @ ( bNF_sndOp_b_a_b @ P2 @ Q2 ) ) ).

% csquare_fstOp_sndOp
thf(fact_665_csquare__fstOp__sndOp,axiom,
    ! [F2: ( b > a > $o ) > product_prod_b_a > $o,P2: b > a > $o,Q2: a > a > $o] : ( bNF_cs6158631073445945778od_a_a @ ( collec548942215411776465od_b_a @ ( F2 @ ( relcompp_b_a_a @ P2 @ Q2 ) ) ) @ product_snd_b_a @ product_fst_a_a @ ( bNF_fstOp_b_a_a @ P2 @ Q2 ) @ ( bNF_sndOp_b_a_a @ P2 @ Q2 ) ) ).

% csquare_fstOp_sndOp
thf(fact_666_csquare__fstOp__sndOp,axiom,
    ! [F2: ( b > a > $o ) > product_prod_b_a > $o,P2: b > b > $o,Q2: b > a > $o] : ( bNF_cs7711678339289956657od_b_a @ ( collec548942215411776465od_b_a @ ( F2 @ ( relcompp_b_b_a @ P2 @ Q2 ) ) ) @ product_snd_b_b @ product_fst_b_a @ ( bNF_fstOp_b_b_a @ P2 @ Q2 ) @ ( bNF_sndOp_b_b_a @ P2 @ Q2 ) ) ).

% csquare_fstOp_sndOp
thf(fact_667_csquare__fstOp__sndOp,axiom,
    ! [F2: ( b > b > $o ) > product_prod_b_b > $o,P2: b > b > $o,Q2: b > b > $o] : ( bNF_cs8936588183760751089od_b_b @ ( collec548942219715005266od_b_b @ ( F2 @ ( relcompp_b_b_b @ P2 @ Q2 ) ) ) @ product_snd_b_b @ product_fst_b_b @ ( bNF_fstOp_b_b_b @ P2 @ Q2 ) @ ( bNF_sndOp_b_b_b @ P2 @ Q2 ) ) ).

% csquare_fstOp_sndOp
thf(fact_668_csquare__fstOp__sndOp,axiom,
    ! [F2: ( a > b > $o ) > product_prod_a_b > $o,P2: a > a > $o,Q2: a > b > $o] : ( bNF_cs5923635793044121010od_a_b @ ( collec3336397801687681299od_a_b @ ( F2 @ ( relcompp_a_a_b @ P2 @ Q2 ) ) ) @ product_snd_a_a @ product_fst_a_b @ ( bNF_fstOp_a_a_b @ P2 @ Q2 ) @ ( bNF_sndOp_a_a_b @ P2 @ Q2 ) ) ).

% csquare_fstOp_sndOp
thf(fact_669_csquare__fstOp__sndOp,axiom,
    ! [F2: ( a > a > $o ) > product_prod_a_a > $o,P2: a > a > $o,Q2: a > a > $o] : ( bNF_cs4698725948573326578od_a_a @ ( collec3336397797384452498od_a_a @ ( F2 @ ( relcompp_a_a_a @ P2 @ Q2 ) ) ) @ product_snd_a_a @ product_fst_a_a @ ( bNF_fstOp_a_a_a @ P2 @ Q2 ) @ ( bNF_sndOp_a_a_a @ P2 @ Q2 ) ) ).

% csquare_fstOp_sndOp
thf(fact_670_csquare__fstOp__sndOp,axiom,
    ! [F2: ( b > list_o > $o ) > produc7228699028472828166list_o > $o,P2: b > a > $o,Q2: a > list_o > $o] : ( bNF_cs1821474949591695794list_o @ ( collec9149580111206035825list_o @ ( F2 @ ( relcompp_b_a_list_o @ P2 @ Q2 ) ) ) @ product_snd_b_a @ product_fst_a_list_o @ ( bNF_fstOp_b_a_list_o @ P2 @ Q2 ) @ ( bNF_sndOp_b_a_list_o @ P2 @ Q2 ) ) ).

% csquare_fstOp_sndOp
thf(fact_671_csquare__fstOp__sndOp,axiom,
    ! [F2: ( a > list_o > $o ) > produc5884233991663340231list_o > $o,P2: a > a > $o,Q2: a > list_o > $o] : ( bNF_cs7935384970986586354list_o @ ( collec7805115074396547890list_o @ ( F2 @ ( relcompp_a_a_list_o @ P2 @ Q2 ) ) ) @ product_snd_a_a @ product_fst_a_list_o @ ( bNF_fstOp_a_a_list_o @ P2 @ Q2 ) @ ( bNF_sndOp_a_a_list_o @ P2 @ Q2 ) ) ).

% csquare_fstOp_sndOp
thf(fact_672_image2__eqI,axiom,
    ! [B: b,F2: a > b,X3: a,C: b,G: a > b,A4: set_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C
          = ( G @ X3 ) )
       => ( ( member_a @ X3 @ A4 )
         => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B @ C ) @ ( bNF_Gr8202675903479541476_a_b_b @ A4 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_673_image2__eqI,axiom,
    ! [B: b,F2: b > b,X3: b,C: b,G: b > b,A4: set_b] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C
          = ( G @ X3 ) )
       => ( ( member_b @ X3 @ A4 )
         => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B @ C ) @ ( bNF_Gr4575522481812896357_b_b_b @ A4 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_674_image2__eqI,axiom,
    ! [B: b,F2: a > b,X3: a,C: a,G: a > a,A4: set_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C
          = ( G @ X3 ) )
       => ( ( member_a @ X3 @ A4 )
         => ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ B @ C ) @ ( bNF_Gr8202675903479541475_a_b_a @ A4 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_675_image2__eqI,axiom,
    ! [B: b,F2: b > b,X3: b,C: a,G: b > a,A4: set_b] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C
          = ( G @ X3 ) )
       => ( ( member_b @ X3 @ A4 )
         => ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ B @ C ) @ ( bNF_Gr4575522481812896356_b_b_a @ A4 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_676_image2__eqI,axiom,
    ! [B: a,F2: a > a,X3: a,C: b,G: a > b,A4: set_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C
          = ( G @ X3 ) )
       => ( ( member_a @ X3 @ A4 )
         => ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ B @ C ) @ ( bNF_Gr1766759448597441701_a_a_b @ A4 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_677_image2__eqI,axiom,
    ! [B: a,F2: b > a,X3: b,C: b,G: b > b,A4: set_b] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C
          = ( G @ X3 ) )
       => ( ( member_b @ X3 @ A4 )
         => ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ B @ C ) @ ( bNF_Gr7362978063785572390_b_a_b @ A4 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_678_image2__eqI,axiom,
    ! [B: a,F2: a > a,X3: a,C: a,G: a > a,A4: set_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C
          = ( G @ X3 ) )
       => ( ( member_a @ X3 @ A4 )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C ) @ ( bNF_Gr1766759448597441700_a_a_a @ A4 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_679_image2__eqI,axiom,
    ! [B: a,F2: b > a,X3: b,C: a,G: b > a,A4: set_b] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C
          = ( G @ X3 ) )
       => ( ( member_b @ X3 @ A4 )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C ) @ ( bNF_Gr7362978063785572389_b_a_a @ A4 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_680_image2__eqI,axiom,
    ! [B: b,F2: set_a > b,X3: set_a,C: b,G: set_a > b,A4: set_set_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C
          = ( G @ X3 ) )
       => ( ( member_set_a @ X3 @ A4 )
         => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B @ C ) @ ( bNF_Gr3840943654187618372_a_b_b @ A4 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_681_image2__eqI,axiom,
    ! [B: b,F2: set_a > b,X3: set_a,C: a,G: set_a > a,A4: set_set_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C
          = ( G @ X3 ) )
       => ( ( member_set_a @ X3 @ A4 )
         => ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ B @ C ) @ ( bNF_Gr3840943654187618371_a_b_a @ A4 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_682_rewriteR__comp__comp2,axiom,
    ! [G: a > a,H: a > a,R1: a > a,R2: a > a,F2: a > a,L: a > a] :
      ( ( ( comp_a_a_a @ G @ H )
        = ( comp_a_a_a @ R1 @ R2 ) )
     => ( ( ( comp_a_a_a @ F2 @ R1 )
          = L )
       => ( ( comp_a_a_a @ ( comp_a_a_a @ F2 @ G ) @ H )
          = ( comp_a_a_a @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_683_rewriteR__comp__comp2,axiom,
    ! [G: b > a,H: product_prod_a_b > b,R1: a > a,R2: product_prod_a_b > a,F2: a > b,L: a > b] :
      ( ( ( comp_b4526394929206263814od_a_b @ G @ H )
        = ( comp_a586446342581427589od_a_b @ R1 @ R2 ) )
     => ( ( ( comp_a_b_a @ F2 @ R1 )
          = L )
       => ( ( comp_b3886954628874447685od_a_b @ ( comp_a_b_b @ F2 @ G ) @ H )
          = ( comp_a9170378079104387268od_a_b @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_684_rewriteR__comp__comp2,axiom,
    ! [G: product_prod_a_b > a,H: a > product_prod_a_b,R1: a > a,R2: a > a,F2: a > b,L: a > b] :
      ( ( ( comp_P3199017072728932485_b_a_a @ G @ H )
        = ( comp_a_a_a @ R1 @ R2 ) )
     => ( ( ( comp_a_b_a @ F2 @ R1 )
          = L )
       => ( ( comp_P411561490756256452_b_b_a @ ( comp_a9170378079104387268od_a_b @ F2 @ G ) @ H )
          = ( comp_a_b_a @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_685_rewriteR__comp__comp2,axiom,
    ! [G: product_prod_a_b > a,H: a > product_prod_a_b,R1: a > a,R2: a > a,F2: a > a,L: a > a] :
      ( ( ( comp_P3199017072728932485_b_a_a @ G @ H )
        = ( comp_a_a_a @ R1 @ R2 ) )
     => ( ( ( comp_a_a_a @ F2 @ R1 )
          = L )
       => ( ( comp_P3199017072728932485_b_a_a @ ( comp_a586446342581427589od_a_b @ F2 @ G ) @ H )
          = ( comp_a_a_a @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_686_rewriteR__comp__comp2,axiom,
    ! [G: b > b,H: product_prod_a_b > b,R1: b > b,R2: product_prod_a_b > b,F2: b > b,L: b > b] :
      ( ( ( comp_b3886954628874447685od_a_b @ G @ H )
        = ( comp_b3886954628874447685od_a_b @ R1 @ R2 ) )
     => ( ( ( comp_b_b_b @ F2 @ R1 )
          = L )
       => ( ( comp_b3886954628874447685od_a_b @ ( comp_b_b_b @ F2 @ G ) @ H )
          = ( comp_b3886954628874447685od_a_b @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_687_rewriteR__comp__comp2,axiom,
    ! [G: b > b,H: product_prod_a_b > b,R1: a > b,R2: product_prod_a_b > a,F2: b > a,L: a > a] :
      ( ( ( comp_b3886954628874447685od_a_b @ G @ H )
        = ( comp_a9170378079104387268od_a_b @ R1 @ R2 ) )
     => ( ( ( comp_b_a_a @ F2 @ R1 )
          = L )
       => ( ( comp_b4526394929206263814od_a_b @ ( comp_b_a_b @ F2 @ G ) @ H )
          = ( comp_a586446342581427589od_a_b @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_688_rewriteR__comp__comp2,axiom,
    ! [G: b > b,H: product_prod_a_b > b,R1: a > b,R2: product_prod_a_b > a,F2: b > b,L: a > b] :
      ( ( ( comp_b3886954628874447685od_a_b @ G @ H )
        = ( comp_a9170378079104387268od_a_b @ R1 @ R2 ) )
     => ( ( ( comp_b_b_a @ F2 @ R1 )
          = L )
       => ( ( comp_b3886954628874447685od_a_b @ ( comp_b_b_b @ F2 @ G ) @ H )
          = ( comp_a9170378079104387268od_a_b @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_689_rewriteR__comp__comp2,axiom,
    ! [G: a > b,H: product_prod_a_b > a,R1: b > b,R2: product_prod_a_b > b,F2: b > b,L: b > b] :
      ( ( ( comp_a9170378079104387268od_a_b @ G @ H )
        = ( comp_b3886954628874447685od_a_b @ R1 @ R2 ) )
     => ( ( ( comp_b_b_b @ F2 @ R1 )
          = L )
       => ( ( comp_a9170378079104387268od_a_b @ ( comp_b_b_a @ F2 @ G ) @ H )
          = ( comp_b3886954628874447685od_a_b @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_690_rewriteR__comp__comp2,axiom,
    ! [G: a > b,H: product_prod_a_b > a,R1: b > b,R2: product_prod_a_b > b,F2: b > a,L: b > a] :
      ( ( ( comp_a9170378079104387268od_a_b @ G @ H )
        = ( comp_b3886954628874447685od_a_b @ R1 @ R2 ) )
     => ( ( ( comp_b_a_b @ F2 @ R1 )
          = L )
       => ( ( comp_a586446342581427589od_a_b @ ( comp_b_a_a @ F2 @ G ) @ H )
          = ( comp_b4526394929206263814od_a_b @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_691_rewriteR__comp__comp2,axiom,
    ! [G: a > b,H: product_prod_a_b > a,R1: a > b,R2: product_prod_a_b > a,F2: b > b,L: a > b] :
      ( ( ( comp_a9170378079104387268od_a_b @ G @ H )
        = ( comp_a9170378079104387268od_a_b @ R1 @ R2 ) )
     => ( ( ( comp_b_b_a @ F2 @ R1 )
          = L )
       => ( ( comp_a9170378079104387268od_a_b @ ( comp_b_b_a @ F2 @ G ) @ H )
          = ( comp_a9170378079104387268od_a_b @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_692_rewriteL__comp__comp2,axiom,
    ! [F2: a > a,G: a > a,L1: a > a,L2: a > a,H: a > a,R: a > a] :
      ( ( ( comp_a_a_a @ F2 @ G )
        = ( comp_a_a_a @ L1 @ L2 ) )
     => ( ( ( comp_a_a_a @ L2 @ H )
          = R )
       => ( ( comp_a_a_a @ F2 @ ( comp_a_a_a @ G @ H ) )
          = ( comp_a_a_a @ L1 @ R ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_693_rewriteL__comp__comp2,axiom,
    ! [F2: b > a,G: b > b,L1: a > a,L2: b > a,H: product_prod_a_b > b,R: product_prod_a_b > a] :
      ( ( ( comp_b_a_b @ F2 @ G )
        = ( comp_a_a_b @ L1 @ L2 ) )
     => ( ( ( comp_b4526394929206263814od_a_b @ L2 @ H )
          = R )
       => ( ( comp_b4526394929206263814od_a_b @ F2 @ ( comp_b3886954628874447685od_a_b @ G @ H ) )
          = ( comp_a586446342581427589od_a_b @ L1 @ R ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_694_rewriteL__comp__comp2,axiom,
    ! [F2: b > b,G: b > b,L1: a > b,L2: b > a,H: product_prod_a_b > b,R: product_prod_a_b > a] :
      ( ( ( comp_b_b_b @ F2 @ G )
        = ( comp_a_b_b @ L1 @ L2 ) )
     => ( ( ( comp_b4526394929206263814od_a_b @ L2 @ H )
          = R )
       => ( ( comp_b3886954628874447685od_a_b @ F2 @ ( comp_b3886954628874447685od_a_b @ G @ H ) )
          = ( comp_a9170378079104387268od_a_b @ L1 @ R ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_695_rewriteL__comp__comp2,axiom,
    ! [F2: b > b,G: b > b,L1: b > b,L2: b > b,H: product_prod_a_b > b,R: product_prod_a_b > b] :
      ( ( ( comp_b_b_b @ F2 @ G )
        = ( comp_b_b_b @ L1 @ L2 ) )
     => ( ( ( comp_b3886954628874447685od_a_b @ L2 @ H )
          = R )
       => ( ( comp_b3886954628874447685od_a_b @ F2 @ ( comp_b3886954628874447685od_a_b @ G @ H ) )
          = ( comp_b3886954628874447685od_a_b @ L1 @ R ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_696_rewriteL__comp__comp2,axiom,
    ! [F2: a > b,G: b > a,L1: b > b,L2: b > b,H: product_prod_a_b > b,R: product_prod_a_b > b] :
      ( ( ( comp_a_b_b @ F2 @ G )
        = ( comp_b_b_b @ L1 @ L2 ) )
     => ( ( ( comp_b3886954628874447685od_a_b @ L2 @ H )
          = R )
       => ( ( comp_a9170378079104387268od_a_b @ F2 @ ( comp_b4526394929206263814od_a_b @ G @ H ) )
          = ( comp_b3886954628874447685od_a_b @ L1 @ R ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_697_rewriteL__comp__comp2,axiom,
    ! [F2: a > a,G: b > a,L1: b > a,L2: b > b,H: product_prod_a_b > b,R: product_prod_a_b > b] :
      ( ( ( comp_a_a_b @ F2 @ G )
        = ( comp_b_a_b @ L1 @ L2 ) )
     => ( ( ( comp_b3886954628874447685od_a_b @ L2 @ H )
          = R )
       => ( ( comp_a586446342581427589od_a_b @ F2 @ ( comp_b4526394929206263814od_a_b @ G @ H ) )
          = ( comp_b4526394929206263814od_a_b @ L1 @ R ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_698_rewriteL__comp__comp2,axiom,
    ! [F2: b > b,G: a > b,L1: b > b,L2: a > b,H: product_prod_a_b > a,R: product_prod_a_b > b] :
      ( ( ( comp_b_b_a @ F2 @ G )
        = ( comp_b_b_a @ L1 @ L2 ) )
     => ( ( ( comp_a9170378079104387268od_a_b @ L2 @ H )
          = R )
       => ( ( comp_b3886954628874447685od_a_b @ F2 @ ( comp_a9170378079104387268od_a_b @ G @ H ) )
          = ( comp_b3886954628874447685od_a_b @ L1 @ R ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_699_rewriteL__comp__comp2,axiom,
    ! [F2: a > b,G: a > a,L1: b > b,L2: a > b,H: product_prod_a_b > a,R: product_prod_a_b > b] :
      ( ( ( comp_a_b_a @ F2 @ G )
        = ( comp_b_b_a @ L1 @ L2 ) )
     => ( ( ( comp_a9170378079104387268od_a_b @ L2 @ H )
          = R )
       => ( ( comp_a9170378079104387268od_a_b @ F2 @ ( comp_a586446342581427589od_a_b @ G @ H ) )
          = ( comp_b3886954628874447685od_a_b @ L1 @ R ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_700_rewriteL__comp__comp2,axiom,
    ! [F2: b > b,G: a > b,L1: a > b,L2: a > a,H: product_prod_a_b > a,R: product_prod_a_b > a] :
      ( ( ( comp_b_b_a @ F2 @ G )
        = ( comp_a_b_a @ L1 @ L2 ) )
     => ( ( ( comp_a586446342581427589od_a_b @ L2 @ H )
          = R )
       => ( ( comp_b3886954628874447685od_a_b @ F2 @ ( comp_a9170378079104387268od_a_b @ G @ H ) )
          = ( comp_a9170378079104387268od_a_b @ L1 @ R ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_701_rewriteL__comp__comp2,axiom,
    ! [F2: a > b,G: a > a,L1: a > b,L2: a > a,H: product_prod_a_b > a,R: product_prod_a_b > a] :
      ( ( ( comp_a_b_a @ F2 @ G )
        = ( comp_a_b_a @ L1 @ L2 ) )
     => ( ( ( comp_a586446342581427589od_a_b @ L2 @ H )
          = R )
       => ( ( comp_a9170378079104387268od_a_b @ F2 @ ( comp_a586446342581427589od_a_b @ G @ H ) )
          = ( comp_a9170378079104387268od_a_b @ L1 @ R ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_702_rewriteR__comp__comp,axiom,
    ! [G: a > a,H: a > a,R: a > a,F2: a > a] :
      ( ( ( comp_a_a_a @ G @ H )
        = R )
     => ( ( comp_a_a_a @ ( comp_a_a_a @ F2 @ G ) @ H )
        = ( comp_a_a_a @ F2 @ R ) ) ) ).

% rewriteR_comp_comp
thf(fact_703_rewriteR__comp__comp,axiom,
    ! [G: product_prod_a_b > a,H: a > product_prod_a_b,R: a > a,F2: a > a] :
      ( ( ( comp_P3199017072728932485_b_a_a @ G @ H )
        = R )
     => ( ( comp_P3199017072728932485_b_a_a @ ( comp_a586446342581427589od_a_b @ F2 @ G ) @ H )
        = ( comp_a_a_a @ F2 @ R ) ) ) ).

% rewriteR_comp_comp
thf(fact_704_rewriteR__comp__comp,axiom,
    ! [G: b > a,H: product_prod_a_b > b,R: product_prod_a_b > a,F2: a > b] :
      ( ( ( comp_b4526394929206263814od_a_b @ G @ H )
        = R )
     => ( ( comp_b3886954628874447685od_a_b @ ( comp_a_b_b @ F2 @ G ) @ H )
        = ( comp_a9170378079104387268od_a_b @ F2 @ R ) ) ) ).

% rewriteR_comp_comp
thf(fact_705_rewriteR__comp__comp,axiom,
    ! [G: b > b,H: product_prod_a_b > b,R: product_prod_a_b > b,F2: b > b] :
      ( ( ( comp_b3886954628874447685od_a_b @ G @ H )
        = R )
     => ( ( comp_b3886954628874447685od_a_b @ ( comp_b_b_b @ F2 @ G ) @ H )
        = ( comp_b3886954628874447685od_a_b @ F2 @ R ) ) ) ).

% rewriteR_comp_comp
thf(fact_706_rewriteR__comp__comp,axiom,
    ! [G: a > b,H: product_prod_a_b > a,R: product_prod_a_b > b,F2: b > b] :
      ( ( ( comp_a9170378079104387268od_a_b @ G @ H )
        = R )
     => ( ( comp_a9170378079104387268od_a_b @ ( comp_b_b_a @ F2 @ G ) @ H )
        = ( comp_b3886954628874447685od_a_b @ F2 @ R ) ) ) ).

% rewriteR_comp_comp
thf(fact_707_rewriteR__comp__comp,axiom,
    ! [G: a > b,H: product_prod_a_b > a,R: product_prod_a_b > b,F2: b > a] :
      ( ( ( comp_a9170378079104387268od_a_b @ G @ H )
        = R )
     => ( ( comp_a586446342581427589od_a_b @ ( comp_b_a_a @ F2 @ G ) @ H )
        = ( comp_b4526394929206263814od_a_b @ F2 @ R ) ) ) ).

% rewriteR_comp_comp
thf(fact_708_rewriteR__comp__comp,axiom,
    ! [G: a > a,H: product_prod_a_b > a,R: product_prod_a_b > a,F2: a > b] :
      ( ( ( comp_a586446342581427589od_a_b @ G @ H )
        = R )
     => ( ( comp_a9170378079104387268od_a_b @ ( comp_a_b_a @ F2 @ G ) @ H )
        = ( comp_a9170378079104387268od_a_b @ F2 @ R ) ) ) ).

% rewriteR_comp_comp
thf(fact_709_rewriteR__comp__comp,axiom,
    ! [G: a > a,H: product_prod_a_b > a,R: product_prod_a_b > a,F2: a > a] :
      ( ( ( comp_a586446342581427589od_a_b @ G @ H )
        = R )
     => ( ( comp_a586446342581427589od_a_b @ ( comp_a_a_a @ F2 @ G ) @ H )
        = ( comp_a586446342581427589od_a_b @ F2 @ R ) ) ) ).

% rewriteR_comp_comp
thf(fact_710_rewriteR__comp__comp,axiom,
    ! [G: product_prod_a_b > b,H: product_prod_a_b > product_prod_a_b,R: product_prod_a_b > b,F2: b > b] :
      ( ( ( comp_P2009515992434452078od_a_b @ G @ H )
        = R )
     => ( ( comp_P2009515992434452078od_a_b @ ( comp_b3886954628874447685od_a_b @ F2 @ G ) @ H )
        = ( comp_b3886954628874447685od_a_b @ F2 @ R ) ) ) ).

% rewriteR_comp_comp
thf(fact_711_rewriteR__comp__comp,axiom,
    ! [G: product_prod_a_b > a,H: product_prod_a_b > product_prod_a_b,R: product_prod_a_b > a,F2: a > b] :
      ( ( ( comp_P2648956292766268207od_a_b @ G @ H )
        = R )
     => ( ( comp_P2009515992434452078od_a_b @ ( comp_a9170378079104387268od_a_b @ F2 @ G ) @ H )
        = ( comp_a9170378079104387268od_a_b @ F2 @ R ) ) ) ).

% rewriteR_comp_comp
thf(fact_712_rewriteL__comp__comp,axiom,
    ! [F2: a > a,G: a > a,L: a > a,H: a > a] :
      ( ( ( comp_a_a_a @ F2 @ G )
        = L )
     => ( ( comp_a_a_a @ F2 @ ( comp_a_a_a @ G @ H ) )
        = ( comp_a_a_a @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_713_rewriteL__comp__comp,axiom,
    ! [F2: b > a,G: a > b,L: a > a,H: product_prod_a_b > a] :
      ( ( ( comp_b_a_a @ F2 @ G )
        = L )
     => ( ( comp_b4526394929206263814od_a_b @ F2 @ ( comp_a9170378079104387268od_a_b @ G @ H ) )
        = ( comp_a586446342581427589od_a_b @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_714_rewriteL__comp__comp,axiom,
    ! [F2: b > b,G: b > b,L: b > b,H: product_prod_a_b > b] :
      ( ( ( comp_b_b_b @ F2 @ G )
        = L )
     => ( ( comp_b3886954628874447685od_a_b @ F2 @ ( comp_b3886954628874447685od_a_b @ G @ H ) )
        = ( comp_b3886954628874447685od_a_b @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_715_rewriteL__comp__comp,axiom,
    ! [F2: b > b,G: a > b,L: a > b,H: product_prod_a_b > a] :
      ( ( ( comp_b_b_a @ F2 @ G )
        = L )
     => ( ( comp_b3886954628874447685od_a_b @ F2 @ ( comp_a9170378079104387268od_a_b @ G @ H ) )
        = ( comp_a9170378079104387268od_a_b @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_716_rewriteL__comp__comp,axiom,
    ! [F2: a > b,G: b > a,L: b > b,H: product_prod_a_b > b] :
      ( ( ( comp_a_b_b @ F2 @ G )
        = L )
     => ( ( comp_a9170378079104387268od_a_b @ F2 @ ( comp_b4526394929206263814od_a_b @ G @ H ) )
        = ( comp_b3886954628874447685od_a_b @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_717_rewriteL__comp__comp,axiom,
    ! [F2: a > b,G: a > a,L: a > b,H: product_prod_a_b > a] :
      ( ( ( comp_a_b_a @ F2 @ G )
        = L )
     => ( ( comp_a9170378079104387268od_a_b @ F2 @ ( comp_a586446342581427589od_a_b @ G @ H ) )
        = ( comp_a9170378079104387268od_a_b @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_718_rewriteL__comp__comp,axiom,
    ! [F2: a > a,G: product_prod_a_b > a,L: product_prod_a_b > a,H: a > product_prod_a_b] :
      ( ( ( comp_a586446342581427589od_a_b @ F2 @ G )
        = L )
     => ( ( comp_a_a_a @ F2 @ ( comp_P3199017072728932485_b_a_a @ G @ H ) )
        = ( comp_P3199017072728932485_b_a_a @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_719_rewriteL__comp__comp,axiom,
    ! [F2: a > a,G: a > a,L: a > a,H: product_prod_a_b > a] :
      ( ( ( comp_a_a_a @ F2 @ G )
        = L )
     => ( ( comp_a586446342581427589od_a_b @ F2 @ ( comp_a586446342581427589od_a_b @ G @ H ) )
        = ( comp_a586446342581427589od_a_b @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_720_rewriteL__comp__comp,axiom,
    ! [F2: product_prod_a_a > b,G: a > product_prod_a_a,L: a > b,H: product_prod_a_b > a] :
      ( ( ( comp_P3190265798616278979_a_b_a @ F2 @ G )
        = L )
     => ( ( comp_P6252944881140724845od_a_b @ F2 @ ( comp_a1036870397537576092od_a_b @ G @ H ) )
        = ( comp_a9170378079104387268od_a_b @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_721_rewriteL__comp__comp,axiom,
    ! [F2: product_prod_a_a > a,G: a > product_prod_a_a,L: a > a,H: product_prod_a_b > a] :
      ( ( ( comp_P5977721380588955012_a_a_a @ F2 @ G )
        = L )
     => ( ( comp_P6892385181472540974od_a_b @ F2 @ ( comp_a1036870397537576092od_a_b @ G @ H ) )
        = ( comp_a586446342581427589od_a_b @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_722_convol__o,axiom,
    ! [F2: a > a,G: a > a,H: a > a] :
      ( ( comp_a5847137845442436850_a_a_a @ ( bNF_convol_a_a_a @ F2 @ G ) @ H )
      = ( bNF_convol_a_a_a @ ( comp_a_a_a @ F2 @ H ) @ ( comp_a_a_a @ G @ H ) ) ) ).

% convol_o
thf(fact_723_convol__o,axiom,
    ! [F2: b > b,G: b > b,H: product_prod_a_b > b] :
      ( ( comp_b6573281471806798941od_a_b @ ( bNF_convol_b_b_b @ F2 @ G ) @ H )
      = ( bNF_co937151728994699583_b_b_b @ ( comp_b3886954628874447685od_a_b @ F2 @ H ) @ ( comp_b3886954628874447685od_a_b @ G @ H ) ) ) ).

% convol_o
thf(fact_724_convol__o,axiom,
    ! [F2: a > b,G: a > b,H: product_prod_a_b > a] :
      ( ( comp_a3640768364209717980od_a_b @ ( bNF_convol_a_b_b @ F2 @ G ) @ H )
      = ( bNF_co937151728994699583_b_b_b @ ( comp_a9170378079104387268od_a_b @ F2 @ H ) @ ( comp_a9170378079104387268od_a_b @ G @ H ) ) ) ).

% convol_o
thf(fact_725_convol__o,axiom,
    ! [F2: a > b,G: a > a,H: product_prod_a_b > a] :
      ( ( comp_a4976818984162412317od_a_b @ ( bNF_convol_a_b_a @ F2 @ G ) @ H )
      = ( bNF_co937151728994699582_b_b_a @ ( comp_a9170378079104387268od_a_b @ F2 @ H ) @ ( comp_a586446342581427589od_a_b @ G @ H ) ) ) ).

% convol_o
thf(fact_726_convol__o,axiom,
    ! [F2: a > a,G: a > b,H: product_prod_a_b > a] :
      ( ( comp_a8924191814439657563od_a_b @ ( bNF_convol_a_a_b @ F2 @ G ) @ H )
      = ( bNF_co3724607310967375616_b_a_b @ ( comp_a586446342581427589od_a_b @ F2 @ H ) @ ( comp_a9170378079104387268od_a_b @ G @ H ) ) ) ).

% convol_o
thf(fact_727_convol__o,axiom,
    ! [F2: a > a,G: a > a,H: product_prod_a_b > a] :
      ( ( comp_a1036870397537576092od_a_b @ ( bNF_convol_a_a_a @ F2 @ G ) @ H )
      = ( bNF_co3724607310967375615_b_a_a @ ( comp_a586446342581427589od_a_b @ F2 @ H ) @ ( comp_a586446342581427589od_a_b @ G @ H ) ) ) ).

% convol_o
thf(fact_728_convol__o,axiom,
    ! [F2: a > product_prod_a_a,G: a > b,H: product_prod_a_b > a] :
      ( ( comp_a5901997731762583044od_a_b @ ( bNF_co6372728083680879981_a_a_b @ F2 @ G ) @ H )
      = ( bNF_co7393721345620556055_a_a_b @ ( comp_a1036870397537576092od_a_b @ F2 @ H ) @ ( comp_a9170378079104387268od_a_b @ G @ H ) ) ) ).

% convol_o
thf(fact_729_convol__o,axiom,
    ! [F2: a > product_prod_a_a,G: a > a,H: product_prod_a_b > a] :
      ( ( comp_a7238048351715277381od_a_b @ ( bNF_co6372728083680879980_a_a_a @ F2 @ G ) @ H )
      = ( bNF_co7393721345620556054_a_a_a @ ( comp_a1036870397537576092od_a_b @ F2 @ H ) @ ( comp_a586446342581427589od_a_b @ G @ H ) ) ) ).

% convol_o
thf(fact_730_convol__o,axiom,
    ! [F2: a > b,G: a > product_prod_a_a,H: product_prod_a_b > a] :
      ( ( comp_a6375159522516334516od_a_b @ ( bNF_co472596276184825789od_a_a @ F2 @ G ) @ H )
      = ( bNF_co1493589538124501863od_a_a @ ( comp_a9170378079104387268od_a_b @ F2 @ H ) @ ( comp_a1036870397537576092od_a_b @ G @ H ) ) ) ).

% convol_o
thf(fact_731_convol__o,axiom,
    ! [F2: a > a,G: a > product_prod_a_a,H: product_prod_a_b > a] :
      ( ( comp_a3442646414919253555od_a_b @ ( bNF_co1112036576516641918od_a_a @ F2 @ G ) @ H )
      = ( bNF_co2133029838456317992od_a_a @ ( comp_a586446342581427589od_a_b @ F2 @ H ) @ ( comp_a1036870397537576092od_a_b @ G @ H ) ) ) ).

% convol_o
thf(fact_732_pick__middlep,axiom,
    ! [P2: a > product_prod_a_b > $o,Q2: product_prod_a_b > b > $o,A: a,C: b] :
      ( ( relcom9179970356352878820_a_b_b @ P2 @ Q2 @ A @ C )
     => ( ( P2 @ A @ ( bNF_pi1436070092841995377_a_b_b @ P2 @ Q2 @ A @ C ) )
        & ( Q2 @ ( bNF_pi1436070092841995377_a_b_b @ P2 @ Q2 @ A @ C ) @ C ) ) ) ).

% pick_middlep
thf(fact_733_pick__middlep,axiom,
    ! [P2: product_prod_a_a > produc2233330766403592615od_a_b > $o,Q2: produc2233330766403592615od_a_b > product_prod_b_b > $o,A: product_prod_a_a,C: product_prod_b_b] :
      ( ( relcom8632546625042571830od_b_b @ P2 @ Q2 @ A @ C )
     => ( ( P2 @ A @ ( bNF_pi4789607327318135235od_b_b @ P2 @ Q2 @ A @ C ) )
        & ( Q2 @ ( bNF_pi4789607327318135235od_b_b @ P2 @ Q2 @ A @ C ) @ C ) ) ) ).

% pick_middlep
thf(fact_734_map__prod__o__convol,axiom,
    ! [H1: a > a,H2: b > b,F2: a > a,G: a > b] :
      ( ( comp_P7273998795022290779_a_b_a @ ( produc1231242867592151606_a_b_b @ H1 @ H2 ) @ ( bNF_convol_a_a_b @ F2 @ G ) )
      = ( bNF_convol_a_a_b @ ( comp_a_a_a @ H1 @ F2 ) @ ( comp_b_b_a @ H2 @ G ) ) ) ).

