%------------------------------------------------------------------------------
% 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    : Separation_Logic_Unbounded/0002_Distributivity/prob_00309_008663__6763200_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1860 ( 457 unt; 577 typ;   0 def)
%            Number of atoms       : 4139 (1577 equ;   0 cnn)
%            Maximal formula atoms :   12 (   3 avg)
%            Number of connectives : 16102 ( 514   ~;  25   |; 266   &;13442   @)
%                                         (   0 <=>;1855  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   28 (   8 avg)
%            Number of types       :   80 (  79 usr)
%            Number of type conns  : 3204 (3204   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  501 ( 498 usr;  34 con; 0-7 aty)
%            Number of variables   : 4915 ( 116   ^;4669   !; 130   ?;4915   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 16:05:06.560
%------------------------------------------------------------------------------
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% Explicit typings (498)
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Option__Ooption_Itf__a_J_001t__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J,type,
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thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Option__Ooption_Itf__a_J_001tf__a_001_062_Itf__d_Mtf__c_J,type,
    bNF_Gr7827691902141730104_a_d_c: set_option_a > ( option_a > a ) > ( option_a > d > c ) > set_Pr5506164808955685639_a_d_c ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_001t__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J,type,
    bNF_Gr5816929820373306605_a_d_c: set_Pr7868159745199425715_a_d_c > ( produc6150846815813599699_a_d_c > product_prod_a_d_c ) > ( produc6150846815813599699_a_d_c > option2362401199305441953_a_d_c ) > set_Pr7868159745199425715_a_d_c ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_001tf__a_001_062_Itf__d_Mtf__c_J,type,
    bNF_Gr3902548584958070633_a_d_c: set_Pr7868159745199425715_a_d_c > ( produc6150846815813599699_a_d_c > a ) > ( produc6150846815813599699_a_d_c > d > c ) > set_Pr5506164808955685639_a_d_c ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001tf__a_001t__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J,type,
    bNF_Gr2096653962509776792_a_d_c: set_a > ( a > product_prod_a_d_c ) > ( a > option2362401199305441953_a_d_c ) > set_Pr7868159745199425715_a_d_c ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001tf__a_001tf__a_001_062_Itf__d_Mtf__c_J,type,
    bNF_Gr4650041265119207678_a_d_c: set_a > ( a > a ) > ( a > d > c ) > set_Pr5506164808955685639_a_d_c ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Option__Ooption_Itf__a_J,type,
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thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J,type,
    bNF_We920288868220767679_a_d_c: set_Pr8390707888732215751_a_d_c > $o ).

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_001t__Option__Ooption_Itf__a_J,type,
    bNF_We2467337426749329402tion_a: set_Pr7585778909603769095tion_a > set_option_a > option_a > $o ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J,type,
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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_001t__Option__Ooption_Itf__a_J,type,
    bNF_We4567742444881707410tion_a: set_Pr7585778909603769095tion_a > option_a > option_a > option_a ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J,type,
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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_001t__Option__Ooption_Itf__a_J,type,
    bNF_We6579146059749918992tion_a: set_Pr7585778909603769095tion_a > set_option_a > option_a ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J,type,
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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_BNF__Wellorder__Relation_Owo__rel_Osuc_001t__Option__Ooption_Itf__a_J,type,
    bNF_We5356091070762920229tion_a: set_Pr7585778909603769095tion_a > set_option_a > option_a ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Osuc_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J,type,
    bNF_We1545902585215803188_a_d_c: set_Pr8390707888732215751_a_d_c > set_Pr7868159745199425715_a_d_c > produc6150846815813599699_a_d_c ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Osuc_001tf__a,type,
    bNF_We6154283375207884895_suc_a: set_Product_prod_a_a > set_a > a ).

thf(sy_c_Equiv__Relations_Oequiv_001t__Option__Ooption_Itf__a_J,type,
    equiv_equiv_option_a: set_option_a > set_Pr7585778909603769095tion_a > $o ).

thf(sy_c_Equiv__Relations_Oequiv_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J,type,
    equiv_1650145883741090185_a_d_c: set_Pr7868159745199425715_a_d_c > set_Pr8390707888732215751_a_d_c > $o ).

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_Oproj_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
    equiv_6865337221296424970tion_a: set_Pr7585778909603769095tion_a > option_a > set_option_a ).

thf(sy_c_Equiv__Relations_Oproj_001tf__a_001tf__a,type,
    equiv_proj_a_a: set_Product_prod_a_a > a > set_a ).

thf(sy_c_Equiv__Relations_Oquotient_001t__Option__Ooption_Itf__a_J,type,
    equiv_2859340374733651339tion_a: set_option_a > set_Pr7585778909603769095tion_a > set_set_option_a ).

thf(sy_c_Equiv__Relations_Oquotient_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J,type,
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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_001t__Option__Ooption_It__Option__Ooption_Itf__a_J_J,type,
    finite8114217219359860531tion_a: set_option_option_a > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Option__Ooption_Itf__a_J,type,
    finite1674126218327898605tion_a: set_option_a > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J,type,
    finite6538345537044216828_a_d_c: set_Pr7868159745199425715_a_d_c > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J,type,
    finite5780817786756972904_a_d_c: set_Pr5506164808955685639_a_d_c > $o ).

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

thf(sy_c_Fun_Obij__betw_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
    bij_be5431266891817924854tion_a: ( option_a > option_a ) > set_option_a > set_option_a > $o ).

thf(sy_c_Fun_Obij__betw_001t__Option__Ooption_Itf__a_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J,type,
    bij_be8275362299291636869_a_d_c: ( option_a > produc6150846815813599699_a_d_c ) > set_option_a > set_Pr7868159745199425715_a_d_c > $o ).

thf(sy_c_Fun_Obij__betw_001t__Option__Ooption_Itf__a_J_001tf__a,type,
    bij_betw_option_a_a: ( option_a > a ) > set_option_a > set_a > $o ).

thf(sy_c_Fun_Obij__betw_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_001t__Option__Ooption_Itf__a_J,type,
    bij_be4831480119166671655tion_a: ( produc6150846815813599699_a_d_c > option_a ) > set_Pr7868159745199425715_a_d_c > set_option_a > $o ).

thf(sy_c_Fun_Obij__betw_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J,type,
    bij_be734363561790614582_a_d_c: ( produc6150846815813599699_a_d_c > produc6150846815813599699_a_d_c ) > set_Pr7868159745199425715_a_d_c > set_Pr7868159745199425715_a_d_c > $o ).

thf(sy_c_Fun_Obij__betw_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_001tf__a,type,
    bij_be6485230601037006177_d_c_a: ( produc6150846815813599699_a_d_c > a ) > set_Pr7868159745199425715_a_d_c > set_a > $o ).

thf(sy_c_Fun_Obij__betw_001tf__a_001t__Option__Ooption_Itf__a_J,type,
    bij_betw_a_option_a: ( a > option_a ) > set_a > set_option_a > $o ).

thf(sy_c_Fun_Obij__betw_001tf__a_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J,type,
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thf(sy_c_Fun_Obij__betw_001tf__a_001tf__a,type,
    bij_betw_a_a: ( a > a ) > set_a > set_a > $o ).

thf(sy_c_Fun_Ocomp_001t__Option__Ooption_It__Option__Ooption_Itf__a_J_J_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
    comp_o1254687777855551975tion_a: ( option_option_a > option_a ) > ( option_a > option_option_a ) > option_a > option_a ).

thf(sy_c_Fun_Ocomp_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_001t__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_001t__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J,type,
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thf(sy_c_Fun_Ocomp_001t__Option__Ooption_Itf__a_J_001t__Nat__Onat_001t__Option__Ooption_Itf__a_J,type,
    comp_o8583038678572498833tion_a: ( option_a > nat ) > ( option_a > option_a ) > option_a > nat ).

thf(sy_c_Fun_Ocomp_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
    comp_o3154387707078715297tion_a: ( option_a > option_a ) > ( option_a > option_a ) > option_a > option_a ).

thf(sy_c_Fun_Ocomp_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J_001tf__a,type,
    comp_o6087033147929006299on_a_a: ( option_a > option_a ) > ( a > option_a ) > a > option_a ).

thf(sy_c_Fun_Ocomp_001t__Option__Ooption_Itf__a_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_001tf__a,type,
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thf(sy_c_Fun_Ocomp_001t__Option__Ooption_Itf__a_J_001tf__a_001t__Option__Ooption_Itf__a_J,type,
    comp_o3864519266390211175tion_a: ( option_a > a ) > ( option_a > option_a ) > option_a > a ).

thf(sy_c_Fun_Ocomp_001t__Option__Ooption_Itf__a_J_001tf__a_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J,type,
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thf(sy_c_Fun_Ocomp_001t__Option__Ooption_Itf__a_J_001tf__a_001tf__a,type,
    comp_option_a_a_a: ( option_a > a ) > ( a > option_a ) > a > a ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
    comp_s1419921648917501825_set_a: ( set_option_a > set_a ) > ( set_a > set_option_a ) > set_a > set_a ).

thf(sy_c_Fun_Ocomp_001tf__a_001_Eo_001tf__a,type,
    comp_a_o_a: ( a > $o ) > ( a > a ) > a > $o ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Nat__Onat_001tf__a,type,
    comp_a_nat_a: ( a > nat ) > ( a > a ) > a > nat ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
    comp_a6249931511552232923tion_a: ( a > option_a ) > ( option_a > a ) > option_a > option_a ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Option__Ooption_Itf__a_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J,type,
    comp_a2470762695209442538_a_d_c: ( a > option_a ) > ( produc6150846815813599699_a_d_c > a ) > produc6150846815813599699_a_d_c > option_a ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Option__Ooption_Itf__a_J_001tf__a,type,
    comp_a_option_a_a: ( a > option_a ) > ( a > a ) > a > option_a ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_001t__Option__Ooption_Itf__a_J,type,
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thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001t__Option__Ooption_Itf__a_J,type,
    comp_a_a_option_a: ( a > a ) > ( option_a > a ) > option_a > a ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001tf__a,type,
    comp_a_a_a: ( a > a ) > ( a > a ) > a > a ).

thf(sy_c_Fun_Ocomp_001tf__c_001tf__c_001tf__d,type,
    comp_c_c_d: ( c > c ) > ( d > c ) > d > c ).

thf(sy_c_Fun_Ofun__upd_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J,type,
    fun_up1917685652830187432_a_d_c: ( option_a > option2362401199305441953_a_d_c ) > option_a > option2362401199305441953_a_d_c > option_a > option2362401199305441953_a_d_c ).

thf(sy_c_Fun_Ofun__upd_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
    fun_up1079276522633388797tion_a: ( option_a > option_a ) > option_a > option_a > option_a > option_a ).

thf(sy_c_Fun_Ofun__upd_001t__Option__Ooption_Itf__a_J_001tf__a,type,
    fun_upd_option_a_a: ( option_a > a ) > option_a > a > option_a > a ).

thf(sy_c_Fun_Ofun__upd_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J,type,
    fun_up4469873178457468855_a_d_c: ( produc6150846815813599699_a_d_c > option2362401199305441953_a_d_c ) > produc6150846815813599699_a_d_c > option2362401199305441953_a_d_c > produc6150846815813599699_a_d_c > option2362401199305441953_a_d_c ).

thf(sy_c_Fun_Ofun__upd_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_001t__Option__Ooption_Itf__a_J,type,
    fun_up4051926500471473070tion_a: ( produc6150846815813599699_a_d_c > option_a ) > produc6150846815813599699_a_d_c > option_a > produc6150846815813599699_a_d_c > option_a ).

thf(sy_c_Fun_Ofun__upd_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_001tf__a,type,
    fun_up3467885399696393320_d_c_a: ( produc6150846815813599699_a_d_c > a ) > produc6150846815813599699_a_d_c > a > produc6150846815813599699_a_d_c > a ).

thf(sy_c_Fun_Ofun__upd_001t__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_001t__Option__Ooption_It__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J,type,
    fun_up7471186249482589761_a_d_c: ( product_prod_a_d_c > option5918686196580481265_a_d_c ) > product_prod_a_d_c > option5918686196580481265_a_d_c > product_prod_a_d_c > option5918686196580481265_a_d_c ).

thf(sy_c_Fun_Ofun__upd_001tf__a_001t__Option__Ooption_I_062_Itf__d_Mtf__c_J_J,type,
    fun_upd_a_option_d_c: ( a > option_d_c ) > a > option_d_c > a > option_d_c ).

thf(sy_c_Fun_Ofun__upd_001tf__a_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J,type,
    fun_up2590722781756628834_a_d_c: ( a > option2362401199305441953_a_d_c ) > a > option2362401199305441953_a_d_c > a > option2362401199305441953_a_d_c ).

thf(sy_c_Fun_Ofun__upd_001tf__a_001t__Option__Ooption_Itf__a_J,type,
    fun_upd_a_option_a: ( a > option_a ) > a > option_a > a > option_a ).

thf(sy_c_Fun_Ofun__upd_001tf__a_001tf__a,type,
    fun_upd_a_a: ( a > a ) > a > a > a > a ).

thf(sy_c_Fun_Ofun__upd_001tf__d_001tf__c,type,
    fun_upd_d_c: ( d > c ) > d > c > d > c ).

thf(sy_c_Fun_Oid_001t__Option__Ooption_Itf__a_J,type,
    id_option_a: option_a > option_a ).

thf(sy_c_Fun_Oid_001t__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J,type,
    id_Pro4159844622578998816_a_d_c: product_prod_a_d_c > product_prod_a_d_c ).

thf(sy_c_Fun_Oid_001t__Set__Oset_Itf__a_J,type,
    id_set_a: set_a > set_a ).