% map_prod_o_convol
thf(fact_735_map__prod__o__convol,axiom,
    ! [H1: a > a,H2: a > a,F2: a > a,G: a > a] :
      ( ( comp_P2929704647478774427_a_a_a @ ( produc4018698449564827638_a_a_a @ H1 @ H2 ) @ ( bNF_convol_a_a_a @ F2 @ G ) )
      = ( bNF_convol_a_a_a @ ( comp_a_a_a @ H1 @ F2 ) @ ( comp_a_a_a @ H2 @ G ) ) ) ).

% map_prod_o_convol
thf(fact_736_map__prod__o__convol,axiom,
    ! [H1: b > b,H2: b > b,F2: product_prod_a_b > b,G: product_prod_a_b > b] :
      ( ( comp_P7896456401148956485od_a_b @ ( produc8315423520075080822_b_b_b @ H1 @ H2 ) @ ( bNF_co937151728994699583_b_b_b @ F2 @ G ) )
      = ( bNF_co937151728994699583_b_b_b @ ( comp_b3886954628874447685od_a_b @ H1 @ F2 ) @ ( comp_b3886954628874447685od_a_b @ H2 @ G ) ) ) ).

% map_prod_o_convol
thf(fact_737_map__prod__o__convol,axiom,
    ! [H1: b > b,H2: a > b,F2: product_prod_a_b > b,G: product_prod_a_b > a] :
      ( ( comp_P5447003528530642500od_a_b @ ( produc1879507065192981047_b_a_b @ H1 @ H2 ) @ ( bNF_co937151728994699582_b_b_a @ F2 @ G ) )
      = ( bNF_co937151728994699583_b_b_b @ ( comp_b3886954628874447685od_a_b @ H1 @ F2 ) @ ( comp_a9170378079104387268od_a_b @ H2 @ G ) ) ) ).

% map_prod_o_convol
thf(fact_738_map__prod__o__convol,axiom,
    ! [H1: b > b,H2: a > a,F2: product_prod_a_b > b,G: product_prod_a_b > a] :
      ( ( comp_P6783054148483336837od_a_b @ ( produc1879507065192981046_b_a_a @ H1 @ H2 ) @ ( bNF_co937151728994699582_b_b_a @ F2 @ G ) )
      = ( bNF_co937151728994699582_b_b_a @ ( comp_b3886954628874447685od_a_b @ H1 @ F2 ) @ ( comp_a586446342581427589od_a_b @ H2 @ G ) ) ) ).

% map_prod_o_convol
thf(fact_739_map__prod__o__convol,axiom,
    ! [H1: a > b,H2: b > b,F2: product_prod_a_b > a,G: product_prod_a_b > b] :
      ( ( comp_P6332851843019826822od_a_b @ ( produc6827461482780282295_b_b_b @ H1 @ H2 ) @ ( bNF_co3724607310967375616_b_a_b @ F2 @ G ) )
      = ( bNF_co937151728994699583_b_b_b @ ( comp_a9170378079104387268od_a_b @ H1 @ F2 ) @ ( comp_b3886954628874447685od_a_b @ H2 @ G ) ) ) ).

% map_prod_o_convol
thf(fact_740_map__prod__o__convol,axiom,
    ! [H1: a > b,H2: a > b,F2: product_prod_a_b > a,G: product_prod_a_b > a] :
      ( ( comp_P3883398970401512837od_a_b @ ( produc391545027898182520_b_a_b @ H1 @ H2 ) @ ( bNF_co3724607310967375615_b_a_a @ F2 @ G ) )
      = ( bNF_co937151728994699583_b_b_b @ ( comp_a9170378079104387268od_a_b @ H1 @ F2 ) @ ( comp_a9170378079104387268od_a_b @ H2 @ G ) ) ) ).

% map_prod_o_convol
thf(fact_741_map__prod__o__convol,axiom,
    ! [H1: a > b,H2: a > a,F2: product_prod_a_b > a,G: product_prod_a_b > a] :
      ( ( comp_P5219449590354207174od_a_b @ ( produc391545027898182519_b_a_a @ H1 @ H2 ) @ ( bNF_co3724607310967375615_b_a_a @ F2 @ G ) )
      = ( bNF_co937151728994699582_b_b_a @ ( comp_a9170378079104387268od_a_b @ H1 @ F2 ) @ ( comp_a586446342581427589od_a_b @ H2 @ G ) ) ) ).

% map_prod_o_convol
thf(fact_742_map__prod__o__convol,axiom,
    ! [H1: a > a,H2: a > b,F2: product_prod_a_b > a,G: product_prod_a_b > a] :
      ( ( comp_P9166822420631452420od_a_b @ ( produc4018698449564827639_a_a_b @ H1 @ H2 ) @ ( bNF_co3724607310967375615_b_a_a @ F2 @ G ) )
      = ( bNF_co3724607310967375616_b_a_b @ ( comp_a586446342581427589od_a_b @ H1 @ F2 ) @ ( comp_a9170378079104387268od_a_b @ H2 @ G ) ) ) ).

% map_prod_o_convol
thf(fact_743_map__prod__o__convol,axiom,
    ! [H1: a > a,H2: b > b,F2: product_prod_a_b > a,G: product_prod_a_b > b] :
      ( ( comp_P2392903256394990597od_a_b @ ( produc1231242867592151606_a_b_b @ H1 @ H2 ) @ ( bNF_co3724607310967375616_b_a_b @ F2 @ G ) )
      = ( bNF_co3724607310967375616_b_a_b @ ( comp_a586446342581427589od_a_b @ H1 @ F2 ) @ ( comp_b3886954628874447685od_a_b @ H2 @ G ) ) ) ).

% map_prod_o_convol
thf(fact_744_sndOp__in,axiom,
    ! [Ac2: produc5884233991663340231list_o,P2: a > a > $o,Q2: a > list_o > $o] :
      ( ( member7948383622993546480list_o @ Ac2 @ ( collec7805115074396547890list_o @ ( produc8342644129638033356st_o_o @ ( relcompp_a_a_list_o @ P2 @ Q2 ) ) ) )
     => ( member7948383622993546480list_o @ ( bNF_sndOp_a_a_list_o @ P2 @ Q2 @ Ac2 ) @ ( collec7805115074396547890list_o @ ( produc8342644129638033356st_o_o @ Q2 ) ) ) ) ).

% sndOp_in
thf(fact_745_sndOp__in,axiom,
    ! [Ac2: product_prod_a_a,P2: a > a > $o,Q2: a > a > $o] :
      ( ( member1426531477525435216od_a_a @ Ac2 @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ ( relcompp_a_a_a @ P2 @ Q2 ) ) ) )
     => ( member1426531477525435216od_a_a @ ( bNF_sndOp_a_a_a @ P2 @ Q2 @ Ac2 ) @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ Q2 ) ) ) ) ).

% sndOp_in
thf(fact_746_sndOp__in,axiom,
    ! [Ac2: produc2233330766403592615od_a_b,P2: product_prod_a_b > product_prod_a_b > $o,Q2: product_prod_a_b > product_prod_a_b > $o] :
      ( ( member5065438833300252112od_a_b @ Ac2 @ ( collec7968472090841016338od_a_b @ ( produc5413562937804622316_a_b_o @ ( relcom7003123350501562167od_a_b @ P2 @ Q2 ) ) ) )
     => ( member5065438833300252112od_a_b @ ( bNF_sn3422074382465070516od_a_b @ P2 @ Q2 @ Ac2 ) @ ( collec7968472090841016338od_a_b @ ( produc5413562937804622316_a_b_o @ Q2 ) ) ) ) ).

% sndOp_in
thf(fact_747_sndOp__in,axiom,
    ! [Ac2: product_prod_nat_nat,P2: nat > nat > $o,Q2: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ Ac2 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( relcompp_nat_nat_nat @ P2 @ Q2 ) ) ) )
     => ( member8440522571783428010at_nat @ ( bNF_sn8916957246138178918at_nat @ P2 @ Q2 @ Ac2 ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ Q2 ) ) ) ) ).

% sndOp_in
thf(fact_748_sndOp__in,axiom,
    ! [Ac2: produc2933732385751300263od_b_b,P2: product_prod_a_a > produc2233330766403592615od_a_b > $o,Q2: produc2233330766403592615od_a_b > product_prod_b_b > $o] :
      ( ( member5765840452647959760od_b_b @ Ac2 @ ( collec8668873710188723986od_b_b @ ( produc2709058680136180588_b_b_o @ ( relcom8632546625042571830od_b_b @ P2 @ Q2 ) ) ) )
     => ( member3803802450123998288od_b_b @ ( bNF_sn6902690285988656819od_b_b @ P2 @ Q2 @ Ac2 ) @ ( collec4814917678252319378od_b_b @ ( produc4340341981123801068_b_b_o @ Q2 ) ) ) ) ).

% sndOp_in
thf(fact_749_sndOp__in,axiom,
    ! [Ac2: product_prod_a_b,P2: a > product_prod_a_b > $o,Q2: product_prod_a_b > b > $o] :
      ( ( member1426531481828664017od_a_b @ Ac2 @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ ( relcom9179970356352878820_a_b_b @ P2 @ Q2 ) ) ) )
     => ( member2194267521500829095_a_b_b @ ( bNF_sn8185316101191594337_a_b_b @ P2 @ Q2 @ Ac2 ) @ ( collec4168926463066647017_a_b_b @ ( produc8203505519722885333_b_b_o @ Q2 ) ) ) ) ).

% sndOp_in
thf(fact_750_sndOp__in,axiom,
    ! [Ac2: product_prod_a_b,P2: a > a > $o,Q2: a > b > $o] :
      ( ( member1426531481828664017od_a_b @ Ac2 @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ ( relcompp_a_a_b @ P2 @ Q2 ) ) ) )
     => ( member1426531481828664017od_a_b @ ( bNF_sndOp_a_a_b @ P2 @ Q2 @ Ac2 ) @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ Q2 ) ) ) ) ).

% sndOp_in
thf(fact_751_fstOp__in,axiom,
    ! [Ac2: product_prod_a_a,P2: a > a > $o,Q2: a > a > $o] :
      ( ( member1426531477525435216od_a_a @ Ac2 @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ ( relcompp_a_a_a @ P2 @ Q2 ) ) ) )
     => ( member1426531477525435216od_a_a @ ( bNF_fstOp_a_a_a @ P2 @ Q2 @ Ac2 ) @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_752_fstOp__in,axiom,
    ! [Ac2: product_prod_nat_nat,P2: nat > nat > $o,Q2: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ Ac2 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( relcompp_nat_nat_nat @ P2 @ Q2 ) ) ) )
     => ( member8440522571783428010at_nat @ ( bNF_fs6799373675401686564at_nat @ P2 @ Q2 @ Ac2 ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_753_fstOp__in,axiom,
    ! [Ac2: product_prod_a_a,P2: a > b > $o,Q2: b > a > $o] :
      ( ( member1426531477525435216od_a_a @ Ac2 @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ ( relcompp_a_b_a @ P2 @ Q2 ) ) ) )
     => ( member1426531481828664017od_a_b @ ( bNF_fstOp_a_b_a @ P2 @ Q2 @ Ac2 ) @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_754_fstOp__in,axiom,
    ! [Ac2: product_prod_a_b,P2: a > a > $o,Q2: a > b > $o] :
      ( ( member1426531481828664017od_a_b @ Ac2 @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ ( relcompp_a_a_b @ P2 @ Q2 ) ) ) )
     => ( member1426531477525435216od_a_a @ ( bNF_fstOp_a_a_b @ P2 @ Q2 @ Ac2 ) @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_755_fstOp__in,axiom,
    ! [Ac2: product_prod_a_b,P2: a > b > $o,Q2: b > b > $o] :
      ( ( member1426531481828664017od_a_b @ Ac2 @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ ( relcompp_a_b_b @ P2 @ Q2 ) ) ) )
     => ( member1426531481828664017od_a_b @ ( bNF_fstOp_a_b_b @ P2 @ Q2 @ Ac2 ) @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_756_fstOp__in,axiom,
    ! [Ac2: produc5884233991663340231list_o,P2: a > a > $o,Q2: a > list_o > $o] :
      ( ( member7948383622993546480list_o @ Ac2 @ ( collec7805115074396547890list_o @ ( produc8342644129638033356st_o_o @ ( relcompp_a_a_list_o @ P2 @ Q2 ) ) ) )
     => ( member1426531477525435216od_a_a @ ( bNF_fstOp_a_a_list_o @ P2 @ Q2 @ Ac2 ) @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_757_fstOp__in,axiom,
    ! [Ac2: product_prod_a_a,P2: a > list_o > $o,Q2: list_o > a > $o] :
      ( ( member1426531477525435216od_a_a @ Ac2 @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ ( relcompp_a_list_o_a @ P2 @ Q2 ) ) ) )
     => ( member7948383622993546480list_o @ ( bNF_fstOp_a_list_o_a @ P2 @ Q2 @ Ac2 ) @ ( collec7805115074396547890list_o @ ( produc8342644129638033356st_o_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_758_fstOp__in,axiom,
    ! [Ac2: produc5884233991663340231list_o,P2: a > b > $o,Q2: b > list_o > $o] :
      ( ( member7948383622993546480list_o @ Ac2 @ ( collec7805115074396547890list_o @ ( produc8342644129638033356st_o_o @ ( relcompp_a_b_list_o @ P2 @ Q2 ) ) ) )
     => ( member1426531481828664017od_a_b @ ( bNF_fstOp_a_b_list_o @ P2 @ Q2 @ Ac2 ) @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_759_fstOp__in,axiom,
    ! [Ac2: product_prod_a_b,P2: a > list_o > $o,Q2: list_o > b > $o] :
      ( ( member1426531481828664017od_a_b @ Ac2 @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ ( relcompp_a_list_o_b @ P2 @ Q2 ) ) ) )
     => ( member7948383622993546480list_o @ ( bNF_fstOp_a_list_o_b @ P2 @ Q2 @ Ac2 ) @ ( collec7805115074396547890list_o @ ( produc8342644129638033356st_o_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_760_fstOp__in,axiom,
    ! [Ac2: produc5884233991663340231list_o,P2: a > list_o > $o,Q2: list_o > list_o > $o] :
      ( ( member7948383622993546480list_o @ Ac2 @ ( collec7805115074396547890list_o @ ( produc8342644129638033356st_o_o @ ( relcom246367653029328397list_o @ P2 @ Q2 ) ) ) )
     => ( member7948383622993546480list_o @ ( bNF_fs1531470065415673164list_o @ P2 @ Q2 @ Ac2 ) @ ( collec7805115074396547890list_o @ ( produc8342644129638033356st_o_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_761_map__prod__o__convol__id,axiom,
    ! [F2: product_prod_a_b > b,G: product_prod_a_b > a,X3: product_prod_a_b] :
      ( ( comp_P2391947450525258140od_a_b @ ( produc4881195996318342733_b_a_a @ F2 @ id_a ) @ ( bNF_co6758072345015569365_a_b_a @ id_Product_prod_a_b @ G ) @ X3 )
      = ( bNF_co937151728994699582_b_b_a @ ( comp_b3886954628874447685od_a_b @ id_b @ F2 ) @ G @ X3 ) ) ).

% map_prod_o_convol_id
thf(fact_762_map__prod__o__convol__id,axiom,
    ! [F2: product_prod_a_b > a,G: product_prod_a_b > a,X3: product_prod_a_b] :
      ( ( comp_P7675370900755197723od_a_b @ ( produc8508349417984987852_a_a_a @ F2 @ id_a ) @ ( bNF_co6758072345015569365_a_b_a @ id_Product_prod_a_b @ G ) @ X3 )
      = ( bNF_co3724607310967375615_b_a_a @ ( comp_a586446342581427589od_a_b @ id_a @ F2 ) @ G @ X3 ) ) ).

% map_prod_o_convol_id
thf(fact_763_map__prod__o__convol__id,axiom,
    ! [F2: a > a,G: a > b,X3: a] :
      ( ( comp_P7273998795022290779_a_b_a @ ( produc1231242867592151606_a_b_b @ F2 @ id_b ) @ ( bNF_convol_a_a_b @ id_a @ G ) @ X3 )
      = ( bNF_convol_a_a_b @ ( comp_a_a_a @ id_a @ F2 ) @ G @ X3 ) ) ).

% map_prod_o_convol_id
thf(fact_764_map__prod__o__convol__id,axiom,
    ! [F2: a > a,G: a > a,X3: a] :
      ( ( comp_P2929704647478774427_a_a_a @ ( produc4018698449564827638_a_a_a @ F2 @ id_a ) @ ( bNF_convol_a_a_a @ id_a @ G ) @ X3 )
      = ( bNF_convol_a_a_a @ ( comp_a_a_a @ id_a @ F2 ) @ G @ X3 ) ) ).

% map_prod_o_convol_id
thf(fact_765_swap__comp__swap,axiom,
    ( ( comp_P1507054941905806275od_a_b @ product_swap_b_a @ product_swap_a_b )
    = id_Product_prod_a_b ) ).

% swap_comp_swap
thf(fact_766_swap__comp__swap,axiom,
    ( ( comp_P4881446876696616325od_b_a @ product_swap_a_b @ product_swap_b_a )
    = id_Product_prod_b_a ) ).

% swap_comp_swap
thf(fact_767_relcompp_Ocases,axiom,
    ! [R: a > product_prod_a_b > $o,S2: product_prod_a_b > b > $o,A1: a,A22: b] :
      ( ( relcom9179970356352878820_a_b_b @ R @ S2 @ A1 @ A22 )
     => ~ ! [B3: product_prod_a_b] :
            ( ( R @ A1 @ B3 )
           => ~ ( S2 @ B3 @ A22 ) ) ) ).

% relcompp.cases
thf(fact_768_relcompp_Ocases,axiom,
    ! [R: product_prod_a_a > produc2233330766403592615od_a_b > $o,S2: produc2233330766403592615od_a_b > product_prod_b_b > $o,A1: product_prod_a_a,A22: product_prod_b_b] :
      ( ( relcom8632546625042571830od_b_b @ R @ S2 @ A1 @ A22 )
     => ~ ! [B3: produc2233330766403592615od_a_b] :
            ( ( R @ A1 @ B3 )
           => ~ ( S2 @ B3 @ A22 ) ) ) ).

% relcompp.cases
thf(fact_769_relcompp_Osimps,axiom,
    ( relcom9179970356352878820_a_b_b
    = ( ^ [R3: a > product_prod_a_b > $o,S: product_prod_a_b > b > $o,A12: a,A23: b] :
        ? [A2: a,B4: product_prod_a_b,C2: b] :
          ( ( A12 = A2 )
          & ( A23 = C2 )
          & ( R3 @ A2 @ B4 )
          & ( S @ B4 @ C2 ) ) ) ) ).

% relcompp.simps
thf(fact_770_relcompp_Osimps,axiom,
    ( relcom8632546625042571830od_b_b
    = ( ^ [R3: product_prod_a_a > produc2233330766403592615od_a_b > $o,S: produc2233330766403592615od_a_b > product_prod_b_b > $o,A12: product_prod_a_a,A23: product_prod_b_b] :
        ? [A2: product_prod_a_a,B4: produc2233330766403592615od_a_b,C2: product_prod_b_b] :
          ( ( A12 = A2 )
          & ( A23 = C2 )
          & ( R3 @ A2 @ B4 )
          & ( S @ B4 @ C2 ) ) ) ) ).

% relcompp.simps
thf(fact_771_relcomppE,axiom,
    ! [R: a > product_prod_a_b > $o,S2: product_prod_a_b > b > $o,A: a,C: b] :
      ( ( relcom9179970356352878820_a_b_b @ R @ S2 @ A @ C )
     => ~ ! [B3: product_prod_a_b] :
            ( ( R @ A @ B3 )
           => ~ ( S2 @ B3 @ C ) ) ) ).

% relcomppE
thf(fact_772_relcomppE,axiom,
    ! [R: product_prod_a_a > produc2233330766403592615od_a_b > $o,S2: produc2233330766403592615od_a_b > product_prod_b_b > $o,A: product_prod_a_a,C: product_prod_b_b] :
      ( ( relcom8632546625042571830od_b_b @ R @ S2 @ A @ C )
     => ~ ! [B3: produc2233330766403592615od_a_b] :
            ( ( R @ A @ B3 )
           => ~ ( S2 @ B3 @ C ) ) ) ).

% relcomppE
thf(fact_773_id__apply,axiom,
    ( id_a
    = ( ^ [X: a] : X ) ) ).

% id_apply
thf(fact_774_fun_Omap__id,axiom,
    ! [T2: product_prod_a_b > b] :
      ( ( comp_b3886954628874447685od_a_b @ id_b @ T2 )
      = T2 ) ).

% fun.map_id
thf(fact_775_fun_Omap__id,axiom,
    ! [T2: product_prod_a_b > a] :
      ( ( comp_a586446342581427589od_a_b @ id_a @ T2 )
      = T2 ) ).

% fun.map_id
thf(fact_776_fun_Omap__id,axiom,
    ! [T2: a > a] :
      ( ( comp_a_a_a @ id_a @ T2 )
      = T2 ) ).

% fun.map_id
thf(fact_777_id__comp,axiom,
    ! [G: product_prod_a_b > b] :
      ( ( comp_b3886954628874447685od_a_b @ id_b @ G )
      = G ) ).

% id_comp
thf(fact_778_id__comp,axiom,
    ! [G: product_prod_a_b > a] :
      ( ( comp_a586446342581427589od_a_b @ id_a @ G )
      = G ) ).

% id_comp
thf(fact_779_id__comp,axiom,
    ! [G: a > a] :
      ( ( comp_a_a_a @ id_a @ G )
      = G ) ).

% id_comp
thf(fact_780_comp__id,axiom,
    ! [F2: a > a] :
      ( ( comp_a_a_a @ F2 @ id_a )
      = F2 ) ).

% comp_id
thf(fact_781_case__prod__Pair,axiom,
    ( ( produc921320349745534075od_b_b @ product_Pair_b_b )
    = id_Product_prod_b_b ) ).

% case_prod_Pair
thf(fact_782_case__prod__Pair,axiom,
    ( ( produc1560760645774121403od_b_a @ product_Pair_b_a )
    = id_Product_prod_b_a ) ).

% case_prod_Pair
thf(fact_783_case__prod__Pair,axiom,
    ( ( produc8935064565815367931list_o @ produc6899623729582506167list_o )
    = id_Pro7663745361186064696list_o ) ).

% case_prod_Pair
thf(fact_784_case__prod__Pair,axiom,
    ( ( produc8992199381948149691od_a_b @ product_Pair_a_b )
    = id_Product_prod_a_b ) ).

% case_prod_Pair
thf(fact_785_case__prod__Pair,axiom,
    ( ( produc408267641121961211od_a_a @ product_Pair_a_a )
    = id_Product_prod_a_a ) ).

% case_prod_Pair
thf(fact_786_apfst__id,axiom,
    ( ( product_apfst_a_a_b @ id_a )
    = id_Product_prod_a_b ) ).

% apfst_id
thf(fact_787_apsnd__id,axiom,
    ( ( product_apsnd_b_b_a @ id_b )
    = id_Product_prod_a_b ) ).

% apsnd_id
thf(fact_788_fun_Omap__id0,axiom,
    ( ( comp_b3886954628874447685od_a_b @ id_b )
    = id_Pro6323836995665883403_a_b_b ) ).

% fun.map_id0
thf(fact_789_fun_Omap__id0,axiom,
    ( ( comp_a586446342581427589od_a_b @ id_a )
    = id_Pro6323836991362654602_a_b_a ) ).

% fun.map_id0
thf(fact_790_fun_Omap__id0,axiom,
    ( ( comp_a_a_a @ id_a )
    = id_a_a ) ).

% fun.map_id0
thf(fact_791_id__def,axiom,
    ( id_a
    = ( ^ [X: a] : X ) ) ).

% id_def
thf(fact_792_eq__id__iff,axiom,
    ! [F2: a > a] :
      ( ( ! [X: a] :
            ( ( F2 @ X )
            = X ) )
      = ( F2 = id_a ) ) ).

% eq_id_iff
thf(fact_793_prod_Omap__id0,axiom,
    ( ( produc1231242867592151606_a_b_b @ id_a @ id_b )
    = id_Product_prod_a_b ) ).

% prod.map_id0
thf(fact_794_prod_Omap__id0,axiom,
    ( ( produc4018698449564827638_a_a_a @ id_a @ id_a )
    = id_Product_prod_a_a ) ).

% prod.map_id0
thf(fact_795_comp__eq__id__dest,axiom,
    ! [A: b > b,B: product_prod_a_b > b,C: product_prod_a_b > b,V: product_prod_a_b] :
      ( ( ( comp_b3886954628874447685od_a_b @ A @ B )
        = ( comp_b3886954628874447685od_a_b @ id_b @ C ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_id_dest
thf(fact_796_comp__eq__id__dest,axiom,
    ! [A: a > product_prod_a_a,B: product_prod_a_b > a,C: product_prod_a_b > product_prod_a_a,V: product_prod_a_b] :
      ( ( ( comp_a1036870397537576092od_a_b @ A @ B )
        = ( comp_P1279501003729370949od_a_b @ id_Product_prod_a_a @ C ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_id_dest
thf(fact_797_comp__eq__id__dest,axiom,
    ! [A: a > b,B: product_prod_a_b > a,C: product_prod_a_b > b,V: product_prod_a_b] :
      ( ( ( comp_a9170378079104387268od_a_b @ A @ B )
        = ( comp_b3886954628874447685od_a_b @ id_b @ C ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_id_dest
thf(fact_798_comp__eq__id__dest,axiom,
    ! [A: a > a,B: product_prod_a_b > a,C: product_prod_a_b > a,V: product_prod_a_b] :
      ( ( ( comp_a586446342581427589od_a_b @ A @ B )
        = ( comp_a586446342581427589od_a_b @ id_a @ C ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_id_dest
thf(fact_799_comp__eq__id__dest,axiom,
    ! [A: a > a,B: a > a,C: a > a,V: a] :
      ( ( ( comp_a_a_a @ A @ B )
        = ( comp_a_a_a @ id_a @ C ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_id_dest
thf(fact_800_pointfree__idE,axiom,
    ! [F2: a > a,G: a > a,X3: a] :
      ( ( ( comp_a_a_a @ F2 @ G )
        = id_a )
     => ( ( F2 @ ( G @ X3 ) )
        = X3 ) ) ).

% pointfree_idE
thf(fact_801_inj__on__id,axiom,
    ! [A4: set_a] : ( inj_on_a_a @ id_a @ A4 ) ).

% inj_on_id
thf(fact_802_prod_Omap__id,axiom,
    ! [T2: product_prod_a_b] :
      ( ( produc1231242867592151606_a_b_b @ id_a @ id_b @ T2 )
      = T2 ) ).

% prod.map_id
thf(fact_803_prod_Omap__id,axiom,
    ! [T2: product_prod_a_a] :
      ( ( produc4018698449564827638_a_a_a @ id_a @ id_a @ T2 )
      = T2 ) ).

% prod.map_id
thf(fact_804_apfst__def,axiom,
    ( product_apfst_a_a_b
    = ( ^ [F: a > a] : ( produc1231242867592151606_a_b_b @ F @ id_b ) ) ) ).

% apfst_def
thf(fact_805_apfst__def,axiom,
    ( product_apfst_a_a_a
    = ( ^ [F: a > a] : ( produc4018698449564827638_a_a_a @ F @ id_a ) ) ) ).

% apfst_def
thf(fact_806_apsnd__def,axiom,
    ( product_apsnd_b_b_a
    = ( produc1231242867592151606_a_b_b @ id_a ) ) ).

% apsnd_def
thf(fact_807_apsnd__def,axiom,
    ( product_apsnd_a_a_a
    = ( produc4018698449564827638_a_a_a @ id_a ) ) ).

% apsnd_def
thf(fact_808_convol__mem__GrpI,axiom,
    ! [X3: a,A4: set_a,G: a > list_o] :
      ( ( member_a @ X3 @ A4 )
     => ( member7948383622993546480list_o @ ( bNF_co6617811479755555573list_o @ id_a @ G @ X3 ) @ ( collec7805115074396547890list_o @ ( produc8342644129638033356st_o_o @ ( bNF_Grp_a_list_o @ A4 @ G ) ) ) ) ) ).

% convol_mem_GrpI
thf(fact_809_convol__mem__GrpI,axiom,
    ! [X3: a,A4: set_a,G: a > a] :
      ( ( member_a @ X3 @ A4 )
     => ( member1426531477525435216od_a_a @ ( bNF_convol_a_a_a @ id_a @ G @ X3 ) @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ ( bNF_Grp_a_a @ A4 @ G ) ) ) ) ) ).

% convol_mem_GrpI
thf(fact_810_convol__mem__GrpI,axiom,
    ! [X3: product_prod_a_b,A4: set_Product_prod_a_b,G: product_prod_a_b > a] :
      ( ( member1426531481828664017od_a_b @ X3 @ A4 )
     => ( member2194267517197600294_a_b_a @ ( bNF_co6758072345015569365_a_b_a @ id_Product_prod_a_b @ G @ X3 ) @ ( collec4168926458763418216_a_b_a @ ( produc1879355882332503830_b_a_o @ ( bNF_Gr7842136747927947300_a_b_a @ A4 @ G ) ) ) ) ) ).

% convol_mem_GrpI
thf(fact_811_convol__mem__GrpI,axiom,
    ! [X3: product_prod_a_b,A4: set_Product_prod_a_b,G: product_prod_a_b > b] :
      ( ( member1426531481828664017od_a_b @ X3 @ A4 )
     => ( member2194267521500829095_a_b_b @ ( bNF_co6758072345015569366_a_b_b @ id_Product_prod_a_b @ G @ X3 ) @ ( collec4168926463066647017_a_b_b @ ( produc8203505519722885333_b_b_o @ ( bNF_Gr7842136747927947301_a_b_b @ A4 @ G ) ) ) ) ) ).

% convol_mem_GrpI
thf(fact_812_convol__mem__GrpI,axiom,
    ! [X3: produc2233330766403592615od_a_b,A4: set_Pr5493900063595384711od_a_b,G: produc2233330766403592615od_a_b > product_prod_a_a] :
      ( ( member5065438833300252112od_a_b @ X3 @ A4 )
     => ( member4368417410682638992od_a_a @ ( bNF_co9004882301282121278od_a_a @ id_Pro8263049352299524632od_a_b @ G @ X3 ) @ ( collec5379532638810960082od_a_a @ ( produc5892353234140082604_a_a_o @ ( bNF_Gr248057877874286094od_a_a @ A4 @ G ) ) ) ) ) ).

% convol_mem_GrpI
thf(fact_813_convol__mem__GrpI,axiom,
    ! [X3: produc2233330766403592615od_a_b,A4: set_Pr5493900063595384711od_a_b,G: produc2233330766403592615od_a_b > product_prod_b_b] :
      ( ( member5065438833300252112od_a_b @ X3 @ A4 )
     => ( member3803802450123998288od_b_b @ ( bNF_co6217426723612674046od_b_b @ id_Pro8263049352299524632od_a_b @ G @ X3 ) @ ( collec4814917678252319378od_b_b @ ( produc4340341981123801068_b_b_o @ ( bNF_Gr6683974337059614670od_b_b @ A4 @ G ) ) ) ) ) ).

% convol_mem_GrpI
thf(fact_814_convol__mem__GrpI,axiom,
    ! [X3: product_prod_a_b,A4: set_Product_prod_a_b,G: product_prod_a_b > product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ X3 @ A4 )
     => ( member5065438833300252112od_a_b @ ( bNF_co4288629676869875327od_a_b @ id_Product_prod_a_b @ G @ X3 ) @ ( collec7968472090841016338od_a_b @ ( produc5413562937804622316_a_b_o @ ( bNF_Gr7211640137128594382od_a_b @ A4 @ G ) ) ) ) ) ).

% convol_mem_GrpI
thf(fact_815_convol__mem__GrpI,axiom,
    ! [X3: nat,A4: set_nat,G: nat > nat] :
      ( ( member_nat @ X3 @ A4 )
     => ( member8440522571783428010at_nat @ ( bNF_co805650143699787099at_nat @ id_nat @ G @ X3 ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( bNF_Grp_nat_nat @ A4 @ G ) ) ) ) ) ).

% convol_mem_GrpI
thf(fact_816_convol__mem__GrpI,axiom,
    ! [X3: a,A4: set_a,G: a > b] :
      ( ( member_a @ X3 @ A4 )
     => ( member1426531481828664017od_a_b @ ( bNF_convol_a_a_b @ id_a @ G @ X3 ) @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ ( bNF_Grp_a_b @ A4 @ G ) ) ) ) ) ).

% convol_mem_GrpI
thf(fact_817_relcompp__assoc,axiom,
    ! [R: a > product_prod_a_b > $o,S2: product_prod_a_b > b > $o,T2: b > b > $o] :
      ( ( relcompp_a_b_b @ ( relcom9179970356352878820_a_b_b @ R @ S2 ) @ T2 )
      = ( relcom9179970356352878820_a_b_b @ R @ ( relcom4380043002271685111_b_b_b @ S2 @ T2 ) ) ) ).

% relcompp_assoc
thf(fact_818_relcompp__assoc,axiom,
    ! [R: product_prod_a_a > produc2233330766403592615od_a_b > $o,S2: produc2233330766403592615od_a_b > product_prod_b_b > $o,T2: product_prod_b_b > product_prod_b_b > $o] :
      ( ( relcom5706163482535408374od_b_b @ ( relcom8632546625042571830od_b_b @ R @ S2 ) @ T2 )
      = ( relcom8632546625042571830od_b_b @ R @ ( relcom9094268924332824950od_b_b @ S2 @ T2 ) ) ) ).

% relcompp_assoc
thf(fact_819_relcompp__assoc,axiom,
    ! [R: a > a > $o,S2: a > product_prod_a_b > $o,T2: product_prod_a_b > b > $o] :
      ( ( relcom9179970356352878820_a_b_b @ ( relcom4554927854096856247od_a_b @ R @ S2 ) @ T2 )
      = ( relcompp_a_a_b @ R @ ( relcom9179970356352878820_a_b_b @ S2 @ T2 ) ) ) ).

% relcompp_assoc
thf(fact_820_relcompp__assoc,axiom,
    ! [R: a > product_prod_a_b > $o,S2: product_prod_a_b > product_prod_a_b > $o,T2: product_prod_a_b > b > $o] :
      ( ( relcom9179970356352878820_a_b_b @ ( relcom3565115020915045773od_a_b @ R @ S2 ) @ T2 )
      = ( relcom9179970356352878820_a_b_b @ R @ ( relcom1914922001497678990_a_b_b @ S2 @ T2 ) ) ) ).

% relcompp_assoc
thf(fact_821_relcompp__assoc,axiom,
    ! [R: product_prod_a_a > product_prod_a_a > $o,S2: product_prod_a_a > produc2233330766403592615od_a_b > $o,T2: produc2233330766403592615od_a_b > product_prod_b_b > $o] :
      ( ( relcom8632546625042571830od_b_b @ ( relcom7688116606491486582od_a_b @ R @ S2 ) @ T2 )
      = ( relcom3102265515863266486od_b_b @ R @ ( relcom8632546625042571830od_b_b @ S2 @ T2 ) ) ) ).

% relcompp_assoc
thf(fact_822_relcompp__assoc,axiom,
    ! [R: product_prod_a_a > produc2233330766403592615od_a_b > $o,S2: produc2233330766403592615od_a_b > produc2233330766403592615od_a_b > $o,T2: produc2233330766403592615od_a_b > product_prod_b_b > $o] :
      ( ( relcom8632546625042571830od_b_b @ ( relcom6321393211869853430od_a_b @ R @ S2 ) @ T2 )
      = ( relcom8632546625042571830od_b_b @ R @ ( relcom8581073455426504886od_b_b @ S2 @ T2 ) ) ) ).

% relcompp_assoc
thf(fact_823_relcompp__apply,axiom,
    ( relcom9179970356352878820_a_b_b
    = ( ^ [R4: a > product_prod_a_b > $o,S3: product_prod_a_b > b > $o,A2: a,C2: b] :
        ? [B4: product_prod_a_b] :
          ( ( R4 @ A2 @ B4 )
          & ( S3 @ B4 @ C2 ) ) ) ) ).

% relcompp_apply
thf(fact_824_relcompp__apply,axiom,
    ( relcom8632546625042571830od_b_b
    = ( ^ [R4: product_prod_a_a > produc2233330766403592615od_a_b > $o,S3: produc2233330766403592615od_a_b > product_prod_b_b > $o,A2: product_prod_a_a,C2: product_prod_b_b] :
        ? [B4: produc2233330766403592615od_a_b] :
          ( ( R4 @ A2 @ B4 )
          & ( S3 @ B4 @ C2 ) ) ) ) ).

% relcompp_apply
thf(fact_825_relcomppI,axiom,
    ! [R: a > product_prod_a_b > $o,A: a,B: product_prod_a_b,S2: product_prod_a_b > b > $o,C: b] :
      ( ( R @ A @ B )
     => ( ( S2 @ B @ C )
       => ( relcom9179970356352878820_a_b_b @ R @ S2 @ A @ C ) ) ) ).

% relcomppI
thf(fact_826_relcomppI,axiom,
    ! [R: product_prod_a_a > produc2233330766403592615od_a_b > $o,A: product_prod_a_a,B: produc2233330766403592615od_a_b,S2: produc2233330766403592615od_a_b > product_prod_b_b > $o,C: product_prod_b_b] :
      ( ( R @ A @ B )
     => ( ( S2 @ B @ C )
       => ( relcom8632546625042571830od_b_b @ R @ S2 @ A @ C ) ) ) ).

% relcomppI
thf(fact_827_left__right__inverse__eq,axiom,
    ! [F2: a > a,G: a > a,H: a > a] :
      ( ( ( comp_a_a_a @ F2 @ G )
        = id_a )
     => ( ( ( comp_a_a_a @ G @ H )
          = id_a )
       => ( F2 = H ) ) ) ).

% left_right_inverse_eq
thf(fact_828_isomorphism__expand,axiom,
    ! [F2: a > a,G: a > a] :
      ( ( ( ( comp_a_a_a @ F2 @ G )
          = id_a )
        & ( ( comp_a_a_a @ G @ F2 )
          = id_a ) )
      = ( ! [X: a] :
            ( ( F2 @ ( G @ X ) )
            = X )
        & ! [X: a] :
            ( ( G @ ( F2 @ X ) )
            = X ) ) ) ).

% isomorphism_expand
thf(fact_829_type__copy__map__cong0,axiom,
    ! [M: a > a,G: a > a,X3: a,N: a > a,H: a > a,F2: a > a] :
      ( ( ( M @ ( G @ X3 ) )
        = ( N @ ( H @ X3 ) ) )
     => ( ( comp_a_a_a @ ( comp_a_a_a @ F2 @ M ) @ G @ X3 )
        = ( comp_a_a_a @ ( comp_a_a_a @ F2 @ N ) @ H @ X3 ) ) ) ).

% type_copy_map_cong0
thf(fact_830_type__copy__map__cong0,axiom,
    ! [M: product_prod_a_b > a,G: a > product_prod_a_b,X3: a,N: a > a,H: a > a,F2: a > a] :
      ( ( ( M @ ( G @ X3 ) )
        = ( N @ ( H @ X3 ) ) )
     => ( ( comp_P3199017072728932485_b_a_a @ ( comp_a586446342581427589od_a_b @ F2 @ M ) @ G @ X3 )
        = ( comp_a_a_a @ ( comp_a_a_a @ F2 @ N ) @ H @ X3 ) ) ) ).

% type_copy_map_cong0
thf(fact_831_type__copy__map__cong0,axiom,
    ! [M: a > a,G: product_prod_a_b > a,X3: product_prod_a_b,N: a > a,H: product_prod_a_b > a,F2: a > a] :
      ( ( ( M @ ( G @ X3 ) )
        = ( N @ ( H @ X3 ) ) )
     => ( ( comp_a586446342581427589od_a_b @ ( comp_a_a_a @ F2 @ M ) @ G @ X3 )
        = ( comp_a586446342581427589od_a_b @ ( comp_a_a_a @ F2 @ N ) @ H @ X3 ) ) ) ).

% type_copy_map_cong0
thf(fact_832_type__copy__map__cong0,axiom,
    ! [M: a > a,G: a > a,X3: a,N: product_prod_a_b > a,H: a > product_prod_a_b,F2: a > a] :
      ( ( ( M @ ( G @ X3 ) )
        = ( N @ ( H @ X3 ) ) )
     => ( ( comp_a_a_a @ ( comp_a_a_a @ F2 @ M ) @ G @ X3 )
        = ( comp_P3199017072728932485_b_a_a @ ( comp_a586446342581427589od_a_b @ F2 @ N ) @ H @ X3 ) ) ) ).

% type_copy_map_cong0
thf(fact_833_type__copy__map__cong0,axiom,
    ! [M: product_prod_a_b > b,G: product_prod_a_b > product_prod_a_b,X3: product_prod_a_b,N: b > b,H: product_prod_a_b > b,F2: b > b] :
      ( ( ( M @ ( G @ X3 ) )
        = ( N @ ( H @ X3 ) ) )
     => ( ( comp_P2009515992434452078od_a_b @ ( comp_b3886954628874447685od_a_b @ F2 @ M ) @ G @ X3 )
        = ( comp_b3886954628874447685od_a_b @ ( comp_b_b_b @ F2 @ N ) @ H @ X3 ) ) ) ).

% type_copy_map_cong0
thf(fact_834_type__copy__map__cong0,axiom,
    ! [M: product_prod_a_b > b,G: product_prod_a_b > product_prod_a_b,X3: product_prod_a_b,N: a > b,H: product_prod_a_b > a,F2: b > b] :
      ( ( ( M @ ( G @ X3 ) )
        = ( N @ ( H @ X3 ) ) )
     => ( ( comp_P2009515992434452078od_a_b @ ( comp_b3886954628874447685od_a_b @ F2 @ M ) @ G @ X3 )
        = ( comp_a9170378079104387268od_a_b @ ( comp_b_b_a @ F2 @ N ) @ H @ X3 ) ) ) ).

% type_copy_map_cong0
thf(fact_835_type__copy__map__cong0,axiom,
    ! [M: product_prod_a_b > a,G: product_prod_a_b > product_prod_a_b,X3: product_prod_a_b,N: b > a,H: product_prod_a_b > b,F2: a > b] :
      ( ( ( M @ ( G @ X3 ) )
        = ( N @ ( H @ X3 ) ) )
     => ( ( comp_P2009515992434452078od_a_b @ ( comp_a9170378079104387268od_a_b @ F2 @ M ) @ G @ X3 )
        = ( comp_b3886954628874447685od_a_b @ ( comp_a_b_b @ F2 @ N ) @ H @ X3 ) ) ) ).

% type_copy_map_cong0
thf(fact_836_type__copy__map__cong0,axiom,
    ! [M: product_prod_a_b > a,G: product_prod_a_b > product_prod_a_b,X3: product_prod_a_b,N: a > a,H: product_prod_a_b > a,F2: a > b] :
      ( ( ( M @ ( G @ X3 ) )
        = ( N @ ( H @ X3 ) ) )
     => ( ( comp_P2009515992434452078od_a_b @ ( comp_a9170378079104387268od_a_b @ F2 @ M ) @ G @ X3 )
        = ( comp_a9170378079104387268od_a_b @ ( comp_a_b_a @ F2 @ N ) @ H @ X3 ) ) ) ).

% type_copy_map_cong0
thf(fact_837_type__copy__map__cong0,axiom,
    ! [M: product_prod_a_b > a,G: product_prod_a_b > product_prod_a_b,X3: product_prod_a_b,N: a > a,H: product_prod_a_b > a,F2: a > a] :
      ( ( ( M @ ( G @ X3 ) )
        = ( N @ ( H @ X3 ) ) )
     => ( ( comp_P2648956292766268207od_a_b @ ( comp_a586446342581427589od_a_b @ F2 @ M ) @ G @ X3 )
        = ( comp_a586446342581427589od_a_b @ ( comp_a_a_a @ F2 @ N ) @ H @ X3 ) ) ) ).

% type_copy_map_cong0
thf(fact_838_type__copy__map__cong0,axiom,
    ! [M: b > b,G: product_prod_a_b > b,X3: product_prod_a_b,N: product_prod_a_b > b,H: product_prod_a_b > product_prod_a_b,F2: b > b] :
      ( ( ( M @ ( G @ X3 ) )
        = ( N @ ( H @ X3 ) ) )
     => ( ( comp_b3886954628874447685od_a_b @ ( comp_b_b_b @ F2 @ M ) @ G @ X3 )
        = ( comp_P2009515992434452078od_a_b @ ( comp_b3886954628874447685od_a_b @ F2 @ N ) @ H @ X3 ) ) ) ).

% type_copy_map_cong0
thf(fact_839_function__factors__right,axiom,
    ! [G: b > b,F2: product_prod_a_b > b] :
      ( ( ! [X: product_prod_a_b] :
          ? [Y: b] :
            ( ( G @ Y )
            = ( F2 @ X ) ) )
      = ( ? [H3: product_prod_a_b > b] :
            ( F2
            = ( comp_b3886954628874447685od_a_b @ G @ H3 ) ) ) ) ).

% function_factors_right
thf(fact_840_function__factors__right,axiom,
    ! [G: a > product_prod_a_a,F2: product_prod_a_b > product_prod_a_a] :
      ( ( ! [X: product_prod_a_b] :
          ? [Y: a] :
            ( ( G @ Y )
            = ( F2 @ X ) ) )
      = ( ? [H3: product_prod_a_b > a] :
            ( F2
            = ( comp_a1036870397537576092od_a_b @ G @ H3 ) ) ) ) ).

% function_factors_right
thf(fact_841_function__factors__right,axiom,
    ! [G: a > b,F2: product_prod_a_b > b] :
      ( ( ! [X: product_prod_a_b] :
          ? [Y: a] :
            ( ( G @ Y )
            = ( F2 @ X ) ) )
      = ( ? [H3: product_prod_a_b > a] :
            ( F2
            = ( comp_a9170378079104387268od_a_b @ G @ H3 ) ) ) ) ).

% function_factors_right
thf(fact_842_function__factors__right,axiom,
    ! [G: a > a,F2: product_prod_a_b > a] :
      ( ( ! [X: product_prod_a_b] :
          ? [Y: a] :
            ( ( G @ Y )
            = ( F2 @ X ) ) )
      = ( ? [H3: product_prod_a_b > a] :
            ( F2
            = ( comp_a586446342581427589od_a_b @ G @ H3 ) ) ) ) ).

% function_factors_right
thf(fact_843_function__factors__right,axiom,
    ! [G: a > a,F2: a > a] :
      ( ( ! [X: a] :
          ? [Y: a] :
            ( ( G @ Y )
            = ( F2 @ X ) ) )
      = ( ? [H3: a > a] :
            ( F2
            = ( comp_a_a_a @ G @ H3 ) ) ) ) ).

% function_factors_right
thf(fact_844_function__factors__left,axiom,
    ! [G: product_prod_a_b > b,F2: product_prod_a_b > b] :
      ( ( ! [X: product_prod_a_b,Y: product_prod_a_b] :
            ( ( ( G @ X )
              = ( G @ Y ) )
           => ( ( F2 @ X )
              = ( F2 @ Y ) ) ) )
      = ( ? [H3: b > b] :
            ( F2
            = ( comp_b3886954628874447685od_a_b @ H3 @ G ) ) ) ) ).

% function_factors_left
thf(fact_845_function__factors__left,axiom,
    ! [G: product_prod_a_b > a,F2: product_prod_a_b > product_prod_a_a] :
      ( ( ! [X: product_prod_a_b,Y: product_prod_a_b] :
            ( ( ( G @ X )
              = ( G @ Y ) )
           => ( ( F2 @ X )
              = ( F2 @ Y ) ) ) )
      = ( ? [H3: a > product_prod_a_a] :
            ( F2
            = ( comp_a1036870397537576092od_a_b @ H3 @ G ) ) ) ) ).

% function_factors_left
thf(fact_846_function__factors__left,axiom,
    ! [G: product_prod_a_b > a,F2: product_prod_a_b > b] :
      ( ( ! [X: product_prod_a_b,Y: product_prod_a_b] :
            ( ( ( G @ X )
              = ( G @ Y ) )
           => ( ( F2 @ X )
              = ( F2 @ Y ) ) ) )
      = ( ? [H3: a > b] :
            ( F2
            = ( comp_a9170378079104387268od_a_b @ H3 @ G ) ) ) ) ).

% function_factors_left
thf(fact_847_function__factors__left,axiom,
    ! [G: product_prod_a_b > a,F2: product_prod_a_b > a] :
      ( ( ! [X: product_prod_a_b,Y: product_prod_a_b] :
            ( ( ( G @ X )
              = ( G @ Y ) )
           => ( ( F2 @ X )
              = ( F2 @ Y ) ) ) )
      = ( ? [H3: a > a] :
            ( F2
            = ( comp_a586446342581427589od_a_b @ H3 @ G ) ) ) ) ).

% function_factors_left
thf(fact_848_function__factors__left,axiom,
    ! [G: a > a,F2: a > a] :
      ( ( ! [X: a,Y: a] :
            ( ( ( G @ X )
              = ( G @ Y ) )
           => ( ( F2 @ X )
              = ( F2 @ Y ) ) ) )
      = ( ? [H3: a > a] :
            ( F2
            = ( comp_a_a_a @ H3 @ G ) ) ) ) ).

% function_factors_left
thf(fact_849_case__prod__curry,axiom,
    ! [F2: product_prod_b_a > product_prod_a_b] :
      ( ( produc4348216232050026237od_a_b @ ( produc8334494339191655838od_a_b @ F2 ) )
      = F2 ) ).

% case_prod_curry
thf(fact_850_case__prod__curry,axiom,
    ! [F2: product_prod_a_b > product_prod_b_a] :
      ( ( produc6204743795672244857od_b_a @ ( produc967649865959098650od_b_a @ F2 ) )
      = F2 ) ).

% case_prod_curry
thf(fact_851_case__prod__curry,axiom,
    ! [F2: product_prod_a_b > $o] :
      ( ( produc3537405659489547051_a_b_o @ ( product_curry_a_b_o @ F2 ) )
      = F2 ) ).

% case_prod_curry
thf(fact_852_case__prod__curry,axiom,
    ! [F2: product_prod_a_b > b] :
      ( ( produc6028431345588019474_a_b_b @ ( product_curry_a_b_b @ F2 ) )
      = F2 ) ).

% case_prod_curry
thf(fact_853_case__prod__curry,axiom,
    ! [F2: product_prod_a_b > a] :
      ( ( produc6028431345588019473_a_b_a @ ( product_curry_a_b_a @ F2 ) )
      = F2 ) ).

% case_prod_curry
thf(fact_854_curry__case__prod,axiom,
    ! [F2: b > a > product_prod_a_b] :
      ( ( produc8334494339191655838od_a_b @ ( produc4348216232050026237od_a_b @ F2 ) )
      = F2 ) ).

% curry_case_prod
thf(fact_855_curry__case__prod,axiom,
    ! [F2: a > b > product_prod_b_a] :
      ( ( produc967649865959098650od_b_a @ ( produc6204743795672244857od_b_a @ F2 ) )
      = F2 ) ).

% curry_case_prod
thf(fact_856_curry__case__prod,axiom,
    ! [F2: a > b > $o] :
      ( ( product_curry_a_b_o @ ( produc3537405659489547051_a_b_o @ F2 ) )
      = F2 ) ).

% curry_case_prod
thf(fact_857_curry__case__prod,axiom,
    ! [F2: a > b > b] :
      ( ( product_curry_a_b_b @ ( produc6028431345588019474_a_b_b @ F2 ) )
      = F2 ) ).

% curry_case_prod
thf(fact_858_curry__case__prod,axiom,
    ! [F2: a > b > a] :
      ( ( product_curry_a_b_a @ ( produc6028431345588019473_a_b_a @ F2 ) )
      = F2 ) ).

% curry_case_prod
thf(fact_859_curryI,axiom,
    ! [F2: product_prod_b_b > $o,A: b,B: b] :
      ( ( F2 @ ( product_Pair_b_b @ A @ B ) )
     => ( product_curry_b_b_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_860_curryI,axiom,
    ! [F2: product_prod_b_a > $o,A: b,B: a] :
      ( ( F2 @ ( product_Pair_b_a @ A @ B ) )
     => ( product_curry_b_a_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_861_curryI,axiom,
    ! [F2: produc5884233991663340231list_o > $o,A: a,B: list_o] :
      ( ( F2 @ ( produc6899623729582506167list_o @ A @ B ) )
     => ( produc1366198832687666157st_o_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_862_curryI,axiom,
    ! [F2: product_prod_a_b > $o,A: a,B: b] :
      ( ( F2 @ ( product_Pair_a_b @ A @ B ) )
     => ( product_curry_a_b_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_863_curryI,axiom,
    ! [F2: product_prod_a_a > $o,A: a,B: a] :
      ( ( F2 @ ( product_Pair_a_a @ A @ B ) )
     => ( product_curry_a_a_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_864_curryE,axiom,
    ! [F2: product_prod_b_b > $o,A: b,B: b] :
      ( ( product_curry_b_b_o @ F2 @ A @ B )
     => ( F2 @ ( product_Pair_b_b @ A @ B ) ) ) ).

% curryE
thf(fact_865_curryE,axiom,
    ! [F2: product_prod_b_a > $o,A: b,B: a] :
      ( ( product_curry_b_a_o @ F2 @ A @ B )
     => ( F2 @ ( product_Pair_b_a @ A @ B ) ) ) ).

% curryE
thf(fact_866_curryE,axiom,
    ! [F2: produc5884233991663340231list_o > $o,A: a,B: list_o] :
      ( ( produc1366198832687666157st_o_o @ F2 @ A @ B )
     => ( F2 @ ( produc6899623729582506167list_o @ A @ B ) ) ) ).

% curryE
thf(fact_867_curryE,axiom,
    ! [F2: product_prod_a_b > $o,A: a,B: b] :
      ( ( product_curry_a_b_o @ F2 @ A @ B )
     => ( F2 @ ( product_Pair_a_b @ A @ B ) ) ) ).

% curryE
thf(fact_868_curryE,axiom,
    ! [F2: product_prod_a_a > $o,A: a,B: a] :
      ( ( product_curry_a_a_o @ F2 @ A @ B )
     => ( F2 @ ( product_Pair_a_a @ A @ B ) ) ) ).

% curryE
thf(fact_869_curryD,axiom,
    ! [F2: product_prod_b_b > $o,A: b,B: b] :
      ( ( product_curry_b_b_o @ F2 @ A @ B )
     => ( F2 @ ( product_Pair_b_b @ A @ B ) ) ) ).

% curryD
thf(fact_870_curryD,axiom,
    ! [F2: product_prod_b_a > $o,A: b,B: a] :
      ( ( product_curry_b_a_o @ F2 @ A @ B )
     => ( F2 @ ( product_Pair_b_a @ A @ B ) ) ) ).

% curryD
thf(fact_871_curryD,axiom,
    ! [F2: produc5884233991663340231list_o > $o,A: a,B: list_o] :
      ( ( produc1366198832687666157st_o_o @ F2 @ A @ B )
     => ( F2 @ ( produc6899623729582506167list_o @ A @ B ) ) ) ).

% curryD
thf(fact_872_curryD,axiom,
    ! [F2: product_prod_a_b > $o,A: a,B: b] :
      ( ( product_curry_a_b_o @ F2 @ A @ B )
     => ( F2 @ ( product_Pair_a_b @ A @ B ) ) ) ).

% curryD
thf(fact_873_curryD,axiom,
    ! [F2: product_prod_a_a > $o,A: a,B: a] :
      ( ( product_curry_a_a_o @ F2 @ A @ B )
     => ( F2 @ ( product_Pair_a_a @ A @ B ) ) ) ).

% curryD
thf(fact_874_ssubst__Pair__rhs,axiom,
    ! [R: b,S2: b,R5: set_Product_prod_b_b,S4: b] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ R @ S2 ) @ R5 )
     => ( ( S4 = S2 )
       => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ R @ S4 ) @ R5 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_875_ssubst__Pair__rhs,axiom,
    ! [R: b,S2: a,R5: set_Product_prod_b_a,S4: a] :
      ( ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ R @ S2 ) @ R5 )
     => ( ( S4 = S2 )
       => ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ R @ S4 ) @ R5 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_876_ssubst__Pair__rhs,axiom,
    ! [R: a,S2: list_o,R5: set_Pr3077528766752018087list_o,S4: list_o] :
      ( ( member7948383622993546480list_o @ ( produc6899623729582506167list_o @ R @ S2 ) @ R5 )
     => ( ( S4 = S2 )
       => ( member7948383622993546480list_o @ ( produc6899623729582506167list_o @ R @ S4 ) @ R5 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_877_ssubst__Pair__rhs,axiom,
    ! [R: a,S2: b,R5: set_Product_prod_a_b,S4: b] :
      ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ R @ S2 ) @ R5 )
     => ( ( S4 = S2 )
       => ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ R @ S4 ) @ R5 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_878_ssubst__Pair__rhs,axiom,
    ! [R: a,S2: a,R5: set_Product_prod_a_a,S4: a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ R @ S2 ) @ R5 )
     => ( ( S4 = S2 )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ R @ S4 ) @ R5 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_879_Grp__fst__snd,axiom,
    ! [R5: product_prod_a_b > product_prod_a_b > $o] :
      ( ( relcom1980427227864585976od_a_b @ ( conver607836479408588470od_a_b @ ( bNF_Gr248057882177514895od_a_b @ ( collec7968472090841016338od_a_b @ ( produc5413562937804622316_a_b_o @ R5 ) ) @ produc5013601669340818563od_a_b ) ) @ ( bNF_Gr248057882177514895od_a_b @ ( collec7968472090841016338od_a_b @ ( produc5413562937804622316_a_b_o @ R5 ) ) @ produc8875788424405714885od_a_b ) )
      = R5 ) ).

% Grp_fst_snd
thf(fact_880_Grp__fst__snd,axiom,
    ! [R5: nat > nat > $o] :
      ( ( relcom8572034174491624592at_nat @ ( conver4691796276509297820at_nat @ ( bNF_Gr193424670081168899at_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R5 ) ) @ product_fst_nat_nat ) ) @ ( bNF_Gr193424670081168899at_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R5 ) ) @ product_snd_nat_nat ) )
      = R5 ) ).

% Grp_fst_snd
thf(fact_881_Grp__fst__snd,axiom,
    ! [R5: a > list_o > $o] :
      ( ( relcom3008056482987546148list_o @ ( conver1512411698036489708st_o_a @ ( bNF_Gr7486154629356127173st_o_a @ ( collec7805115074396547890list_o @ ( produc8342644129638033356st_o_o @ R5 ) ) @ product_fst_a_list_o ) ) @ ( bNF_Gr1706342031205223781list_o @ ( collec7805115074396547890list_o @ ( produc8342644129638033356st_o_o @ R5 ) ) @ product_snd_a_list_o ) )
      = R5 ) ).

% Grp_fst_snd
thf(fact_882_Grp__fst__snd,axiom,
    ! [R5: b > a > $o] :
      ( ( relcom905042485061280806_b_a_a @ ( conver6154593848572614414_b_a_b @ ( bNF_Gr4850632326866288871_b_a_b @ ( collec548942215411776465od_b_a @ ( produc7672072387910335853_b_a_o @ R5 ) ) @ product_fst_b_a ) ) @ ( bNF_Gr4850632326866288870_b_a_a @ ( collec548942215411776465od_b_a @ ( produc7672072387910335853_b_a_o @ R5 ) ) @ product_snd_b_a ) )
      = R5 ) ).

% Grp_fst_snd
thf(fact_883_Grp__fst__snd,axiom,
    ! [R5: b > b > $o] :
      ( ( relcom269393484456294118_b_b_b @ ( conver5518944847967627725_b_b_b @ ( bNF_Gr4214983326261302182_b_b_b @ ( collec548942219715005266od_b_b @ ( produc4772849988445941548_b_b_o @ R5 ) ) @ product_fst_b_b ) ) @ ( bNF_Gr4214983326261302182_b_b_b @ ( collec548942219715005266od_b_b @ ( produc4772849988445941548_b_b_o @ R5 ) ) @ product_snd_b_b ) )
      = R5 ) ).

% Grp_fst_snd
thf(fact_884_Grp__fst__snd,axiom,
    ! [R5: a > a > $o] :
      ( ( relcom592247320103089700_a_a_a @ ( conver558375233384483724_a_a_a @ ( bNF_Gr8477785748532933989_a_a_a @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ R5 ) ) @ product_fst_a_a ) ) @ ( bNF_Gr8477785748532933989_a_a_a @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ R5 ) ) @ product_snd_a_a ) )
      = R5 ) ).

% Grp_fst_snd
thf(fact_885_Grp__fst__snd,axiom,
    ! [R5: a > b > $o] :
      ( ( relcom9179970356352878820_a_b_b @ ( conver9146098269634272843_a_b_a @ ( bNF_Gr7842136747927947300_a_b_a @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ R5 ) ) @ product_fst_a_b ) ) @ ( bNF_Gr7842136747927947301_a_b_b @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ R5 ) ) @ product_snd_a_b ) )
      = R5 ) ).

% Grp_fst_snd
thf(fact_886_pair__in__swap__image,axiom,
    ! [Y4: list_o,X3: a,A4: set_Pr3077528766752018087list_o] :
      ( ( member8444411472706555312st_o_a @ ( produc6614149361254850935st_o_a @ Y4 @ X3 ) @ ( image_2556850475766464645st_o_a @ produc370354279364378199list_o @ A4 ) )
      = ( member7948383622993546480list_o @ ( produc6899623729582506167list_o @ X3 @ Y4 ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_887_pair__in__swap__image,axiom,
    ! [Y4: b,X3: b,A4: set_Product_prod_b_b] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y4 @ X3 ) @ ( image_4453096554207573957od_b_b @ product_swap_b_b @ A4 ) )
      = ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X3 @ Y4 ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_888_pair__in__swap__image,axiom,
    ! [Y4: a,X3: list_o,A4: set_Pr5014569662245023591st_o_a] :
      ( ( member7948383622993546480list_o @ ( produc6899623729582506167list_o @ Y4 @ X3 ) @ ( image_7451805610179047173list_o @ produc84879911036722967st_o_a @ A4 ) )
      = ( member8444411472706555312st_o_a @ ( produc6614149361254850935st_o_a @ X3 @ Y4 ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_889_pair__in__swap__image,axiom,
    ! [Y4: a,X3: a,A4: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X3 ) @ ( image_4636654165204879301od_a_a @ product_swap_a_a @ A4 ) )
      = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_890_pair__in__swap__image,axiom,
    ! [Y4: a,X3: b,A4: set_Product_prod_b_a] :
      ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ Y4 @ X3 ) @ ( image_8576602756132944327od_a_b @ product_swap_b_a @ A4 ) )
      = ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ X3 @ Y4 ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_891_pair__in__swap__image,axiom,
    ! [Y4: b,X3: a,A4: set_Product_prod_a_b] :
      ( ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ Y4 @ X3 ) @ ( image_513147963279508931od_b_a @ product_swap_a_b @ A4 ) )
      = ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X3 @ Y4 ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_892_in__inv__image,axiom,
    ! [X3: b,Y4: b,R: set_Product_prod_b_b,F2: b > b] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X3 @ Y4 ) @ ( inv_image_b_b @ R @ F2 ) )
      = ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) @ R ) ) ).

% in_inv_image
thf(fact_893_in__inv__image,axiom,
    ! [X3: b,Y4: b,R: set_Product_prod_a_a,F2: b > a] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X3 @ Y4 ) @ ( inv_image_a_b @ R @ F2 ) )
      = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) @ R ) ) ).

% in_inv_image
thf(fact_894_in__inv__image,axiom,
    ! [X3: a,Y4: a,R: set_Product_prod_b_b,F2: a > b] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ ( inv_image_b_a @ R @ F2 ) )
      = ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) @ R ) ) ).

% in_inv_image
thf(fact_895_in__inv__image,axiom,
    ! [X3: a,Y4: a,R: set_Product_prod_a_a,F2: a > a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ ( inv_image_a_a @ R @ F2 ) )
      = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) @ R ) ) ).

% in_inv_image
thf(fact_896_image__id,axiom,
    ( ( image_a_a2 @ id_a )
    = id_set_a ) ).

% image_id
thf(fact_897_map__prod__imageI,axiom,
    ! [A: b,B: b,R5: set_Product_prod_b_b,F2: b > b,G: b > b] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ R5 )
     => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_4453096554207573957od_b_b @ ( produc8315423520075080822_b_b_b @ F2 @ G ) @ R5 ) ) ) ).

% map_prod_imageI
thf(fact_898_map__prod__imageI,axiom,
    ! [A: b,B: b,R5: set_Product_prod_b_b,F2: b > b,G: b > a] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ R5 )
     => ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_4453096549904345156od_b_a @ ( produc8315423520075080821_b_b_a @ F2 @ G ) @ R5 ) ) ) ).

% map_prod_imageI
thf(fact_899_map__prod__imageI,axiom,
    ! [A: b,B: b,R5: set_Product_prod_b_b,F2: b > a,G: b > b] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ R5 )
     => ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_7240552136180249990od_a_b @ ( produc2719204904886950133_a_b_b @ F2 @ G ) @ R5 ) ) ) ).

% map_prod_imageI
thf(fact_900_map__prod__imageI,axiom,
    ! [A: b,B: b,R5: set_Product_prod_b_b,F2: b > a,G: b > a] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ R5 )
     => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_7240552131877021189od_a_a @ ( produc2719204904886950132_a_b_a @ F2 @ G ) @ R5 ) ) ) ).

% map_prod_imageI
thf(fact_901_map__prod__imageI,axiom,
    ! [A: b,B: a,R5: set_Product_prod_b_a,F2: b > b,G: a > b] :
      ( ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ A @ B ) @ R5 )
     => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_5789147174160268294od_b_b @ ( produc1879507065192981047_b_a_b @ F2 @ G ) @ R5 ) ) ) ).

% map_prod_imageI
thf(fact_902_map__prod__imageI,axiom,
    ! [A: b,B: a,R5: set_Product_prod_b_a,F2: b > b,G: a > a] :
      ( ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ A @ B ) @ R5 )
     => ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_5789147169857039493od_b_a @ ( produc1879507065192981046_b_a_a @ F2 @ G ) @ R5 ) ) ) ).

% map_prod_imageI
thf(fact_903_map__prod__imageI,axiom,
    ! [A: b,B: a,R5: set_Product_prod_b_a,F2: b > a,G: a > b] :
      ( ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ A @ B ) @ R5 )
     => ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_8576602756132944327od_a_b @ ( produc5506660486859626166_a_a_b @ F2 @ G ) @ R5 ) ) ) ).

% map_prod_imageI
thf(fact_904_map__prod__imageI,axiom,
    ! [A: b,B: a,R5: set_Product_prod_b_a,F2: b > a,G: a > a] :
      ( ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ A @ B ) @ R5 )
     => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_8576602751829715526od_a_a @ ( produc5506660486859626165_a_a_a @ F2 @ G ) @ R5 ) ) ) ).

% map_prod_imageI
thf(fact_905_map__prod__imageI,axiom,
    ! [A: a,B: b,R5: set_Product_prod_a_b,F2: a > b,G: b > b] :
      ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ A @ B ) @ R5 )
     => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_513147967582737732od_b_b @ ( produc6827461482780282295_b_b_b @ F2 @ G ) @ R5 ) ) ) ).

% map_prod_imageI
thf(fact_906_map__prod__imageI,axiom,
    ! [A: a,B: b,R5: set_Product_prod_a_b,F2: a > b,G: b > a] :
      ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ A @ B ) @ R5 )
     => ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_513147963279508931od_b_a @ ( produc6827461482780282294_b_b_a @ F2 @ G ) @ R5 ) ) ) ).

% map_prod_imageI
thf(fact_907_converse__relcompp,axiom,
    ! [R: a > product_prod_a_b > $o,S2: product_prod_a_b > b > $o] :
      ( ( conversep_a_b @ ( relcom9179970356352878820_a_b_b @ R @ S2 ) )
      = ( relcom3896546906122939236_a_b_a @ ( conver9146098269634272844_a_b_b @ S2 ) @ ( conver4521055767378250271od_a_b @ R ) ) ) ).

% converse_relcompp
thf(fact_908_converse__relcompp,axiom,
    ! [R: product_prod_a_a > produc2233330766403592615od_a_b > $o,S2: produc2233330766403592615od_a_b > product_prod_b_b > $o] :
      ( ( conver1659967761066010997od_b_b @ ( relcom8632546625042571830od_b_b @ R @ S2 ) )
      = ( relcom7263568941997752886od_a_a @ ( conver7043752934290688245od_b_b @ S2 ) @ ( conver6099322915739602997od_a_b @ R ) ) ) ).

% converse_relcompp
thf(fact_909_converse__relcompp,axiom,
    ! [R: b > product_prod_a_b > $o,S2: product_prod_a_b > a > $o] :
      ( ( conversep_b_a @ ( relcom3896546906122939236_a_b_a @ R @ S2 ) )
      = ( relcom9179970356352878820_a_b_b @ ( conver9146098269634272843_a_b_a @ S2 ) @ ( conver3881615467046434142od_a_b @ R ) ) ) ).

% converse_relcompp
thf(fact_910_converse__relcompp,axiom,
    ! [R: product_prod_b_b > produc2233330766403592615od_a_b > $o,S2: produc2233330766403592615od_a_b > product_prod_a_a > $o] :
      ( ( conver7051321305407600117od_a_a @ ( relcom7263568941997752886od_a_a @ R @ S2 ) )
      = ( relcom8632546625042571830od_b_b @ ( conver607836475105359669od_a_a @ S2 ) @ ( conver1467941833701682805od_a_b @ R ) ) ) ).

% converse_relcompp
thf(fact_911_converse__relcompp,axiom,
    ! [R: product_prod_a_b > a > $o,S2: a > a > $o] :
      ( ( conver9146098269634272843_a_b_a @ ( relcom7167498584244361143_b_a_a @ R @ S2 ) )
      = ( relcom4554927854096856247od_a_b @ ( conversep_a_a @ S2 ) @ ( conver9146098269634272843_a_b_a @ R ) ) ) ).