thf(sy_c_Fun_Oid_001tf__a,type,
    id_a: a > a ).

thf(sy_c_Fun_Oinj__on_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_It__Option__Ooption_Itf__a_J_J,type,
    inj_on2224753519991154999tion_a: ( option_a > option_option_a ) > set_option_a > $o ).

thf(sy_c_Fun_Oinj__on_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
    inj_on8559383841115902449tion_a: ( option_a > option_a ) > set_option_a > $o ).

thf(sy_c_Fun_Oinj__on_001t__Option__Ooption_Itf__a_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J,type,
    inj_on7085348629561674240_a_d_c: ( option_a > produc6150846815813599699_a_d_c ) > set_option_a > $o ).

thf(sy_c_Fun_Oinj__on_001t__Option__Ooption_Itf__a_J_001tf__a,type,
    inj_on_option_a_a: ( option_a > a ) > set_option_a > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_001t__Option__Ooption_Itf__a_J,type,
    inj_on3641466449436709026tion_a: ( produc6150846815813599699_a_d_c > option_a ) > set_Pr7868159745199425715_a_d_c > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J,type,
    inj_on5053348487737048625_a_d_c: ( produc6150846815813599699_a_d_c > produc6150846815813599699_a_d_c ) > set_Pr7868159745199425715_a_d_c > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_001tf__a,type,
    inj_on6701986109746958172_d_c_a: ( produc6150846815813599699_a_d_c > a ) > set_Pr7868159745199425715_a_d_c > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J,type,
    inj_on4952834208159004389_a_d_c: ( product_prod_a_d_c > option2362401199305441953_a_d_c ) > set_Pr5506164808955685639_a_d_c > $o ).

thf(sy_c_Fun_Oinj__on_001tf__a_001t__Option__Ooption_Itf__a_J,type,
    inj_on_a_option_a: ( a > option_a ) > set_a > $o ).

thf(sy_c_Fun_Oinj__on_001tf__a_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J,type,
    inj_on3709328703910593990_a_d_c: ( a > produc6150846815813599699_a_d_c ) > set_a > $o ).

thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
    inj_on_a_a: ( a > a ) > set_a > $o ).

thf(sy_c_Fun_Oinj__on_001tf__d_001tf__c,type,
    inj_on_d_c: ( d > c ) > set_d > $o ).

thf(sy_c_Fun_Ooverride__on_001tf__d_001tf__c,type,
    override_on_d_c: ( d > c ) > ( d > c ) > set_d > d > c ).

thf(sy_c_Fun_Othe__inv__into_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
    the_in2538339130118444083tion_a: set_option_a > ( option_a > option_a ) > option_a > option_a ).

thf(sy_c_Fun_Othe__inv__into_001t__Option__Ooption_Itf__a_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J,type,
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thf(sy_c_Fun_Othe__inv__into_001t__Option__Ooption_Itf__a_J_001tf__a,type,
    the_in1757154643552616557on_a_a: set_option_a > ( option_a > a ) > a > option_a ).

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    the_in1584024643688058340tion_a: set_Pr7868159745199425715_a_d_c > ( produc6150846815813599699_a_d_c > option_a ) > option_a > produc6150846815813599699_a_d_c ).

thf(sy_c_Fun_Othe__inv__into_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J,type,
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thf(sy_c_Fun_Othe__inv__into_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_001tf__a,type,
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thf(sy_c_Fun_Othe__inv__into_001tf__a_001t__Option__Ooption_Itf__a_J,type,
    the_in8758012798868597241tion_a: set_a > ( a > option_a ) > option_a > a ).

thf(sy_c_Fun_Othe__inv__into_001tf__a_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J,type,
    the_in8611454470807686664_a_d_c: set_a > ( a > produc6150846815813599699_a_d_c ) > produc6150846815813599699_a_d_c > a ).

thf(sy_c_Fun_Othe__inv__into_001tf__a_001tf__a,type,
    the_inv_into_a_a: set_a > ( a > a ) > a > a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
    minus_1574173051537231627tion_a: set_option_a > set_option_a > set_option_a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_J,type,
    minus_6512073291116468334tion_a: set_Pr7585778909603769095tion_a > set_Pr7585778909603769095tion_a > set_Pr7585778909603769095tion_a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_J_J,type,
    minus_1498755887985271854_a_d_c: set_Pr8390707888732215751_a_d_c > set_Pr8390707888732215751_a_d_c > set_Pr8390707888732215751_a_d_c ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_J,type,
    minus_1640812273454246426_a_d_c: set_Pr7868159745199425715_a_d_c > set_Pr7868159745199425715_a_d_c > set_Pr7868159745199425715_a_d_c ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
    minus_6817036919807184750od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
    minus_minus_set_a: set_a > set_a > set_a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__d_J,type,
    minus_minus_set_d: set_d > set_d > set_d ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
    uminus6205308855922866075tion_a: set_option_a > set_option_a ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
    uminus_uminus_set_a: set_a > set_a ).

thf(sy_c_If_001t__Option__Ooption_Itf__a_J,type,
    if_option_a: $o > option_a > option_a > option_a ).

thf(sy_c_If_001tf__c,type,
    if_c: $o > c > c > c ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
    inf_inf_set_option_a: set_option_a > set_option_a > set_option_a ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
    inf_inf_set_a: set_a > set_a > set_a ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
    sup_sup_set_option_a: set_option_a > set_option_a > set_option_a ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
    sup_sup_set_a: set_a > set_a > set_a ).

thf(sy_c_List_Ofind_001tf__a,type,
    find_a: ( a > $o ) > list_a > option_a ).

thf(sy_c_List_Olist_OCons_001_062_Itf__d_Mtf__c_J,type,
    cons_d_c: ( d > c ) > list_d_c > list_d_c ).

thf(sy_c_List_Olist_OCons_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J,type,
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thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J,type,
    cons_P5713675596995722633_a_d_c: produc6150846815813599699_a_d_c > list_P8522657617046293465_a_d_c > list_P8522657617046293465_a_d_c ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J,type,
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thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    cons_P7316939126706565853od_a_a: product_prod_a_a > list_P1396940483166286381od_a_a > list_P1396940483166286381od_a_a ).

thf(sy_c_List_Olist_OCons_001tf__a,type,
    cons_a: a > list_a > list_a ).

thf(sy_c_List_Olist_ONil_001tf__a,type,
    nil_a: list_a ).

thf(sy_c_List_Olist_Oset_001_062_Itf__d_Mtf__c_J,type,
    set_d_c2: list_d_c > set_d_c ).

thf(sy_c_List_Olist_Oset_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J,type,
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thf(sy_c_List_Olist_Oset_001t__Option__Ooption_Itf__a_J,type,
    set_option_a2: list_option_a > set_option_a ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J,type,
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thf(sy_c_UnboundedLogic_Ologic_001tf__a_001tf__b,type,
    logic_a_b: ( a > a > option_a ) > ( b > a > a ) > ( b > b > b ) > ( b > b > b ) > ( b > b ) > b > ( a > $o ) > $o ).

thf(sy_c_UnboundedLogic_Ologic_Oentails_001tf__a_001tf__b_001tf__c_001tf__d,type,
    entails_a_b_c_d: ( a > a > option_a ) > ( b > a > a ) > ( a > $o ) > assertion_a_b_c_d > ( ( d > c ) > set_a ) > assertion_a_b_c_d > $o ).

thf(sy_c_UnboundedLogic_Ologic_Oequal__outside_001tf__d_001tf__c,type,
    equal_outside_d_c: ( d > c ) > ( d > c ) > set_d > $o ).

thf(sy_c_UnboundedLogic_Ologic_Oequivalent_001tf__a_001tf__b_001tf__c_001tf__d,type,
    equivalent_a_b_c_d: ( a > a > option_a ) > ( b > a > a ) > ( a > $o ) > assertion_a_b_c_d > ( ( d > c ) > set_a ) > assertion_a_b_c_d > $o ).

thf(sy_c_UnboundedLogic_Ologic_Oframe__property_001tf__a_001tf__d_001tf__c,type,
    frame_property_a_d_c: ( a > a > option_a ) > ( a > $o ) > set_Pr7868159745199425715_a_d_c > $o ).

thf(sy_c_UnboundedLogic_Ologic_Ointuitionistic_001tf__a_001tf__b_001tf__d_001tf__c,type,
    intuit4720955538653295669_b_d_c: ( a > a > option_a ) > ( b > a > a ) > ( a > $o ) > ( d > c ) > ( ( d > c ) > set_a ) > assertion_a_b_c_d > $o ).

thf(sy_c_UnboundedLogic_Ologic_Omodified_001tf__a_001tf__d_001tf__c,type,
    modified_a_d_c: set_Pr7868159745199425715_a_d_c > set_d ).

thf(sy_c_UnboundedLogic_Ologic_Onot__in__fv_001tf__a_001tf__b_001tf__c_001tf__d,type,
    not_in_fv_a_b_c_d: ( a > a > option_a ) > ( b > a > a ) > ( a > $o ) > assertion_a_b_c_d > set_d > $o ).

thf(sy_c_UnboundedLogic_Ologic_Opure_001tf__a_001tf__b_001tf__c_001tf__d,type,
    pure_a_b_c_d: ( a > a > option_a ) > ( b > a > a ) > ( a > $o ) > assertion_a_b_c_d > $o ).

thf(sy_c_UnboundedLogic_Ologic_Osafe_001tf__a_001tf__d_001tf__c,type,
    safe_a_d_c: set_Pr7868159745199425715_a_d_c > product_prod_a_d_c > $o ).

thf(sy_c_UnboundedLogic_Ologic_Osafety__monotonicity_001tf__a_001tf__d_001tf__c,type,
    safety7280469885071620222_a_d_c: ( a > a > option_a ) > ( a > $o ) > set_Pr7868159745199425715_a_d_c > $o ).

thf(sy_c_UnboundedLogic_Ologic_Osat_001tf__a_001tf__b_001tf__d_001tf__c,type,
    sat_a_b_d_c: ( a > a > option_a ) > ( b > a > a ) > ( a > $o ) > a > ( d > c ) > ( ( d > c ) > set_a ) > assertion_a_b_c_d > $o ).

thf(sy_c_UnboundedLogic_Ologic_Ovalid__command_001tf__a_001tf__d_001tf__c,type,
    valid_command_a_d_c: ( a > $o ) > set_Pr7868159745199425715_a_d_c > $o ).

thf(sy_c_UnboundedLogic_Ologic_Ovalid__hoare__triple_001tf__a_001tf__b_001tf__c_001tf__d,type,
    valid_8824771084768397689_b_c_d: ( a > a > option_a ) > ( b > a > a ) > ( a > $o ) > assertion_a_b_c_d > set_Pr7868159745199425715_a_d_c > assertion_a_b_c_d > ( ( d > c ) > set_a ) > $o ).

thf(sy_c_UnboundedLogic_Opre__logic_Ocompatible_001tf__a,type,
    pre_compatible_a: ( a > a > option_a ) > a > a > $o ).

thf(sy_c_UnboundedLogic_Opre__logic_Olarger_001t__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J,type,
    pre_la7052613405163955386_a_d_c: ( product_prod_a_d_c > product_prod_a_d_c > option2362401199305441953_a_d_c ) > product_prod_a_d_c > product_prod_a_d_c > $o ).

thf(sy_c_UnboundedLogic_Opre__logic_Olarger_001tf__a,type,
    pre_larger_a: ( a > a > option_a ) > a > a > $o ).

thf(sy_c_Wellfounded_Ofinite__psubset_001t__Option__Ooption_Itf__a_J,type,
    finite4966134214920407047tion_a: set_Pr452613198074451719tion_a ).

thf(sy_c_Wellfounded_Ofinite__psubset_001tf__a,type,
    finite_psubset_a: set_Pr5845495582615845127_set_a ).

thf(sy_c_Wellfounded_Olex__prod_001t__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J,type,
    lex_pr7013848916100950307_a_d_c: set_Pr2251131264138062051_a_d_c > set_Pr2205332559432586627_a_d_c > set_Pr8390707888732215751_a_d_c ).

thf(sy_c_Wellfounded_Olex__prod_001tf__a_001_062_Itf__d_Mtf__c_J,type,
    lex_prod_a_d_c: set_Product_prod_a_a > set_Pr1239784501399087881_c_d_c > set_Pr2251131264138062051_a_d_c ).

thf(sy_c_Wellfounded_Omax__ext_001t__Option__Ooption_Itf__a_J,type,
    max_ext_option_a: set_Pr7585778909603769095tion_a > set_Pr452613198074451719tion_a ).

thf(sy_c_Wellfounded_Omax__ext_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J,type,
    max_ex8685825336233442702_a_d_c: set_Pr8390707888732215751_a_d_c > set_Pr1328490333363696071_a_d_c ).

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__Option__Ooption_Itf__a_J,type,
    wf_option_a: set_Pr7585778909603769095tion_a > $o ).

thf(sy_c_Wellfounded_Owf_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J,type,
    wf_Pro1713206634478224572_a_d_c: set_Pr8390707888732215751_a_d_c > $o ).