% converse_relcompp
thf(fact_912_converse__relcompp,axiom,
    ! [R: product_prod_a_b > product_prod_a_b > $o,S2: product_prod_a_b > a > $o] :
      ( ( conver9146098269634272843_a_b_a @ ( relcom1914922001497678989_a_b_a @ R @ S2 ) )
      = ( relcom3565115020915045773od_a_b @ ( conver9146098269634272843_a_b_a @ S2 ) @ ( conver3111372723085992693od_a_b @ R ) ) ) ).

% converse_relcompp
thf(fact_913_converse__relcompp,axiom,
    ! [R: produc2233330766403592615od_a_b > product_prod_a_a > $o,S2: product_prod_a_a > product_prod_a_a > $o] :
      ( ( conver607836475105359669od_a_a @ ( relcom54454498475354486od_a_a @ R @ S2 ) )
      = ( relcom7688116606491486582od_a_b @ ( conver4447423338735458229od_a_a @ S2 ) @ ( conver607836475105359669od_a_a @ R ) ) ) ).

% converse_relcompp
thf(fact_914_converse__relcompp,axiom,
    ! [R: produc2233330766403592615od_a_b > produc2233330766403592615od_a_b > $o,S2: produc2233330766403592615od_a_b > product_prod_a_a > $o] :
      ( ( conver607836475105359669od_a_a @ ( relcom2145156996241176310od_a_a @ R @ S2 ) )
      = ( relcom6321393211869853430od_a_b @ ( conver607836475105359669od_a_a @ S2 ) @ ( conver8842143164975968181od_a_b @ R ) ) ) ).

% converse_relcompp
thf(fact_915_image__eq__imp__comp,axiom,
    ! [F2: a > a,A4: set_a,G: a > a,B5: set_a,H: a > a] :
      ( ( ( image_a_a2 @ F2 @ A4 )
        = ( image_a_a2 @ G @ B5 ) )
     => ( ( image_a_a2 @ ( comp_a_a_a @ H @ F2 ) @ A4 )
        = ( image_a_a2 @ ( comp_a_a_a @ H @ G ) @ B5 ) ) ) ).

% image_eq_imp_comp
thf(fact_916_image__eq__imp__comp,axiom,
    ! [F2: a > set_b,A4: set_a,G: a > set_b,B5: set_a,H: set_b > a] :
      ( ( ( image_a_set_b @ F2 @ A4 )
        = ( image_a_set_b @ G @ B5 ) )
     => ( ( image_a_a2 @ ( comp_set_b_a_a @ H @ F2 ) @ A4 )
        = ( image_a_a2 @ ( comp_set_b_a_a @ H @ G ) @ B5 ) ) ) ).

% image_eq_imp_comp
thf(fact_917_image__eq__imp__comp,axiom,
    ! [F2: a > a,A4: set_a,G: a > a,B5: set_a,H: a > set_b] :
      ( ( ( image_a_a2 @ F2 @ A4 )
        = ( image_a_a2 @ G @ B5 ) )
     => ( ( image_a_set_b @ ( comp_a_set_b_a @ H @ F2 ) @ A4 )
        = ( image_a_set_b @ ( comp_a_set_b_a @ H @ G ) @ B5 ) ) ) ).

% image_eq_imp_comp
thf(fact_918_image__eq__imp__comp,axiom,
    ! [F2: a > b,A4: set_a,G: product_prod_a_b > b,B5: set_Product_prod_a_b,H: b > a] :
      ( ( ( image_a_b @ F2 @ A4 )
        = ( image_2802296252294471260_a_b_b @ G @ B5 ) )
     => ( ( image_a_a2 @ ( comp_b_a_a @ H @ F2 ) @ A4 )
        = ( image_2802296252294471259_a_b_a @ ( comp_b4526394929206263814od_a_b @ H @ G ) @ B5 ) ) ) ).

% image_eq_imp_comp
thf(fact_919_image__eq__imp__comp,axiom,
    ! [F2: product_prod_a_b > b,A4: set_Product_prod_a_b,G: a > b,B5: set_a,H: b > a] :
      ( ( ( image_2802296252294471260_a_b_b @ F2 @ A4 )
        = ( image_a_b @ G @ B5 ) )
     => ( ( image_2802296252294471259_a_b_a @ ( comp_b4526394929206263814od_a_b @ H @ F2 ) @ A4 )
        = ( image_a_a2 @ ( comp_b_a_a @ H @ G ) @ B5 ) ) ) ).

% image_eq_imp_comp
thf(fact_920_image__eq__imp__comp,axiom,
    ! [F2: product_prod_a_b > a,A4: set_Product_prod_a_b,G: a > a,B5: set_a,H: a > b] :
      ( ( ( image_2802296252294471259_a_b_a @ F2 @ A4 )
        = ( image_a_a2 @ G @ B5 ) )
     => ( ( image_2802296252294471260_a_b_b @ ( comp_a9170378079104387268od_a_b @ H @ F2 ) @ A4 )
        = ( image_a_b @ ( comp_a_b_a @ H @ G ) @ B5 ) ) ) ).

% image_eq_imp_comp
thf(fact_921_image__eq__imp__comp,axiom,
    ! [F2: product_prod_a_b > a,A4: set_Product_prod_a_b,G: a > a,B5: set_a,H: a > a] :
      ( ( ( image_2802296252294471259_a_b_a @ F2 @ A4 )
        = ( image_a_a2 @ G @ B5 ) )
     => ( ( image_2802296252294471259_a_b_a @ ( comp_a586446342581427589od_a_b @ H @ F2 ) @ A4 )
        = ( image_a_a2 @ ( comp_a_a_a @ H @ G ) @ B5 ) ) ) ).

% image_eq_imp_comp
thf(fact_922_image__eq__imp__comp,axiom,
    ! [F2: a > set_b,A4: set_a,G: a > set_b,B5: set_a,H: set_b > set_b] :
      ( ( ( image_a_set_b @ F2 @ A4 )
        = ( image_a_set_b @ G @ B5 ) )
     => ( ( image_a_set_b @ ( comp_set_b_set_b_a @ H @ F2 ) @ A4 )
        = ( image_a_set_b @ ( comp_set_b_set_b_a @ H @ G ) @ B5 ) ) ) ).

% image_eq_imp_comp
thf(fact_923_image__eq__imp__comp,axiom,
    ! [F2: a > a,A4: set_a,G: product_prod_a_b > a,B5: set_Product_prod_a_b,H: a > b] :
      ( ( ( image_a_a2 @ F2 @ A4 )
        = ( image_2802296252294471259_a_b_a @ G @ B5 ) )
     => ( ( image_a_b @ ( comp_a_b_a @ H @ F2 ) @ A4 )
        = ( image_2802296252294471260_a_b_b @ ( comp_a9170378079104387268od_a_b @ H @ G ) @ B5 ) ) ) ).

% image_eq_imp_comp
thf(fact_924_image__eq__imp__comp,axiom,
    ! [F2: a > a,A4: set_a,G: product_prod_a_b > a,B5: set_Product_prod_a_b,H: a > a] :
      ( ( ( image_a_a2 @ F2 @ A4 )
        = ( image_2802296252294471259_a_b_a @ G @ B5 ) )
     => ( ( image_a_a2 @ ( comp_a_a_a @ H @ F2 ) @ A4 )
        = ( image_2802296252294471259_a_b_a @ ( comp_a586446342581427589od_a_b @ H @ G ) @ B5 ) ) ) ).

% image_eq_imp_comp
thf(fact_925_image__comp,axiom,
    ! [F2: a > a,G: a > a,R: set_a] :
      ( ( image_a_a2 @ F2 @ ( image_a_a2 @ G @ R ) )
      = ( image_a_a2 @ ( comp_a_a_a @ F2 @ G ) @ R ) ) ).

% image_comp
thf(fact_926_image__comp,axiom,
    ! [F2: set_b > a,G: a > set_b,R: set_a] :
      ( ( image_set_b_a @ F2 @ ( image_a_set_b @ G @ R ) )
      = ( image_a_a2 @ ( comp_set_b_a_a @ F2 @ G ) @ R ) ) ).

% image_comp
thf(fact_927_image__comp,axiom,
    ! [F2: a > set_b,G: a > a,R: set_a] :
      ( ( image_a_set_b @ F2 @ ( image_a_a2 @ G @ R ) )
      = ( image_a_set_b @ ( comp_a_set_b_a @ F2 @ G ) @ R ) ) ).

% image_comp
thf(fact_928_image__comp,axiom,
    ! [F2: b > b,G: product_prod_a_b > b,R: set_Product_prod_a_b] :
      ( ( image_b_b @ F2 @ ( image_2802296252294471260_a_b_b @ G @ R ) )
      = ( image_2802296252294471260_a_b_b @ ( comp_b3886954628874447685od_a_b @ F2 @ G ) @ R ) ) ).

% image_comp
thf(fact_929_image__comp,axiom,
    ! [F2: b > a,G: product_prod_a_b > b,R: set_Product_prod_a_b] :
      ( ( image_b_a @ F2 @ ( image_2802296252294471260_a_b_b @ G @ R ) )
      = ( image_2802296252294471259_a_b_a @ ( comp_b4526394929206263814od_a_b @ F2 @ G ) @ R ) ) ).

% image_comp
thf(fact_930_image__comp,axiom,
    ! [F2: a > b,G: product_prod_a_b > a,R: set_Product_prod_a_b] :
      ( ( image_a_b @ F2 @ ( image_2802296252294471259_a_b_a @ G @ R ) )
      = ( image_2802296252294471260_a_b_b @ ( comp_a9170378079104387268od_a_b @ F2 @ G ) @ R ) ) ).

% image_comp
thf(fact_931_image__comp,axiom,
    ! [F2: set_b > set_b,G: a > set_b,R: set_a] :
      ( ( image_set_b_set_b @ F2 @ ( image_a_set_b @ G @ R ) )
      = ( image_a_set_b @ ( comp_set_b_set_b_a @ F2 @ G ) @ R ) ) ).

% image_comp
thf(fact_932_image__comp,axiom,
    ! [F2: product_prod_a_b > a,G: a > product_prod_a_b,R: set_a] :
      ( ( image_2802296252294471259_a_b_a @ F2 @ ( image_7400625786893224495od_a_b @ G @ R ) )
      = ( image_a_a2 @ ( comp_P3199017072728932485_b_a_a @ F2 @ G ) @ R ) ) ).

% image_comp
thf(fact_933_image__comp,axiom,
    ! [F2: a > a,G: product_prod_a_b > a,R: set_Product_prod_a_b] :
      ( ( image_a_a2 @ F2 @ ( image_2802296252294471259_a_b_a @ G @ R ) )
      = ( image_2802296252294471259_a_b_a @ ( comp_a586446342581427589od_a_b @ F2 @ G ) @ R ) ) ).

% image_comp
thf(fact_934_image__comp,axiom,
    ! [F2: produc5884233991663340231list_o > a,G: a > produc5884233991663340231list_o,R: set_a] :
      ( ( image_8953000392185216508st_o_a @ F2 @ ( image_1329905666413968462list_o @ G @ R ) )
      = ( image_a_a2 @ ( comp_P941169184177061668_o_a_a @ F2 @ G ) @ R ) ) ).

% image_comp
thf(fact_935_inj__on__image__iff,axiom,
    ! [A4: set_b,G: b > option_list_o,F2: b > b] :
      ( ! [X2: b] :
          ( ( member_b @ X2 @ A4 )
         => ! [Xa: b] :
              ( ( member_b @ Xa @ A4 )
             => ( ( ( G @ ( F2 @ X2 ) )
                  = ( G @ ( F2 @ Xa ) ) )
                = ( ( G @ X2 )
                  = ( G @ Xa ) ) ) ) )
     => ( ( inj_on_b_b @ F2 @ A4 )
       => ( ( inj_on8261448415883032086list_o @ G @ ( image_b_b @ F2 @ A4 ) )
          = ( inj_on8261448415883032086list_o @ G @ A4 ) ) ) ) ).

% inj_on_image_iff
thf(fact_936_inj__on__image__iff,axiom,
    ! [A4: set_Product_prod_a_b,G: product_prod_a_b > a,F2: product_prod_a_b > product_prod_a_b] :
      ( ! [X2: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X2 @ A4 )
         => ! [Xa: product_prod_a_b] :
              ( ( member1426531481828664017od_a_b @ Xa @ A4 )
             => ( ( ( G @ ( F2 @ X2 ) )
                  = ( G @ ( F2 @ Xa ) ) )
                = ( ( G @ X2 )
                  = ( G @ Xa ) ) ) ) )
     => ( ( inj_on1230094055151127153od_a_b @ F2 @ A4 )
       => ( ( inj_on4343330552946057671_a_b_a @ G @ ( image_3300603549555413765od_a_b @ F2 @ A4 ) )
          = ( inj_on4343330552946057671_a_b_a @ G @ A4 ) ) ) ) ).

% inj_on_image_iff
thf(fact_937_inj__on__image__iff,axiom,
    ! [A4: set_a,G: a > option_list_o,F2: a > a] :
      ( ! [X2: a] :
          ( ( member_a @ X2 @ A4 )
         => ! [Xa: a] :
              ( ( member_a @ Xa @ A4 )
             => ( ( ( G @ ( F2 @ X2 ) )
                  = ( G @ ( F2 @ Xa ) ) )
                = ( ( G @ X2 )
                  = ( G @ Xa ) ) ) ) )
     => ( ( inj_on_a_a @ F2 @ A4 )
       => ( ( inj_on374126998980950615list_o @ G @ ( image_a_a2 @ F2 @ A4 ) )
          = ( inj_on374126998980950615list_o @ G @ A4 ) ) ) ) ).

% inj_on_image_iff
thf(fact_938_inj__on__image__iff,axiom,
    ! [A4: set_a,G: a > a,F2: a > a] :
      ( ! [X2: a] :
          ( ( member_a @ X2 @ A4 )
         => ! [Xa: a] :
              ( ( member_a @ Xa @ A4 )
             => ( ( ( G @ ( F2 @ X2 ) )
                  = ( G @ ( F2 @ Xa ) ) )
                = ( ( G @ X2 )
                  = ( G @ Xa ) ) ) ) )
     => ( ( inj_on_a_a @ F2 @ A4 )
       => ( ( inj_on_a_a @ G @ ( image_a_a2 @ F2 @ A4 ) )
          = ( inj_on_a_a @ G @ A4 ) ) ) ) ).

% inj_on_image_iff
thf(fact_939_fcomp__comp,axiom,
    ( fcomp_6394288868438237061_b_b_b
    = ( ^ [F: product_prod_a_b > b,G2: b > b] : ( comp_b3886954628874447685od_a_b @ G2 @ F ) ) ) ).

% fcomp_comp
thf(fact_940_fcomp__comp,axiom,
    ( fcomp_5720495361292945518od_a_a
    = ( ^ [F: product_prod_a_b > a,G2: a > product_prod_a_a] : ( comp_a1036870397537576092od_a_b @ G2 @ F ) ) ) ).

% fcomp_comp
thf(fact_941_fcomp__comp,axiom,
    ( fcomp_9181744450410913094_b_a_b
    = ( ^ [F: product_prod_a_b > a,G2: a > b] : ( comp_a9170378079104387268od_a_b @ G2 @ F ) ) ) ).

% fcomp_comp
thf(fact_942_fcomp__comp,axiom,
    ( fcomp_9181744450410913093_b_a_a
    = ( ^ [F: product_prod_a_b > a,G2: a > a] : ( comp_a586446342581427589od_a_b @ G2 @ F ) ) ) ).

% fcomp_comp
thf(fact_943_fcomp__comp,axiom,
    ( fcomp_a_a_a
    = ( ^ [F: a > a,G2: a > a] : ( comp_a_a_a @ G2 @ F ) ) ) ).

% fcomp_comp
thf(fact_944_OO__Grp__cong,axiom,
    ! [A4: set_a,B5: set_a,F2: a > b,G: a > b] :
      ( ( A4 = B5 )
     => ( ( relcompp_b_a_b @ ( conversep_a_b @ ( bNF_Grp_a_b @ A4 @ F2 ) ) @ ( bNF_Grp_a_b @ A4 @ G ) )
        = ( relcompp_b_a_b @ ( conversep_a_b @ ( bNF_Grp_a_b @ B5 @ F2 ) ) @ ( bNF_Grp_a_b @ B5 @ G ) ) ) ) ).

% OO_Grp_cong
thf(fact_945_OO__Grp__cong,axiom,
    ! [A4: set_Product_prod_a_b,B5: set_Product_prod_a_b,F2: product_prod_a_b > b,G: product_prod_a_b > a] :
      ( ( A4 = B5 )
     => ( ( relcom3896546906122939236_a_b_a @ ( conver9146098269634272844_a_b_b @ ( bNF_Gr7842136747927947301_a_b_b @ A4 @ F2 ) ) @ ( bNF_Gr7842136747927947300_a_b_a @ A4 @ G ) )
        = ( relcom3896546906122939236_a_b_a @ ( conver9146098269634272844_a_b_b @ ( bNF_Gr7842136747927947301_a_b_b @ B5 @ F2 ) ) @ ( bNF_Gr7842136747927947300_a_b_a @ B5 @ G ) ) ) ) ).

% OO_Grp_cong
thf(fact_946_OO__Grp__cong,axiom,
    ! [A4: set_Product_prod_a_b,B5: set_Product_prod_a_b,F2: product_prod_a_b > b,G: product_prod_a_b > b] :
      ( ( A4 = B5 )
     => ( ( relcom3896546906122939237_a_b_b @ ( conver9146098269634272844_a_b_b @ ( bNF_Gr7842136747927947301_a_b_b @ A4 @ F2 ) ) @ ( bNF_Gr7842136747927947301_a_b_b @ A4 @ G ) )
        = ( relcom3896546906122939237_a_b_b @ ( conver9146098269634272844_a_b_b @ ( bNF_Gr7842136747927947301_a_b_b @ B5 @ F2 ) ) @ ( bNF_Gr7842136747927947301_a_b_b @ B5 @ G ) ) ) ) ).

% OO_Grp_cong
thf(fact_947_OO__Grp__cong,axiom,
    ! [A4: set_Pr5493900063595384711od_a_b,B5: set_Pr5493900063595384711od_a_b,F2: produc2233330766403592615od_a_b > product_prod_b_b,G: produc2233330766403592615od_a_b > product_prod_a_a] :
      ( ( A4 = B5 )
     => ( ( relcom7263568941997752886od_a_a @ ( conver7043752934290688245od_b_b @ ( bNF_Gr6683974337059614670od_b_b @ A4 @ F2 ) ) @ ( bNF_Gr248057877874286094od_a_a @ A4 @ G ) )
        = ( relcom7263568941997752886od_a_a @ ( conver7043752934290688245od_b_b @ ( bNF_Gr6683974337059614670od_b_b @ B5 @ F2 ) ) @ ( bNF_Gr248057877874286094od_a_a @ B5 @ G ) ) ) ) ).

% OO_Grp_cong
thf(fact_948_OO__Grp__cong,axiom,
    ! [A4: set_Pr5493900063595384711od_a_b,B5: set_Pr5493900063595384711od_a_b,F2: produc2233330766403592615od_a_b > product_prod_b_b,G: produc2233330766403592615od_a_b > product_prod_b_b] :
      ( ( A4 = B5 )
     => ( ( relcom4476113364328305654od_b_b @ ( conver7043752934290688245od_b_b @ ( bNF_Gr6683974337059614670od_b_b @ A4 @ F2 ) ) @ ( bNF_Gr6683974337059614670od_b_b @ A4 @ G ) )
        = ( relcom4476113364328305654od_b_b @ ( conver7043752934290688245od_b_b @ ( bNF_Gr6683974337059614670od_b_b @ B5 @ F2 ) ) @ ( bNF_Gr6683974337059614670od_b_b @ B5 @ G ) ) ) ) ).

% OO_Grp_cong
thf(fact_949_OO__Grp__cong,axiom,
    ! [A4: set_Product_prod_a_b,B5: set_Product_prod_a_b,F2: product_prod_a_b > a,G: product_prod_a_b > a] :
      ( ( A4 = B5 )
     => ( ( relcom9179970356352878819_a_b_a @ ( conver9146098269634272843_a_b_a @ ( bNF_Gr7842136747927947300_a_b_a @ A4 @ F2 ) ) @ ( bNF_Gr7842136747927947300_a_b_a @ A4 @ G ) )
        = ( relcom9179970356352878819_a_b_a @ ( conver9146098269634272843_a_b_a @ ( bNF_Gr7842136747927947300_a_b_a @ B5 @ F2 ) ) @ ( bNF_Gr7842136747927947300_a_b_a @ B5 @ G ) ) ) ) ).

% OO_Grp_cong
thf(fact_950_OO__Grp__cong,axiom,
    ! [A4: set_Product_prod_a_b,B5: set_Product_prod_a_b,F2: product_prod_a_b > a,G: product_prod_a_b > b] :
      ( ( A4 = B5 )
     => ( ( relcom9179970356352878820_a_b_b @ ( conver9146098269634272843_a_b_a @ ( bNF_Gr7842136747927947300_a_b_a @ A4 @ F2 ) ) @ ( bNF_Gr7842136747927947301_a_b_b @ A4 @ G ) )
        = ( relcom9179970356352878820_a_b_b @ ( conver9146098269634272843_a_b_a @ ( bNF_Gr7842136747927947300_a_b_a @ B5 @ F2 ) ) @ ( bNF_Gr7842136747927947301_a_b_b @ B5 @ G ) ) ) ) ).

% OO_Grp_cong
thf(fact_951_OO__Grp__cong,axiom,
    ! [A4: set_Pr5493900063595384711od_a_b,B5: set_Pr5493900063595384711od_a_b,F2: produc2233330766403592615od_a_b > product_prod_a_a,G: produc2233330766403592615od_a_b > product_prod_a_a] :
      ( ( A4 = B5 )
     => ( ( relcom2196630165857243254od_a_a @ ( conver607836475105359669od_a_a @ ( bNF_Gr248057877874286094od_a_a @ A4 @ F2 ) ) @ ( bNF_Gr248057877874286094od_a_a @ A4 @ G ) )
        = ( relcom2196630165857243254od_a_a @ ( conver607836475105359669od_a_a @ ( bNF_Gr248057877874286094od_a_a @ B5 @ F2 ) ) @ ( bNF_Gr248057877874286094od_a_a @ B5 @ G ) ) ) ) ).

% OO_Grp_cong
thf(fact_952_OO__Grp__cong,axiom,
    ! [A4: set_Pr5493900063595384711od_a_b,B5: set_Pr5493900063595384711od_a_b,F2: produc2233330766403592615od_a_b > product_prod_a_a,G: produc2233330766403592615od_a_b > product_prod_b_b] :
      ( ( A4 = B5 )
     => ( ( relcom8632546625042571830od_b_b @ ( conver607836475105359669od_a_a @ ( bNF_Gr248057877874286094od_a_a @ A4 @ F2 ) ) @ ( bNF_Gr6683974337059614670od_b_b @ A4 @ G ) )
        = ( relcom8632546625042571830od_b_b @ ( conver607836475105359669od_a_a @ ( bNF_Gr248057877874286094od_a_a @ B5 @ F2 ) ) @ ( bNF_Gr6683974337059614670od_b_b @ B5 @ G ) ) ) ) ).

% OO_Grp_cong
thf(fact_953_comp__inj__on__iff,axiom,
    ! [F2: a > a,A4: set_a,F3: a > a] :
      ( ( inj_on_a_a @ F2 @ A4 )
     => ( ( inj_on_a_a @ F3 @ ( image_a_a2 @ F2 @ A4 ) )
        = ( inj_on_a_a @ ( comp_a_a_a @ F3 @ F2 ) @ A4 ) ) ) ).

% comp_inj_on_iff
thf(fact_954_comp__inj__on__iff,axiom,
    ! [F2: a > set_b,A4: set_a,F3: set_b > a] :
      ( ( inj_on_a_set_b @ F2 @ A4 )
     => ( ( inj_on_set_b_a @ F3 @ ( image_a_set_b @ F2 @ A4 ) )
        = ( inj_on_a_a @ ( comp_set_b_a_a @ F3 @ F2 ) @ A4 ) ) ) ).

% comp_inj_on_iff
thf(fact_955_comp__inj__on__iff,axiom,
    ! [F2: product_prod_a_b > b,A4: set_Product_prod_a_b,F3: b > b] :
      ( ( inj_on4343330552946057672_a_b_b @ F2 @ A4 )
     => ( ( inj_on_b_b @ F3 @ ( image_2802296252294471260_a_b_b @ F2 @ A4 ) )
        = ( inj_on4343330552946057672_a_b_b @ ( comp_b3886954628874447685od_a_b @ F3 @ F2 ) @ A4 ) ) ) ).

% comp_inj_on_iff
thf(fact_956_comp__inj__on__iff,axiom,
    ! [F2: product_prod_a_b > b,A4: set_Product_prod_a_b,F3: b > a] :
      ( ( inj_on4343330552946057672_a_b_b @ F2 @ A4 )
     => ( ( inj_on_b_a @ F3 @ ( image_2802296252294471260_a_b_b @ F2 @ A4 ) )
        = ( inj_on4343330552946057671_a_b_a @ ( comp_b4526394929206263814od_a_b @ F3 @ F2 ) @ A4 ) ) ) ).

% comp_inj_on_iff
thf(fact_957_comp__inj__on__iff,axiom,
    ! [F2: b > b,A4: set_b,F3: b > option_list_o] :
      ( ( inj_on_b_b @ F2 @ A4 )
     => ( ( inj_on8261448415883032086list_o @ F3 @ ( image_b_b @ F2 @ A4 ) )
        = ( inj_on8261448415883032086list_o @ ( comp_b2182212639404545655st_o_b @ F3 @ F2 ) @ A4 ) ) ) ).

% comp_inj_on_iff
thf(fact_958_comp__inj__on__iff,axiom,
    ! [F2: a > b,A4: set_a,F3: b > option_list_o] :
      ( ( inj_on_a_b @ F2 @ A4 )
     => ( ( inj_on8261448415883032086list_o @ F3 @ ( image_a_b @ F2 @ A4 ) )
        = ( inj_on374126998980950615list_o @ ( comp_b2182212639404545654st_o_a @ F3 @ F2 ) @ A4 ) ) ) ).

% comp_inj_on_iff
thf(fact_959_comp__inj__on__iff,axiom,
    ! [F2: b > a,A4: set_b,F3: a > option_list_o] :
      ( ( inj_on_b_a @ F2 @ A4 )
     => ( ( inj_on374126998980950615list_o @ F3 @ ( image_b_a @ F2 @ A4 ) )
        = ( inj_on8261448415883032086list_o @ ( comp_a6425641528110818422st_o_b @ F3 @ F2 ) @ A4 ) ) ) ).

% comp_inj_on_iff
thf(fact_960_comp__inj__on__iff,axiom,
    ! [F2: a > product_prod_a_b,A4: set_a,F3: product_prod_a_b > a] :
      ( ( inj_on8941660087544810907od_a_b @ F2 @ A4 )
     => ( ( inj_on4343330552946057671_a_b_a @ F3 @ ( image_7400625786893224495od_a_b @ F2 @ A4 ) )
        = ( inj_on_a_a @ ( comp_P3199017072728932485_b_a_a @ F3 @ F2 ) @ A4 ) ) ) ).

% comp_inj_on_iff
thf(fact_961_comp__inj__on__iff,axiom,
    ! [F2: a > option_list_o,A4: set_a,F3: option_list_o > a] :
      ( ( inj_on374126998980950615list_o @ F2 @ A4 )
     => ( ( inj_on2088879393520314763st_o_a @ F3 @ ( image_5155782636146240747list_o @ F2 @ A4 ) )
        = ( inj_on_a_a @ ( comp_o695885166685416001_o_a_a @ F3 @ F2 ) @ A4 ) ) ) ).

% comp_inj_on_iff
thf(fact_962_comp__inj__on__iff,axiom,
    ! [F2: a > a,A4: set_a,F3: a > option_list_o] :
      ( ( inj_on_a_a @ F2 @ A4 )
     => ( ( inj_on374126998980950615list_o @ F3 @ ( image_a_a2 @ F2 @ A4 ) )
        = ( inj_on374126998980950615list_o @ ( comp_a6425641528110818421st_o_a @ F3 @ F2 ) @ A4 ) ) ) ).

% comp_inj_on_iff
thf(fact_963_inj__on__imageI,axiom,
    ! [G: a > a,F2: a > a,A4: set_a] :
      ( ( inj_on_a_a @ ( comp_a_a_a @ G @ F2 ) @ A4 )
     => ( inj_on_a_a @ G @ ( image_a_a2 @ F2 @ A4 ) ) ) ).

% inj_on_imageI
thf(fact_964_inj__on__imageI,axiom,
    ! [G: set_b > a,F2: a > set_b,A4: set_a] :
      ( ( inj_on_a_a @ ( comp_set_b_a_a @ G @ F2 ) @ A4 )
     => ( inj_on_set_b_a @ G @ ( image_a_set_b @ F2 @ A4 ) ) ) ).

% inj_on_imageI
thf(fact_965_inj__on__imageI,axiom,
    ! [G: b > b,F2: product_prod_a_b > b,A4: set_Product_prod_a_b] :
      ( ( inj_on4343330552946057672_a_b_b @ ( comp_b3886954628874447685od_a_b @ G @ F2 ) @ A4 )
     => ( inj_on_b_b @ G @ ( image_2802296252294471260_a_b_b @ F2 @ A4 ) ) ) ).

% inj_on_imageI
thf(fact_966_inj__on__imageI,axiom,
    ! [G: a > b,F2: product_prod_a_b > a,A4: set_Product_prod_a_b] :
      ( ( inj_on4343330552946057672_a_b_b @ ( comp_a9170378079104387268od_a_b @ G @ F2 ) @ A4 )
     => ( inj_on_a_b @ G @ ( image_2802296252294471259_a_b_a @ F2 @ A4 ) ) ) ).

% inj_on_imageI
thf(fact_967_inj__on__imageI,axiom,
    ! [G: b > option_list_o,F2: b > b,A4: set_b] :
      ( ( inj_on8261448415883032086list_o @ ( comp_b2182212639404545655st_o_b @ G @ F2 ) @ A4 )
     => ( inj_on8261448415883032086list_o @ G @ ( image_b_b @ F2 @ A4 ) ) ) ).

% inj_on_imageI
thf(fact_968_inj__on__imageI,axiom,
    ! [G: a > option_list_o,F2: b > a,A4: set_b] :
      ( ( inj_on8261448415883032086list_o @ ( comp_a6425641528110818422st_o_b @ G @ F2 ) @ A4 )
     => ( inj_on374126998980950615list_o @ G @ ( image_b_a @ F2 @ A4 ) ) ) ).

% inj_on_imageI
thf(fact_969_inj__on__imageI,axiom,
    ! [G: b > option_list_o,F2: a > b,A4: set_a] :
      ( ( inj_on374126998980950615list_o @ ( comp_b2182212639404545654st_o_a @ G @ F2 ) @ A4 )
     => ( inj_on8261448415883032086list_o @ G @ ( image_a_b @ F2 @ A4 ) ) ) ).

% inj_on_imageI
thf(fact_970_inj__on__imageI,axiom,
    ! [G: a > option_list_o,F2: a > a,A4: set_a] :
      ( ( inj_on374126998980950615list_o @ ( comp_a6425641528110818421st_o_a @ G @ F2 ) @ A4 )
     => ( inj_on374126998980950615list_o @ G @ ( image_a_a2 @ F2 @ A4 ) ) ) ).

% inj_on_imageI
thf(fact_971_inj__on__imageI,axiom,
    ! [G: product_prod_a_b > a,F2: a > product_prod_a_b,A4: set_a] :
      ( ( inj_on_a_a @ ( comp_P3199017072728932485_b_a_a @ G @ F2 ) @ A4 )
     => ( inj_on4343330552946057671_a_b_a @ G @ ( image_7400625786893224495od_a_b @ F2 @ A4 ) ) ) ).

% inj_on_imageI
thf(fact_972_inj__on__imageI,axiom,
    ! [G: b > a,F2: product_prod_a_b > b,A4: set_Product_prod_a_b] :
      ( ( inj_on4343330552946057671_a_b_a @ ( comp_b4526394929206263814od_a_b @ G @ F2 ) @ A4 )
     => ( inj_on_b_a @ G @ ( image_2802296252294471260_a_b_b @ F2 @ A4 ) ) ) ).

% inj_on_imageI
thf(fact_973_comp__inj__on,axiom,
    ! [F2: a > a,A4: set_a,G: a > a] :
      ( ( inj_on_a_a @ F2 @ A4 )
     => ( ( inj_on_a_a @ G @ ( image_a_a2 @ F2 @ A4 ) )
       => ( inj_on_a_a @ ( comp_a_a_a @ G @ F2 ) @ A4 ) ) ) ).

% comp_inj_on
thf(fact_974_comp__inj__on,axiom,
    ! [F2: a > set_b,A4: set_a,G: set_b > a] :
      ( ( inj_on_a_set_b @ F2 @ A4 )
     => ( ( inj_on_set_b_a @ G @ ( image_a_set_b @ F2 @ A4 ) )
       => ( inj_on_a_a @ ( comp_set_b_a_a @ G @ F2 ) @ A4 ) ) ) ).

% comp_inj_on
thf(fact_975_comp__inj__on,axiom,
    ! [F2: product_prod_a_b > b,A4: set_Product_prod_a_b,G: b > b] :
      ( ( inj_on4343330552946057672_a_b_b @ F2 @ A4 )
     => ( ( inj_on_b_b @ G @ ( image_2802296252294471260_a_b_b @ F2 @ A4 ) )
       => ( inj_on4343330552946057672_a_b_b @ ( comp_b3886954628874447685od_a_b @ G @ F2 ) @ A4 ) ) ) ).

% comp_inj_on
thf(fact_976_comp__inj__on,axiom,
    ! [F2: product_prod_a_b > b,A4: set_Product_prod_a_b,G: b > a] :
      ( ( inj_on4343330552946057672_a_b_b @ F2 @ A4 )
     => ( ( inj_on_b_a @ G @ ( image_2802296252294471260_a_b_b @ F2 @ A4 ) )
       => ( inj_on4343330552946057671_a_b_a @ ( comp_b4526394929206263814od_a_b @ G @ F2 ) @ A4 ) ) ) ).

% comp_inj_on
thf(fact_977_comp__inj__on,axiom,
    ! [F2: b > b,A4: set_b,G: b > option_list_o] :
      ( ( inj_on_b_b @ F2 @ A4 )
     => ( ( inj_on8261448415883032086list_o @ G @ ( image_b_b @ F2 @ A4 ) )
       => ( inj_on8261448415883032086list_o @ ( comp_b2182212639404545655st_o_b @ G @ F2 ) @ A4 ) ) ) ).

% comp_inj_on
thf(fact_978_comp__inj__on,axiom,
    ! [F2: a > b,A4: set_a,G: b > option_list_o] :
      ( ( inj_on_a_b @ F2 @ A4 )
     => ( ( inj_on8261448415883032086list_o @ G @ ( image_a_b @ F2 @ A4 ) )
       => ( inj_on374126998980950615list_o @ ( comp_b2182212639404545654st_o_a @ G @ F2 ) @ A4 ) ) ) ).

% comp_inj_on
thf(fact_979_comp__inj__on,axiom,
    ! [F2: b > a,A4: set_b,G: a > option_list_o] :
      ( ( inj_on_b_a @ F2 @ A4 )
     => ( ( inj_on374126998980950615list_o @ G @ ( image_b_a @ F2 @ A4 ) )
       => ( inj_on8261448415883032086list_o @ ( comp_a6425641528110818422st_o_b @ G @ F2 ) @ A4 ) ) ) ).

% comp_inj_on
thf(fact_980_comp__inj__on,axiom,
    ! [F2: a > product_prod_a_b,A4: set_a,G: product_prod_a_b > a] :
      ( ( inj_on8941660087544810907od_a_b @ F2 @ A4 )
     => ( ( inj_on4343330552946057671_a_b_a @ G @ ( image_7400625786893224495od_a_b @ F2 @ A4 ) )
       => ( inj_on_a_a @ ( comp_P3199017072728932485_b_a_a @ G @ F2 ) @ A4 ) ) ) ).

% comp_inj_on
thf(fact_981_comp__inj__on,axiom,
    ! [F2: a > option_list_o,A4: set_a,G: option_list_o > a] :
      ( ( inj_on374126998980950615list_o @ F2 @ A4 )
     => ( ( inj_on2088879393520314763st_o_a @ G @ ( image_5155782636146240747list_o @ F2 @ A4 ) )
       => ( inj_on_a_a @ ( comp_o695885166685416001_o_a_a @ G @ F2 ) @ A4 ) ) ) ).

% comp_inj_on
thf(fact_982_comp__inj__on,axiom,
    ! [F2: a > a,A4: set_a,G: a > option_list_o] :
      ( ( inj_on_a_a @ F2 @ A4 )
     => ( ( inj_on374126998980950615list_o @ G @ ( image_a_a2 @ F2 @ A4 ) )
       => ( inj_on374126998980950615list_o @ ( comp_a6425641528110818421st_o_a @ G @ F2 ) @ A4 ) ) ) ).

% comp_inj_on
thf(fact_983_prod__fun__imageE,axiom,
    ! [C: product_prod_b_b,F2: b > b,G: b > b,R5: set_Product_prod_b_b] :
      ( ( member7862447936710763792od_b_b @ C @ ( image_4453096554207573957od_b_b @ ( produc8315423520075080822_b_b_b @ F2 @ G ) @ R5 ) )
     => ~ ! [X2: b,Y2: b] :
            ( ( C
              = ( product_Pair_b_b @ ( F2 @ X2 ) @ ( G @ Y2 ) ) )
           => ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X2 @ Y2 ) @ R5 ) ) ) ).

% prod_fun_imageE
thf(fact_984_prod__fun__imageE,axiom,
    ! [C: product_prod_b_b,F2: b > b,G: a > b,R5: set_Product_prod_b_a] :
      ( ( member7862447936710763792od_b_b @ C @ ( image_5789147174160268294od_b_b @ ( produc1879507065192981047_b_a_b @ F2 @ G ) @ R5 ) )
     => ~ ! [X2: b,Y2: a] :
            ( ( C
              = ( product_Pair_b_b @ ( F2 @ X2 ) @ ( G @ Y2 ) ) )
           => ~ ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ X2 @ Y2 ) @ R5 ) ) ) ).

% prod_fun_imageE
thf(fact_985_prod__fun__imageE,axiom,
    ! [C: product_prod_b_b,F2: a > b,G: b > b,R5: set_Product_prod_a_b] :
      ( ( member7862447936710763792od_b_b @ C @ ( image_513147967582737732od_b_b @ ( produc6827461482780282295_b_b_b @ F2 @ G ) @ R5 ) )
     => ~ ! [X2: a,Y2: b] :
            ( ( C
              = ( product_Pair_b_b @ ( F2 @ X2 ) @ ( G @ Y2 ) ) )
           => ~ ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X2 @ Y2 ) @ R5 ) ) ) ).

% prod_fun_imageE
thf(fact_986_prod__fun__imageE,axiom,
    ! [C: product_prod_b_b,F2: a > b,G: a > b,R5: set_Product_prod_a_a] :
      ( ( member7862447936710763792od_b_b @ C @ ( image_1849198587535432069od_b_b @ ( produc391545027898182520_b_a_b @ F2 @ G ) @ R5 ) )
     => ~ ! [X2: a,Y2: a] :
            ( ( C
              = ( product_Pair_b_b @ ( F2 @ X2 ) @ ( G @ Y2 ) ) )
           => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ R5 ) ) ) ).

% prod_fun_imageE
thf(fact_987_prod__fun__imageE,axiom,
    ! [C: product_prod_b_a,F2: b > b,G: b > a,R5: set_Product_prod_b_b] :
      ( ( member7862447932407534991od_b_a @ C @ ( image_4453096549904345156od_b_a @ ( produc8315423520075080821_b_b_a @ F2 @ G ) @ R5 ) )
     => ~ ! [X2: b,Y2: b] :
            ( ( C
              = ( product_Pair_b_a @ ( F2 @ X2 ) @ ( G @ Y2 ) ) )
           => ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X2 @ Y2 ) @ R5 ) ) ) ).

% prod_fun_imageE
thf(fact_988_prod__fun__imageE,axiom,
    ! [C: product_prod_b_a,F2: b > b,G: a > a,R5: set_Product_prod_b_a] :
      ( ( member7862447932407534991od_b_a @ C @ ( image_5789147169857039493od_b_a @ ( produc1879507065192981046_b_a_a @ F2 @ G ) @ R5 ) )
     => ~ ! [X2: b,Y2: a] :
            ( ( C
              = ( product_Pair_b_a @ ( F2 @ X2 ) @ ( G @ Y2 ) ) )
           => ~ ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ X2 @ Y2 ) @ R5 ) ) ) ).

% prod_fun_imageE
thf(fact_989_prod__fun__imageE,axiom,
    ! [C: product_prod_b_a,F2: a > b,G: b > a,R5: set_Product_prod_a_b] :
      ( ( member7862447932407534991od_b_a @ C @ ( image_513147963279508931od_b_a @ ( produc6827461482780282294_b_b_a @ F2 @ G ) @ R5 ) )
     => ~ ! [X2: a,Y2: b] :
            ( ( C
              = ( product_Pair_b_a @ ( F2 @ X2 ) @ ( G @ Y2 ) ) )
           => ~ ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X2 @ Y2 ) @ R5 ) ) ) ).

% prod_fun_imageE
thf(fact_990_prod__fun__imageE,axiom,
    ! [C: product_prod_b_a,F2: a > b,G: a > a,R5: set_Product_prod_a_a] :
      ( ( member7862447932407534991od_b_a @ C @ ( image_1849198583232203268od_b_a @ ( produc391545027898182519_b_a_a @ F2 @ G ) @ R5 ) )
     => ~ ! [X2: a,Y2: a] :
            ( ( C
              = ( product_Pair_b_a @ ( F2 @ X2 ) @ ( G @ Y2 ) ) )
           => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ R5 ) ) ) ).

% prod_fun_imageE
thf(fact_991_prod__fun__imageE,axiom,
    ! [C: product_prod_a_b,F2: b > a,G: b > b,R5: set_Product_prod_b_b] :
      ( ( member1426531481828664017od_a_b @ C @ ( image_7240552136180249990od_a_b @ ( produc2719204904886950133_a_b_b @ F2 @ G ) @ R5 ) )
     => ~ ! [X2: b,Y2: b] :
            ( ( C
              = ( product_Pair_a_b @ ( F2 @ X2 ) @ ( G @ Y2 ) ) )
           => ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X2 @ Y2 ) @ R5 ) ) ) ).

% prod_fun_imageE
thf(fact_992_prod__fun__imageE,axiom,
    ! [C: product_prod_a_b,F2: b > a,G: a > b,R5: set_Product_prod_b_a] :
      ( ( member1426531481828664017od_a_b @ C @ ( image_8576602756132944327od_a_b @ ( produc5506660486859626166_a_a_b @ F2 @ G ) @ R5 ) )
     => ~ ! [X2: b,Y2: a] :
            ( ( C
              = ( product_Pair_a_b @ ( F2 @ X2 ) @ ( G @ Y2 ) ) )
           => ~ ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ X2 @ Y2 ) @ R5 ) ) ) ).

% prod_fun_imageE
thf(fact_993_Inf_OINF__image,axiom,
    ! [Inf: set_a > a,G: a > a,F2: a > a,A4: set_a] :
      ( ( Inf @ ( image_a_a2 @ G @ ( image_a_a2 @ F2 @ A4 ) ) )
      = ( Inf @ ( image_a_a2 @ ( comp_a_a_a @ G @ F2 ) @ A4 ) ) ) ).

% Inf.INF_image
thf(fact_994_Inf_OINF__image,axiom,
    ! [Inf: set_a > a,G: set_b > a,F2: a > set_b,A4: set_a] :
      ( ( Inf @ ( image_set_b_a @ G @ ( image_a_set_b @ F2 @ A4 ) ) )
      = ( Inf @ ( image_a_a2 @ ( comp_set_b_a_a @ G @ F2 ) @ A4 ) ) ) ).

% Inf.INF_image
thf(fact_995_Inf_OINF__image,axiom,
    ! [Inf: set_set_b > set_b,G: a > set_b,F2: a > a,A4: set_a] :
      ( ( Inf @ ( image_a_set_b @ G @ ( image_a_a2 @ F2 @ A4 ) ) )
      = ( Inf @ ( image_a_set_b @ ( comp_a_set_b_a @ G @ F2 ) @ A4 ) ) ) ).

% Inf.INF_image
thf(fact_996_Inf_OINF__image,axiom,
    ! [Inf: set_b > b,G: b > b,F2: product_prod_a_b > b,A4: set_Product_prod_a_b] :
      ( ( Inf @ ( image_b_b @ G @ ( image_2802296252294471260_a_b_b @ F2 @ A4 ) ) )
      = ( Inf @ ( image_2802296252294471260_a_b_b @ ( comp_b3886954628874447685od_a_b @ G @ F2 ) @ A4 ) ) ) ).

% Inf.INF_image
thf(fact_997_Inf_OINF__image,axiom,
    ! [Inf: set_a > a,G: b > a,F2: product_prod_a_b > b,A4: set_Product_prod_a_b] :
      ( ( Inf @ ( image_b_a @ G @ ( image_2802296252294471260_a_b_b @ F2 @ A4 ) ) )
      = ( Inf @ ( image_2802296252294471259_a_b_a @ ( comp_b4526394929206263814od_a_b @ G @ F2 ) @ A4 ) ) ) ).

% Inf.INF_image
thf(fact_998_Inf_OINF__image,axiom,
    ! [Inf: set_b > b,G: a > b,F2: product_prod_a_b > a,A4: set_Product_prod_a_b] :
      ( ( Inf @ ( image_a_b @ G @ ( image_2802296252294471259_a_b_a @ F2 @ A4 ) ) )
      = ( Inf @ ( image_2802296252294471260_a_b_b @ ( comp_a9170378079104387268od_a_b @ G @ F2 ) @ A4 ) ) ) ).

% Inf.INF_image
thf(fact_999_Inf_OINF__image,axiom,
    ! [Inf: set_set_b > set_b,G: set_b > set_b,F2: a > set_b,A4: set_a] :
      ( ( Inf @ ( image_set_b_set_b @ G @ ( image_a_set_b @ F2 @ A4 ) ) )
      = ( Inf @ ( image_a_set_b @ ( comp_set_b_set_b_a @ G @ F2 ) @ A4 ) ) ) ).

% Inf.INF_image
thf(fact_1000_Inf_OINF__image,axiom,
    ! [Inf: set_a > a,G: product_prod_a_b > a,F2: a > product_prod_a_b,A4: set_a] :
      ( ( Inf @ ( image_2802296252294471259_a_b_a @ G @ ( image_7400625786893224495od_a_b @ F2 @ A4 ) ) )
      = ( Inf @ ( image_a_a2 @ ( comp_P3199017072728932485_b_a_a @ G @ F2 ) @ A4 ) ) ) ).

% Inf.INF_image
thf(fact_1001_Inf_OINF__image,axiom,
    ! [Inf: set_a > a,G: a > a,F2: product_prod_a_b > a,A4: set_Product_prod_a_b] :
      ( ( Inf @ ( image_a_a2 @ G @ ( image_2802296252294471259_a_b_a @ F2 @ A4 ) ) )
      = ( Inf @ ( image_2802296252294471259_a_b_a @ ( comp_a586446342581427589od_a_b @ G @ F2 ) @ A4 ) ) ) ).

% Inf.INF_image
thf(fact_1002_Inf_OINF__image,axiom,
    ! [Inf: set_a > a,G: produc5884233991663340231list_o > a,F2: a > produc5884233991663340231list_o,A4: set_a] :
      ( ( Inf @ ( image_8953000392185216508st_o_a @ G @ ( image_1329905666413968462list_o @ F2 @ A4 ) ) )
      = ( Inf @ ( image_a_a2 @ ( comp_P941169184177061668_o_a_a @ G @ F2 ) @ A4 ) ) ) ).

% Inf.INF_image
thf(fact_1003_Sup_OSUP__image,axiom,
    ! [Sup: set_a > a,G: a > a,F2: a > a,A4: set_a] :
      ( ( Sup @ ( image_a_a2 @ G @ ( image_a_a2 @ F2 @ A4 ) ) )
      = ( Sup @ ( image_a_a2 @ ( comp_a_a_a @ G @ F2 ) @ A4 ) ) ) ).

% Sup.SUP_image
thf(fact_1004_Sup_OSUP__image,axiom,
    ! [Sup: set_a > a,G: set_b > a,F2: a > set_b,A4: set_a] :
      ( ( Sup @ ( image_set_b_a @ G @ ( image_a_set_b @ F2 @ A4 ) ) )
      = ( Sup @ ( image_a_a2 @ ( comp_set_b_a_a @ G @ F2 ) @ A4 ) ) ) ).

% Sup.SUP_image
thf(fact_1005_Sup_OSUP__image,axiom,
    ! [Sup: set_set_b > set_b,G: a > set_b,F2: a > a,A4: set_a] :
      ( ( Sup @ ( image_a_set_b @ G @ ( image_a_a2 @ F2 @ A4 ) ) )
      = ( Sup @ ( image_a_set_b @ ( comp_a_set_b_a @ G @ F2 ) @ A4 ) ) ) ).

% Sup.SUP_image
thf(fact_1006_Sup_OSUP__image,axiom,
    ! [Sup: set_b > b,G: b > b,F2: product_prod_a_b > b,A4: set_Product_prod_a_b] :
      ( ( Sup @ ( image_b_b @ G @ ( image_2802296252294471260_a_b_b @ F2 @ A4 ) ) )
      = ( Sup @ ( image_2802296252294471260_a_b_b @ ( comp_b3886954628874447685od_a_b @ G @ F2 ) @ A4 ) ) ) ).

% Sup.SUP_image
thf(fact_1007_Sup_OSUP__image,axiom,
    ! [Sup: set_a > a,G: b > a,F2: product_prod_a_b > b,A4: set_Product_prod_a_b] :
      ( ( Sup @ ( image_b_a @ G @ ( image_2802296252294471260_a_b_b @ F2 @ A4 ) ) )
      = ( Sup @ ( image_2802296252294471259_a_b_a @ ( comp_b4526394929206263814od_a_b @ G @ F2 ) @ A4 ) ) ) ).

% Sup.SUP_image
thf(fact_1008_Sup_OSUP__image,axiom,
    ! [Sup: set_b > b,G: a > b,F2: product_prod_a_b > a,A4: set_Product_prod_a_b] :
      ( ( Sup @ ( image_a_b @ G @ ( image_2802296252294471259_a_b_a @ F2 @ A4 ) ) )
      = ( Sup @ ( image_2802296252294471260_a_b_b @ ( comp_a9170378079104387268od_a_b @ G @ F2 ) @ A4 ) ) ) ).

% Sup.SUP_image
thf(fact_1009_Sup_OSUP__image,axiom,
    ! [Sup: set_set_b > set_b,G: set_b > set_b,F2: a > set_b,A4: set_a] :
      ( ( Sup @ ( image_set_b_set_b @ G @ ( image_a_set_b @ F2 @ A4 ) ) )
      = ( Sup @ ( image_a_set_b @ ( comp_set_b_set_b_a @ G @ F2 ) @ A4 ) ) ) ).

% Sup.SUP_image
thf(fact_1010_Sup_OSUP__image,axiom,
    ! [Sup: set_a > a,G: product_prod_a_b > a,F2: a > product_prod_a_b,A4: set_a] :
      ( ( Sup @ ( image_2802296252294471259_a_b_a @ G @ ( image_7400625786893224495od_a_b @ F2 @ A4 ) ) )
      = ( Sup @ ( image_a_a2 @ ( comp_P3199017072728932485_b_a_a @ G @ F2 ) @ A4 ) ) ) ).

% Sup.SUP_image
thf(fact_1011_Sup_OSUP__image,axiom,
    ! [Sup: set_a > a,G: a > a,F2: product_prod_a_b > a,A4: set_Product_prod_a_b] :
      ( ( Sup @ ( image_a_a2 @ G @ ( image_2802296252294471259_a_b_a @ F2 @ A4 ) ) )
      = ( Sup @ ( image_2802296252294471259_a_b_a @ ( comp_a586446342581427589od_a_b @ G @ F2 ) @ A4 ) ) ) ).

% Sup.SUP_image
thf(fact_1012_Sup_OSUP__image,axiom,
    ! [Sup: set_a > a,G: produc5884233991663340231list_o > a,F2: a > produc5884233991663340231list_o,A4: set_a] :
      ( ( Sup @ ( image_8953000392185216508st_o_a @ G @ ( image_1329905666413968462list_o @ F2 @ A4 ) ) )
      = ( Sup @ ( image_a_a2 @ ( comp_P941169184177061668_o_a_a @ G @ F2 ) @ A4 ) ) ) ).

% Sup.SUP_image
thf(fact_1013_Grp__UNIV__id,axiom,
    ! [F2: product_prod_a_b > product_prod_a_b] :
      ( ( F2 = id_Product_prod_a_b )
     => ( ( relcom7003123350501562167od_a_b @ ( conver3111372723085992693od_a_b @ ( bNF_Gr7211640137128594382od_a_b @ top_to8134405472303993176od_a_b @ F2 ) ) @ ( bNF_Gr7211640137128594382od_a_b @ top_to8134405472303993176od_a_b @ F2 ) )
        = ( bNF_Gr7211640137128594382od_a_b @ top_to8134405472303993176od_a_b @ F2 ) ) ) ).

% Grp_UNIV_id
thf(fact_1014_Grp__UNIV__id,axiom,
    ! [F2: a > a] :
      ( ( F2 = id_a )
     => ( ( relcompp_a_a_a @ ( conversep_a_a @ ( bNF_Grp_a_a @ top_top_set_a @ F2 ) ) @ ( bNF_Grp_a_a @ top_top_set_a @ F2 ) )
        = ( bNF_Grp_a_a @ top_top_set_a @ F2 ) ) ) ).

% Grp_UNIV_id
thf(fact_1015_Grp__UNIV__id,axiom,
    ! [F2: b > b] :
      ( ( F2 = id_b )
     => ( ( relcompp_b_b_b @ ( conversep_b_b @ ( bNF_Grp_b_b @ top_top_set_b @ F2 ) ) @ ( bNF_Grp_b_b @ top_top_set_b @ F2 ) )
        = ( bNF_Grp_b_b @ top_top_set_b @ F2 ) ) ) ).

% Grp_UNIV_id
thf(fact_1016_Collect__split__mono__strong,axiom,
    ! [X5: set_Product_prod_a_b,A4: set_Pr5493900063595384711od_a_b,Y5: set_Product_prod_a_b,P2: product_prod_a_b > product_prod_a_b > $o,Q2: product_prod_a_b > product_prod_a_b > $o] :
      ( ( X5
        = ( image_2926283117629707974od_a_b @ produc5013601669340818563od_a_b @ A4 ) )
     => ( ( Y5
          = ( image_2926283117629707974od_a_b @ produc8875788424405714885od_a_b @ A4 ) )
       => ( ! [X2: product_prod_a_b] :
              ( ( member1426531481828664017od_a_b @ X2 @ X5 )
             => ! [Xa: product_prod_a_b] :
                  ( ( member1426531481828664017od_a_b @ Xa @ Y5 )
                 => ( ( P2 @ X2 @ Xa )
                   => ( Q2 @ X2 @ Xa ) ) ) )
         => ( ( ord_le362614178720401191od_a_b @ A4 @ ( collec7968472090841016338od_a_b @ ( produc5413562937804622316_a_b_o @ P2 ) ) )
           => ( ord_le362614178720401191od_a_b @ A4 @ ( collec7968472090841016338od_a_b @ ( produc5413562937804622316_a_b_o @ Q2 ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_1017_Collect__split__mono__strong,axiom,
    ! [X5: set_nat,A4: set_Pr1261947904930325089at_nat,Y5: set_nat,P2: nat > nat > $o,Q2: nat > nat > $o] :
      ( ( X5
        = ( image_2486076414777270412at_nat @ product_fst_nat_nat @ A4 ) )
     => ( ( Y5
          = ( image_2486076414777270412at_nat @ product_snd_nat_nat @ A4 ) )
       => ( ! [X2: nat] :
              ( ( member_nat @ X2 @ X5 )
             => ! [Xa: nat] :
                  ( ( member_nat @ Xa @ Y5 )
                 => ( ( P2 @ X2 @ Xa )
                   => ( Q2 @ X2 @ Xa ) ) ) )
         => ( ( ord_le3146513528884898305at_nat @ A4 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ P2 ) ) )
           => ( ord_le3146513528884898305at_nat @ A4 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ Q2 ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_1018_Collect__split__mono__strong,axiom,
    ! [X5: set_a,A4: set_Pr3077528766752018087list_o,Y5: set_list_o,P2: a > list_o > $o,Q2: a > list_o > $o] :
      ( ( X5
        = ( image_8953000392185216508st_o_a @ product_fst_a_list_o @ A4 ) )
     => ( ( Y5
          = ( image_8759964538478586780list_o @ product_snd_a_list_o @ A4 ) )
       => ( ! [X2: a] :
              ( ( member_a @ X2 @ X5 )
             => ! [Xa: list_o] :
                  ( ( member_list_o @ Xa @ Y5 )
                 => ( ( P2 @ X2 @ Xa )
                   => ( Q2 @ X2 @ Xa ) ) ) )
         => ( ( ord_le8560161592425875015list_o @ A4 @ ( collec7805115074396547890list_o @ ( produc8342644129638033356st_o_o @ P2 ) ) )
           => ( ord_le8560161592425875015list_o @ A4 @ ( collec7805115074396547890list_o @ ( produc8342644129638033356st_o_o @ Q2 ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_1019_Collect__split__mono__strong,axiom,
    ! [X5: set_b,A4: set_Product_prod_b_a,Y5: set_a,P2: b > a > $o,Q2: b > a > $o] :
      ( ( X5
        = ( image_9034163868087588638_b_a_b @ product_fst_b_a @ A4 ) )
     => ( ( Y5
          = ( image_9034163868087588637_b_a_a @ product_snd_b_a @ A4 ) )
       => ( ! [X2: b] :
              ( ( member_b @ X2 @ X5 )
             => ! [Xa: a] :
                  ( ( member_a @ Xa @ Y5 )
                 => ( ( P2 @ X2 @ Xa )
                   => ( Q2 @ X2 @ Xa ) ) ) )
         => ( ( ord_le111053957804629862od_b_a @ A4 @ ( collec548942215411776465od_b_a @ ( produc7672072387910335853_b_a_o @ P2 ) ) )
           => ( ord_le111053957804629862od_b_a @ A4 @ ( collec548942215411776465od_b_a @ ( produc7672072387910335853_b_a_o @ Q2 ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_1020_Collect__split__mono__strong,axiom,
    ! [X5: set_b,A4: set_Product_prod_b_b,Y5: set_b,P2: b > b > $o,Q2: b > b > $o] :
      ( ( X5
        = ( image_8398514867482601949_b_b_b @ product_fst_b_b @ A4 ) )
     => ( ( Y5
          = ( image_8398514867482601949_b_b_b @ product_snd_b_b @ A4 ) )
       => ( ! [X2: b] :
              ( ( member_b @ X2 @ X5 )
             => ! [Xa: b] :
                  ( ( member_b @ Xa @ Y5 )
                 => ( ( P2 @ X2 @ Xa )
                   => ( Q2 @ X2 @ Xa ) ) ) )
         => ( ( ord_le182087997850975847od_b_b @ A4 @ ( collec548942219715005266od_b_b @ ( produc4772849988445941548_b_b_o @ P2 ) ) )
           => ( ord_le182087997850975847od_b_b @ A4 @ ( collec548942219715005266od_b_b @ ( produc4772849988445941548_b_b_o @ Q2 ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_1021_Collect__split__mono__strong,axiom,
    ! [X5: set_a,A4: set_Product_prod_a_a,Y5: set_a,P2: a > a > $o,Q2: a > a > $o] :
      ( ( X5
        = ( image_3437945252899457948_a_a_a @ product_fst_a_a @ A4 ) )
     => ( ( Y5
          = ( image_3437945252899457948_a_a_a @ product_snd_a_a @ A4 ) )
       => ( ! [X2: a] :
              ( ( member_a @ X2 @ X5 )
             => ! [Xa: a] :
                  ( ( member_a @ Xa @ Y5 )
                 => ( ( P2 @ X2 @ Xa )
                   => ( Q2 @ X2 @ Xa ) ) ) )
         => ( ( ord_le746702958409616551od_a_a @ A4 @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ P2 ) ) )
           => ( ord_le746702958409616551od_a_a @ A4 @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ Q2 ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_1022_Collect__split__mono__strong,axiom,
    ! [X5: set_a,A4: set_Product_prod_a_b,Y5: set_b,P2: a > b > $o,Q2: a > b > $o] :
      ( ( X5
        = ( image_2802296252294471259_a_b_a @ product_fst_a_b @ A4 ) )
     => ( ( Y5
          = ( image_2802296252294471260_a_b_b @ product_snd_a_b @ A4 ) )
       => ( ! [X2: a] :
              ( ( member_a @ X2 @ X5 )
             => ! [Xa: b] :
                  ( ( member_b @ Xa @ Y5 )
                 => ( ( P2 @ X2 @ Xa )
                   => ( Q2 @ X2 @ Xa ) ) ) )
         => ( ( ord_le817736998455962536od_a_b @ A4 @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ P2 ) ) )
           => ( ord_le817736998455962536od_a_b @ A4 @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ Q2 ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_1023_the__inv__into__comp,axiom,
    ! [F2: a > b,G: a > a,A4: set_a,X3: b] :
      ( ( inj_on_a_b @ F2 @ ( image_a_a2 @ G @ A4 ) )
     => ( ( inj_on_a_a @ G @ A4 )
       => ( ( member_b @ X3 @ ( image_a_b @ F2 @ ( image_a_a2 @ G @ A4 ) ) )
         => ( ( the_inv_into_a_b @ A4 @ ( comp_a_b_a @ F2 @ G ) @ X3 )
            = ( comp_a_a_b @ ( the_inv_into_a_a @ A4 @ G ) @ ( the_inv_into_a_b @ ( image_a_a2 @ G @ A4 ) @ F2 ) @ X3 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1024_the__inv__into__comp,axiom,
    ! [F2: a > a,G: a > a,A4: set_a,X3: a] :
      ( ( inj_on_a_a @ F2 @ ( image_a_a2 @ G @ A4 ) )
     => ( ( inj_on_a_a @ G @ A4 )
       => ( ( member_a @ X3 @ ( image_a_a2 @ F2 @ ( image_a_a2 @ G @ A4 ) ) )
         => ( ( the_inv_into_a_a @ A4 @ ( comp_a_a_a @ F2 @ G ) @ X3 )
            = ( comp_a_a_a @ ( the_inv_into_a_a @ A4 @ G ) @ ( the_inv_into_a_a @ ( image_a_a2 @ G @ A4 ) @ F2 ) @ X3 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1025_the__inv__into__comp,axiom,
    ! [F2: set_b > b,G: a > set_b,A4: set_a,X3: b] :
      ( ( inj_on_set_b_b @ F2 @ ( image_a_set_b @ G @ A4 ) )
     => ( ( inj_on_a_set_b @ G @ A4 )
       => ( ( member_b @ X3 @ ( image_set_b_b @ F2 @ ( image_a_set_b @ G @ A4 ) ) )
         => ( ( the_inv_into_a_b @ A4 @ ( comp_set_b_b_a @ F2 @ G ) @ X3 )
            = ( comp_set_b_a_b @ ( the_inv_into_a_set_b @ A4 @ G ) @ ( the_inv_into_set_b_b @ ( image_a_set_b @ G @ A4 ) @ F2 ) @ X3 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1026_the__inv__into__comp,axiom,
    ! [F2: a > set_b,G: a > a,A4: set_a,X3: set_b] :
      ( ( inj_on_a_set_b @ F2 @ ( image_a_a2 @ G @ A4 ) )
     => ( ( inj_on_a_a @ G @ A4 )
       => ( ( member_set_b @ X3 @ ( image_a_set_b @ F2 @ ( image_a_a2 @ G @ A4 ) ) )
         => ( ( the_inv_into_a_set_b @ A4 @ ( comp_a_set_b_a @ F2 @ G ) @ X3 )
            = ( comp_a_a_set_b @ ( the_inv_into_a_a @ A4 @ G ) @ ( the_inv_into_a_set_b @ ( image_a_a2 @ G @ A4 ) @ F2 ) @ X3 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1027_the__inv__into__comp,axiom,
    ! [F2: a > set_a,G: a > a,A4: set_a,X3: set_a] :
      ( ( inj_on_a_set_a @ F2 @ ( image_a_a2 @ G @ A4 ) )
     => ( ( inj_on_a_a @ G @ A4 )
       => ( ( member_set_a @ X3 @ ( image_a_set_a @ F2 @ ( image_a_a2 @ G @ A4 ) ) )
         => ( ( the_inv_into_a_set_a @ A4 @ ( comp_a_set_a_a @ F2 @ G ) @ X3 )
            = ( comp_a_a_set_a @ ( the_inv_into_a_a @ A4 @ G ) @ ( the_inv_into_a_set_a @ ( image_a_a2 @ G @ A4 ) @ F2 ) @ X3 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1028_the__inv__into__comp,axiom,
    ! [F2: set_b > a,G: a > set_b,A4: set_a,X3: a] :
      ( ( inj_on_set_b_a @ F2 @ ( image_a_set_b @ G @ A4 ) )
     => ( ( inj_on_a_set_b @ G @ A4 )
       => ( ( member_a @ X3 @ ( image_set_b_a @ F2 @ ( image_a_set_b @ G @ A4 ) ) )
         => ( ( the_inv_into_a_a @ A4 @ ( comp_set_b_a_a @ F2 @ G ) @ X3 )
            = ( comp_set_b_a_a @ ( the_inv_into_a_set_b @ A4 @ G ) @ ( the_inv_into_set_b_a @ ( image_a_set_b @ G @ A4 ) @ F2 ) @ X3 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1029_the__inv__into__comp,axiom,
    ! [F2: b > a,G: product_prod_a_b > b,A4: set_Product_prod_a_b,X3: a] :
      ( ( inj_on_b_a @ F2 @ ( image_2802296252294471260_a_b_b @ G @ A4 ) )
     => ( ( inj_on4343330552946057672_a_b_b @ G @ A4 )
       => ( ( member_a @ X3 @ ( image_b_a @ F2 @ ( image_2802296252294471260_a_b_b @ G @ A4 ) ) )
         => ( ( the_in1061761310735001609_a_b_a @ A4 @ ( comp_b4526394929206263814od_a_b @ F2 @ G ) @ X3 )
            = ( comp_b9151437431462286386_a_b_a @ ( the_in1061761310735001610_a_b_b @ A4 @ G ) @ ( the_inv_into_b_a @ ( image_2802296252294471260_a_b_b @ G @ A4 ) @ F2 ) @ X3 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1030_the__inv__into__comp,axiom,
    ! [F2: option_list_o > a,G: b > option_list_o,A4: set_b,X3: a] :
      ( ( inj_on2088879393520314763st_o_a @ F2 @ ( image_3819732016193546410list_o @ G @ A4 ) )
     => ( ( inj_on8261448415883032086list_o @ G @ A4 )
       => ( ( member_a @ X3 @ ( image_6870535030685604895st_o_a @ F2 @ ( image_3819732016193546410list_o @ G @ A4 ) ) )
         => ( ( the_inv_into_b_a @ A4 @ ( comp_o695885166685416002_o_a_b @ F2 @ G ) @ X3 )
            = ( comp_o7131801621567515776_o_b_a @ ( the_in1091250017314962264list_o @ A4 @ G ) @ ( the_in4142053031807020749st_o_a @ ( image_3819732016193546410list_o @ G @ A4 ) @ F2 ) @ X3 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1031_the__inv__into__comp,axiom,
    ! [F2: b > product_prod_a_b,G: b > b,A4: set_b,X3: product_prod_a_b] :
      ( ( inj_on8302219787212994778od_a_b @ F2 @ ( image_b_b @ G @ A4 ) )
     => ( ( inj_on_b_b @ G @ A4 )
       => ( ( member1426531481828664017od_a_b @ X3 @ ( image_6761185486561408366od_a_b @ F2 @ ( image_b_b @ G @ A4 ) ) )
         => ( ( the_in5020650545001938716od_a_b @ A4 @ ( comp_b9151437431462286387_a_b_b @ F2 @ G ) @ X3 )
            = ( comp_b3886954628874447685od_a_b @ ( the_inv_into_b_b @ A4 @ G ) @ ( the_in5020650545001938716od_a_b @ ( image_b_b @ G @ A4 ) @ F2 ) @ X3 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1032_the__inv__into__comp,axiom,
    ! [F2: a > product_prod_a_b,G: b > a,A4: set_b,X3: product_prod_a_b] :
      ( ( inj_on8941660087544810907od_a_b @ F2 @ ( image_b_a @ G @ A4 ) )
     => ( ( inj_on_b_a @ G @ A4 )
       => ( ( member1426531481828664017od_a_b @ X3 @ ( image_7400625786893224495od_a_b @ F2 @ ( image_b_a @ G @ A4 ) ) )
         => ( ( the_in5020650545001938716od_a_b @ A4 @ ( comp_a5211488844837450162_a_b_b @ F2 @ G ) @ X3 )
            = ( comp_a9170378079104387268od_a_b @ ( the_inv_into_b_a @ A4 @ G ) @ ( the_in5660090845333754845od_a_b @ ( image_b_a @ G @ A4 ) @ F2 ) @ X3 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1033_the__inv__into__onto,axiom,
    ! [F2: a > a,A4: set_a] :
      ( ( inj_on_a_a @ F2 @ A4 )
     => ( ( image_a_a2 @ ( the_inv_into_a_a @ A4 @ F2 ) @ ( image_a_a2 @ F2 @ A4 ) )
        = A4 ) ) ).