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

thf(sy_c_Wfrec_Osame__fst_001t__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J,type,
    same_f4766842598631432416_a_d_c: ( product_prod_a_d_c > $o ) > ( product_prod_a_d_c > set_Pr2205332559432586627_a_d_c ) > set_Pr8390707888732215751_a_d_c ).

thf(sy_c_Wfrec_Osame__fst_001tf__a_001_062_Itf__d_Mtf__c_J,type,
    same_fst_a_d_c: ( a > $o ) > ( a > set_Pr1239784501399087881_c_d_c ) > set_Pr2251131264138062051_a_d_c ).

thf(sy_c_member_001_062_Itf__d_Mtf__c_J,type,
    member_d_c: ( d > c ) > set_d_c > $o ).

thf(sy_c_member_001t__Option__Ooption_It__Option__Ooption_Itf__a_J_J,type,
    member5113800082084363315tion_a: option_option_a > set_option_option_a > $o ).

thf(sy_c_member_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_J,type,
    member4494181189314409282_a_d_c: option4105738954736457113_a_d_c > set_op4102275208527013369_a_d_c > $o ).

thf(sy_c_member_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J,type,
    member1874962706598213560_a_d_c: option2362401199305441953_a_d_c > set_op3089619689617166423_a_d_c > $o ).

thf(sy_c_member_001t__Option__Ooption_Itf__a_J,type,
    member_option_a: option_a > set_option_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_Itf__d_Mtf__c_J_M_062_Itf__d_Mtf__c_J_J,type,
    member3094720156254151762_c_d_c: product_prod_d_c_d_c > set_Pr1239784501399087881_c_d_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_Itf__d_Mtf__c_J_Mtf__a_J,type,
    member3659673801472585190_d_c_a: product_prod_d_c_a > set_Pr6048285399905979781_d_c_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J,type,
    member7044238567156598732_a_d_c: produc6655796332578646563_a_d_c > set_Pr2205332559432586627_a_d_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_Mt__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J,type,
    member8774983680413349372_a_d_c: produc3649636187029808851_a_d_c > set_Pr7666514177632660915_a_d_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J,type,
    member5498148017924304208tion_a: produc3509355604313844263tion_a > set_Pr7585778909603769095tion_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_J,type,
    member6161829431480814559_a_d_c: produc4766651102893408054_a_d_c > set_Pr8153776533466944534_a_d_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mtf__a_J,type,
    member6056235002698166154on_a_a: produc3083010940779526881on_a_a > set_Pr6308966090954093121on_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_Mt__Option__Ooption_Itf__a_J_J,type,
    member1057245869843309569tion_a: produc8885439578110678872tion_a > set_Pr5439073823713721144tion_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_J,type,
    member3575411683413719056_a_d_c: produc5969672217230369255_a_d_c > set_Pr8390707888732215751_a_d_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_Mtf__a_J,type,
    member2251515730839929787_d_c_a: produc2466114957033899410_d_c_a > set_Pr1516632107838038002_d_c_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J,type,
    member2052822272342364412_a_d_c: produc6150846815813599699_a_d_c > set_Pr7868159745199425715_a_d_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J,type,
    member2237522729980725036_a_d_c: produc20764062985668227_a_d_c > set_Pr2251131264138062051_a_d_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Option__Ooption_Itf__a_J_J_Mt__Set__Oset_It__Option__Ooption_Itf__a_J_J_J,type,
    member5358692782348450128tion_a: produc8652252815484796455tion_a > set_Pr452613198074451719tion_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_J_J,type,
    member2235620415397146640_a_d_c: produc5088816726977995239_a_d_c > set_Pr1328490333363696071_a_d_c > $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_M_062_Itf__d_Mtf__c_J_J,type,
    member8140300833826434152_a_d_c: product_prod_a_d_c > set_Pr5506164808955685639_a_d_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Option__Ooption_Itf__a_J_J,type,
    member6937434987665551382tion_a: produc3964210925746912109tion_a > set_Pr3411724424142761165tion_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_J,type,
    member4023273573022285733_a_d_c: produc4237872799216255356_a_d_c > set_Pr3248570043410859740_a_d_c > $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__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
    member_set_option_a: set_option_a > set_set_option_a > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_J,type,
    member7718464485209093084_a_d_c: set_Pr7868159745199425715_a_d_c > set_se8930433228465195923_a_d_c > $o ).

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

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

thf(sy_c_member_001tf__c,type,
    member_c: c > set_c > $o ).

thf(sy_c_member_001tf__d,type,
    member_d: d > set_d > $o ).

thf(sy_v_A,type,
    a2: assertion_a_b_c_d ).

thf(sy_v_B,type,
    b2: assertion_a_b_c_d ).

thf(sy_v__092_060Delta_062_H____,type,
    delta: ( d > c ) > set_a ).

thf(sy_v__092_060sigma_062____,type,
    sigma: a ).

thf(sy_v_mult,type,
    mult: b > a > a ).

thf(sy_v_one,type,
    one: b ).

thf(sy_v_p,type,
    p: b ).

thf(sy_v_plus,type,
    plus: a > a > option_a ).

thf(sy_v_s____,type,
    s: d > c ).

thf(sy_v_sadd,type,
    sadd: b > b > b ).

thf(sy_v_sinv,type,
    sinv: b > b ).

thf(sy_v_smult,type,
    smult: b > b > b ).

thf(sy_v_valid,type,
    valid: a > $o ).

% Relevant facts (1277)
thf(fact_0_equal__outside__def,axiom,
    ( equal_outside_d_c
    = ( ^ [S: d > c,S2: d > c,S3: set_d] :
        ! [X: d] :
          ( ~ ( member_d @ X @ S3 )
         => ( ( S @ X )
            = ( S2 @ X ) ) ) ) ) ).

% equal_outside_def
thf(fact_1_commutative,axiom,
    ! [A: a,B: a] :
      ( ( plus @ A @ B )
      = ( plus @ B @ A ) ) ).

% commutative
thf(fact_2_can__divide,axiom,
    ! [P: b,A: a,B: a] :
      ( ( ( mult @ P @ A )
        = ( mult @ P @ B ) )
     => ( A = B ) ) ).

% can_divide
thf(fact_3_sat_Osimps_I1_J,axiom,
    ! [Sigma: a,S4: d > c,Delta: ( d > c ) > set_a,P: b,A2: assertion_a_b_c_d] :
      ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( mult_b_a_c_d @ P @ A2 ) )
      = ( ? [A3: a] :
            ( ( Sigma
              = ( mult @ P @ A3 ) )
            & ( sat_a_b_d_c @ plus @ mult @ valid @ A3 @ S4 @ Delta @ A2 ) ) ) ) ).

% sat.simps(1)
thf(fact_4_sat__mult,axiom,
    ! [Sigma: a,P: b,S4: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
      ( ! [A4: a] :
          ( ( Sigma
            = ( mult @ P @ A4 ) )
         => ( sat_a_b_d_c @ plus @ mult @ valid @ A4 @ S4 @ Delta @ A2 ) )
     => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( mult_b_a_c_d @ P @ A2 ) ) ) ).

% sat_mult
thf(fact_5_sat_Osimps_I7_J,axiom,
    ! [Sigma: a,S4: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( and_a_b_c_d @ A2 @ B2 ) )
      = ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ A2 )
        & ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ B2 ) ) ) ).

% sat.simps(7)
thf(fact_6_unique__inv,axiom,
    ! [A: a,P: b,B: a] :
      ( ( A
        = ( mult @ P @ B ) )
      = ( B
        = ( mult @ ( sinv @ P ) @ A ) ) ) ).

% unique_inv
thf(fact_7__092_060open_062_092_060sigma_062_M_As_M_A_092_060Delta_062_A_092_060Turnstile_062_AAnd_A_IMult_Ap_AA_J_A_IMult_Ap_AB_J_092_060close_062,axiom,
    sat_a_b_d_c @ plus @ mult @ valid @ sigma @ s @ delta @ ( and_a_b_c_d @ ( mult_b_a_c_d @ p @ a2 ) @ ( mult_b_a_c_d @ p @ b2 ) ) ).

% \<open>\<sigma>, s, \<Delta> \<Turnstile> And (Mult p A) (Mult p B)\<close>
thf(fact_8_pure__def,axiom,
    ! [A2: assertion_a_b_c_d] :
      ( ( pure_a_b_c_d @ plus @ mult @ valid @ A2 )
      = ( ! [Sigma2: a,Sigma3: a,S: d > c,Delta2: ( d > c ) > set_a,Delta3: ( d > c ) > set_a] :
            ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma2 @ S @ Delta2 @ A2 )
            = ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma3 @ S @ Delta3 @ A2 ) ) ) ) ).

% pure_def
thf(fact_9_one__neutral,axiom,
    ! [A: a] :
      ( ( mult @ one @ A )
      = A ) ).

% one_neutral
thf(fact_10_equivalentI,axiom,
    ! [Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ! [Sigma4: a,S5: d > c] :
          ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma4 @ S5 @ Delta @ A2 )
         => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma4 @ S5 @ Delta @ B2 ) )
     => ( ! [Sigma4: a,S5: d > c] :
            ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma4 @ S5 @ Delta @ B2 )
           => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma4 @ S5 @ Delta @ A2 ) )
       => ( equivalent_a_b_c_d @ plus @ mult @ valid @ A2 @ Delta @ B2 ) ) ) ).

% equivalentI
thf(fact_11_sat_Osimps_I10_J,axiom,
    ! [Sigma: a,S4: d > c,Delta: ( d > c ) > set_a] :
      ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ pred_a_b_c_d )
      = ( member_a @ Sigma @ ( Delta @ S4 ) ) ) ).

% sat.simps(10)
thf(fact_12_dot__and1,axiom,
    ! [P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( entails_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ P @ ( and_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( and_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ).

% dot_and1
thf(fact_13_not__in__fv__def,axiom,
    ! [A2: assertion_a_b_c_d,S6: set_d] :
      ( ( not_in_fv_a_b_c_d @ plus @ mult @ valid @ A2 @ S6 )
      = ( ! [Sigma2: a,S: d > c,Delta2: ( d > c ) > set_a,S2: d > c] :
            ( ( equal_outside_d_c @ S @ S2 @ S6 )
           => ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma2 @ S @ Delta2 @ A2 )
              = ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma2 @ S2 @ Delta2 @ A2 ) ) ) ) ) ).

% not_in_fv_def
thf(fact_14_DotPos,axiom,
    ! [A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a,B2: assertion_a_b_c_d,Pi: b] :
      ( ( entails_a_b_c_d @ plus @ mult @ valid @ A2 @ Delta @ B2 )
      = ( entails_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ Pi @ A2 ) @ Delta @ ( mult_b_a_c_d @ Pi @ B2 ) ) ) ).

% DotPos
thf(fact_15_entailsI,axiom,
    ! [Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ! [Sigma4: a,S5: d > c] :
          ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma4 @ S5 @ Delta @ A2 )
         => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma4 @ S5 @ Delta @ B2 ) )
     => ( entails_a_b_c_d @ plus @ mult @ valid @ A2 @ Delta @ B2 ) ) ).

% entailsI
thf(fact_16_entails__def,axiom,
    ! [A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a,B2: assertion_a_b_c_d] :
      ( ( entails_a_b_c_d @ plus @ mult @ valid @ A2 @ Delta @ B2 )
      = ( ! [Sigma2: a,S: d > c] :
            ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma2 @ S @ Delta @ A2 )
           => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma2 @ S @ Delta @ B2 ) ) ) ) ).

% entails_def
thf(fact_17_sat_Osimps_I5_J,axiom,
    ! [Sigma: a,S4: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( imp_a_b_c_d @ A2 @ B2 ) )
      = ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ A2 )
       => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ B2 ) ) ) ).

% sat.simps(5)
thf(fact_18_sat__imp,axiom,
    ! [Sigma: a,S4: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ A2 )
       => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ B2 ) )
     => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( imp_a_b_c_d @ A2 @ B2 ) ) ) ).

% sat_imp
thf(fact_19_sat_Osimps_I11_J,axiom,
    ! [Sigma: a,S4: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
      ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( bounded_a_b_c_d @ A2 ) )
      = ( ( valid @ Sigma )
       => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ A2 ) ) ) ).

% sat.simps(11)
thf(fact_20_sat_Osimps_I6_J,axiom,
    ! [Sigma: a,S4: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( or_a_b_c_d @ A2 @ B2 ) )
      = ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ A2 )
        | ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ B2 ) ) ) ).

% sat.simps(6)
thf(fact_21_sat_Osimps_I4_J,axiom,
    ! [Sigma: a,S4: d > c,Delta: ( d > c ) > set_a,B: ( d > c ) > a > $o] :
      ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( sem_d_c_a_b @ B ) )
      = ( B @ S4 @ Sigma ) ) ).

% sat.simps(4)
thf(fact_22_equivalent__def,axiom,
    ! [A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a,B2: assertion_a_b_c_d] :
      ( ( equivalent_a_b_c_d @ plus @ mult @ valid @ A2 @ Delta @ B2 )
      = ( ( entails_a_b_c_d @ plus @ mult @ valid @ A2 @ Delta @ B2 )
        & ( entails_a_b_c_d @ plus @ mult @ valid @ B2 @ Delta @ A2 ) ) ) ).

% equivalent_def
thf(fact_23_dot__or2,axiom,
    ! [P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( entails_a_b_c_d @ plus @ mult @ valid @ ( or_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) @ Delta @ ( mult_b_a_c_d @ P @ ( or_a_b_c_d @ A2 @ B2 ) ) ) ).

% dot_or2
thf(fact_24_dot__or1,axiom,
    ! [P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( entails_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ P @ ( or_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( or_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ).

% dot_or1
thf(fact_25_DotOr,axiom,
    ! [P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( equivalent_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ P @ ( or_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( or_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ).

% DotOr
thf(fact_26_hoare__triple__input,axiom,
    ! [P2: assertion_a_b_c_d,C: set_Pr7868159745199425715_a_d_c,Q: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( valid_8824771084768397689_b_c_d @ plus @ mult @ valid @ P2 @ C @ Q @ Delta )
      = ( valid_8824771084768397689_b_c_d @ plus @ mult @ valid @ ( bounded_a_b_c_d @ P2 ) @ C @ Q @ Delta ) ) ).