% the_inv_into_onto
thf(fact_1034_the__inv__into__onto,axiom,
    ! [F2: a > set_b,A4: set_a] :
      ( ( inj_on_a_set_b @ F2 @ A4 )
     => ( ( image_set_b_a @ ( the_inv_into_a_set_b @ A4 @ F2 ) @ ( image_a_set_b @ F2 @ A4 ) )
        = A4 ) ) ).

% the_inv_into_onto
thf(fact_1035_the__inv__into__onto,axiom,
    ! [F2: set_b > a,A4: set_set_b] :
      ( ( inj_on_set_b_a @ F2 @ A4 )
     => ( ( image_a_set_b @ ( the_inv_into_set_b_a @ A4 @ F2 ) @ ( image_set_b_a @ F2 @ A4 ) )
        = A4 ) ) ).

% the_inv_into_onto
thf(fact_1036_the__inv__into__onto,axiom,
    ! [F2: product_prod_a_b > b,A4: set_Product_prod_a_b] :
      ( ( inj_on4343330552946057672_a_b_b @ F2 @ A4 )
     => ( ( image_6761185486561408366od_a_b @ ( the_in1061761310735001610_a_b_b @ A4 @ F2 ) @ ( image_2802296252294471260_a_b_b @ F2 @ A4 ) )
        = A4 ) ) ).

% the_inv_into_onto
thf(fact_1037_the__inv__into__onto,axiom,
    ! [F2: b > product_prod_a_b,A4: set_b] :
      ( ( inj_on8302219787212994778od_a_b @ F2 @ A4 )
     => ( ( image_2802296252294471260_a_b_b @ ( the_in5020650545001938716od_a_b @ A4 @ F2 ) @ ( image_6761185486561408366od_a_b @ F2 @ A4 ) )
        = A4 ) ) ).

% the_inv_into_onto
thf(fact_1038_the__inv__into__onto,axiom,
    ! [F2: a > product_prod_a_b,A4: set_a] :
      ( ( inj_on8941660087544810907od_a_b @ F2 @ A4 )
     => ( ( image_2802296252294471259_a_b_a @ ( the_in5660090845333754845od_a_b @ A4 @ F2 ) @ ( image_7400625786893224495od_a_b @ F2 @ A4 ) )
        = A4 ) ) ).

% the_inv_into_onto
thf(fact_1039_the__inv__into__onto,axiom,
    ! [F2: b > option_list_o,A4: set_b] :
      ( ( inj_on8261448415883032086list_o @ F2 @ A4 )
     => ( ( image_6870535030685604896st_o_b @ ( the_in1091250017314962264list_o @ A4 @ F2 ) @ ( image_3819732016193546410list_o @ F2 @ A4 ) )
        = A4 ) ) ).

% the_inv_into_onto
thf(fact_1040_the__inv__into__onto,axiom,
    ! [F2: a > option_list_o,A4: set_a] :
      ( ( inj_on374126998980950615list_o @ F2 @ A4 )
     => ( ( image_6870535030685604895st_o_a @ ( the_in2427300637267656601list_o @ A4 @ F2 ) @ ( image_5155782636146240747list_o @ F2 @ A4 ) )
        = A4 ) ) ).

% the_inv_into_onto
thf(fact_1041_the__inv__into__onto,axiom,
    ! [F2: product_prod_a_b > a,A4: set_Product_prod_a_b] :
      ( ( inj_on4343330552946057671_a_b_a @ F2 @ A4 )
     => ( ( image_7400625786893224495od_a_b @ ( the_in1061761310735001609_a_b_a @ A4 @ F2 ) @ ( image_2802296252294471259_a_b_a @ F2 @ A4 ) )
        = A4 ) ) ).

% the_inv_into_onto
thf(fact_1042_the__inv__into__onto,axiom,
    ! [F2: a > produc5884233991663340231list_o,A4: set_a] :
      ( ( inj_on6352985132387159994list_o @ F2 @ A4 )
     => ( ( image_8953000392185216508st_o_a @ ( the_in7435632087746858492list_o @ A4 @ F2 ) @ ( image_1329905666413968462list_o @ F2 @ A4 ) )
        = A4 ) ) ).

% the_inv_into_onto
thf(fact_1043_surj__swap,axiom,
    ( ( image_8576602756132944327od_a_b @ product_swap_b_a @ top_to7427722431652660502od_b_a )
    = top_to8134405472303993176od_a_b ) ).

% surj_swap
thf(fact_1044_surj__swap,axiom,
    ( ( image_513147963279508931od_b_a @ product_swap_a_b @ top_to8134405472303993176od_a_b )
    = top_to7427722431652660502od_b_a ) ).

% surj_swap
thf(fact_1045_convol__image__vimage2p,axiom,
    ! [F2: nat > nat,G: nat > nat,R5: nat > nat > $o] : ( ord_le3146513528884898305at_nat @ ( image_5168914502847457605at_nat @ ( bNF_co8276296344278409722at_nat @ ( comp_n2711240574518091068at_nat @ F2 @ product_fst_nat_nat ) @ ( comp_n2711240574518091068at_nat @ G @ product_snd_nat_nat ) ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( bNF_vi6667456707345531737_nat_o @ F2 @ G @ R5 ) ) ) ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R5 ) ) ) ).

% convol_image_vimage2p
thf(fact_1046_convol__image__vimage2p,axiom,
    ! [F2: b > nat,G: a > nat,R5: nat > nat > $o] : ( ord_le3146513528884898305at_nat @ ( image_9012383631328784800at_nat @ ( bNF_co1811873346508511455at_nat @ ( comp_b1535156575568618168od_b_a @ F2 @ product_fst_b_a ) @ ( comp_a294031402070750073od_b_a @ G @ product_snd_b_a ) ) @ ( collec548942215411776465od_b_a @ ( produc7672072387910335853_b_a_o @ ( bNF_vi3961714438943086494_nat_o @ F2 @ G @ R5 ) ) ) ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R5 ) ) ) ).

% convol_image_vimage2p
thf(fact_1047_convol__image__vimage2p,axiom,
    ! [F2: b > nat,G: b > nat,R5: nat > nat > $o] : ( ord_le3146513528884898305at_nat @ ( image_1030136767971877087at_nat @ ( bNF_co3087766193879408736at_nat @ ( comp_b1535156579871846969od_b_b @ F2 @ product_fst_b_b ) @ ( comp_b1535156579871846969od_b_b @ G @ product_snd_b_b ) ) @ ( collec548942219715005266od_b_b @ ( produc4772849988445941548_b_b_o @ ( bNF_vi1076643929659559261_nat_o @ F2 @ G @ R5 ) ) ) ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R5 ) ) ) ).

% convol_image_vimage2p
thf(fact_1048_convol__image__vimage2p,axiom,
    ! [F2: a > nat,G: a > nat,R5: nat > nat > $o] : ( ord_le3146513528884898305at_nat @ ( image_257580707435209375at_nat @ ( bNF_co8117653604839296672at_nat @ ( comp_a3081486984043426106od_a_a @ F2 @ product_fst_a_a ) @ ( comp_a3081486984043426106od_a_a @ G @ product_snd_a_a ) ) @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ ( bNF_vi7065238521659171741_nat_o @ F2 @ G @ R5 ) ) ) ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R5 ) ) ) ).

% convol_image_vimage2p
thf(fact_1049_convol__image__vimage2p,axiom,
    ! [F2: nat > a,G: nat > a,R5: a > a > $o] : ( ord_le746702958409616551od_a_a @ ( image_4894260356024038251od_a_a @ ( bNF_co4831131333957098072at_a_a @ ( comp_n7987077848806063030at_nat @ F2 @ product_fst_nat_nat ) @ ( comp_n7987077848806063030at_nat @ G @ product_snd_nat_nat ) ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( bNF_vi53994321937086493at_a_o @ F2 @ G @ R5 ) ) ) ) @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ R5 ) ) ) ).

% convol_image_vimage2p
thf(fact_1050_convol__image__vimage2p,axiom,
    ! [F2: b > a,G: a > a,R5: a > a > $o] : ( ord_le746702958409616551od_a_a @ ( image_8576602751829715526od_a_a @ ( bNF_co7991273656122196669_a_a_a @ ( comp_b1738939342930358980od_b_a @ F2 @ product_fst_b_a ) @ ( comp_a7022362793160298563od_b_a @ G @ product_snd_b_a ) ) @ ( collec548942215411776465od_b_a @ ( produc7672072387910335853_b_a_o @ ( bNF_vi277092895069316382_a_a_o @ F2 @ G @ R5 ) ) ) ) @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ R5 ) ) ) ).

% convol_image_vimage2p
thf(fact_1051_convol__image__vimage2p,axiom,
    ! [F2: b > a,G: b > a,R5: a > a > $o] : ( ord_le746702958409616551od_a_a @ ( image_7240552131877021189od_a_a @ ( bNF_co5212569348262174142_b_a_a @ ( comp_b1738939347233587781od_b_b @ F2 @ product_fst_b_b ) @ ( comp_b1738939347233587781od_b_b @ G @ product_snd_b_b ) ) @ ( collec548942219715005266od_b_b @ ( produc4772849988445941548_b_b_o @ ( bNF_vi1512537224025710879_b_a_o @ F2 @ G @ R5 ) ) ) ) @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ R5 ) ) ) ).

% convol_image_vimage2p
thf(fact_1052_convol__image__vimage2p,axiom,
    ! [F2: a > a,G: a > a,R5: a > a > $o] : ( ord_le746702958409616551od_a_a @ ( image_4636654165204879301od_a_a @ ( bNF_co6503311618827398142_a_a_a @ ( comp_a586446338278198788od_a_a @ F2 @ product_fst_a_a ) @ ( comp_a586446338278198788od_a_a @ G @ product_snd_a_a ) ) @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ ( bNF_vi8224572084553194909_a_a_o @ F2 @ G @ R5 ) ) ) ) @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ R5 ) ) ) ).

% convol_image_vimage2p
thf(fact_1053_convol__image__vimage2p,axiom,
    ! [F2: nat > a,G: nat > b,R5: a > b > $o] : ( ord_le817736998455962536od_a_b @ ( image_4894260360327267052od_a_b @ ( bNF_co4831131333957098073at_a_b @ ( comp_n7987077848806063030at_nat @ F2 @ product_fst_nat_nat ) @ ( comp_n8333496002694486645at_nat @ G @ product_snd_nat_nat ) ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( bNF_vi6378143959327467996at_b_o @ F2 @ G @ R5 ) ) ) ) @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ R5 ) ) ) ).

% convol_image_vimage2p
thf(fact_1054_convol__image__vimage2p,axiom,
    ! [F2: b > a,G: a > b,R5: a > b > $o] : ( ord_le817736998455962536od_a_b @ ( image_8576602756132944327od_a_b @ ( bNF_co7991273656122196670_a_a_b @ ( comp_b1738939342930358980od_b_a @ F2 @ product_fst_b_a ) @ ( comp_a6382922492828482434od_b_a @ G @ product_snd_b_a ) ) @ ( collec548942215411776465od_b_a @ ( produc7672072387910335853_b_a_o @ ( bNF_vi6601242532459697885_a_b_o @ F2 @ G @ R5 ) ) ) ) @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ R5 ) ) ) ).

% convol_image_vimage2p
thf(fact_1055_range__fst,axiom,
    ( ( image_3437945252899457948_a_a_a @ product_fst_a_a @ top_to8063371432257647191od_a_a )
    = top_top_set_a ) ).

% range_fst
thf(fact_1056_range__fst,axiom,
    ( ( image_8953000392185216508st_o_a @ product_fst_a_list_o @ top_to7613047688909597687list_o )
    = top_top_set_a ) ).

% range_fst
thf(fact_1057_range__fst,axiom,
    ( ( image_9034163868087588638_b_a_b @ product_fst_b_a @ top_to7427722431652660502od_b_a )
    = top_top_set_b ) ).

% range_fst
thf(fact_1058_range__fst,axiom,
    ( ( image_8398514867482601949_b_b_b @ product_fst_b_b @ top_to7498756471699006487od_b_b )
    = top_top_set_b ) ).

% range_fst
thf(fact_1059_range__fst,axiom,
    ( ( image_2802296252294471259_a_b_a @ product_fst_a_b @ top_to8134405472303993176od_a_b )
    = top_top_set_a ) ).

% range_fst
thf(fact_1060_range__snd,axiom,
    ( ( image_9034163868087588637_b_a_a @ product_snd_b_a @ top_to7427722431652660502od_b_a )
    = top_top_set_a ) ).

% range_snd
thf(fact_1061_range__snd,axiom,
    ( ( image_3437945252899457948_a_a_a @ product_snd_a_a @ top_to8063371432257647191od_a_a )
    = top_top_set_a ) ).

% range_snd
thf(fact_1062_range__snd,axiom,
    ( ( image_8398514867482601949_b_b_b @ product_snd_b_b @ top_to7498756471699006487od_b_b )
    = top_top_set_b ) ).

% range_snd
thf(fact_1063_range__snd,axiom,
    ( ( image_2802296252294471260_a_b_b @ product_snd_a_b @ top_to8134405472303993176od_a_b )
    = top_top_set_b ) ).

% range_snd
thf(fact_1064_inj__apfst,axiom,
    ! [F2: a > option_list_o] :
      ( ( inj_on5101338214896250123st_o_b @ ( produc7556171008775904733st_o_b @ F2 ) @ top_to8134405472303993176od_a_b )
      = ( inj_on374126998980950615list_o @ F2 @ top_top_set_a ) ) ).

% inj_apfst
thf(fact_1065_inj__apfst,axiom,
    ! [F2: a > a] :
      ( ( inj_on1230094055151127153od_a_b @ ( product_apfst_a_a_b @ F2 ) @ top_to8134405472303993176od_a_b )
      = ( inj_on_a_a @ F2 @ top_top_set_a ) ) ).

% inj_apfst
thf(fact_1066_inj__apsnd,axiom,
    ! [F2: b > option_list_o] :
      ( ( inj_on112764807354443094list_o @ ( produc1993804381082613787st_o_a @ F2 ) @ top_to8134405472303993176od_a_b )
      = ( inj_on8261448415883032086list_o @ F2 @ top_top_set_b ) ) ).

% inj_apsnd
thf(fact_1067_inj__apsnd,axiom,
    ! [F2: b > b] :
      ( ( inj_on1230094055151127153od_a_b @ ( product_apsnd_b_b_a @ F2 ) @ top_to8134405472303993176od_a_b )
      = ( inj_on_b_b @ F2 @ top_top_set_b ) ) ).

% inj_apsnd
thf(fact_1068_surj__Compl__image__subset,axiom,
    ! [F2: a > a,A4: set_a] :
      ( ( ( image_a_a2 @ F2 @ top_top_set_a )
        = top_top_set_a )
     => ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ ( image_a_a2 @ F2 @ A4 ) ) @ ( image_a_a2 @ F2 @ ( uminus_uminus_set_a @ A4 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_1069_surj__Compl__image__subset,axiom,
    ! [F2: b > a,A4: set_b] :
      ( ( ( image_b_a @ F2 @ top_top_set_b )
        = top_top_set_a )
     => ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ ( image_b_a @ F2 @ A4 ) ) @ ( image_b_a @ F2 @ ( uminus_uminus_set_b @ A4 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_1070_surj__Compl__image__subset,axiom,
    ! [F2: a > b,A4: set_a] :
      ( ( ( image_a_b @ F2 @ top_top_set_a )
        = top_top_set_b )
     => ( ord_less_eq_set_b @ ( uminus_uminus_set_b @ ( image_a_b @ F2 @ A4 ) ) @ ( image_a_b @ F2 @ ( uminus_uminus_set_a @ A4 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_1071_surj__Compl__image__subset,axiom,
    ! [F2: b > b,A4: set_b] :
      ( ( ( image_b_b @ F2 @ top_top_set_b )
        = top_top_set_b )
     => ( ord_less_eq_set_b @ ( uminus_uminus_set_b @ ( image_b_b @ F2 @ A4 ) ) @ ( image_b_b @ F2 @ ( uminus_uminus_set_b @ A4 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_1072_surj__Compl__image__subset,axiom,
    ! [F2: a > set_b,A4: set_a] :
      ( ( ( image_a_set_b @ F2 @ top_top_set_a )
        = top_top_set_set_b )
     => ( ord_le3795704787696855135_set_b @ ( uminus6174936397961129654_set_b @ ( image_a_set_b @ F2 @ A4 ) ) @ ( image_a_set_b @ F2 @ ( uminus_uminus_set_a @ A4 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_1073_surj__Compl__image__subset,axiom,
    ! [F2: a > list_o,A4: set_a] :
      ( ( ( image_a_list_o @ F2 @ top_top_set_a )
        = top_top_set_list_o )
     => ( ord_le6901083488122529182list_o @ ( uminus56943061532027893list_o @ ( image_a_list_o @ F2 @ A4 ) ) @ ( image_a_list_o @ F2 @ ( uminus_uminus_set_a @ A4 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_1074_surj__Compl__image__subset,axiom,
    ! [F2: b > list_o,A4: set_b] :
      ( ( ( image_b_list_o @ F2 @ top_top_set_b )
        = top_top_set_list_o )
     => ( ord_le6901083488122529182list_o @ ( uminus56943061532027893list_o @ ( image_b_list_o @ F2 @ A4 ) ) @ ( image_b_list_o @ F2 @ ( uminus_uminus_set_b @ A4 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_1075_surj__Compl__image__subset,axiom,
    ! [F2: a > product_prod_a_b,A4: set_a] :
      ( ( ( image_7400625786893224495od_a_b @ F2 @ top_top_set_a )
        = top_to8134405472303993176od_a_b )
     => ( ord_le817736998455962536od_a_b @ ( uminus5601964437033819903od_a_b @ ( image_7400625786893224495od_a_b @ F2 @ A4 ) ) @ ( image_7400625786893224495od_a_b @ F2 @ ( uminus_uminus_set_a @ A4 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_1076_surj__Compl__image__subset,axiom,
    ! [F2: b > product_prod_a_b,A4: set_b] :
      ( ( ( image_6761185486561408366od_a_b @ F2 @ top_top_set_b )
        = top_to8134405472303993176od_a_b )
     => ( ord_le817736998455962536od_a_b @ ( uminus5601964437033819903od_a_b @ ( image_6761185486561408366od_a_b @ F2 @ A4 ) ) @ ( image_6761185486561408366od_a_b @ F2 @ ( uminus_uminus_set_b @ A4 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_1077_surj__Compl__image__subset,axiom,
    ! [F2: product_prod_a_b > a,A4: set_Product_prod_a_b] :
      ( ( ( image_2802296252294471259_a_b_a @ F2 @ top_to8134405472303993176od_a_b )
        = top_top_set_a )
     => ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ ( image_2802296252294471259_a_b_a @ F2 @ A4 ) ) @ ( image_2802296252294471259_a_b_a @ F2 @ ( uminus5601964437033819903od_a_b @ A4 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_1078_the__inv__f__f,axiom,
    ! [F2: product_prod_a_b > a,X3: product_prod_a_b] :
      ( ( inj_on4343330552946057671_a_b_a @ F2 @ top_to8134405472303993176od_a_b )
     => ( ( the_in1061761310735001609_a_b_a @ top_to8134405472303993176od_a_b @ F2 @ ( F2 @ X3 ) )
        = X3 ) ) ).

% the_inv_f_f
thf(fact_1079_the__inv__f__f,axiom,
    ! [F2: a > option_list_o,X3: a] :
      ( ( inj_on374126998980950615list_o @ F2 @ top_top_set_a )
     => ( ( the_in2427300637267656601list_o @ top_top_set_a @ F2 @ ( F2 @ X3 ) )
        = X3 ) ) ).

% the_inv_f_f
thf(fact_1080_the__inv__f__f,axiom,
    ! [F2: a > a,X3: a] :
      ( ( inj_on_a_a @ F2 @ top_top_set_a )
     => ( ( the_inv_into_a_a @ top_top_set_a @ F2 @ ( F2 @ X3 ) )
        = X3 ) ) ).

% the_inv_f_f
thf(fact_1081_the__inv__f__f,axiom,
    ! [F2: b > option_list_o,X3: b] :
      ( ( inj_on8261448415883032086list_o @ F2 @ top_top_set_b )
     => ( ( the_in1091250017314962264list_o @ top_top_set_b @ F2 @ ( F2 @ X3 ) )
        = X3 ) ) ).

% the_inv_f_f
thf(fact_1082_inj__image__Compl__subset,axiom,
    ! [F2: a > a,A4: set_a] :
      ( ( inj_on_a_a @ F2 @ top_top_set_a )
     => ( ord_less_eq_set_a @ ( image_a_a2 @ F2 @ ( uminus_uminus_set_a @ A4 ) ) @ ( uminus_uminus_set_a @ ( image_a_a2 @ F2 @ A4 ) ) ) ) ).

% inj_image_Compl_subset
thf(fact_1083_inj__image__Compl__subset,axiom,
    ! [F2: b > a,A4: set_b] :
      ( ( inj_on_b_a @ F2 @ top_top_set_b )
     => ( ord_less_eq_set_a @ ( image_b_a @ F2 @ ( uminus_uminus_set_b @ A4 ) ) @ ( uminus_uminus_set_a @ ( image_b_a @ F2 @ A4 ) ) ) ) ).

% inj_image_Compl_subset
thf(fact_1084_inj__image__Compl__subset,axiom,
    ! [F2: a > b,A4: set_a] :
      ( ( inj_on_a_b @ F2 @ top_top_set_a )
     => ( ord_less_eq_set_b @ ( image_a_b @ F2 @ ( uminus_uminus_set_a @ A4 ) ) @ ( uminus_uminus_set_b @ ( image_a_b @ F2 @ A4 ) ) ) ) ).

% inj_image_Compl_subset
thf(fact_1085_inj__image__Compl__subset,axiom,
    ! [F2: b > b,A4: set_b] :
      ( ( inj_on_b_b @ F2 @ top_top_set_b )
     => ( ord_less_eq_set_b @ ( image_b_b @ F2 @ ( uminus_uminus_set_b @ A4 ) ) @ ( uminus_uminus_set_b @ ( image_b_b @ F2 @ A4 ) ) ) ) ).

% inj_image_Compl_subset
thf(fact_1086_inj__image__Compl__subset,axiom,
    ! [F2: a > set_b,A4: set_a] :
      ( ( inj_on_a_set_b @ F2 @ top_top_set_a )
     => ( ord_le3795704787696855135_set_b @ ( image_a_set_b @ F2 @ ( uminus_uminus_set_a @ A4 ) ) @ ( uminus6174936397961129654_set_b @ ( image_a_set_b @ F2 @ A4 ) ) ) ) ).

% inj_image_Compl_subset
thf(fact_1087_inj__image__Compl__subset,axiom,
    ! [F2: a > list_o,A4: set_a] :
      ( ( inj_on_a_list_o @ F2 @ top_top_set_a )
     => ( ord_le6901083488122529182list_o @ ( image_a_list_o @ F2 @ ( uminus_uminus_set_a @ A4 ) ) @ ( uminus56943061532027893list_o @ ( image_a_list_o @ F2 @ A4 ) ) ) ) ).

% inj_image_Compl_subset
thf(fact_1088_inj__image__Compl__subset,axiom,
    ! [F2: b > list_o,A4: set_b] :
      ( ( inj_on_b_list_o @ F2 @ top_top_set_b )
     => ( ord_le6901083488122529182list_o @ ( image_b_list_o @ F2 @ ( uminus_uminus_set_b @ A4 ) ) @ ( uminus56943061532027893list_o @ ( image_b_list_o @ F2 @ A4 ) ) ) ) ).

% inj_image_Compl_subset
thf(fact_1089_inj__image__Compl__subset,axiom,
    ! [F2: a > option_list_o,A4: set_a] :
      ( ( inj_on374126998980950615list_o @ F2 @ top_top_set_a )
     => ( ord_le1162937763994921316list_o @ ( image_5155782636146240747list_o @ F2 @ ( uminus_uminus_set_a @ A4 ) ) @ ( uminus2228965239982383419list_o @ ( image_5155782636146240747list_o @ F2 @ A4 ) ) ) ) ).

% inj_image_Compl_subset
thf(fact_1090_inj__image__Compl__subset,axiom,
    ! [F2: b > option_list_o,A4: set_b] :
      ( ( inj_on8261448415883032086list_o @ F2 @ top_top_set_b )
     => ( ord_le1162937763994921316list_o @ ( image_3819732016193546410list_o @ F2 @ ( uminus_uminus_set_b @ A4 ) ) @ ( uminus2228965239982383419list_o @ ( image_3819732016193546410list_o @ F2 @ A4 ) ) ) ) ).

% inj_image_Compl_subset
thf(fact_1091_inj__image__Compl__subset,axiom,
    ! [F2: a > product_prod_a_b,A4: set_a] :
      ( ( inj_on8941660087544810907od_a_b @ F2 @ top_top_set_a )
     => ( ord_le817736998455962536od_a_b @ ( image_7400625786893224495od_a_b @ F2 @ ( uminus_uminus_set_a @ A4 ) ) @ ( uminus5601964437033819903od_a_b @ ( image_7400625786893224495od_a_b @ F2 @ A4 ) ) ) ) ).

% inj_image_Compl_subset
thf(fact_1092_inj__image__subset__iff,axiom,
    ! [F2: a > a,A4: set_a,B5: set_a] :
      ( ( inj_on_a_a @ F2 @ top_top_set_a )
     => ( ( ord_less_eq_set_a @ ( image_a_a2 @ F2 @ A4 ) @ ( image_a_a2 @ F2 @ B5 ) )
        = ( ord_less_eq_set_a @ A4 @ B5 ) ) ) ).

% inj_image_subset_iff
thf(fact_1093_inj__image__subset__iff,axiom,
    ! [F2: b > a,A4: set_b,B5: set_b] :
      ( ( inj_on_b_a @ F2 @ top_top_set_b )
     => ( ( ord_less_eq_set_a @ ( image_b_a @ F2 @ A4 ) @ ( image_b_a @ F2 @ B5 ) )
        = ( ord_less_eq_set_b @ A4 @ B5 ) ) ) ).

% inj_image_subset_iff
thf(fact_1094_inj__image__subset__iff,axiom,
    ! [F2: a > b,A4: set_a,B5: set_a] :
      ( ( inj_on_a_b @ F2 @ top_top_set_a )
     => ( ( ord_less_eq_set_b @ ( image_a_b @ F2 @ A4 ) @ ( image_a_b @ F2 @ B5 ) )
        = ( ord_less_eq_set_a @ A4 @ B5 ) ) ) ).

% inj_image_subset_iff
thf(fact_1095_inj__image__subset__iff,axiom,
    ! [F2: b > b,A4: set_b,B5: set_b] :
      ( ( inj_on_b_b @ F2 @ top_top_set_b )
     => ( ( ord_less_eq_set_b @ ( image_b_b @ F2 @ A4 ) @ ( image_b_b @ F2 @ B5 ) )
        = ( ord_less_eq_set_b @ A4 @ B5 ) ) ) ).

% inj_image_subset_iff
thf(fact_1096_inj__image__subset__iff,axiom,
    ! [F2: a > set_b,A4: set_a,B5: set_a] :
      ( ( inj_on_a_set_b @ F2 @ top_top_set_a )
     => ( ( ord_le3795704787696855135_set_b @ ( image_a_set_b @ F2 @ A4 ) @ ( image_a_set_b @ F2 @ B5 ) )
        = ( ord_less_eq_set_a @ A4 @ B5 ) ) ) ).

% inj_image_subset_iff
thf(fact_1097_inj__image__subset__iff,axiom,
    ! [F2: list_o > a,A4: set_list_o,B5: set_list_o] :
      ( ( inj_on_list_o_a @ F2 @ top_top_set_list_o )
     => ( ( ord_less_eq_set_a @ ( image_list_o_a @ F2 @ A4 ) @ ( image_list_o_a @ F2 @ B5 ) )
        = ( ord_le6901083488122529182list_o @ A4 @ B5 ) ) ) ).

% inj_image_subset_iff
thf(fact_1098_inj__image__subset__iff,axiom,
    ! [F2: list_o > b,A4: set_list_o,B5: set_list_o] :
      ( ( inj_on_list_o_b @ F2 @ top_top_set_list_o )
     => ( ( ord_less_eq_set_b @ ( image_list_o_b @ F2 @ A4 ) @ ( image_list_o_b @ F2 @ B5 ) )
        = ( ord_le6901083488122529182list_o @ A4 @ B5 ) ) ) ).

% inj_image_subset_iff
thf(fact_1099_inj__image__subset__iff,axiom,
    ! [F2: a > list_o,A4: set_a,B5: set_a] :
      ( ( inj_on_a_list_o @ F2 @ top_top_set_a )
     => ( ( ord_le6901083488122529182list_o @ ( image_a_list_o @ F2 @ A4 ) @ ( image_a_list_o @ F2 @ B5 ) )
        = ( ord_less_eq_set_a @ A4 @ B5 ) ) ) ).

% inj_image_subset_iff
thf(fact_1100_inj__image__subset__iff,axiom,
    ! [F2: b > list_o,A4: set_b,B5: set_b] :
      ( ( inj_on_b_list_o @ F2 @ top_top_set_b )
     => ( ( ord_le6901083488122529182list_o @ ( image_b_list_o @ F2 @ A4 ) @ ( image_b_list_o @ F2 @ B5 ) )
        = ( ord_less_eq_set_b @ A4 @ B5 ) ) ) ).

% inj_image_subset_iff
thf(fact_1101_inj__image__subset__iff,axiom,
    ! [F2: a > option_list_o,A4: set_a,B5: set_a] :
      ( ( inj_on374126998980950615list_o @ F2 @ top_top_set_a )
     => ( ( ord_le1162937763994921316list_o @ ( image_5155782636146240747list_o @ F2 @ A4 ) @ ( image_5155782636146240747list_o @ F2 @ B5 ) )
        = ( ord_less_eq_set_a @ A4 @ B5 ) ) ) ).

% inj_image_subset_iff
thf(fact_1102_fun_Oinj__map,axiom,
    ! [F2: a > product_prod_a_a] :
      ( ( inj_on8941660083241582106od_a_a @ F2 @ top_top_set_a )
     => ( inj_on6028998115040070034od_a_a @ ( comp_a1036870397537576092od_a_b @ F2 ) @ top_to8812525692031969313_a_b_a ) ) ).

% fun.inj_map
thf(fact_1103_fun_Oinj__map,axiom,
    ! [F2: a > b] :
      ( ( inj_on_a_b @ F2 @ top_top_set_a )
     => ( inj_on511531098731434858_a_b_b @ ( comp_a9170378079104387268od_a_b @ F2 ) @ top_to8812525692031969313_a_b_a ) ) ).

% fun.inj_map
thf(fact_1104_fun_Oinj__map,axiom,
    ! [F2: a > a] :
      ( ( inj_on_a_a @ F2 @ top_top_set_a )
     => ( inj_on511531094428206057_a_b_a @ ( comp_a586446342581427589od_a_b @ F2 ) @ top_to8812525692031969313_a_b_a ) ) ).

% fun.inj_map
thf(fact_1105_fun_Oinj__map,axiom,
    ! [F2: a > a] :
      ( ( inj_on_a_a @ F2 @ top_top_set_a )
     => ( inj_on_a_a_a_a @ ( comp_a_a_a @ F2 ) @ top_top_set_a_a ) ) ).

% fun.inj_map
thf(fact_1106_fun_Oinj__map,axiom,
    ! [F2: b > b] :
      ( ( inj_on_b_b @ F2 @ top_top_set_b )
     => ( inj_on2376818068274541547_a_b_b @ ( comp_b3886954628874447685od_a_b @ F2 ) @ top_to8883559732078315298_a_b_b ) ) ).

% fun.inj_map
thf(fact_1107_subrelI,axiom,
    ! [R: set_Product_prod_b_b,S2: set_Product_prod_b_b] :
      ( ! [X2: b,Y2: b] :
          ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X2 @ Y2 ) @ R )
         => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X2 @ Y2 ) @ S2 ) )
     => ( ord_le182087997850975847od_b_b @ R @ S2 ) ) ).

% subrelI
thf(fact_1108_subrelI,axiom,
    ! [R: set_Product_prod_b_a,S2: set_Product_prod_b_a] :
      ( ! [X2: b,Y2: a] :
          ( ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ X2 @ Y2 ) @ R )
         => ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ X2 @ Y2 ) @ S2 ) )
     => ( ord_le111053957804629862od_b_a @ R @ S2 ) ) ).

% subrelI
thf(fact_1109_subrelI,axiom,
    ! [R: set_Pr3077528766752018087list_o,S2: set_Pr3077528766752018087list_o] :
      ( ! [X2: a,Y2: list_o] :
          ( ( member7948383622993546480list_o @ ( produc6899623729582506167list_o @ X2 @ Y2 ) @ R )
         => ( member7948383622993546480list_o @ ( produc6899623729582506167list_o @ X2 @ Y2 ) @ S2 ) )
     => ( ord_le8560161592425875015list_o @ R @ S2 ) ) ).

% subrelI
thf(fact_1110_subrelI,axiom,
    ! [R: set_Product_prod_a_b,S2: set_Product_prod_a_b] :
      ( ! [X2: a,Y2: b] :
          ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X2 @ Y2 ) @ R )
         => ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X2 @ Y2 ) @ S2 ) )
     => ( ord_le817736998455962536od_a_b @ R @ S2 ) ) ).

% subrelI
thf(fact_1111_subrelI,axiom,
    ! [R: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
      ( ! [X2: a,Y2: a] :
          ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ R )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ S2 ) )
     => ( ord_le746702958409616551od_a_a @ R @ S2 ) ) ).