% hoare_triple_input
thf(fact_27_DotWand,axiom,
    ! [P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( equivalent_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ P @ ( wand_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( wand_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ).

% DotWand
thf(fact_28_dot__wand2,axiom,
    ! [P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( entails_a_b_c_d @ plus @ mult @ valid @ ( wand_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) @ Delta @ ( mult_b_a_c_d @ P @ ( wand_a_b_c_d @ A2 @ B2 ) ) ) ).

% dot_wand2
thf(fact_29_dot__wand1,axiom,
    ! [P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( entails_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ P @ ( wand_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( wand_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ).

% dot_wand1
thf(fact_30_DotStar,axiom,
    ! [P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( equivalent_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ P @ ( star_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( star_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ).

% DotStar
thf(fact_31_DotDot,axiom,
    ! [P: b,Q2: b,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( equivalent_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ P @ ( mult_b_a_c_d @ Q2 @ A2 ) ) @ Delta @ ( mult_b_a_c_d @ ( smult @ P @ Q2 ) @ A2 ) ) ).

% DotDot
thf(fact_32_dot__star2,axiom,
    ! [P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( entails_a_b_c_d @ plus @ mult @ valid @ ( star_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) @ Delta @ ( mult_b_a_c_d @ P @ ( star_a_b_c_d @ A2 @ B2 ) ) ) ).

% dot_star2
thf(fact_33_dot__star1,axiom,
    ! [P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( entails_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ P @ ( star_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( star_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ).

% dot_star1
thf(fact_34_dot__mult2,axiom,
    ! [P: b,Q2: b,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( entails_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ ( smult @ P @ Q2 ) @ A2 ) @ Delta @ ( mult_b_a_c_d @ P @ ( mult_b_a_c_d @ Q2 @ A2 ) ) ) ).

% dot_mult2
thf(fact_35_dot__mult1,axiom,
    ! [P: b,Q2: b,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( entails_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ P @ ( mult_b_a_c_d @ Q2 @ A2 ) ) @ Delta @ ( mult_b_a_c_d @ ( smult @ P @ Q2 ) @ A2 ) ) ).

% dot_mult1
thf(fact_36_WildOr,axiom,
    ! [A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( equivalent_a_b_c_d @ plus @ mult @ valid @ ( wildcard_a_b_c_d @ ( or_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( or_a_b_c_d @ ( wildcard_a_b_c_d @ A2 ) @ ( wildcard_a_b_c_d @ B2 ) ) ) ).

% WildOr
thf(fact_37_can__factorize,axiom,
    ! [Q2: b,P: b] :
    ? [R: b] :
      ( Q2
      = ( smult @ R @ P ) ) ).

% can_factorize
thf(fact_38_smult__asso,axiom,
    ! [P: b,Q2: b,R2: b] :
      ( ( smult @ ( smult @ P @ Q2 ) @ R2 )
      = ( smult @ P @ ( smult @ Q2 @ R2 ) ) ) ).

% smult_asso
thf(fact_39_smult__comm,axiom,
    ! [P: b,Q2: b] :
      ( ( smult @ P @ Q2 )
      = ( smult @ Q2 @ P ) ) ).

% smult_comm
thf(fact_40_double__mult,axiom,
    ! [P: b,Q2: b,A: a] :
      ( ( mult @ P @ ( mult @ Q2 @ A ) )
      = ( mult @ ( smult @ P @ Q2 ) @ A ) ) ).

% double_mult
thf(fact_41_sone__neutral,axiom,
    ! [P: b] :
      ( ( smult @ one @ P )
      = P ) ).

% sone_neutral
thf(fact_42_sat_Osimps_I12_J,axiom,
    ! [Sigma: a,S4: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
      ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( wildcard_a_b_c_d @ A2 ) )
      = ( ? [A3: a,P3: b] :
            ( ( Sigma
              = ( mult @ P3 @ A3 ) )
            & ( sat_a_b_d_c @ plus @ mult @ valid @ A3 @ S4 @ Delta @ A2 ) ) ) ) ).

% sat.simps(12)
thf(fact_43_sinv__inverse,axiom,
    ! [P: b] :
      ( ( smult @ P @ ( sinv @ P ) )
      = one ) ).

% sinv_inverse
thf(fact_44_WildPos,axiom,
    ! [A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a,B2: assertion_a_b_c_d] :
      ( ( entails_a_b_c_d @ plus @ mult @ valid @ A2 @ Delta @ B2 )
     => ( entails_a_b_c_d @ plus @ mult @ valid @ ( wildcard_a_b_c_d @ A2 ) @ Delta @ ( wildcard_a_b_c_d @ B2 ) ) ) ).

% WildPos
thf(fact_45_mem__Collect__eq,axiom,
    ! [A: a,P2: a > $o] :
      ( ( member_a @ A @ ( collect_a @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_46_mem__Collect__eq,axiom,
    ! [A: produc6150846815813599699_a_d_c,P2: produc6150846815813599699_a_d_c > $o] :
      ( ( member2052822272342364412_a_d_c @ A @ ( collec1118585765074254142_a_d_c @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
    ! [A: option_a,P2: option_a > $o] :
      ( ( member_option_a @ A @ ( collect_option_a @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_48_Collect__mem__eq,axiom,
    ! [A2: set_a] :
      ( ( collect_a
        @ ^ [X: a] : ( member_a @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_49_Collect__mem__eq,axiom,
    ! [A2: set_Pr7868159745199425715_a_d_c] :
      ( ( collec1118585765074254142_a_d_c
        @ ^ [X: produc6150846815813599699_a_d_c] : ( member2052822272342364412_a_d_c @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_50_Collect__mem__eq,axiom,
    ! [A2: set_option_a] :
      ( ( collect_option_a
        @ ^ [X: option_a] : ( member_option_a @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_51_Collect__cong,axiom,
    ! [P2: option_a > $o,Q: option_a > $o] :
      ( ! [X2: option_a] :
          ( ( P2 @ X2 )
          = ( Q @ X2 ) )
     => ( ( collect_option_a @ P2 )
        = ( collect_option_a @ Q ) ) ) ).

% Collect_cong
thf(fact_52_WildWild,axiom,
    ! [A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( equivalent_a_b_c_d @ plus @ mult @ valid @ ( wildcard_a_b_c_d @ ( wildcard_a_b_c_d @ A2 ) ) @ Delta @ ( wildcard_a_b_c_d @ A2 ) ) ).

% WildWild
thf(fact_53_DotWild,axiom,
    ! [P: b,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( equivalent_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ P @ ( wildcard_a_b_c_d @ A2 ) ) @ Delta @ ( wildcard_a_b_c_d @ A2 ) ) ).

% DotWild
thf(fact_54_WildDot,axiom,
    ! [P: b,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( equivalent_a_b_c_d @ plus @ mult @ valid @ ( wildcard_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) ) @ Delta @ ( wildcard_a_b_c_d @ A2 ) ) ).

% WildDot
thf(fact_55_WildStar1,axiom,
    ! [A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( entails_a_b_c_d @ plus @ mult @ valid @ ( wildcard_a_b_c_d @ ( star_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( star_a_b_c_d @ ( wildcard_a_b_c_d @ A2 ) @ ( wildcard_a_b_c_d @ B2 ) ) ) ).

% WildStar1
thf(fact_56_hoare__triple__output,axiom,
    ! [C: set_Pr7868159745199425715_a_d_c,P2: assertion_a_b_c_d,Q: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( valid_command_a_d_c @ valid @ C )
     => ( ( valid_8824771084768397689_b_c_d @ plus @ mult @ valid @ P2 @ C @ Q @ Delta )
        = ( valid_8824771084768397689_b_c_d @ plus @ mult @ valid @ P2 @ C @ ( bounded_a_b_c_d @ Q ) @ Delta ) ) ) ).

% hoare_triple_output
thf(fact_57_smult__distrib,axiom,
    ! [P: b,Q2: b,R2: b] :
      ( ( smult @ P @ ( sadd @ Q2 @ R2 ) )
      = ( sadd @ ( smult @ P @ Q2 ) @ ( smult @ P @ R2 ) ) ) ).

% smult_distrib
thf(fact_58_logic__axioms,axiom,
    logic_a_b @ plus @ mult @ smult @ sadd @ sinv @ one @ valid ).

% logic_axioms
thf(fact_59_sat_Osimps_I3_J,axiom,
    ! [Sigma: a,S4: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( wand_a_b_c_d @ A2 @ B2 ) )
      = ( ! [A3: a,Sigma3: a] :
            ( ( ( sat_a_b_d_c @ plus @ mult @ valid @ A3 @ S4 @ Delta @ A2 )
              & ( ( some_a @ Sigma3 )
                = ( plus @ Sigma @ A3 ) ) )
           => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma3 @ S4 @ Delta @ B2 ) ) ) ) ).

% sat.simps(3)
thf(fact_60_sat__wand,axiom,
    ! [S4: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,Sigma: a,B2: assertion_a_b_c_d] :
      ( ! [A4: a,Sigma5: a] :
          ( ( ( sat_a_b_d_c @ plus @ mult @ valid @ A4 @ S4 @ Delta @ A2 )
            & ( ( some_a @ Sigma5 )
              = ( plus @ Sigma @ A4 ) ) )
         => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma5 @ S4 @ Delta @ B2 ) )
     => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( wand_a_b_c_d @ A2 @ B2 ) ) ) ).

% sat_wand
thf(fact_61_sat_Osimps_I2_J,axiom,
    ! [Sigma: a,S4: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( star_a_b_c_d @ A2 @ B2 ) )
      = ( ? [A3: a,B3: a] :
            ( ( ( some_a @ Sigma )
              = ( plus @ A3 @ B3 ) )
            & ( sat_a_b_d_c @ plus @ mult @ valid @ A3 @ S4 @ Delta @ A2 )
            & ( sat_a_b_d_c @ plus @ mult @ valid @ B3 @ S4 @ Delta @ B2 ) ) ) ) ).

% sat.simps(2)
thf(fact_62_valid__mono,axiom,
    ! [A: a,B: a] :
      ( ( ( valid @ A )
        & ( pre_larger_a @ plus @ A @ B ) )
     => ( valid @ B ) ) ).

% valid_mono
thf(fact_63_larger__same,axiom,
    ! [A: a,B: a,P: b] :
      ( ( pre_larger_a @ plus @ A @ B )
      = ( pre_larger_a @ plus @ ( mult @ P @ A ) @ ( mult @ P @ B ) ) ) ).

% larger_same
thf(fact_64_compatible__iff,axiom,
    ! [A: a,B: a,P: b] :
      ( ( pre_compatible_a @ plus @ A @ B )
      = ( pre_compatible_a @ plus @ ( mult @ P @ A ) @ ( mult @ P @ B ) ) ) ).

% compatible_iff
thf(fact_65_compatible__imp,axiom,
    ! [A: a,B: a,P: b] :
      ( ( pre_compatible_a @ plus @ A @ B )
     => ( pre_compatible_a @ plus @ ( mult @ P @ A ) @ ( mult @ P @ B ) ) ) ).

% compatible_imp
thf(fact_66_compatible__multiples,axiom,
    ! [P: b,A: a,Q2: b,B: a] :
      ( ( pre_compatible_a @ plus @ ( mult @ P @ A ) @ ( mult @ Q2 @ B ) )
     => ( pre_compatible_a @ plus @ A @ B ) ) ).

% compatible_multiples
thf(fact_67_assertion_Oinject_I1_J,axiom,
    ! [X1: ( d > c ) > a > $o,Y1: ( d > c ) > a > $o] :
      ( ( ( sem_d_c_a_b @ X1 )
        = ( sem_d_c_a_b @ Y1 ) )
      = ( X1 = Y1 ) ) ).

% assertion.inject(1)
thf(fact_68_sadd__comm,axiom,
    ! [P: b,Q2: b] :
      ( ( sadd @ P @ Q2 )
      = ( sadd @ Q2 @ P ) ) ).

% sadd_comm
thf(fact_69_move__sum,axiom,
    ! [A: a,A1: a,A22: a,B: a,B1: a,B22: a,X3: a,X1: a,X22: a] :
      ( ( ( some_a @ A )
        = ( plus @ A1 @ A22 ) )
     => ( ( ( some_a @ B )
          = ( plus @ B1 @ B22 ) )
       => ( ( ( some_a @ X3 )
            = ( plus @ A @ B ) )
         => ( ( ( some_a @ X1 )
              = ( plus @ A1 @ B1 ) )
           => ( ( ( some_a @ X22 )
                = ( plus @ A22 @ B22 ) )
             => ( ( some_a @ X3 )
                = ( plus @ X1 @ X22 ) ) ) ) ) ) ) ).

% move_sum
thf(fact_70_asso1,axiom,
    ! [A: a,B: a,Ab: a,C: a,Bc: a] :
      ( ( ( ( plus @ A @ B )
          = ( some_a @ Ab ) )
        & ( ( plus @ B @ C )
          = ( some_a @ Bc ) ) )
     => ( ( plus @ Ab @ C )
        = ( plus @ A @ Bc ) ) ) ).

% asso1
thf(fact_71_asso3,axiom,
    ! [A: a,B: a,C: a,Bc: a] :
      ( ~ ( pre_compatible_a @ plus @ A @ B )
     => ( ( ( plus @ B @ C )
          = ( some_a @ Bc ) )
       => ~ ( pre_compatible_a @ plus @ A @ Bc ) ) ) ).

% asso3
thf(fact_72_asso2,axiom,
    ! [A: a,B: a,Ab: a,C: a] :
      ( ( ( ( plus @ A @ B )
          = ( some_a @ Ab ) )
        & ~ ( pre_compatible_a @ plus @ B @ C ) )
     => ~ ( pre_compatible_a @ plus @ Ab @ C ) ) ).

% asso2
thf(fact_73_sum__both__larger,axiom,
    ! [X4: a,A5: a,B4: a,X3: a,A: a,B: a] :
      ( ( ( some_a @ X4 )
        = ( plus @ A5 @ B4 ) )
     => ( ( ( some_a @ X3 )
          = ( plus @ A @ B ) )
       => ( ( pre_larger_a @ plus @ A5 @ A )
         => ( ( pre_larger_a @ plus @ B4 @ B )
           => ( pre_larger_a @ plus @ X4 @ X3 ) ) ) ) ) ).

% sum_both_larger
thf(fact_74_larger__first__sum,axiom,
    ! [Y: a,A: a,B: a,X3: a] :
      ( ( ( some_a @ Y )
        = ( plus @ A @ B ) )
     => ( ( pre_larger_a @ plus @ X3 @ Y )
       => ? [A6: a] :
            ( ( ( some_a @ X3 )
              = ( plus @ A6 @ B ) )
            & ( pre_larger_a @ plus @ A6 @ A ) ) ) ) ).

% larger_first_sum
thf(fact_75_larger__def,axiom,
    ! [A: a,B: a] :
      ( ( pre_larger_a @ plus @ A @ B )
      = ( ? [C2: a] :
            ( ( some_a @ A )
            = ( plus @ B @ C2 ) ) ) ) ).

% larger_def
thf(fact_76_larger__implies__compatible,axiom,
    ! [X3: a,Y: a] :
      ( ( pre_larger_a @ plus @ X3 @ Y )
     => ( pre_compatible_a @ plus @ X3 @ Y ) ) ).

% larger_implies_compatible
thf(fact_77_compatible__smaller,axiom,
    ! [A: a,B: a,X3: a] :
      ( ( pre_larger_a @ plus @ A @ B )
     => ( ( pre_compatible_a @ plus @ X3 @ A )
       => ( pre_compatible_a @ plus @ X3 @ B ) ) ) ).

% compatible_smaller
thf(fact_78_plus__mult,axiom,
    ! [A: a,B: a,C: a,P: b] :
      ( ( ( some_a @ A )
        = ( plus @ B @ C ) )
     => ( ( some_a @ ( mult @ P @ A ) )
        = ( plus @ ( mult @ P @ B ) @ ( mult @ P @ C ) ) ) ) ).

% plus_mult
thf(fact_79_assertion_Oinject_I2_J,axiom,
    ! [X21: b,X222: assertion_a_b_c_d,Y21: b,Y22: assertion_a_b_c_d] :
      ( ( ( mult_b_a_c_d @ X21 @ X222 )
        = ( mult_b_a_c_d @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% assertion.inject(2)
thf(fact_80_assertion_Oinject_I11_J,axiom,
    ! [X12: assertion_a_b_c_d,Y12: assertion_a_b_c_d] :
      ( ( ( wildcard_a_b_c_d @ X12 )
        = ( wildcard_a_b_c_d @ Y12 ) )
      = ( X12 = Y12 ) ) ).

% assertion.inject(11)
thf(fact_81_assertion_Oinject_I3_J,axiom,
    ! [X31: assertion_a_b_c_d,X32: assertion_a_b_c_d,Y31: assertion_a_b_c_d,Y32: assertion_a_b_c_d] :
      ( ( ( star_a_b_c_d @ X31 @ X32 )
        = ( star_a_b_c_d @ Y31 @ Y32 ) )
      = ( ( X31 = Y31 )
        & ( X32 = Y32 ) ) ) ).

% assertion.inject(3)
thf(fact_82_assertion_Oinject_I5_J,axiom,
    ! [X51: assertion_a_b_c_d,X52: assertion_a_b_c_d,Y51: assertion_a_b_c_d,Y52: assertion_a_b_c_d] :
      ( ( ( or_a_b_c_d @ X51 @ X52 )
        = ( or_a_b_c_d @ Y51 @ Y52 ) )
      = ( ( X51 = Y51 )
        & ( X52 = Y52 ) ) ) ).

% assertion.inject(5)
thf(fact_83_assertion_Oinject_I4_J,axiom,
    ! [X41: assertion_a_b_c_d,X42: assertion_a_b_c_d,Y41: assertion_a_b_c_d,Y42: assertion_a_b_c_d] :
      ( ( ( wand_a_b_c_d @ X41 @ X42 )
        = ( wand_a_b_c_d @ Y41 @ Y42 ) )
      = ( ( X41 = Y41 )
        & ( X42 = Y42 ) ) ) ).

% assertion.inject(4)
thf(fact_84_assertion_Oinject_I6_J,axiom,
    ! [X61: assertion_a_b_c_d,X62: assertion_a_b_c_d,Y61: assertion_a_b_c_d,Y62: assertion_a_b_c_d] :
      ( ( ( and_a_b_c_d @ X61 @ X62 )
        = ( and_a_b_c_d @ Y61 @ Y62 ) )
      = ( ( X61 = Y61 )
        & ( X62 = Y62 ) ) ) ).

% assertion.inject(6)
thf(fact_85_assertion_Oinject_I10_J,axiom,
    ! [X11: assertion_a_b_c_d,Y11: assertion_a_b_c_d] :
      ( ( ( bounded_a_b_c_d @ X11 )
        = ( bounded_a_b_c_d @ Y11 ) )
      = ( X11 = Y11 ) ) ).

% assertion.inject(10)
thf(fact_86_assertion_Oinject_I7_J,axiom,
    ! [X71: assertion_a_b_c_d,X72: assertion_a_b_c_d,Y71: assertion_a_b_c_d,Y72: assertion_a_b_c_d] :
      ( ( ( imp_a_b_c_d @ X71 @ X72 )
        = ( imp_a_b_c_d @ Y71 @ Y72 ) )
      = ( ( X71 = Y71 )
        & ( X72 = Y72 ) ) ) ).

% assertion.inject(7)
thf(fact_87_distrib__mult,axiom,
    ! [P: b,Q2: b,X3: a] :
      ( ( some_a @ ( mult @ ( sadd @ P @ Q2 ) @ X3 ) )
      = ( plus @ ( mult @ P @ X3 ) @ ( mult @ Q2 @ X3 ) ) ) ).

% distrib_mult
thf(fact_88_intuitionistic__def,axiom,
    ! [S4: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
      ( ( intuit4720955538653295669_b_d_c @ plus @ mult @ valid @ S4 @ Delta @ A2 )
      = ( ! [A3: a,B3: a] :
            ( ( ( pre_larger_a @ plus @ A3 @ B3 )
              & ( sat_a_b_d_c @ plus @ mult @ valid @ B3 @ S4 @ Delta @ A2 ) )
           => ( sat_a_b_d_c @ plus @ mult @ valid @ A3 @ S4 @ Delta @ A2 ) ) ) ) ).

% intuitionistic_def
thf(fact_89_intuitionisticI,axiom,
    ! [S4: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
      ( ! [A4: a,B5: a] :
          ( ( ( pre_larger_a @ plus @ A4 @ B5 )
            & ( sat_a_b_d_c @ plus @ mult @ valid @ B5 @ S4 @ Delta @ A2 ) )
         => ( sat_a_b_d_c @ plus @ mult @ valid @ A4 @ S4 @ Delta @ A2 ) )
     => ( intuit4720955538653295669_b_d_c @ plus @ mult @ valid @ S4 @ Delta @ A2 ) ) ).

% intuitionisticI
thf(fact_90_logic_Oasso1,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,Ab: a,C: a,Bc: a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( ( Plus @ A @ B )
            = ( some_a @ Ab ) )
          & ( ( Plus @ B @ C )
            = ( some_a @ Bc ) ) )
       => ( ( Plus @ Ab @ C )
          = ( Plus @ A @ Bc ) ) ) ) ).

% logic.asso1
thf(fact_91_logic_Oasso2,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,Ab: a,C: a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( ( Plus @ A @ B )
            = ( some_a @ Ab ) )
          & ~ ( pre_compatible_a @ Plus @ B @ C ) )
       => ~ ( pre_compatible_a @ Plus @ Ab @ C ) ) ) ).

% logic.asso2
thf(fact_92_logic_Oasso3,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,C: a,Bc: a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ~ ( pre_compatible_a @ Plus @ A @ B )
       => ( ( ( Plus @ B @ C )
            = ( some_a @ Bc ) )
         => ~ ( pre_compatible_a @ Plus @ A @ Bc ) ) ) ) ).

% logic.asso3
thf(fact_93_logic_Omove__sum,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,A1: a,A22: a,B: a,B1: a,B22: a,X3: a,X1: a,X22: a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( some_a @ A )
          = ( Plus @ A1 @ A22 ) )
       => ( ( ( some_a @ B )
            = ( Plus @ B1 @ B22 ) )
         => ( ( ( some_a @ X3 )
              = ( Plus @ A @ B ) )
           => ( ( ( some_a @ X1 )
                = ( Plus @ A1 @ B1 ) )
             => ( ( ( some_a @ X22 )
                  = ( Plus @ A22 @ B22 ) )
               => ( ( some_a @ X3 )
                  = ( Plus @ X1 @ X22 ) ) ) ) ) ) ) ) ).

% logic.move_sum
thf(fact_94_logic_Oplus__mult,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,C: a,P: b] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( some_a @ A )
          = ( Plus @ B @ C ) )
       => ( ( some_a @ ( Mult @ P @ A ) )
          = ( Plus @ ( Mult @ P @ B ) @ ( Mult @ P @ C ) ) ) ) ) ).

% logic.plus_mult
thf(fact_95_logic_Osadd__comm,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q2: b] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Sadd @ P @ Q2 )
        = ( Sadd @ Q2 @ P ) ) ) ).

% logic.sadd_comm
thf(fact_96_logic_Ocan__divide,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A: a,B: a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( Mult @ P @ A )
          = ( Mult @ P @ B ) )
       => ( A = B ) ) ) ).

% logic.can_divide
thf(fact_97_logic_Osmult__asso,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q2: b,R2: b] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Smult @ ( Smult @ P @ Q2 ) @ R2 )
        = ( Smult @ P @ ( Smult @ Q2 @ R2 ) ) ) ) ).

% logic.smult_asso
thf(fact_98_logic_Osmult__comm,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q2: b] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Smult @ P @ Q2 )
        = ( Smult @ Q2 @ P ) ) ) ).

% logic.smult_comm
thf(fact_99_logic_Ounique__inv,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,P: b,B: a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( A
          = ( Mult @ P @ B ) )
        = ( B
          = ( Mult @ ( Sinv @ P ) @ A ) ) ) ) ).

% logic.unique_inv
thf(fact_100_logic_Ovalid__mono,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( Valid @ A )
          & ( pre_larger_a @ Plus @ A @ B ) )
       => ( Valid @ B ) ) ) ).

% logic.valid_mono
thf(fact_101_logic_Ocommutative,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Plus @ A @ B )
        = ( Plus @ B @ A ) ) ) ).

% logic.commutative
thf(fact_102_logic_Odouble__mult,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q2: b,A: a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Mult @ P @ ( Mult @ Q2 @ A ) )
        = ( Mult @ ( Smult @ P @ Q2 ) @ A ) ) ) ).

% logic.double_mult
thf(fact_103_logic_Olarger__same,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,P: b] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( pre_larger_a @ Plus @ A @ B )
        = ( pre_larger_a @ Plus @ ( Mult @ P @ A ) @ ( Mult @ P @ B ) ) ) ) ).

% logic.larger_same
thf(fact_104_logic_Oone__neutral,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Mult @ One @ A )
        = A ) ) ).

% logic.one_neutral
thf(fact_105_logic_Odistrib__mult,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q2: b,X3: a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( some_a @ ( Mult @ ( Sadd @ P @ Q2 ) @ X3 ) )
        = ( Plus @ ( Mult @ P @ X3 ) @ ( Mult @ Q2 @ X3 ) ) ) ) ).

% logic.distrib_mult
thf(fact_106_logic_Osinv__inverse,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Smult @ P @ ( Sinv @ P ) )
        = One ) ) ).

% logic.sinv_inverse
thf(fact_107_logic_Osone__neutral,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Smult @ One @ P )
        = P ) ) ).

% logic.sone_neutral
thf(fact_108_logic_Ocan__factorize,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Q2: b,P: b] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ? [R: b] :
          ( Q2
          = ( Smult @ R @ P ) ) ) ).

% logic.can_factorize
thf(fact_109_logic_Osmult__distrib,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q2: b,R2: b] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Smult @ P @ ( Sadd @ Q2 @ R2 ) )
        = ( Sadd @ ( Smult @ P @ Q2 ) @ ( Smult @ P @ R2 ) ) ) ) ).

% logic.smult_distrib
thf(fact_110_logic_Ocompatible__iff,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,P: b] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( pre_compatible_a @ Plus @ A @ B )
        = ( pre_compatible_a @ Plus @ ( Mult @ P @ A ) @ ( Mult @ P @ B ) ) ) ) ).

% logic.compatible_iff
thf(fact_111_logic_Ocompatible__imp,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,P: b] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( pre_compatible_a @ Plus @ A @ B )
       => ( pre_compatible_a @ Plus @ ( Mult @ P @ A ) @ ( Mult @ P @ B ) ) ) ) ).

% logic.compatible_imp
thf(fact_112_pre__logic_Olarger__def,axiom,
    ( pre_la7052613405163955386_a_d_c
    = ( ^ [Plus2: product_prod_a_d_c > product_prod_a_d_c > option2362401199305441953_a_d_c,A3: product_prod_a_d_c,B3: product_prod_a_d_c] :
        ? [C2: product_prod_a_d_c] :
          ( ( some_P377817780860425132_a_d_c @ A3 )
          = ( Plus2 @ B3 @ C2 ) ) ) ) ).