% subrelI
thf(fact_1112_the__inv__into__into,axiom,
    ! [F2: a > a,A4: set_a,X3: a,B5: set_a] :
      ( ( inj_on_a_a @ F2 @ A4 )
     => ( ( member_a @ X3 @ ( image_a_a2 @ F2 @ A4 ) )
       => ( ( ord_less_eq_set_a @ A4 @ B5 )
         => ( member_a @ ( the_inv_into_a_a @ A4 @ F2 @ X3 ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_1113_the__inv__into__into,axiom,
    ! [F2: a > b,A4: set_a,X3: b,B5: set_a] :
      ( ( inj_on_a_b @ F2 @ A4 )
     => ( ( member_b @ X3 @ ( image_a_b @ F2 @ A4 ) )
       => ( ( ord_less_eq_set_a @ A4 @ B5 )
         => ( member_a @ ( the_inv_into_a_b @ A4 @ F2 @ X3 ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_1114_the__inv__into__into,axiom,
    ! [F2: b > a,A4: set_b,X3: a,B5: set_b] :
      ( ( inj_on_b_a @ F2 @ A4 )
     => ( ( member_a @ X3 @ ( image_b_a @ F2 @ A4 ) )
       => ( ( ord_less_eq_set_b @ A4 @ B5 )
         => ( member_b @ ( the_inv_into_b_a @ A4 @ F2 @ X3 ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_1115_the__inv__into__into,axiom,
    ! [F2: b > b,A4: set_b,X3: b,B5: set_b] :
      ( ( inj_on_b_b @ F2 @ A4 )
     => ( ( member_b @ X3 @ ( image_b_b @ F2 @ A4 ) )
       => ( ( ord_less_eq_set_b @ A4 @ B5 )
         => ( member_b @ ( the_inv_into_b_b @ A4 @ F2 @ X3 ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_1116_the__inv__into__into,axiom,
    ! [F2: set_a > a,A4: set_set_a,X3: a,B5: set_set_a] :
      ( ( inj_on_set_a_a @ F2 @ A4 )
     => ( ( member_a @ X3 @ ( image_set_a_a @ F2 @ A4 ) )
       => ( ( ord_le3724670747650509150_set_a @ A4 @ B5 )
         => ( member_set_a @ ( the_inv_into_set_a_a @ A4 @ F2 @ X3 ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_1117_the__inv__into__into,axiom,
    ! [F2: set_a > b,A4: set_set_a,X3: b,B5: set_set_a] :
      ( ( inj_on_set_a_b @ F2 @ A4 )
     => ( ( member_b @ X3 @ ( image_set_a_b @ F2 @ A4 ) )
       => ( ( ord_le3724670747650509150_set_a @ A4 @ B5 )
         => ( member_set_a @ ( the_inv_into_set_a_b @ A4 @ F2 @ X3 ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_1118_the__inv__into__into,axiom,
    ! [F2: a > set_b,A4: set_a,X3: set_b,B5: set_a] :
      ( ( inj_on_a_set_b @ F2 @ A4 )
     => ( ( member_set_b @ X3 @ ( image_a_set_b @ F2 @ A4 ) )
       => ( ( ord_less_eq_set_a @ A4 @ B5 )
         => ( member_a @ ( the_inv_into_a_set_b @ A4 @ F2 @ X3 ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_1119_the__inv__into__into,axiom,
    ! [F2: a > set_a,A4: set_a,X3: set_a,B5: set_a] :
      ( ( inj_on_a_set_a @ F2 @ A4 )
     => ( ( member_set_a @ X3 @ ( image_a_set_a @ F2 @ A4 ) )
       => ( ( ord_less_eq_set_a @ A4 @ B5 )
         => ( member_a @ ( the_inv_into_a_set_a @ A4 @ F2 @ X3 ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_1120_the__inv__into__into,axiom,
    ! [F2: b > set_a,A4: set_b,X3: set_a,B5: set_b] :
      ( ( inj_on_b_set_a @ F2 @ A4 )
     => ( ( member_set_a @ X3 @ ( image_b_set_a @ F2 @ A4 ) )
       => ( ( ord_less_eq_set_b @ A4 @ B5 )
         => ( member_b @ ( the_inv_into_b_set_a @ A4 @ F2 @ X3 ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_1121_the__inv__into__into,axiom,
    ! [F2: list_o > a,A4: set_list_o,X3: a,B5: set_list_o] :
      ( ( inj_on_list_o_a @ F2 @ A4 )
     => ( ( member_a @ X3 @ ( image_list_o_a @ F2 @ A4 ) )
       => ( ( ord_le6901083488122529182list_o @ A4 @ B5 )
         => ( member_list_o @ ( the_in5258099993405090195st_o_a @ A4 @ F2 @ X3 ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_1122_surj__def,axiom,
    ! [F2: a > a] :
      ( ( ( image_a_a2 @ F2 @ top_top_set_a )
        = top_top_set_a )
      = ( ! [Y: a] :
          ? [X: a] :
            ( Y
            = ( F2 @ X ) ) ) ) ).

% surj_def
thf(fact_1123_surj__def,axiom,
    ! [F2: a > b] :
      ( ( ( image_a_b @ F2 @ top_top_set_a )
        = top_top_set_b )
      = ( ! [Y: b] :
          ? [X: a] :
            ( Y
            = ( F2 @ X ) ) ) ) ).

% surj_def
thf(fact_1124_surj__def,axiom,
    ! [F2: b > a] :
      ( ( ( image_b_a @ F2 @ top_top_set_b )
        = top_top_set_a )
      = ( ! [Y: a] :
          ? [X: b] :
            ( Y
            = ( F2 @ X ) ) ) ) ).

% surj_def
thf(fact_1125_surj__def,axiom,
    ! [F2: b > b] :
      ( ( ( image_b_b @ F2 @ top_top_set_b )
        = top_top_set_b )
      = ( ! [Y: b] :
          ? [X: b] :
            ( Y
            = ( F2 @ X ) ) ) ) ).

% surj_def
thf(fact_1126_surj__def,axiom,
    ! [F2: a > set_b] :
      ( ( ( image_a_set_b @ F2 @ top_top_set_a )
        = top_top_set_set_b )
      = ( ! [Y: set_b] :
          ? [X: a] :
            ( Y
            = ( F2 @ X ) ) ) ) ).

% surj_def
thf(fact_1127_surj__def,axiom,
    ! [F2: product_prod_a_b > a] :
      ( ( ( image_2802296252294471259_a_b_a @ F2 @ top_to8134405472303993176od_a_b )
        = top_top_set_a )
      = ( ! [Y: a] :
          ? [X: product_prod_a_b] :
            ( Y
            = ( F2 @ X ) ) ) ) ).

% surj_def
thf(fact_1128_surj__def,axiom,
    ! [F2: product_prod_a_b > b] :
      ( ( ( image_2802296252294471260_a_b_b @ F2 @ top_to8134405472303993176od_a_b )
        = top_top_set_b )
      = ( ! [Y: b] :
          ? [X: product_prod_a_b] :
            ( Y
            = ( F2 @ X ) ) ) ) ).

% surj_def
thf(fact_1129_surj__def,axiom,
    ! [F2: a > product_prod_a_b] :
      ( ( ( image_7400625786893224495od_a_b @ F2 @ top_top_set_a )
        = top_to8134405472303993176od_a_b )
      = ( ! [Y: product_prod_a_b] :
          ? [X: a] :
            ( Y
            = ( F2 @ X ) ) ) ) ).

% surj_def
thf(fact_1130_surj__def,axiom,
    ! [F2: b > product_prod_a_b] :
      ( ( ( image_6761185486561408366od_a_b @ F2 @ top_top_set_b )
        = top_to8134405472303993176od_a_b )
      = ( ! [Y: product_prod_a_b] :
          ? [X: b] :
            ( Y
            = ( F2 @ X ) ) ) ) ).

% surj_def
thf(fact_1131_surj__def,axiom,
    ! [F2: a > produc5884233991663340231list_o] :
      ( ( ( image_1329905666413968462list_o @ F2 @ top_top_set_a )
        = top_to7613047688909597687list_o )
      = ( ! [Y: produc5884233991663340231list_o] :
          ? [X: a] :
            ( Y
            = ( F2 @ X ) ) ) ) ).

% surj_def
thf(fact_1132_surjI,axiom,
    ! [G: a > a,F2: a > a] :
      ( ! [X2: a] :
          ( ( G @ ( F2 @ X2 ) )
          = X2 )
     => ( ( image_a_a2 @ G @ top_top_set_a )
        = top_top_set_a ) ) ).

% surjI
thf(fact_1133_surjI,axiom,
    ! [G: a > b,F2: b > a] :
      ( ! [X2: b] :
          ( ( G @ ( F2 @ X2 ) )
          = X2 )
     => ( ( image_a_b @ G @ top_top_set_a )
        = top_top_set_b ) ) ).

% surjI
thf(fact_1134_surjI,axiom,
    ! [G: b > product_prod_a_b,F2: product_prod_a_b > b] :
      ( ! [X2: product_prod_a_b] :
          ( ( G @ ( F2 @ X2 ) )
          = X2 )
     => ( ( image_6761185486561408366od_a_b @ G @ top_top_set_b )
        = top_to8134405472303993176od_a_b ) ) ).

% surjI
thf(fact_1135_surjI,axiom,
    ! [G: b > a,F2: a > b] :
      ( ! [X2: a] :
          ( ( G @ ( F2 @ X2 ) )
          = X2 )
     => ( ( image_b_a @ G @ top_top_set_b )
        = top_top_set_a ) ) ).

% surjI
thf(fact_1136_surjI,axiom,
    ! [G: b > b,F2: b > b] :
      ( ! [X2: b] :
          ( ( G @ ( F2 @ X2 ) )
          = X2 )
     => ( ( image_b_b @ G @ top_top_set_b )
        = top_top_set_b ) ) ).

% surjI
thf(fact_1137_inj__on__image__mem__iff,axiom,
    ! [F2: a > a,B5: set_a,A: a,A4: set_a] :
      ( ( inj_on_a_a @ F2 @ B5 )
     => ( ( member_a @ A @ B5 )
       => ( ( ord_less_eq_set_a @ A4 @ B5 )
         => ( ( member_a @ ( F2 @ A ) @ ( image_a_a2 @ F2 @ A4 ) )
            = ( member_a @ A @ A4 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1138_inj__image__mem__iff,axiom,
    ! [F2: a > a,A: a,A4: set_a] :
      ( ( inj_on_a_a @ F2 @ top_top_set_a )
     => ( ( member_a @ ( F2 @ A ) @ ( image_a_a2 @ F2 @ A4 ) )
        = ( member_a @ A @ A4 ) ) ) ).

% inj_image_mem_iff
thf(fact_1139_total__onI,axiom,
    ! [A4: set_a,R: set_Product_prod_a_a] :
      ( ! [X2: a,Y2: a] :
          ( ( member_a @ X2 @ A4 )
         => ( ( member_a @ Y2 @ A4 )
           => ( ( X2 != Y2 )
             => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ R )
                | ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y2 @ X2 ) @ R ) ) ) ) )
     => ( total_on_a @ A4 @ R ) ) ).

% total_onI
thf(fact_1140_subset__Collect__iff,axiom,
    ! [B5: set_a,A4: set_a,P2: a > $o] :
      ( ( ord_less_eq_set_a @ B5 @ A4 )
     => ( ( ord_less_eq_set_a @ B5
          @ ( collect_a
            @ ^ [X: a] :
                ( ( member_a @ X @ A4 )
                & ( P2 @ X ) ) ) )
        = ( ! [X: a] :
              ( ( member_a @ X @ B5 )
             => ( P2 @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_1141_subset__CollectI,axiom,
    ! [B5: set_a,A4: set_a,Q2: a > $o,P2: a > $o] :
      ( ( ord_less_eq_set_a @ B5 @ A4 )
     => ( ! [X2: a] :
            ( ( member_a @ X2 @ B5 )
           => ( ( Q2 @ X2 )
             => ( P2 @ X2 ) ) )
       => ( ord_less_eq_set_a
          @ ( collect_a
            @ ^ [X: a] :
                ( ( member_a @ X @ B5 )
                & ( Q2 @ X ) ) )
          @ ( collect_a
            @ ^ [X: a] :
                ( ( member_a @ X @ A4 )
                & ( P2 @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_1142_fst__def,axiom,
    ( product_fst_a_b
    = ( produc6028431345588019473_a_b_a
      @ ^ [X12: a,X23: b] : X12 ) ) ).

% fst_def
thf(fact_1143_snd__def,axiom,
    ( product_snd_a_b
    = ( produc6028431345588019474_a_b_b
      @ ^ [X12: a,X23: b] : X23 ) ) ).

% snd_def
thf(fact_1144_predicate2__transferD,axiom,
    ! [R12: a > b > $o,R22: a > b > $o,P2: a > a > $o,Q2: b > b > $o,A: product_prod_a_b,A4: set_Product_prod_a_b,B: product_prod_a_b,B5: set_Product_prod_a_b] :
      ( ( bNF_re5830743871565202077_o_b_o @ R12
        @ ( bNF_rel_fun_a_b_o_o @ R22
          @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
        @ P2
        @ Q2 )
     => ( ( member1426531481828664017od_a_b @ A @ A4 )
       => ( ( member1426531481828664017od_a_b @ B @ B5 )
         => ( ( ord_le817736998455962536od_a_b @ A4 @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ R12 ) ) )
           => ( ( ord_le817736998455962536od_a_b @ B5 @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ R22 ) ) )
             => ( ( P2 @ ( product_fst_a_b @ A ) @ ( product_fst_a_b @ B ) )
                = ( Q2 @ ( product_snd_a_b @ A ) @ ( product_snd_a_b @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_1145_fst__diag__fst,axiom,
    ( ( comp_P6892385181472540974od_a_b @ product_fst_a_a
      @ ( comp_a1036870397537576092od_a_b
        @ ^ [X: a] : ( product_Pair_a_a @ X @ X )
        @ product_fst_a_b ) )
    = product_fst_a_b ) ).

% fst_diag_fst
thf(fact_1146_snd__diag__snd,axiom,
    ( ( comp_P5715990135234274861od_a_b @ product_snd_b_b
      @ ( comp_b6573281471806798941od_a_b
        @ ^ [X: b] : ( product_Pair_b_b @ X @ X )
        @ product_snd_a_b ) )
    = product_snd_a_b ) ).

% snd_diag_snd
thf(fact_1147_If__the__inv__into__f__f,axiom,
    ! [I: a,C3: set_a,G: a > a,X3: a] :
      ( ( member_a @ I @ C3 )
     => ( ( inj_on_a_a @ G @ C3 )
       => ( ( comp_a_a_a
            @ ^ [I2: a] : ( if_a @ ( member_a @ I2 @ ( image_a_a2 @ G @ C3 ) ) @ ( the_inv_into_a_a @ C3 @ G @ I2 ) @ X3 )
            @ G
            @ I )
          = ( id_a @ I ) ) ) ) ).

% If_the_inv_into_f_f
thf(fact_1148_snd__fst__flip,axiom,
    ( product_snd_b_a
    = ( comp_P9084872743345139181od_b_a @ product_fst_a_b
      @ ( produc4348216232050026237od_a_b
        @ ^ [X: b,Y: a] : ( product_Pair_a_b @ Y @ X ) ) ) ) ).

% snd_fst_flip
thf(fact_1149_snd__fst__flip,axiom,
    ( product_snd_a_b
    = ( comp_P736046987085771820od_a_b @ product_fst_b_a
      @ ( produc6204743795672244857od_b_a
        @ ^ [X: a,Y: b] : ( product_Pair_b_a @ Y @ X ) ) ) ) ).

% snd_fst_flip
thf(fact_1150_fst__snd__flip,axiom,
    ( product_fst_a_b
    = ( comp_P1375487287417587949od_a_b @ product_snd_b_a
      @ ( produc6204743795672244857od_b_a
        @ ^ [X: a,Y: b] : ( product_Pair_b_a @ Y @ X ) ) ) ) ).

% fst_snd_flip
thf(fact_1151_fst__snd__flip,axiom,
    ( product_fst_b_a
    = ( comp_P8445432443013323052od_b_a @ product_snd_a_b
      @ ( produc4348216232050026237od_a_b
        @ ^ [X: b,Y: a] : ( product_Pair_a_b @ Y @ X ) ) ) ) ).

% fst_snd_flip
thf(fact_1152_snd__diag__fst,axiom,
    ( ( comp_P6892385181472540974od_a_b @ product_snd_a_a
      @ ( comp_a1036870397537576092od_a_b
        @ ^ [X: a] : ( product_Pair_a_a @ X @ X )
        @ product_fst_a_b ) )
    = product_fst_a_b ) ).

% snd_diag_fst
thf(fact_1153_fst__diag__snd,axiom,
    ( ( comp_P5715990135234274861od_a_b @ product_fst_b_b
      @ ( comp_b6573281471806798941od_a_b
        @ ^ [X: b] : ( product_Pair_b_b @ X @ X )
        @ product_snd_a_b ) )
    = product_snd_a_b ) ).

% fst_diag_snd
thf(fact_1154_SigmaI,axiom,
    ! [A: a,A4: set_a,B: a,B5: a > set_a] :
      ( ( member_a @ A @ A4 )
     => ( ( member_a @ B @ ( B5 @ A ) )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( product_Sigma_a_a @ A4 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_1155_mem__Sigma__iff,axiom,
    ! [A: a,B: a,A4: set_a,B5: a > set_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( product_Sigma_a_a @ A4 @ B5 ) )
      = ( ( member_a @ A @ A4 )
        & ( member_a @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1156_SigmaE,axiom,
    ! [C: product_prod_a_a,A4: set_a,B5: a > set_a] :
      ( ( member1426531477525435216od_a_a @ C @ ( product_Sigma_a_a @ A4 @ B5 ) )
     => ~ ! [X2: a] :
            ( ( member_a @ X2 @ A4 )
           => ! [Y2: a] :
                ( ( member_a @ Y2 @ ( B5 @ X2 ) )
               => ( C
                 != ( product_Pair_a_a @ X2 @ Y2 ) ) ) ) ) ).

% SigmaE
thf(fact_1157_SigmaE2,axiom,
    ! [A: a,B: a,A4: set_a,B5: a > set_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( product_Sigma_a_a @ A4 @ B5 ) )
     => ~ ( ( member_a @ A @ A4 )
         => ~ ( member_a @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1158_mem__Times__iff,axiom,
    ! [X3: product_prod_a_a,A4: set_a,B5: set_a] :
      ( ( member1426531477525435216od_a_a @ X3
        @ ( product_Sigma_a_a @ A4
          @ ^ [Uu: a] : B5 ) )
      = ( ( member_a @ ( product_fst_a_a @ X3 ) @ A4 )
        & ( member_a @ ( product_snd_a_a @ X3 ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_1159_mem__Times__iff,axiom,
    ! [X3: product_prod_a_b,A4: set_a,B5: set_b] :
      ( ( member1426531481828664017od_a_b @ X3
        @ ( product_Sigma_a_b @ A4
          @ ^ [Uu: a] : B5 ) )
      = ( ( member_a @ ( product_fst_a_b @ X3 ) @ A4 )
        & ( member_b @ ( product_snd_a_b @ X3 ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_1160_subset__fst__snd,axiom,
    ! [A4: set_Product_prod_a_b] :
      ( ord_le817736998455962536od_a_b @ A4
      @ ( product_Sigma_a_b @ ( image_2802296252294471259_a_b_a @ product_fst_a_b @ A4 )
        @ ^ [Uu: a] : ( image_2802296252294471260_a_b_b @ product_snd_a_b @ A4 ) ) ) ).

% subset_fst_snd
thf(fact_1161_subset__snd__imageI,axiom,
    ! [A4: set_a,B5: set_b,S5: set_Product_prod_a_b,X3: a] :
      ( ( ord_le817736998455962536od_a_b
        @ ( product_Sigma_a_b @ A4
          @ ^ [Uu: a] : B5 )
        @ S5 )
     => ( ( member_a @ X3 @ A4 )
       => ( ord_less_eq_set_b @ B5 @ ( image_2802296252294471260_a_b_b @ product_snd_a_b @ S5 ) ) ) ) ).

% subset_snd_imageI
thf(fact_1162_subset__fst__imageI,axiom,
    ! [A4: set_a,B5: set_b,S5: set_Product_prod_a_b,Y4: b] :
      ( ( ord_le817736998455962536od_a_b
        @ ( product_Sigma_a_b @ A4
          @ ^ [Uu: a] : B5 )
        @ S5 )
     => ( ( member_b @ Y4 @ B5 )
       => ( ord_less_eq_set_a @ A4 @ ( image_2802296252294471259_a_b_a @ product_fst_a_b @ S5 ) ) ) ) ).

% subset_fst_imageI
thf(fact_1163_snd__image__Sigma,axiom,
    ! [A4: set_a,B5: a > set_b] :
      ( ( image_2802296252294471260_a_b_b @ product_snd_a_b @ ( product_Sigma_a_b @ A4 @ B5 ) )
      = ( comple2307003614231284044_set_b @ ( image_a_set_b @ B5 @ A4 ) ) ) ).

% snd_image_Sigma
thf(fact_1164_snd__image__times,axiom,
    ! [A4: set_a,B5: set_b] :
      ( ( ( A4 = bot_bot_set_a )
       => ( ( image_2802296252294471260_a_b_b @ product_snd_a_b
            @ ( product_Sigma_a_b @ A4
              @ ^ [Uu: a] : B5 ) )
          = bot_bot_set_b ) )
      & ( ( A4 != bot_bot_set_a )
       => ( ( image_2802296252294471260_a_b_b @ product_snd_a_b
            @ ( product_Sigma_a_b @ A4
              @ ^ [Uu: a] : B5 ) )
          = B5 ) ) ) ).

% snd_image_times
thf(fact_1165_fst__image__times,axiom,
    ! [B5: set_b,A4: set_a] :
      ( ( ( B5 = bot_bot_set_b )
       => ( ( image_2802296252294471259_a_b_a @ product_fst_a_b
            @ ( product_Sigma_a_b @ A4
              @ ^ [Uu: a] : B5 ) )
          = bot_bot_set_a ) )
      & ( ( B5 != bot_bot_set_b )
       => ( ( image_2802296252294471259_a_b_a @ product_fst_a_b
            @ ( product_Sigma_a_b @ A4
              @ ^ [Uu: a] : B5 ) )
          = A4 ) ) ) ).

% fst_image_times
thf(fact_1166_fst__image__Sigma,axiom,
    ! [A4: set_a,B5: a > set_b] :
      ( ( image_2802296252294471259_a_b_a @ product_fst_a_b @ ( product_Sigma_a_b @ A4 @ B5 ) )
      = ( collect_a
        @ ^ [X: a] :
            ( ( member_a @ X @ A4 )
            & ( ( B5 @ X )
             != bot_bot_set_b ) ) ) ) ).

% fst_image_Sigma
thf(fact_1167_inj__img__insertE,axiom,
    ! [F2: a > a,A4: set_a,X3: a,B5: set_a] :
      ( ( inj_on_a_a @ F2 @ A4 )
     => ( ~ ( member_a @ X3 @ B5 )
       => ( ( ( insert_a @ X3 @ B5 )
            = ( image_a_a2 @ F2 @ A4 ) )
         => ~ ! [X6: a,A7: set_a] :
                ( ~ ( member_a @ X6 @ A7 )
               => ( ( A4
                    = ( insert_a @ X6 @ A7 ) )
                 => ( ( X3
                      = ( F2 @ X6 ) )
                   => ( B5
                     != ( image_a_a2 @ F2 @ A7 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_1168_Id__onI,axiom,
    ! [A: a,A4: set_a] :
      ( ( member_a @ A @ A4 )
     => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A ) @ ( id_on_a @ A4 ) ) ) ).

% Id_onI
thf(fact_1169_refl__onD2,axiom,
    ! [A4: set_a,R: set_Product_prod_a_a,X3: a,Y4: a] :
      ( ( refl_on_a @ A4 @ R )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R )
       => ( member_a @ Y4 @ A4 ) ) ) ).

% refl_onD2
thf(fact_1170_refl__onD1,axiom,
    ! [A4: set_a,R: set_Product_prod_a_a,X3: a,Y4: a] :
      ( ( refl_on_a @ A4 @ R )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R )
       => ( member_a @ X3 @ A4 ) ) ) ).

% refl_onD1
thf(fact_1171_refl__onD,axiom,
    ! [A4: set_a,R: set_Product_prod_a_a,A: a] :
      ( ( refl_on_a @ A4 @ R )
     => ( ( member_a @ A @ A4 )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_1172_Id__on__iff,axiom,
    ! [X3: a,Y4: a,A4: set_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ ( id_on_a @ A4 ) )
      = ( ( X3 = Y4 )
        & ( member_a @ X3 @ A4 ) ) ) ).

% Id_on_iff
thf(fact_1173_Id__on__eqI,axiom,
    ! [A: a,B: a,A4: set_a] :
      ( ( A = B )
     => ( ( member_a @ A @ A4 )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( id_on_a @ A4 ) ) ) ) ).

% Id_on_eqI
thf(fact_1174_Id__onE,axiom,
    ! [C: product_prod_a_a,A4: set_a] :
      ( ( member1426531477525435216od_a_a @ C @ ( id_on_a @ A4 ) )
     => ~ ! [X2: a] :
            ( ( member_a @ X2 @ A4 )
           => ( C
             != ( product_Pair_a_a @ X2 @ X2 ) ) ) ) ).

% Id_onE
thf(fact_1175_refl__onI,axiom,
    ! [R: set_Product_prod_a_a,A4: set_a] :
      ( ( ord_le746702958409616551od_a_a @ R
        @ ( product_Sigma_a_a @ A4
          @ ^ [Uu: a] : A4 ) )
     => ( ! [X2: a] :
            ( ( member_a @ X2 @ A4 )
           => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ X2 ) @ R ) )
       => ( refl_on_a @ A4 @ R ) ) ) ).

% refl_onI
thf(fact_1176_refl__on__domain,axiom,
    ! [A4: set_a,R: set_Product_prod_a_a,A: a,B: a] :
      ( ( refl_on_a @ A4 @ R )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
       => ( ( member_a @ A @ A4 )
          & ( member_a @ B @ A4 ) ) ) ) ).

% refl_on_domain
thf(fact_1177_FieldI1,axiom,
    ! [I: a,J: a,R5: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R5 )
     => ( member_a @ I @ ( field_a @ R5 ) ) ) ).

% FieldI1
thf(fact_1178_FieldI2,axiom,
    ! [I: a,J: a,R5: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R5 )
     => ( member_a @ J @ ( field_a @ R5 ) ) ) ).

% FieldI2
thf(fact_1179_prod_Omap__ident__strong,axiom,
    ! [T2: product_prod_a_a,F1: a > a,F22: a > a] :
      ( ! [Z1: a] :
          ( ( member_a @ Z1 @ ( basic_fsts_a_a @ T2 ) )
         => ( ( F1 @ Z1 )
            = Z1 ) )
     => ( ! [Z22: a] :
            ( ( member_a @ Z22 @ ( basic_snds_a_a @ T2 ) )
           => ( ( F22 @ Z22 )
              = Z22 ) )
       => ( ( produc4018698449564827638_a_a_a @ F1 @ F22 @ T2 )
          = T2 ) ) ) ).

% prod.map_ident_strong
thf(fact_1180_fsts_Ocases,axiom,
    ! [A: a,P: product_prod_a_b] :
      ( ( member_a @ A @ ( basic_fsts_a_b @ P ) )
     => ( A
        = ( product_fst_a_b @ P ) ) ) ).

% fsts.cases
thf(fact_1181_fsts_Osimps,axiom,
    ! [A: a,P: product_prod_a_b] :
      ( ( member_a @ A @ ( basic_fsts_a_b @ P ) )
      = ( A
        = ( product_fst_a_b @ P ) ) ) ).

% fsts.simps
thf(fact_1182_fsts_Ointros,axiom,
    ! [P: product_prod_a_b] : ( member_a @ ( product_fst_a_b @ P ) @ ( basic_fsts_a_b @ P ) ) ).

% fsts.intros
thf(fact_1183_snds_Ointros,axiom,
    ! [P: product_prod_a_b] : ( member_b @ ( product_snd_a_b @ P ) @ ( basic_snds_a_b @ P ) ) ).

% snds.intros
thf(fact_1184_snds_Osimps,axiom,
    ! [A: b,P: product_prod_a_b] :
      ( ( member_b @ A @ ( basic_snds_a_b @ P ) )
      = ( A
        = ( product_snd_a_b @ P ) ) ) ).

% snds.simps
thf(fact_1185_snds_Ocases,axiom,
    ! [A: b,P: product_prod_a_b] :
      ( ( member_b @ A @ ( basic_snds_a_b @ P ) )
     => ( A
        = ( product_snd_a_b @ P ) ) ) ).

% snds.cases
thf(fact_1186_prod__set__defs_I1_J,axiom,
    ( basic_fsts_a_b
    = ( ^ [P3: product_prod_a_b] : ( insert_a @ ( product_fst_a_b @ P3 ) @ bot_bot_set_a ) ) ) ).

% prod_set_defs(1)
thf(fact_1187_prod__set__defs_I2_J,axiom,
    ( basic_snds_a_b
    = ( ^ [P3: product_prod_a_b] : ( insert_b @ ( product_snd_a_b @ P3 ) @ bot_bot_set_b ) ) ) ).

% prod_set_defs(2)
thf(fact_1188_snd__transfer,axiom,
    ! [A4: a > a > $o,B5: b > b > $o] : ( bNF_re8996528489398286621_b_b_b @ ( basic_5660118979342926069_a_b_b @ A4 @ B5 ) @ B5 @ product_snd_a_b @ product_snd_a_b ) ).

% snd_transfer
thf(fact_1189_fst__transfer,axiom,
    ! [A4: a > a > $o,B5: b > b > $o] : ( bNF_re2560612034516186845_b_a_a @ ( basic_5660118979342926069_a_b_b @ A4 @ B5 ) @ A4 @ product_fst_a_b @ product_fst_a_b ) ).

% fst_transfer
thf(fact_1190_rel__prod__sel,axiom,
    ( basic_5660118979342926069_a_b_b
    = ( ^ [R13: a > a > $o,R23: b > b > $o,P3: product_prod_a_b,Q4: product_prod_a_b] :
          ( ( R13 @ ( product_fst_a_b @ P3 ) @ ( product_fst_a_b @ Q4 ) )
          & ( R23 @ ( product_snd_a_b @ P3 ) @ ( product_snd_a_b @ Q4 ) ) ) ) ) ).

% rel_prod_sel
thf(fact_1191_prod_Orel__refl__strong,axiom,
    ! [X3: product_prod_a_a,R1a: a > a > $o,R2a: a > a > $o] :
      ( ! [Z1: a] :
          ( ( member_a @ Z1 @ ( basic_fsts_a_a @ X3 ) )
         => ( R1a @ Z1 @ Z1 ) )
     => ( ! [Z22: a] :
            ( ( member_a @ Z22 @ ( basic_snds_a_a @ X3 ) )
           => ( R2a @ Z22 @ Z22 ) )
       => ( basic_8447574561315602101_a_a_a @ R1a @ R2a @ X3 @ X3 ) ) ) ).

% prod.rel_refl_strong
thf(fact_1192_prod_Orel__mono__strong,axiom,
    ! [R12: a > a > $o,R22: a > a > $o,X3: product_prod_a_a,Y4: product_prod_a_a,R1a: a > a > $o,R2a: a > a > $o] :
      ( ( basic_8447574561315602101_a_a_a @ R12 @ R22 @ X3 @ Y4 )
     => ( ! [Z1: a,Y12: a] :
            ( ( member_a @ Z1 @ ( basic_fsts_a_a @ X3 ) )
           => ( ( member_a @ Y12 @ ( basic_fsts_a_a @ Y4 ) )
             => ( ( R12 @ Z1 @ Y12 )
               => ( R1a @ Z1 @ Y12 ) ) ) )
       => ( ! [Z22: a,Y23: a] :
              ( ( member_a @ Z22 @ ( basic_snds_a_a @ X3 ) )
             => ( ( member_a @ Y23 @ ( basic_snds_a_a @ Y4 ) )
               => ( ( R22 @ Z22 @ Y23 )
                 => ( R2a @ Z22 @ Y23 ) ) ) )
         => ( basic_8447574561315602101_a_a_a @ R1a @ R2a @ X3 @ Y4 ) ) ) ) ).

% prod.rel_mono_strong
thf(fact_1193_prod_Orel__cong,axiom,
    ! [X3: product_prod_a_a,Ya: product_prod_a_a,Y4: product_prod_a_a,Xa2: product_prod_a_a,R12: a > a > $o,R1a: a > a > $o,R22: a > a > $o,R2a: a > a > $o] :
      ( ( X3 = Ya )
     => ( ( Y4 = Xa2 )
       => ( ! [Z1: a,Y12: a] :
              ( ( member_a @ Z1 @ ( basic_fsts_a_a @ Ya ) )
             => ( ( member_a @ Y12 @ ( basic_fsts_a_a @ Xa2 ) )
               => ( ( R12 @ Z1 @ Y12 )
                  = ( R1a @ Z1 @ Y12 ) ) ) )
         => ( ! [Z22: a,Y23: a] :
                ( ( member_a @ Z22 @ ( basic_snds_a_a @ Ya ) )
               => ( ( member_a @ Y23 @ ( basic_snds_a_a @ Xa2 ) )
                 => ( ( R22 @ Z22 @ Y23 )
                    = ( R2a @ Z22 @ Y23 ) ) ) )
           => ( ( basic_8447574561315602101_a_a_a @ R12 @ R22 @ X3 @ Y4 )
              = ( basic_8447574561315602101_a_a_a @ R1a @ R2a @ Ya @ Xa2 ) ) ) ) ) ) ).

% prod.rel_cong
thf(fact_1194_prod_Orel__compp__Grp,axiom,
    ( basic_4820421139648956983_b_a_b
    = ( ^ [R13: a > b > $o,R23: a > b > $o] :
          ( relcom8632546625042571830od_b_b
          @ ( conver607836475105359669od_a_a
            @ ( bNF_Gr248057877874286094od_a_a
              @ ( collec7968472090841016338od_a_b
                @ ^ [X: produc2233330766403592615od_a_b] :
                    ( ( ord_le817736998455962536od_a_b @ ( basic_7688285234108943239od_a_b @ X ) @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ R13 ) ) )
                    & ( ord_le817736998455962536od_a_b @ ( basic_6470750372406100165od_a_b @ X ) @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ R23 ) ) ) ) )
              @ ( produc4058734581585674914_a_b_a @ product_fst_a_b @ product_fst_a_b ) ) )
          @ ( bNF_Gr6683974337059614670od_b_b
            @ ( collec7968472090841016338od_a_b
              @ ^ [X: produc2233330766403592615od_a_b] :
                  ( ( ord_le817736998455962536od_a_b @ ( basic_7688285234108943239od_a_b @ X ) @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ R13 ) ) )
                  & ( ord_le817736998455962536od_a_b @ ( basic_6470750372406100165od_a_b @ X ) @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ R23 ) ) ) ) )
            @ ( produc7998683168210511140_a_b_b @ product_snd_a_b @ product_snd_a_b ) ) ) ) ) ).

% prod.rel_compp_Grp
thf(fact_1195_prod_Oin__rel,axiom,
    ( basic_4820421139648956983_b_a_b
    = ( ^ [R13: a > b > $o,R23: a > b > $o,A2: product_prod_a_a,B4: product_prod_b_b] :
        ? [Z3: produc2233330766403592615od_a_b] :
          ( ( member5065438833300252112od_a_b @ Z3
            @ ( collec7968472090841016338od_a_b
              @ ^ [X: produc2233330766403592615od_a_b] :
                  ( ( ord_le817736998455962536od_a_b @ ( basic_7688285234108943239od_a_b @ X ) @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ R13 ) ) )
                  & ( ord_le817736998455962536od_a_b @ ( basic_6470750372406100165od_a_b @ X ) @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ R23 ) ) ) ) ) )
          & ( ( produc4058734581585674914_a_b_a @ product_fst_a_b @ product_fst_a_b @ Z3 )
            = A2 )
          & ( ( produc7998683168210511140_a_b_b @ product_snd_a_b @ product_snd_a_b @ Z3 )
            = B4 ) ) ) ) ).