% pre_logic.larger_def
thf(fact_113_pre__logic_Olarger__def,axiom,
    ( pre_larger_a
    = ( ^ [Plus2: a > a > option_a,A3: a,B3: a] :
        ? [C2: a] :
          ( ( some_a @ A3 )
          = ( Plus2 @ B3 @ C2 ) ) ) ) ).

% pre_logic.larger_def
thf(fact_114_logic_Osum__both__larger,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X4: a,A5: a,B4: a,X3: a,A: a,B: a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( some_a @ X4 )
          = ( Plus @ A5 @ B4 ) )
       => ( ( ( some_a @ X3 )
            = ( Plus @ A @ B ) )
         => ( ( pre_larger_a @ Plus @ A5 @ A )
           => ( ( pre_larger_a @ Plus @ B4 @ B )
             => ( pre_larger_a @ Plus @ X4 @ X3 ) ) ) ) ) ) ).

% logic.sum_both_larger
thf(fact_115_pre__logic_Olarger_Ocong,axiom,
    pre_larger_a = pre_larger_a ).

% pre_logic.larger.cong
thf(fact_116_logic_Olarger__first__sum,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Y: a,A: a,B: a,X3: a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( some_a @ Y )
          = ( Plus @ A @ B ) )
       => ( ( pre_larger_a @ Plus @ X3 @ Y )
         => ? [A6: a] :
              ( ( ( some_a @ X3 )
                = ( Plus @ A6 @ B ) )
              & ( pre_larger_a @ Plus @ A6 @ A ) ) ) ) ) ).

% logic.larger_first_sum
thf(fact_117_logic_Ocompatible__smaller,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,X3: a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( pre_larger_a @ Plus @ A @ B )
       => ( ( pre_compatible_a @ Plus @ X3 @ A )
         => ( pre_compatible_a @ Plus @ X3 @ B ) ) ) ) ).

% logic.compatible_smaller
thf(fact_118_pre__logic_Ocompatible_Ocong,axiom,
    pre_compatible_a = pre_compatible_a ).

% pre_logic.compatible.cong
thf(fact_119_logic_Ocompatible__multiples,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A: a,Q2: b,B: a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( pre_compatible_a @ Plus @ ( Mult @ P @ A ) @ ( Mult @ Q2 @ B ) )
       => ( pre_compatible_a @ Plus @ A @ B ) ) ) ).

% logic.compatible_multiples
thf(fact_120_logic_Olarger__implies__compatible,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X3: a,Y: a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( pre_larger_a @ Plus @ X3 @ Y )
       => ( pre_compatible_a @ Plus @ X3 @ Y ) ) ) ).

% logic.larger_implies_compatible
thf(fact_121_logic__def,axiom,
    ( logic_a_b
    = ( ^ [Plus2: a > a > option_a,Mult2: b > a > a,Smult2: b > b > b,Sadd2: b > b > b,Sinv2: b > b,One2: b,Valid2: a > $o] :
          ( ! [A3: a,B3: a] :
              ( ( Plus2 @ A3 @ B3 )
              = ( Plus2 @ B3 @ A3 ) )
          & ! [A3: a,B3: a,Ab2: a,C2: a,Bc2: a] :
              ( ( ( ( Plus2 @ A3 @ B3 )
                  = ( some_a @ Ab2 ) )
                & ( ( Plus2 @ B3 @ C2 )
                  = ( some_a @ Bc2 ) ) )
             => ( ( Plus2 @ Ab2 @ C2 )
                = ( Plus2 @ A3 @ Bc2 ) ) )
          & ! [A3: a,B3: a,Ab2: a,C2: a] :
              ( ( ( ( Plus2 @ A3 @ B3 )
                  = ( some_a @ Ab2 ) )
                & ~ ( pre_compatible_a @ Plus2 @ B3 @ C2 ) )
             => ~ ( pre_compatible_a @ Plus2 @ Ab2 @ C2 ) )
          & ! [P3: b] :
              ( ( Smult2 @ P3 @ ( Sinv2 @ P3 ) )
              = One2 )
          & ! [P3: b] :
              ( ( Smult2 @ One2 @ P3 )
              = P3 )
          & ! [P3: b,Q3: b] :
              ( ( Sadd2 @ P3 @ Q3 )
              = ( Sadd2 @ Q3 @ P3 ) )
          & ! [P3: b,Q3: b] :
              ( ( Smult2 @ P3 @ Q3 )
              = ( Smult2 @ Q3 @ P3 ) )
          & ! [P3: b,Q3: b,R3: b] :
              ( ( Smult2 @ P3 @ ( Sadd2 @ Q3 @ R3 ) )
              = ( Sadd2 @ ( Smult2 @ P3 @ Q3 ) @ ( Smult2 @ P3 @ R3 ) ) )
          & ! [P3: b,Q3: b,R3: b] :
              ( ( Smult2 @ ( Smult2 @ P3 @ Q3 ) @ R3 )
              = ( Smult2 @ P3 @ ( Smult2 @ Q3 @ R3 ) ) )
          & ! [P3: b,Q3: b,A3: a] :
              ( ( Mult2 @ P3 @ ( Mult2 @ Q3 @ A3 ) )
              = ( Mult2 @ ( Smult2 @ P3 @ Q3 ) @ A3 ) )
          & ! [A3: a,B3: a,C2: a,P3: b] :
              ( ( ( some_a @ A3 )
                = ( Plus2 @ B3 @ C2 ) )
             => ( ( some_a @ ( Mult2 @ P3 @ A3 ) )
                = ( Plus2 @ ( Mult2 @ P3 @ B3 ) @ ( Mult2 @ P3 @ C2 ) ) ) )
          & ! [P3: b,Q3: b,X: a] :
              ( ( some_a @ ( Mult2 @ ( Sadd2 @ P3 @ Q3 ) @ X ) )
              = ( Plus2 @ ( Mult2 @ P3 @ X ) @ ( Mult2 @ Q3 @ X ) ) )
          & ! [A3: a] :
              ( ( Mult2 @ One2 @ A3 )
              = A3 )
          & ! [A3: a,B3: a] :
              ( ( ( Valid2 @ A3 )
                & ( pre_larger_a @ Plus2 @ A3 @ B3 ) )
             => ( Valid2 @ B3 ) ) ) ) ) ).

% logic_def
thf(fact_122_logic_Osat_Osimps_I2_J,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S4: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( star_a_b_c_d @ A2 @ B2 ) )
        = ( ? [A3: a,B3: a] :
              ( ( ( some_a @ Sigma )
                = ( Plus @ A3 @ B3 ) )
              & ( sat_a_b_d_c @ Plus @ Mult @ Valid @ A3 @ S4 @ Delta @ A2 )
              & ( sat_a_b_d_c @ Plus @ Mult @ Valid @ B3 @ S4 @ Delta @ B2 ) ) ) ) ) ).

% logic.sat.simps(2)
thf(fact_123_logic_Osat__wand,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,S4: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,Sigma: a,B2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ! [A4: a,Sigma5: a] :
            ( ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ A4 @ S4 @ Delta @ A2 )
              & ( ( some_a @ Sigma5 )
                = ( Plus @ Sigma @ A4 ) ) )
           => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma5 @ S4 @ Delta @ B2 ) )
       => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( wand_a_b_c_d @ A2 @ B2 ) ) ) ) ).

% logic.sat_wand
thf(fact_124_logic_Osat_Osimps_I3_J,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S4: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( wand_a_b_c_d @ A2 @ B2 ) )
        = ( ! [A3: a,Sigma3: a] :
              ( ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ A3 @ S4 @ Delta @ A2 )
                & ( ( some_a @ Sigma3 )
                  = ( Plus @ Sigma @ A3 ) ) )
             => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma3 @ S4 @ Delta @ B2 ) ) ) ) ) ).

% logic.sat.simps(3)
thf(fact_125_logic_Oequal__outside__def,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,S4: d > c,S7: d > c,S6: set_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( equal_outside_d_c @ S4 @ S7 @ S6 )
        = ( ! [X: d] :
              ( ~ ( member_d @ X @ S6 )
             => ( ( S4 @ X )
                = ( S7 @ X ) ) ) ) ) ) ).

% logic.equal_outside_def
thf(fact_126_logic_Ohoare__triple__output,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,C: set_Pr7868159745199425715_a_d_c,P2: assertion_a_b_c_d,Q: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( valid_command_a_d_c @ Valid @ C )
       => ( ( valid_8824771084768397689_b_c_d @ Plus @ Mult @ Valid @ P2 @ C @ Q @ Delta )
          = ( valid_8824771084768397689_b_c_d @ Plus @ Mult @ Valid @ P2 @ C @ ( bounded_a_b_c_d @ Q ) @ Delta ) ) ) ) ).

% logic.hoare_triple_output
thf(fact_127_logic_Osat__mult,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,P: b,S4: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ! [A4: a] :
            ( ( Sigma
              = ( Mult @ P @ A4 ) )
           => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ A4 @ S4 @ Delta @ A2 ) )
       => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( mult_b_a_c_d @ P @ A2 ) ) ) ) ).

% logic.sat_mult
thf(fact_128_logic_Osat_Osimps_I1_J,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S4: d > c,Delta: ( d > c ) > set_a,P: b,A2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( mult_b_a_c_d @ P @ A2 ) )
        = ( ? [A3: a] :
              ( ( Sigma
                = ( Mult @ P @ A3 ) )
              & ( sat_a_b_d_c @ Plus @ Mult @ Valid @ A3 @ S4 @ Delta @ A2 ) ) ) ) ) ).

% logic.sat.simps(1)
thf(fact_129_logic_Osat_Osimps_I12_J,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S4: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( wildcard_a_b_c_d @ A2 ) )
        = ( ? [A3: a,P3: b] :
              ( ( Sigma
                = ( Mult @ P3 @ A3 ) )
              & ( sat_a_b_d_c @ Plus @ Mult @ Valid @ A3 @ S4 @ Delta @ A2 ) ) ) ) ) ).

% logic.sat.simps(12)
thf(fact_130_logic_Osat_Osimps_I6_J,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S4: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( or_a_b_c_d @ A2 @ B2 ) )
        = ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S4 @ Delta @ A2 )
          | ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S4 @ Delta @ B2 ) ) ) ) ).

% logic.sat.simps(6)
thf(fact_131_logic_Osat_Osimps_I7_J,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S4: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( and_a_b_c_d @ A2 @ B2 ) )
        = ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S4 @ Delta @ A2 )
          & ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S4 @ Delta @ B2 ) ) ) ) ).

% logic.sat.simps(7)
thf(fact_132_logic_Osat_Osimps_I11_J,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S4: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( bounded_a_b_c_d @ A2 ) )
        = ( ( Valid @ Sigma )
         => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S4 @ Delta @ A2 ) ) ) ) ).

% logic.sat.simps(11)
thf(fact_133_logic_Osat_Osimps_I5_J,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S4: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( imp_a_b_c_d @ A2 @ B2 ) )
        = ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S4 @ Delta @ A2 )
         => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S4 @ Delta @ B2 ) ) ) ) ).

% logic.sat.simps(5)
thf(fact_134_logic_Osat__imp,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S4: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S4 @ Delta @ A2 )
         => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S4 @ Delta @ B2 ) )
       => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( imp_a_b_c_d @ A2 @ B2 ) ) ) ) ).

% logic.sat_imp
thf(fact_135_logic_Osat_Osimps_I4_J,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S4: d > c,Delta: ( d > c ) > set_a,B: ( d > c ) > a > $o] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( sem_d_c_a_b @ B ) )
        = ( B @ S4 @ Sigma ) ) ) ).

% logic.sat.simps(4)
thf(fact_136_logic_Osat_Osimps_I10_J,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S4: d > c,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S4 @ Delta @ pred_a_b_c_d )
        = ( member_a @ Sigma @ ( Delta @ S4 ) ) ) ) ).

% logic.sat.simps(10)
thf(fact_137_logic_OentailsI,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ! [Sigma4: a,S5: d > c] :
            ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma4 @ S5 @ Delta @ A2 )
           => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma4 @ S5 @ Delta @ B2 ) )
       => ( entails_a_b_c_d @ Plus @ Mult @ Valid @ A2 @ Delta @ B2 ) ) ) ).

% logic.entailsI
thf(fact_138_logic_Oentails__def,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a,B2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( entails_a_b_c_d @ Plus @ Mult @ Valid @ A2 @ Delta @ B2 )
        = ( ! [Sigma2: a,S: d > c] :
              ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma2 @ S @ Delta @ A2 )
             => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma2 @ S @ Delta @ B2 ) ) ) ) ) ).

% logic.entails_def
thf(fact_139_logic_OequivalentI,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ! [Sigma4: a,S5: d > c] :
            ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma4 @ S5 @ Delta @ A2 )
           => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma4 @ S5 @ Delta @ B2 ) )
       => ( ! [Sigma4: a,S5: d > c] :
              ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma4 @ S5 @ Delta @ B2 )
             => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma4 @ S5 @ Delta @ A2 ) )
         => ( equivalent_a_b_c_d @ Plus @ Mult @ Valid @ A2 @ Delta @ B2 ) ) ) ) ).

% logic.equivalentI
thf(fact_140_logic_Ohoare__triple__input,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P2: assertion_a_b_c_d,C: set_Pr7868159745199425715_a_d_c,Q: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( valid_8824771084768397689_b_c_d @ Plus @ Mult @ Valid @ P2 @ C @ Q @ Delta )
        = ( valid_8824771084768397689_b_c_d @ Plus @ Mult @ Valid @ ( bounded_a_b_c_d @ P2 ) @ C @ Q @ Delta ) ) ) ).

% logic.hoare_triple_input
thf(fact_141_logic_Oequivalent__def,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a,B2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( equivalent_a_b_c_d @ Plus @ Mult @ Valid @ A2 @ Delta @ B2 )
        = ( ( entails_a_b_c_d @ Plus @ Mult @ Valid @ A2 @ Delta @ B2 )
          & ( entails_a_b_c_d @ Plus @ Mult @ Valid @ B2 @ Delta @ A2 ) ) ) ) ).

% logic.equivalent_def
thf(fact_142_logic_Opure__def,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( pure_a_b_c_d @ Plus @ Mult @ Valid @ A2 )
        = ( ! [Sigma2: a,Sigma3: a,S: d > c,Delta2: ( d > c ) > set_a,Delta3: ( d > c ) > set_a] :
              ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma2 @ S @ Delta2 @ A2 )
              = ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma3 @ S @ Delta3 @ A2 ) ) ) ) ) ).

% logic.pure_def
thf(fact_143_logic_Odot__mult2,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q2: b,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ ( Smult @ P @ Q2 ) @ A2 ) @ Delta @ ( mult_b_a_c_d @ P @ ( mult_b_a_c_d @ Q2 @ A2 ) ) ) ) ).

% logic.dot_mult2
thf(fact_144_logic_Odot__mult1,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q2: b,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ P @ ( mult_b_a_c_d @ Q2 @ A2 ) ) @ Delta @ ( mult_b_a_c_d @ ( Smult @ P @ Q2 ) @ A2 ) ) ) ).

% logic.dot_mult1
thf(fact_145_logic_ODotPos,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a,B2: assertion_a_b_c_d,Pi: b] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( entails_a_b_c_d @ Plus @ Mult @ Valid @ A2 @ Delta @ B2 )
        = ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ Pi @ A2 ) @ Delta @ ( mult_b_a_c_d @ Pi @ B2 ) ) ) ) ).

% logic.DotPos
thf(fact_146_logic_OWildPos,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a,B2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( entails_a_b_c_d @ Plus @ Mult @ Valid @ A2 @ Delta @ B2 )
       => ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( wildcard_a_b_c_d @ A2 ) @ Delta @ ( wildcard_a_b_c_d @ B2 ) ) ) ) ).

% logic.WildPos
thf(fact_147_logic_ODotDot,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q2: b,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( equivalent_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ P @ ( mult_b_a_c_d @ Q2 @ A2 ) ) @ Delta @ ( mult_b_a_c_d @ ( Smult @ P @ Q2 ) @ A2 ) ) ) ).

% logic.DotDot
thf(fact_148_logic_OWildWild,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( equivalent_a_b_c_d @ Plus @ Mult @ Valid @ ( wildcard_a_b_c_d @ ( wildcard_a_b_c_d @ A2 ) ) @ Delta @ ( wildcard_a_b_c_d @ A2 ) ) ) ).

% logic.WildWild
thf(fact_149_logic_Onot__in__fv__def,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_c_d,S6: set_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( not_in_fv_a_b_c_d @ Plus @ Mult @ Valid @ A2 @ S6 )
        = ( ! [Sigma2: a,S: d > c,Delta2: ( d > c ) > set_a,S2: d > c] :
              ( ( equal_outside_d_c @ S @ S2 @ S6 )
             => ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma2 @ S @ Delta2 @ A2 )
                = ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma2 @ S2 @ Delta2 @ A2 ) ) ) ) ) ) ).

% logic.not_in_fv_def
thf(fact_150_logic_Odot__star1,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ P @ ( star_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( star_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ) ).

% logic.dot_star1
thf(fact_151_logic_Odot__star2,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( star_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) @ Delta @ ( mult_b_a_c_d @ P @ ( star_a_b_c_d @ A2 @ B2 ) ) ) ) ).

% logic.dot_star2
thf(fact_152_logic_Odot__or1,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ P @ ( or_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( or_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ) ).

% logic.dot_or1
thf(fact_153_logic_Odot__or2,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( or_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) @ Delta @ ( mult_b_a_c_d @ P @ ( or_a_b_c_d @ A2 @ B2 ) ) ) ) ).

% logic.dot_or2
thf(fact_154_logic_Odot__wand1,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ P @ ( wand_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( wand_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ) ).

% logic.dot_wand1
thf(fact_155_logic_Odot__wand2,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( wand_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) @ Delta @ ( mult_b_a_c_d @ P @ ( wand_a_b_c_d @ A2 @ B2 ) ) ) ) ).

% logic.dot_wand2
thf(fact_156_logic_Odot__and1,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ P @ ( and_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( and_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ) ).

% logic.dot_and1
thf(fact_157_logic_OWildStar1,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( wildcard_a_b_c_d @ ( star_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( star_a_b_c_d @ ( wildcard_a_b_c_d @ A2 ) @ ( wildcard_a_b_c_d @ B2 ) ) ) ) ).

% logic.WildStar1
thf(fact_158_logic_ODotWild,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( equivalent_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ P @ ( wildcard_a_b_c_d @ A2 ) ) @ Delta @ ( wildcard_a_b_c_d @ A2 ) ) ) ).

% logic.DotWild
thf(fact_159_logic_OWildDot,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( equivalent_a_b_c_d @ Plus @ Mult @ Valid @ ( wildcard_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) ) @ Delta @ ( wildcard_a_b_c_d @ A2 ) ) ) ).

% logic.WildDot
thf(fact_160_logic_ODotStar,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( equivalent_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ P @ ( star_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( star_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ) ).

% logic.DotStar
thf(fact_161_logic_ODotOr,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( equivalent_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ P @ ( or_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( or_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ) ).

% logic.DotOr
thf(fact_162_logic_ODotWand,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( equivalent_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ P @ ( wand_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( wand_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ) ).

% logic.DotWand
thf(fact_163_logic_OWildOr,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( equivalent_a_b_c_d @ Plus @ Mult @ Valid @ ( wildcard_a_b_c_d @ ( or_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( or_a_b_c_d @ ( wildcard_a_b_c_d @ A2 ) @ ( wildcard_a_b_c_d @ B2 ) ) ) ) ).

% logic.WildOr
thf(fact_164_logic_Osat_Ocong,axiom,
    sat_a_b_d_c = sat_a_b_d_c ).

% logic.sat.cong
thf(fact_165_logic_Oentails_Ocong,axiom,
    entails_a_b_c_d = entails_a_b_c_d ).

% logic.entails.cong
thf(fact_166_logic_Oequivalent_Ocong,axiom,
    equivalent_a_b_c_d = equivalent_a_b_c_d ).

% logic.equivalent.cong
thf(fact_167_logic_Ovalid__hoare__triple_Ocong,axiom,
    valid_8824771084768397689_b_c_d = valid_8824771084768397689_b_c_d ).

% logic.valid_hoare_triple.cong
thf(fact_168_logic_Onot__in__fv_Ocong,axiom,
    not_in_fv_a_b_c_d = not_in_fv_a_b_c_d ).

% logic.not_in_fv.cong
thf(fact_169_logic_Opure_Ocong,axiom,
    pure_a_b_c_d = pure_a_b_c_d ).

% logic.pure.cong
thf(fact_170_assertion_Odistinct_I41_J,axiom,
    ! [X21: b,X222: assertion_a_b_c_d,X12: assertion_a_b_c_d] :
      ( ( mult_b_a_c_d @ X21 @ X222 )
     != ( wildcard_a_b_c_d @ X12 ) ) ).

% assertion.distinct(41)
thf(fact_171_assertion_Odistinct_I23_J,axiom,
    ! [X21: b,X222: assertion_a_b_c_d,X31: assertion_a_b_c_d,X32: assertion_a_b_c_d] :
      ( ( mult_b_a_c_d @ X21 @ X222 )
     != ( star_a_b_c_d @ X31 @ X32 ) ) ).

% assertion.distinct(23)
thf(fact_172_assertion_Odistinct_I27_J,axiom,
    ! [X21: b,X222: assertion_a_b_c_d,X51: assertion_a_b_c_d,X52: assertion_a_b_c_d] :
      ( ( mult_b_a_c_d @ X21 @ X222 )
     != ( or_a_b_c_d @ X51 @ X52 ) ) ).

% assertion.distinct(27)
thf(fact_173_assertion_Odistinct_I25_J,axiom,
    ! [X21: b,X222: assertion_a_b_c_d,X41: assertion_a_b_c_d,X42: assertion_a_b_c_d] :
      ( ( mult_b_a_c_d @ X21 @ X222 )
     != ( wand_a_b_c_d @ X41 @ X42 ) ) ).

% assertion.distinct(25)
thf(fact_174_assertion_Odistinct_I29_J,axiom,
    ! [X21: b,X222: assertion_a_b_c_d,X61: assertion_a_b_c_d,X62: assertion_a_b_c_d] :
      ( ( mult_b_a_c_d @ X21 @ X222 )
     != ( and_a_b_c_d @ X61 @ X62 ) ) ).

% assertion.distinct(29)
thf(fact_175_assertion_Odistinct_I39_J,axiom,
    ! [X21: b,X222: assertion_a_b_c_d,X11: assertion_a_b_c_d] :
      ( ( mult_b_a_c_d @ X21 @ X222 )
     != ( bounded_a_b_c_d @ X11 ) ) ).

% assertion.distinct(39)
thf(fact_176_assertion_Odistinct_I31_J,axiom,
    ! [X21: b,X222: assertion_a_b_c_d,X71: assertion_a_b_c_d,X72: assertion_a_b_c_d] :
      ( ( mult_b_a_c_d @ X21 @ X222 )
     != ( imp_a_b_c_d @ X71 @ X72 ) ) ).

% assertion.distinct(31)
thf(fact_177_assertion_Odistinct_I1_J,axiom,
    ! [X1: ( d > c ) > a > $o,X21: b,X222: assertion_a_b_c_d] :
      ( ( sem_d_c_a_b @ X1 )
     != ( mult_b_a_c_d @ X21 @ X222 ) ) ).

% assertion.distinct(1)
thf(fact_178_assertion_Odistinct_I59_J,axiom,
    ! [X31: assertion_a_b_c_d,X32: assertion_a_b_c_d,X12: assertion_a_b_c_d] :
      ( ( star_a_b_c_d @ X31 @ X32 )
     != ( wildcard_a_b_c_d @ X12 ) ) ).

% assertion.distinct(59)
thf(fact_179_assertion_Odistinct_I89_J,axiom,
    ! [X51: assertion_a_b_c_d,X52: assertion_a_b_c_d,X12: assertion_a_b_c_d] :
      ( ( or_a_b_c_d @ X51 @ X52 )
     != ( wildcard_a_b_c_d @ X12 ) ) ).

% assertion.distinct(89)
thf(fact_180_assertion_Odistinct_I37_J,axiom,
    ! [X21: b,X222: assertion_a_b_c_d] :
      ( ( mult_b_a_c_d @ X21 @ X222 )
     != pred_a_b_c_d ) ).

% assertion.distinct(37)
thf(fact_181_assertion_Odistinct_I75_J,axiom,
    ! [X41: assertion_a_b_c_d,X42: assertion_a_b_c_d,X12: assertion_a_b_c_d] :
      ( ( wand_a_b_c_d @ X41 @ X42 )
     != ( wildcard_a_b_c_d @ X12 ) ) ).

% assertion.distinct(75)
thf(fact_182_assertion_Odistinct_I45_J,axiom,
    ! [X31: assertion_a_b_c_d,X32: assertion_a_b_c_d,X51: assertion_a_b_c_d,X52: assertion_a_b_c_d] :
      ( ( star_a_b_c_d @ X31 @ X32 )
     != ( or_a_b_c_d @ X51 @ X52 ) ) ).

% assertion.distinct(45)
thf(fact_183_assertion_Odistinct_I43_J,axiom,
    ! [X31: assertion_a_b_c_d,X32: assertion_a_b_c_d,X41: assertion_a_b_c_d,X42: assertion_a_b_c_d] :
      ( ( star_a_b_c_d @ X31 @ X32 )
     != ( wand_a_b_c_d @ X41 @ X42 ) ) ).

% assertion.distinct(43)
thf(fact_184_assertion_Odistinct_I101_J,axiom,
    ! [X61: assertion_a_b_c_d,X62: assertion_a_b_c_d,X12: assertion_a_b_c_d] :
      ( ( and_a_b_c_d @ X61 @ X62 )
     != ( wildcard_a_b_c_d @ X12 ) ) ).

% assertion.distinct(101)
thf(fact_185_assertion_Odistinct_I61_J,axiom,
    ! [X41: assertion_a_b_c_d,X42: assertion_a_b_c_d,X51: assertion_a_b_c_d,X52: assertion_a_b_c_d] :
      ( ( wand_a_b_c_d @ X41 @ X42 )
     != ( or_a_b_c_d @ X51 @ X52 ) ) ).

% assertion.distinct(61)
thf(fact_186_assertion_Odistinct_I131_J,axiom,
    ! [X11: assertion_a_b_c_d,X12: assertion_a_b_c_d] :
      ( ( bounded_a_b_c_d @ X11 )
     != ( wildcard_a_b_c_d @ X12 ) ) ).

% assertion.distinct(131)
thf(fact_187_assertion_Odistinct_I111_J,axiom,
    ! [X71: assertion_a_b_c_d,X72: assertion_a_b_c_d,X12: assertion_a_b_c_d] :
      ( ( imp_a_b_c_d @ X71 @ X72 )
     != ( wildcard_a_b_c_d @ X12 ) ) ).