% prod.in_rel
thf(fact_1196_Linear__order__in__diff__Id,axiom,
    ! [R: set_Product_prod_a_a,A: a,B: a] :
      ( ( order_8768733634509060147r_on_a @ ( field_a @ R ) @ R )
     => ( ( member_a @ A @ ( field_a @ R ) )
       => ( ( member_a @ B @ ( field_a @ R ) )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
            = ( ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ A ) @ ( minus_6817036919807184750od_a_a @ R @ id_a2 ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_1197_wfI,axiom,
    ! [R: set_Product_prod_a_a,A4: set_a,B5: set_a] :
      ( ( ord_le746702958409616551od_a_a @ R
        @ ( product_Sigma_a_a @ A4
          @ ^ [Uu: a] : B5 ) )
     => ( ! [X2: a,P5: a > $o] :
            ( ! [Xa3: a] :
                ( ! [Y2: a] :
                    ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y2 @ Xa3 ) @ R )
                   => ( P5 @ Y2 ) )
               => ( P5 @ Xa3 ) )
           => ( ( member_a @ X2 @ A4 )
             => ( ( member_a @ X2 @ B5 )
               => ( P5 @ X2 ) ) ) )
       => ( wf_a @ R ) ) ) ).

% wfI
thf(fact_1198_wfE__min_H,axiom,
    ! [R5: set_Product_prod_a_a,Q2: set_a] :
      ( ( wf_a @ R5 )
     => ( ( Q2 != bot_bot_set_a )
       => ~ ! [Z4: a] :
              ( ( member_a @ Z4 @ Q2 )
             => ~ ! [Y6: a] :
                    ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y6 @ Z4 ) @ R5 )
                   => ~ ( member_a @ Y6 @ Q2 ) ) ) ) ) ).

% wfE_min'
thf(fact_1199_wf__eq__minimal,axiom,
    ( wf_a
    = ( ^ [R3: set_Product_prod_a_a] :
        ! [Q3: set_a] :
          ( ? [X: a] : ( member_a @ X @ Q3 )
         => ? [X: a] :
              ( ( member_a @ X @ Q3 )
              & ! [Y: a] :
                  ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X ) @ R3 )
                 => ~ ( member_a @ Y @ Q3 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_1200_wfI__min,axiom,
    ! [R5: set_Product_prod_a_a] :
      ( ! [X2: a,Q5: set_a] :
          ( ( member_a @ X2 @ Q5 )
         => ? [Xa3: a] :
              ( ( member_a @ Xa3 @ Q5 )
              & ! [Y2: a] :
                  ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y2 @ Xa3 ) @ R5 )
                 => ~ ( member_a @ Y2 @ Q5 ) ) ) )
     => ( wf_a @ R5 ) ) ).

% wfI_min
thf(fact_1201_wfE__min,axiom,
    ! [R5: set_Product_prod_a_a,X3: a,Q2: set_a] :
      ( ( wf_a @ R5 )
     => ( ( member_a @ X3 @ Q2 )
       => ~ ! [Z4: a] :
              ( ( member_a @ Z4 @ Q2 )
             => ~ ! [Y6: a] :
                    ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y6 @ Z4 ) @ R5 )
                   => ~ ( member_a @ Y6 @ Q2 ) ) ) ) ) ).

% wfE_min
thf(fact_1202_wo__rel_Oin__notinI,axiom,
    ! [R: set_Product_prod_a_a,J: a,I: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R )
     => ( ( ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ J @ I ) @ R )
          | ( J = I ) )
       => ( ( member_a @ I @ ( field_a @ R ) )
         => ( ( member_a @ J @ ( field_a @ R ) )
           => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_1203_well__order__induct__imp,axiom,
    ! [R: set_Product_prod_a_a,P2: a > $o,A: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R )
     => ( ! [X2: a] :
            ( ! [Y6: a] :
                ( ( ( Y6 != X2 )
                  & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y6 @ X2 ) @ R ) )
               => ( ( member_a @ Y6 @ ( field_a @ R ) )
                 => ( P2 @ Y6 ) ) )
           => ( ( member_a @ X2 @ ( field_a @ R ) )
             => ( P2 @ X2 ) ) )
       => ( ( member_a @ A @ ( field_a @ R ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_1204_wo__rel_Omax2__greater,axiom,
    ! [R: set_Product_prod_a_a,A: a,B: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R )
     => ( ( member_a @ A @ ( field_a @ R ) )
       => ( ( member_a @ B @ ( field_a @ R ) )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ ( bNF_We3763454674811381836max2_a @ R @ A @ B ) ) @ R )
            & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ ( bNF_We3763454674811381836max2_a @ R @ A @ B ) ) @ R ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_1205_wo__rel_Omax2__equals2,axiom,
    ! [R: set_Product_prod_a_a,A: a,B: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R )
     => ( ( member_a @ A @ ( field_a @ R ) )
       => ( ( member_a @ B @ ( field_a @ R ) )
         => ( ( ( bNF_We3763454674811381836max2_a @ R @ A @ B )
              = B )
            = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_1206_wo__rel_Omax2__equals1,axiom,
    ! [R: set_Product_prod_a_a,A: a,B: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R )
     => ( ( member_a @ A @ ( field_a @ R ) )
       => ( ( member_a @ B @ ( field_a @ R ) )
         => ( ( ( bNF_We3763454674811381836max2_a @ R @ A @ B )
              = A )
            = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ A ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_1207_wo__rel_Omax2__greater__among,axiom,
    ! [R: set_Product_prod_a_a,A: a,B: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R )
     => ( ( member_a @ A @ ( field_a @ R ) )
       => ( ( member_a @ B @ ( field_a @ R ) )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ ( bNF_We3763454674811381836max2_a @ R @ A @ B ) ) @ R )
            & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ ( bNF_We3763454674811381836max2_a @ R @ A @ B ) ) @ R )
            & ( member_a @ ( bNF_We3763454674811381836max2_a @ R @ A @ B ) @ ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_1208_wo__rel_Ominim__least,axiom,
    ! [R: set_Product_prod_a_a,B5: set_a,B: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R )
     => ( ( ord_less_eq_set_a @ B5 @ ( field_a @ R ) )
       => ( ( member_a @ B @ B5 )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( bNF_We5615626441682584778inim_a @ R @ B5 ) @ B ) @ R ) ) ) ) ).

% wo_rel.minim_least
thf(fact_1209_wo__rel_Oequals__minim,axiom,
    ! [R: set_Product_prod_a_a,B5: set_a,A: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R )
     => ( ( ord_less_eq_set_a @ B5 @ ( field_a @ R ) )
       => ( ( member_a @ A @ B5 )
         => ( ! [B3: a] :
                ( ( member_a @ B3 @ B5 )
               => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B3 ) @ R ) )
           => ( A
              = ( bNF_We5615626441682584778inim_a @ R @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_1210_subset__Image1__Image1__iff,axiom,
    ! [R: set_Product_prod_a_a,A: a,B: a] :
      ( ( order_preorder_on_a @ ( field_a @ R ) @ R )
     => ( ( member_a @ A @ ( field_a @ R ) )
       => ( ( member_a @ B @ ( field_a @ R ) )
         => ( ( ord_less_eq_set_a @ ( image_a_a @ R @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( image_a_a @ R @ ( insert_a @ B @ bot_bot_set_a ) ) )
            = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ A ) @ R ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_1211_ImageI,axiom,
    ! [A: a,B: a,R: set_Product_prod_a_a,A4: set_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
     => ( ( member_a @ A @ A4 )
       => ( member_a @ B @ ( image_a_a @ R @ A4 ) ) ) ) ).

% ImageI
thf(fact_1212_well__order__on__domain,axiom,
    ! [A4: set_a,R: set_Product_prod_a_a,A: a,B: a] :
      ( ( order_6972113574731384220r_on_a @ A4 @ R )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
       => ( ( member_a @ A @ A4 )
          & ( member_a @ B @ A4 ) ) ) ) ).

% well_order_on_domain
thf(fact_1213_rev__ImageI,axiom,
    ! [A: a,A4: set_a,B: a,R: set_Product_prod_a_a] :
      ( ( member_a @ A @ A4 )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
       => ( member_a @ B @ ( image_a_a @ R @ A4 ) ) ) ) ).

% rev_ImageI
thf(fact_1214_ImageE,axiom,
    ! [B: a,R: set_Product_prod_a_a,A4: set_a] :
      ( ( member_a @ B @ ( image_a_a @ R @ A4 ) )
     => ~ ! [X2: a] :
            ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ B ) @ R )
           => ~ ( member_a @ X2 @ A4 ) ) ) ).

% ImageE
thf(fact_1215_equiv__class__eq__iff,axiom,
    ! [A4: set_a,R: set_Product_prod_a_a,X3: a,Y4: a] :
      ( ( equiv_equiv_a @ A4 @ R )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R )
        = ( ( ( image_a_a @ R @ ( insert_a @ X3 @ bot_bot_set_a ) )
            = ( image_a_a @ R @ ( insert_a @ Y4 @ bot_bot_set_a ) ) )
          & ( member_a @ X3 @ A4 )
          & ( member_a @ Y4 @ A4 ) ) ) ) ).

% equiv_class_eq_iff
thf(fact_1216_eq__equiv__class__iff,axiom,
    ! [A4: set_a,R: set_Product_prod_a_a,X3: a,Y4: a] :
      ( ( equiv_equiv_a @ A4 @ R )
     => ( ( member_a @ X3 @ A4 )
       => ( ( member_a @ Y4 @ A4 )
         => ( ( ( image_a_a @ R @ ( insert_a @ X3 @ bot_bot_set_a ) )
              = ( image_a_a @ R @ ( insert_a @ Y4 @ bot_bot_set_a ) ) )
            = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R ) ) ) ) ) ).

% eq_equiv_class_iff
thf(fact_1217_eq__equiv__class,axiom,
    ! [R: set_Product_prod_a_a,A: a,B: a,A4: set_a] :
      ( ( ( image_a_a @ R @ ( insert_a @ A @ bot_bot_set_a ) )
        = ( image_a_a @ R @ ( insert_a @ B @ bot_bot_set_a ) ) )
     => ( ( equiv_equiv_a @ A4 @ R )
       => ( ( member_a @ B @ A4 )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R ) ) ) ) ).

% eq_equiv_class
thf(fact_1218_subset__equiv__class,axiom,
    ! [A4: set_a,R: set_Product_prod_a_a,B: a,A: a] :
      ( ( equiv_equiv_a @ A4 @ R )
     => ( ( ord_less_eq_set_a @ ( image_a_a @ R @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( image_a_a @ R @ ( insert_a @ A @ bot_bot_set_a ) ) )
       => ( ( member_a @ B @ A4 )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R ) ) ) ) ).

% subset_equiv_class
thf(fact_1219_equiv__class__nondisjoint,axiom,
    ! [A4: set_a,R: set_Product_prod_a_a,X3: a,A: a,B: a] :
      ( ( equiv_equiv_a @ A4 @ R )
     => ( ( member_a @ X3 @ ( inf_inf_set_a @ ( image_a_a @ R @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( image_a_a @ R @ ( insert_a @ B @ bot_bot_set_a ) ) ) )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R ) ) ) ).

% equiv_class_nondisjoint
thf(fact_1220_antisym__onD,axiom,
    ! [A4: set_a,R: set_Product_prod_a_a,X3: a,Y4: a] :
      ( ( antisym_on_a @ A4 @ R )
     => ( ( member_a @ X3 @ A4 )
       => ( ( member_a @ Y4 @ A4 )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R )
           => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X3 ) @ R )
             => ( X3 = Y4 ) ) ) ) ) ) ).

% antisym_onD
thf(fact_1221_antisym__onI,axiom,
    ! [A4: set_a,R: set_Product_prod_a_a] :
      ( ! [X2: a,Y2: a] :
          ( ( member_a @ X2 @ A4 )
         => ( ( member_a @ Y2 @ A4 )
           => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ R )
             => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y2 @ X2 ) @ R )
               => ( X2 = Y2 ) ) ) ) )
     => ( antisym_on_a @ A4 @ R ) ) ).

% antisym_onI
thf(fact_1222_in__quotient__imp__closed,axiom,
    ! [A4: set_a,R: set_Product_prod_a_a,X5: set_a,X3: a,Y4: a] :
      ( ( equiv_equiv_a @ A4 @ R )
     => ( ( member_set_a @ X5 @ ( equiv_quotient_a @ A4 @ R ) )
       => ( ( member_a @ X3 @ X5 )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R )
           => ( member_a @ Y4 @ X5 ) ) ) ) ) ).

% in_quotient_imp_closed
thf(fact_1223_quotient__eq__iff,axiom,
    ! [A4: set_a,R: set_Product_prod_a_a,X5: set_a,Y5: set_a,X3: a,Y4: a] :
      ( ( equiv_equiv_a @ A4 @ R )
     => ( ( member_set_a @ X5 @ ( equiv_quotient_a @ A4 @ R ) )
       => ( ( member_set_a @ Y5 @ ( equiv_quotient_a @ A4 @ R ) )
         => ( ( member_a @ X3 @ X5 )
           => ( ( member_a @ Y4 @ Y5 )
             => ( ( X5 = Y5 )
                = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R ) ) ) ) ) ) ) ).

% quotient_eq_iff
thf(fact_1224_quotient__eqI,axiom,
    ! [A4: set_a,R: set_Product_prod_a_a,X5: set_a,Y5: set_a,X3: a,Y4: a] :
      ( ( equiv_equiv_a @ A4 @ R )
     => ( ( member_set_a @ X5 @ ( equiv_quotient_a @ A4 @ R ) )
       => ( ( member_set_a @ Y5 @ ( equiv_quotient_a @ A4 @ R ) )
         => ( ( member_a @ X3 @ X5 )
           => ( ( member_a @ Y4 @ Y5 )
             => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R )
               => ( X5 = Y5 ) ) ) ) ) ) ) ).

% quotient_eqI
thf(fact_1225_eq__equiv__class__iff2,axiom,
    ! [A4: set_a,R: set_Product_prod_a_a,X3: a,Y4: a] :
      ( ( equiv_equiv_a @ A4 @ R )
     => ( ( member_a @ X3 @ A4 )
       => ( ( member_a @ Y4 @ A4 )
         => ( ( ( equiv_quotient_a @ ( insert_a @ X3 @ bot_bot_set_a ) @ R )
              = ( equiv_quotient_a @ ( insert_a @ Y4 @ bot_bot_set_a ) @ R ) )
            = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R ) ) ) ) ) ).

% eq_equiv_class_iff2
thf(fact_1226_dom__if,axiom,
    ! [P2: a > $o,F2: a > option_list_o,G: a > option_list_o] :
      ( ( dom_a_list_o
        @ ^ [X: a] : ( if_option_list_o @ ( P2 @ X ) @ ( F2 @ X ) @ ( G @ X ) ) )
      = ( sup_sup_set_a @ ( inf_inf_set_a @ ( dom_a_list_o @ F2 ) @ ( collect_a @ P2 ) )
        @ ( inf_inf_set_a @ ( dom_a_list_o @ G )
          @ ( collect_a
            @ ^ [X: a] :
                ~ ( P2 @ X ) ) ) ) ) ).

% dom_if
thf(fact_1227_dom__override__on,axiom,
    ! [F2: a > option_list_o,G: a > option_list_o,A4: set_a] :
      ( ( dom_a_list_o @ ( overri6937707214658884900list_o @ F2 @ G @ A4 ) )
      = ( sup_sup_set_a
        @ ( minus_minus_set_a @ ( dom_a_list_o @ F2 )
          @ ( collect_a
            @ ^ [A2: a] : ( member_a @ A2 @ ( minus_minus_set_a @ A4 @ ( dom_a_list_o @ G ) ) ) ) )
        @ ( collect_a
          @ ^ [A2: a] : ( member_a @ A2 @ ( inf_inf_set_a @ A4 @ ( dom_a_list_o @ G ) ) ) ) ) ) ).

% dom_override_on
thf(fact_1228_fst__vimage__eq__Times,axiom,
    ! [S5: set_a] :
      ( ( vimage545595664742944273_a_b_a @ product_fst_a_b @ S5 )
      = ( product_Sigma_a_b @ S5
        @ ^ [Uu: a] : top_top_set_b ) ) ).

% fst_vimage_eq_Times
thf(fact_1229_vimage__fst,axiom,
    ! [A4: set_a] :
      ( ( vimage545595664742944273_a_b_a @ product_fst_a_b @ A4 )
      = ( product_Sigma_a_b @ A4
        @ ^ [Uu: a] : top_top_set_b ) ) ).

% vimage_fst
thf(fact_1230_snd__vimage__eq__Times,axiom,
    ! [S5: set_b] :
      ( ( vimage545595664742944274_a_b_b @ product_snd_a_b @ S5 )
      = ( product_Sigma_a_b @ top_top_set_a
        @ ^ [Uu: a] : S5 ) ) ).

% snd_vimage_eq_Times
thf(fact_1231_vimage__snd,axiom,
    ! [A4: set_b] :
      ( ( vimage545595664742944274_a_b_b @ product_snd_a_b @ A4 )
      = ( product_Sigma_a_b @ top_top_set_a
        @ ^ [Uu: a] : A4 ) ) ).

% vimage_snd
thf(fact_1232_The__case__prod,axiom,
    ! [P2: a > b > $o] :
      ( ( the_Product_prod_a_b @ ( produc3537405659489547051_a_b_o @ P2 ) )
      = ( the_Product_prod_a_b
        @ ^ [Xy: product_prod_a_b] : ( P2 @ ( product_fst_a_b @ Xy ) @ ( product_snd_a_b @ Xy ) ) ) ) ).

% The_case_prod
thf(fact_1233_underS__incl__iff,axiom,
    ! [R: set_Product_prod_a_a,A: a,B: a] :
      ( ( order_8768733634509060147r_on_a @ ( field_a @ R ) @ R )
     => ( ( member_a @ A @ ( field_a @ R ) )
       => ( ( member_a @ B @ ( field_a @ R ) )
         => ( ( ord_less_eq_set_a @ ( order_underS_a @ R @ A ) @ ( order_underS_a @ R @ B ) )
            = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1234_underS__E,axiom,
    ! [I: a,R5: set_Product_prod_a_a,J: a] :
      ( ( member_a @ I @ ( order_underS_a @ R5 @ J ) )
     => ( ( I != J )
        & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R5 ) ) ) ).

% underS_E
thf(fact_1235_underS__I,axiom,
    ! [I: a,J: a,R5: set_Product_prod_a_a] :
      ( ( I != J )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R5 )
       => ( member_a @ I @ ( order_underS_a @ R5 @ J ) ) ) ) ).

% underS_I
thf(fact_1236_trans__onI,axiom,
    ! [A4: set_a,R: set_Product_prod_a_a] :
      ( ! [X2: a,Y2: a,Z4: a] :
          ( ( member_a @ X2 @ A4 )
         => ( ( member_a @ Y2 @ A4 )
           => ( ( member_a @ Z4 @ A4 )
             => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ R )
               => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y2 @ Z4 ) @ R )
                 => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Z4 ) @ R ) ) ) ) ) )
     => ( trans_on_a @ A4 @ R ) ) ).

% trans_onI
thf(fact_1237_trans__onD,axiom,
    ! [A4: set_a,R: set_Product_prod_a_a,X3: a,Y4: a,Z2: a] :
      ( ( trans_on_a @ A4 @ R )
     => ( ( member_a @ X3 @ A4 )
       => ( ( member_a @ Y4 @ A4 )
         => ( ( member_a @ Z2 @ A4 )
           => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R )
             => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ Z2 ) @ R )
               => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Z2 ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_1238_pred__prod__beta,axiom,
    ( basic_pred_prod_a_b
    = ( ^ [P4: a > $o,Q3: b > $o,Xy: product_prod_a_b] :
          ( ( P4 @ ( product_fst_a_b @ Xy ) )
          & ( Q3 @ ( product_snd_a_b @ Xy ) ) ) ) ) ).

% pred_prod_beta
thf(fact_1239_prod_Opred__cong,axiom,
    ! [X3: product_prod_a_a,Ya: product_prod_a_a,P1: a > $o,P1a: a > $o,P22: a > $o,P2a: a > $o] :
      ( ( X3 = Ya )
     => ( ! [Z1: a] :
            ( ( member_a @ Z1 @ ( basic_fsts_a_a @ Ya ) )
           => ( ( P1 @ Z1 )
              = ( P1a @ Z1 ) ) )
       => ( ! [Z22: a] :
              ( ( member_a @ Z22 @ ( basic_snds_a_a @ Ya ) )
             => ( ( P22 @ Z22 )
                = ( P2a @ Z22 ) ) )
         => ( ( basic_pred_prod_a_a @ P1 @ P22 @ X3 )
            = ( basic_pred_prod_a_a @ P1a @ P2a @ Ya ) ) ) ) ) ).

% prod.pred_cong
thf(fact_1240_prod_Opred__mono__strong,axiom,
    ! [P1: a > $o,P22: a > $o,X3: product_prod_a_a,P1a: a > $o,P2a: a > $o] :
      ( ( basic_pred_prod_a_a @ P1 @ P22 @ X3 )
     => ( ! [Z1: a] :
            ( ( member_a @ Z1 @ ( basic_fsts_a_a @ X3 ) )
           => ( ( P1 @ Z1 )
             => ( P1a @ Z1 ) ) )
       => ( ! [Z22: a] :
              ( ( member_a @ Z22 @ ( basic_snds_a_a @ X3 ) )
             => ( ( P22 @ Z22 )
               => ( P2a @ Z22 ) ) )
         => ( basic_pred_prod_a_a @ P1a @ P2a @ X3 ) ) ) ) ).

% prod.pred_mono_strong
thf(fact_1241_snd__eq__Range,axiom,
    ! [R5: set_Product_prod_a_b] :
      ( ( image_2802296252294471260_a_b_b @ product_snd_a_b @ R5 )
      = ( range_a_b @ R5 ) ) ).

% snd_eq_Range
thf(fact_1242_Range__snd,axiom,
    ( range_a_b
    = ( image_2802296252294471260_a_b_b @ product_snd_a_b ) ) ).

% Range_snd
thf(fact_1243_Domain__fst,axiom,
    ( domain_a_b
    = ( image_2802296252294471259_a_b_a @ product_fst_a_b ) ) ).

% Domain_fst
thf(fact_1244_fst__eq__Domain,axiom,
    ! [R5: set_Product_prod_a_b] :
      ( ( image_2802296252294471259_a_b_a @ product_fst_a_b @ R5 )
      = ( domain_a_b @ R5 ) ) ).

% fst_eq_Domain
thf(fact_1245_wo__rel_OisMinim__def,axiom,
    ! [R: set_Product_prod_a_a,A4: set_a,B: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R )
     => ( ( bNF_We6697304935525757620inim_a @ R @ A4 @ B )
        = ( ( member_a @ B @ A4 )
          & ! [X: a] :
              ( ( member_a @ X @ A4 )
             => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ X ) @ R ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_1246_max__ext_Ocases,axiom,
    ! [A1: set_a,A22: set_a,R5: set_Product_prod_a_a] :
      ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ A1 @ A22 ) @ ( max_ext_a @ R5 ) )
     => ~ ( ( finite_finite_a @ A1 )
         => ( ( finite_finite_a @ A22 )
           => ( ( A22 != bot_bot_set_a )
             => ~ ! [X7: a] :
                    ( ( member_a @ X7 @ A1 )
                   => ? [Xa: a] :
                        ( ( member_a @ Xa @ A22 )
                        & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X7 @ Xa ) @ R5 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_1247_finite__inverse__image__gen,axiom,
    ! [A4: set_a,F2: a > a,D2: set_a] :
      ( ( finite_finite_a @ A4 )
     => ( ( inj_on_a_a @ F2 @ D2 )
       => ( finite_finite_a
          @ ( collect_a
            @ ^ [J2: a] :
                ( ( member_a @ J2 @ D2 )
                & ( member_a @ ( F2 @ J2 ) @ A4 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_1248_max__ext_Omax__extI,axiom,
    ! [X5: set_a,Y5: set_a,R5: set_Product_prod_a_a] :
      ( ( finite_finite_a @ X5 )
     => ( ( finite_finite_a @ Y5 )
       => ( ( Y5 != bot_bot_set_a )
         => ( ! [X2: a] :
                ( ( member_a @ X2 @ X5 )
               => ? [Xa3: a] :
                    ( ( member_a @ Xa3 @ Y5 )
                    & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Xa3 ) @ R5 ) ) )
           => ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X5 @ Y5 ) @ ( max_ext_a @ R5 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_1249_max__ext_Osimps,axiom,
    ! [A1: set_a,A22: set_a,R5: set_Product_prod_a_a] :
      ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ A1 @ A22 ) @ ( max_ext_a @ R5 ) )
      = ( ( finite_finite_a @ A1 )
        & ( finite_finite_a @ A22 )
        & ( A22 != bot_bot_set_a )
        & ! [X: a] :
            ( ( member_a @ X @ A1 )
           => ? [Y: a] :
                ( ( member_a @ Y @ A22 )
                & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R5 ) ) ) ) ) ).

% max_ext.simps
thf(fact_1250_irrefl__onI,axiom,
    ! [A4: set_a,R: set_Product_prod_a_a] :
      ( ! [A5: a] :
          ( ( member_a @ A5 @ A4 )
         => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A5 @ A5 ) @ R ) )
     => ( irrefl_on_a @ A4 @ R ) ) ).

% irrefl_onI
thf(fact_1251_irrefl__onD,axiom,
    ! [A4: set_a,R: set_Product_prod_a_a,A: a] :
      ( ( irrefl_on_a @ A4 @ R )
     => ( ( member_a @ A @ A4 )
       => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A ) @ R ) ) ) ).

% irrefl_onD
thf(fact_1252_finite__graph__iff__finite__dom,axiom,
    ! [M2: a > option_list_o] :
      ( ( finite5626129373397972976list_o @ ( graph_a_list_o @ M2 ) )
      = ( finite_finite_a @ ( dom_a_list_o @ M2 ) ) ) ).

% finite_graph_iff_finite_dom
thf(fact_1253_graph__domD,axiom,
    ! [X3: produc5884233991663340231list_o,M2: a > option_list_o] :
      ( ( member7948383622993546480list_o @ X3 @ ( graph_a_list_o @ M2 ) )
     => ( member_a @ ( product_fst_a_list_o @ X3 ) @ ( dom_a_list_o @ M2 ) ) ) ).

% graph_domD
thf(fact_1254_graph__domD,axiom,
    ! [X3: product_prod_a_b,M2: a > option_b] :
      ( ( member1426531481828664017od_a_b @ X3 @ ( graph_a_b @ M2 ) )
     => ( member_a @ ( product_fst_a_b @ X3 ) @ ( dom_a_b @ M2 ) ) ) ).

% graph_domD
thf(fact_1255_inj__on__fst__graph,axiom,
    ! [M2: a > option_b] : ( inj_on4343330552946057671_a_b_a @ product_fst_a_b @ ( graph_a_b @ M2 ) ) ).

% inj_on_fst_graph
thf(fact_1256_fst__graph__eq__dom,axiom,
    ! [M2: a > option_list_o] :
      ( ( image_8953000392185216508st_o_a @ product_fst_a_list_o @ ( graph_a_list_o @ M2 ) )
      = ( dom_a_list_o @ M2 ) ) ).

% fst_graph_eq_dom
thf(fact_1257_fst__graph__eq__dom,axiom,
    ! [M2: a > option_b] :
      ( ( image_2802296252294471259_a_b_a @ product_fst_a_b @ ( graph_a_b @ M2 ) )
      = ( dom_a_b @ M2 ) ) ).

% fst_graph_eq_dom
thf(fact_1258_graph__map__add,axiom,
    ! [M1: a > option_list_o,M22: a > option_list_o] :
      ( ( ( inf_inf_set_a @ ( dom_a_list_o @ M1 ) @ ( dom_a_list_o @ M22 ) )
        = bot_bot_set_a )
     => ( ( graph_a_list_o @ ( map_add_a_list_o @ M1 @ M22 ) )
        = ( sup_su4181551675182618747list_o @ ( graph_a_list_o @ M1 ) @ ( graph_a_list_o @ M22 ) ) ) ) ).

% graph_map_add
thf(fact_1259_graph__eq__to__snd__dom,axiom,
    ( graph_a_list_o
    = ( ^ [M3: a > option_list_o] :
          ( image_1329905666413968462list_o
          @ ^ [X: a] : ( produc6899623729582506167list_o @ X @ ( the_list_o @ ( M3 @ X ) ) )
          @ ( dom_a_list_o @ M3 ) ) ) ) ).

% graph_eq_to_snd_dom
thf(fact_1260_dom__map__add,axiom,
    ! [M2: a > option_list_o,N2: a > option_list_o] :
      ( ( dom_a_list_o @ ( map_add_a_list_o @ M2 @ N2 ) )
      = ( sup_sup_set_a @ ( dom_a_list_o @ N2 ) @ ( dom_a_list_o @ M2 ) ) ) ).

% dom_map_add
thf(fact_1261_inj__on__map__add__dom,axiom,
    ! [M2: a > option_list_o,M4: a > option_list_o] :
      ( ( inj_on374126998980950615list_o @ ( map_add_a_list_o @ M2 @ M4 ) @ ( dom_a_list_o @ M4 ) )
      = ( inj_on374126998980950615list_o @ M4 @ ( dom_a_list_o @ M4 ) ) ) ).

% inj_on_map_add_dom
thf(fact_1262_map__add__dom__app__simps_I3_J,axiom,
    ! [M2: a,L2: a > option_list_o,L1: a > option_list_o] :
      ( ~ ( member_a @ M2 @ ( dom_a_list_o @ L2 ) )
     => ( ( map_add_a_list_o @ L1 @ L2 @ M2 )
        = ( L1 @ M2 ) ) ) ).

% map_add_dom_app_simps(3)
thf(fact_1263_map__add__dom__app__simps_I2_J,axiom,
    ! [M2: a,L1: a > option_list_o,L2: a > option_list_o] :
      ( ~ ( member_a @ M2 @ ( dom_a_list_o @ L1 ) )
     => ( ( map_add_a_list_o @ L1 @ L2 @ M2 )
        = ( L2 @ M2 ) ) ) ).

% map_add_dom_app_simps(2)
thf(fact_1264_map__add__dom__app__simps_I1_J,axiom,
    ! [M2: a,L2: a > option_list_o,L1: a > option_list_o] :
      ( ( member_a @ M2 @ ( dom_a_list_o @ L2 ) )
     => ( ( map_add_a_list_o @ L1 @ L2 @ M2 )
        = ( L2 @ M2 ) ) ) ).

% map_add_dom_app_simps(1)
thf(fact_1265_map__add__comm,axiom,
    ! [M1: a > option_list_o,M22: a > option_list_o] :
      ( ( ( inf_inf_set_a @ ( dom_a_list_o @ M1 ) @ ( dom_a_list_o @ M22 ) )
        = bot_bot_set_a )
     => ( ( map_add_a_list_o @ M1 @ M22 )
        = ( map_add_a_list_o @ M22 @ M1 ) ) ) ).

% map_add_comm
thf(fact_1266_ran__map__add,axiom,
    ! [M1: a > option_list_o,M22: a > option_list_o] :
      ( ( ( inf_inf_set_a @ ( dom_a_list_o @ M1 ) @ ( dom_a_list_o @ M22 ) )
        = bot_bot_set_a )
     => ( ( ran_a_list_o @ ( map_add_a_list_o @ M1 @ M22 ) )
        = ( sup_sup_set_list_o @ ( ran_a_list_o @ M1 ) @ ( ran_a_list_o @ M22 ) ) ) ) ).

% ran_map_add
thf(fact_1267_finite__set__of__finite__maps,axiom,
    ! [A4: set_a,B5: set_list_o] :
      ( ( finite_finite_a @ A4 )
     => ( ( finite_finite_list_o @ B5 )
       => ( finite6109038453002737250list_o
          @ ( collec1572211083203048352list_o
            @ ^ [M3: a > option_list_o] :
                ( ( ( dom_a_list_o @ M3 )
                  = A4 )
                & ( ord_le6901083488122529182list_o @ ( ran_a_list_o @ M3 ) @ B5 ) ) ) ) ) ) ).

% finite_set_of_finite_maps
thf(fact_1268_finite__ran,axiom,
    ! [P: a > option_list_o] :
      ( ( finite_finite_a @ ( dom_a_list_o @ P ) )
     => ( finite_finite_list_o @ ( ran_a_list_o @ P ) ) ) ).

% finite_ran
thf(fact_1269_graph__ranD,axiom,
    ! [X3: product_prod_a_b,M2: a > option_b] :
      ( ( member1426531481828664017od_a_b @ X3 @ ( graph_a_b @ M2 ) )
     => ( member_b @ ( product_snd_a_b @ X3 ) @ ( ran_a_b @ M2 ) ) ) ).

% graph_ranD
thf(fact_1270_snd__graph__ran,axiom,
    ! [M2: a > option_b] :
      ( ( image_2802296252294471260_a_b_b @ product_snd_a_b @ ( graph_a_b @ M2 ) )
      = ( ran_a_b @ M2 ) ) ).

% snd_graph_ran
thf(fact_1271_Not__Domain__rtrancl,axiom,
    ! [X3: a,R5: set_Product_prod_a_a,Y4: a] :
      ( ~ ( member_a @ X3 @ ( domain_a_a @ R5 ) )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ ( transitive_rtrancl_a @ R5 ) )
        = ( X3 = Y4 ) ) ) ).

% Not_Domain_rtrancl
thf(fact_1272_wf,axiom,
    wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).

% wf
thf(fact_1273_nat__descend__induct,axiom,
    ! [N2: nat,P2: nat > $o,M2: nat] :
      ( ! [K2: nat] :
          ( ( ord_less_nat @ N2 @ K2 )
         => ( P2 @ K2 ) )
     => ( ! [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N2 )
           => ( ! [I3: nat] :
                  ( ( ord_less_nat @ K2 @ I3 )
                 => ( P2 @ I3 ) )
             => ( P2 @ K2 ) ) )
       => ( P2 @ M2 ) ) ) ).

% nat_descend_induct
thf(fact_1274_trancl__subset__Sigma__aux,axiom,
    ! [A: a,B: a,R: set_Product_prod_a_a,A4: set_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_rtrancl_a @ R ) )
     => ( ( ord_le746702958409616551od_a_a @ R
          @ ( product_Sigma_a_a @ A4
            @ ^ [Uu: a] : A4 ) )
       => ( ( A = B )
          | ( member_a @ A @ A4 ) ) ) ) ).

% trancl_subset_Sigma_aux

% Helper facts (5)
thf(help_If_2_1_If_001tf__a_T,axiom,
    ! [X3: a,Y4: a] :
      ( ( if_a @ $false @ X3 @ Y4 )
      = Y4 ) ).

thf(help_If_1_1_If_001tf__a_T,axiom,
    ! [X3: a,Y4: a] :
      ( ( if_a @ $true @ X3 @ Y4 )
      = X3 ) ).

thf(help_If_3_1_If_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_T,axiom,
    ! [P2: $o] :
      ( ( P2 = $true )
      | ( P2 = $false ) ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_T,axiom,
    ! [X3: option_list_o,Y4: option_list_o] :
      ( ( if_option_list_o @ $false @ X3 @ Y4 )
      = Y4 ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_T,axiom,
    ! [X3: option_list_o,Y4: option_list_o] :
      ( ( if_option_list_o @ $true @ X3 @ Y4 )
      = X3 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( product_snd_a_b @ x )
    = ( product_snd_a_b @ y ) ) ).

%------------------------------------------------------------------------------