% assertion.distinct(111)
thf(fact_188_assertion_Odistinct_I47_J,axiom,
    ! [X31: assertion_a_b_c_d,X32: assertion_a_b_c_d,X61: assertion_a_b_c_d,X62: assertion_a_b_c_d] :
      ( ( star_a_b_c_d @ X31 @ X32 )
     != ( and_a_b_c_d @ X61 @ X62 ) ) ).

% assertion.distinct(47)
thf(fact_189_assertion_Odistinct_I21_J,axiom,
    ! [X1: ( d > c ) > a > $o,X12: assertion_a_b_c_d] :
      ( ( sem_d_c_a_b @ X1 )
     != ( wildcard_a_b_c_d @ X12 ) ) ).

% assertion.distinct(21)
thf(fact_190_assertion_Odistinct_I57_J,axiom,
    ! [X31: assertion_a_b_c_d,X32: assertion_a_b_c_d,X11: assertion_a_b_c_d] :
      ( ( star_a_b_c_d @ X31 @ X32 )
     != ( bounded_a_b_c_d @ X11 ) ) ).

% assertion.distinct(57)
thf(fact_191_assertion_Odistinct_I77_J,axiom,
    ! [X51: assertion_a_b_c_d,X52: assertion_a_b_c_d,X61: assertion_a_b_c_d,X62: assertion_a_b_c_d] :
      ( ( or_a_b_c_d @ X51 @ X52 )
     != ( and_a_b_c_d @ X61 @ X62 ) ) ).

% assertion.distinct(77)
thf(fact_192_assertion_Odistinct_I49_J,axiom,
    ! [X31: assertion_a_b_c_d,X32: assertion_a_b_c_d,X71: assertion_a_b_c_d,X72: assertion_a_b_c_d] :
      ( ( star_a_b_c_d @ X31 @ X32 )
     != ( imp_a_b_c_d @ X71 @ X72 ) ) ).

% assertion.distinct(49)
thf(fact_193_assertion_Odistinct_I63_J,axiom,
    ! [X41: assertion_a_b_c_d,X42: assertion_a_b_c_d,X61: assertion_a_b_c_d,X62: assertion_a_b_c_d] :
      ( ( wand_a_b_c_d @ X41 @ X42 )
     != ( and_a_b_c_d @ X61 @ X62 ) ) ).

% assertion.distinct(63)
thf(fact_194_assertion_Odistinct_I87_J,axiom,
    ! [X51: assertion_a_b_c_d,X52: assertion_a_b_c_d,X11: assertion_a_b_c_d] :
      ( ( or_a_b_c_d @ X51 @ X52 )
     != ( bounded_a_b_c_d @ X11 ) ) ).

% assertion.distinct(87)
thf(fact_195_assertion_Odistinct_I79_J,axiom,
    ! [X51: assertion_a_b_c_d,X52: assertion_a_b_c_d,X71: assertion_a_b_c_d,X72: assertion_a_b_c_d] :
      ( ( or_a_b_c_d @ X51 @ X52 )
     != ( imp_a_b_c_d @ X71 @ X72 ) ) ).

% assertion.distinct(79)
thf(fact_196_assertion_Odistinct_I73_J,axiom,
    ! [X41: assertion_a_b_c_d,X42: assertion_a_b_c_d,X11: assertion_a_b_c_d] :
      ( ( wand_a_b_c_d @ X41 @ X42 )
     != ( bounded_a_b_c_d @ X11 ) ) ).

% assertion.distinct(73)
thf(fact_197_assertion_Odistinct_I3_J,axiom,
    ! [X1: ( d > c ) > a > $o,X31: assertion_a_b_c_d,X32: assertion_a_b_c_d] :
      ( ( sem_d_c_a_b @ X1 )
     != ( star_a_b_c_d @ X31 @ X32 ) ) ).

% assertion.distinct(3)
thf(fact_198_assertion_Odistinct_I65_J,axiom,
    ! [X41: assertion_a_b_c_d,X42: assertion_a_b_c_d,X71: assertion_a_b_c_d,X72: assertion_a_b_c_d] :
      ( ( wand_a_b_c_d @ X41 @ X42 )
     != ( imp_a_b_c_d @ X71 @ X72 ) ) ).

% assertion.distinct(65)
thf(fact_199_assertion_Odistinct_I129_J,axiom,
    ! [X12: assertion_a_b_c_d] :
      ( pred_a_b_c_d
     != ( wildcard_a_b_c_d @ X12 ) ) ).

% assertion.distinct(129)
thf(fact_200_assertion_Odistinct_I7_J,axiom,
    ! [X1: ( d > c ) > a > $o,X51: assertion_a_b_c_d,X52: assertion_a_b_c_d] :
      ( ( sem_d_c_a_b @ X1 )
     != ( or_a_b_c_d @ X51 @ X52 ) ) ).

% assertion.distinct(7)
thf(fact_201_assertion_Odistinct_I5_J,axiom,
    ! [X1: ( d > c ) > a > $o,X41: assertion_a_b_c_d,X42: assertion_a_b_c_d] :
      ( ( sem_d_c_a_b @ X1 )
     != ( wand_a_b_c_d @ X41 @ X42 ) ) ).

% assertion.distinct(5)
thf(fact_202_assertion_Odistinct_I99_J,axiom,
    ! [X61: assertion_a_b_c_d,X62: assertion_a_b_c_d,X11: assertion_a_b_c_d] :
      ( ( and_a_b_c_d @ X61 @ X62 )
     != ( bounded_a_b_c_d @ X11 ) ) ).

% assertion.distinct(99)
thf(fact_203_assertion_Odistinct_I55_J,axiom,
    ! [X31: assertion_a_b_c_d,X32: assertion_a_b_c_d] :
      ( ( star_a_b_c_d @ X31 @ X32 )
     != pred_a_b_c_d ) ).

% assertion.distinct(55)
thf(fact_204_assertion_Odistinct_I91_J,axiom,
    ! [X61: assertion_a_b_c_d,X62: assertion_a_b_c_d,X71: assertion_a_b_c_d,X72: assertion_a_b_c_d] :
      ( ( and_a_b_c_d @ X61 @ X62 )
     != ( imp_a_b_c_d @ X71 @ X72 ) ) ).

% assertion.distinct(91)
thf(fact_205_assertion_Odistinct_I109_J,axiom,
    ! [X71: assertion_a_b_c_d,X72: assertion_a_b_c_d,X11: assertion_a_b_c_d] :
      ( ( imp_a_b_c_d @ X71 @ X72 )
     != ( bounded_a_b_c_d @ X11 ) ) ).

% assertion.distinct(109)
thf(fact_206_assertion_Odistinct_I85_J,axiom,
    ! [X51: assertion_a_b_c_d,X52: assertion_a_b_c_d] :
      ( ( or_a_b_c_d @ X51 @ X52 )
     != pred_a_b_c_d ) ).

% assertion.distinct(85)
thf(fact_207_assertion_Odistinct_I71_J,axiom,
    ! [X41: assertion_a_b_c_d,X42: assertion_a_b_c_d] :
      ( ( wand_a_b_c_d @ X41 @ X42 )
     != pred_a_b_c_d ) ).

% assertion.distinct(71)
thf(fact_208_assertion_Odistinct_I9_J,axiom,
    ! [X1: ( d > c ) > a > $o,X61: assertion_a_b_c_d,X62: assertion_a_b_c_d] :
      ( ( sem_d_c_a_b @ X1 )
     != ( and_a_b_c_d @ X61 @ X62 ) ) ).

% assertion.distinct(9)
thf(fact_209_assertion_Odistinct_I19_J,axiom,
    ! [X1: ( d > c ) > a > $o,X11: assertion_a_b_c_d] :
      ( ( sem_d_c_a_b @ X1 )
     != ( bounded_a_b_c_d @ X11 ) ) ).

% assertion.distinct(19)
thf(fact_210_assertion_Odistinct_I11_J,axiom,
    ! [X1: ( d > c ) > a > $o,X71: assertion_a_b_c_d,X72: assertion_a_b_c_d] :
      ( ( sem_d_c_a_b @ X1 )
     != ( imp_a_b_c_d @ X71 @ X72 ) ) ).

% assertion.distinct(11)
thf(fact_211_assertion_Odistinct_I97_J,axiom,
    ! [X61: assertion_a_b_c_d,X62: assertion_a_b_c_d] :
      ( ( and_a_b_c_d @ X61 @ X62 )
     != pred_a_b_c_d ) ).

% assertion.distinct(97)
thf(fact_212_assertion_Odistinct_I127_J,axiom,
    ! [X11: assertion_a_b_c_d] :
      ( pred_a_b_c_d
     != ( bounded_a_b_c_d @ X11 ) ) ).

% assertion.distinct(127)
thf(fact_213_assertion_Odistinct_I107_J,axiom,
    ! [X71: assertion_a_b_c_d,X72: assertion_a_b_c_d] :
      ( ( imp_a_b_c_d @ X71 @ X72 )
     != pred_a_b_c_d ) ).

% assertion.distinct(107)
thf(fact_214_assertion_Odistinct_I17_J,axiom,
    ! [X1: ( d > c ) > a > $o] :
      ( ( sem_d_c_a_b @ X1 )
     != pred_a_b_c_d ) ).

% assertion.distinct(17)
thf(fact_215_frame__rule,axiom,
    ! [C: set_Pr7868159745199425715_a_d_c,P2: assertion_a_b_c_d,Q: assertion_a_b_c_d,Delta: ( d > c ) > set_a,R4: assertion_a_b_c_d] :
      ( ( valid_command_a_d_c @ valid @ C )
     => ( ( safety7280469885071620222_a_d_c @ plus @ valid @ C )
       => ( ( frame_property_a_d_c @ plus @ valid @ C )
         => ( ( valid_8824771084768397689_b_c_d @ plus @ mult @ valid @ P2 @ C @ Q @ Delta )
           => ( ( not_in_fv_a_b_c_d @ plus @ mult @ valid @ R4 @ ( modified_a_d_c @ C ) )
             => ( valid_8824771084768397689_b_c_d @ plus @ mult @ valid @ ( star_a_b_c_d @ P2 @ R4 ) @ C @ ( star_a_b_c_d @ Q @ R4 ) @ Delta ) ) ) ) ) ) ).

% frame_rule
thf(fact_216_option_Oinject,axiom,
    ! [X22: product_prod_a_d_c,Y2: product_prod_a_d_c] :
      ( ( ( some_P377817780860425132_a_d_c @ X22 )
        = ( some_P377817780860425132_a_d_c @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% option.inject
thf(fact_217_option_Oinject,axiom,
    ! [X22: a,Y2: a] :
      ( ( ( some_a @ X22 )
        = ( some_a @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% option.inject
thf(fact_218_logic_Ointuitionistic__def,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,S4: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( intuit4720955538653295669_b_d_c @ Plus @ Mult @ Valid @ S4 @ Delta @ A2 )
        = ( ! [A3: a,B3: a] :
              ( ( ( pre_larger_a @ Plus @ A3 @ B3 )
                & ( sat_a_b_d_c @ Plus @ Mult @ Valid @ B3 @ S4 @ Delta @ A2 ) )
             => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ A3 @ S4 @ Delta @ A2 ) ) ) ) ) ).

% logic.intuitionistic_def
thf(fact_219_logic_OintuitionisticI,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,S4: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ! [A4: a,B5: a] :
            ( ( ( pre_larger_a @ Plus @ A4 @ B5 )
              & ( sat_a_b_d_c @ Plus @ Mult @ Valid @ B5 @ S4 @ Delta @ A2 ) )
           => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ A4 @ S4 @ Delta @ A2 ) )
       => ( intuit4720955538653295669_b_d_c @ Plus @ Mult @ Valid @ S4 @ Delta @ A2 ) ) ) ).

% logic.intuitionisticI
thf(fact_220_compatible__def,axiom,
    ! [A: a,B: a] :
      ( ( pre_compatible_a @ plus @ A @ B )
      = ( ( plus @ A @ B )
       != none_a ) ) ).

% compatible_def
thf(fact_221_assertion_Oexhaust,axiom,
    ! [Y: assertion_a_b_c_d] :
      ( ! [X13: ( d > c ) > a > $o] :
          ( Y
         != ( sem_d_c_a_b @ X13 ) )
     => ( ! [X212: b,X223: assertion_a_b_c_d] :
            ( Y
           != ( mult_b_a_c_d @ X212 @ X223 ) )
       => ( ! [X312: assertion_a_b_c_d,X322: assertion_a_b_c_d] :
              ( Y
             != ( star_a_b_c_d @ X312 @ X322 ) )
         => ( ! [X412: assertion_a_b_c_d,X422: assertion_a_b_c_d] :
                ( Y
               != ( wand_a_b_c_d @ X412 @ X422 ) )
           => ( ! [X512: assertion_a_b_c_d,X522: assertion_a_b_c_d] :
                  ( Y
                 != ( or_a_b_c_d @ X512 @ X522 ) )
             => ( ! [X612: assertion_a_b_c_d,X622: assertion_a_b_c_d] :
                    ( Y
                   != ( and_a_b_c_d @ X612 @ X622 ) )
               => ( ! [X712: assertion_a_b_c_d,X722: assertion_a_b_c_d] :
                      ( Y
                     != ( imp_a_b_c_d @ X712 @ X722 ) )
                 => ( ! [X81: d,X82: assertion_a_b_c_d] :
                        ( Y
                       != ( exists_d_a_b_c @ X81 @ X82 ) )
                   => ( ! [X91: d,X92: assertion_a_b_c_d] :
                          ( Y
                         != ( forall_d_a_b_c @ X91 @ X92 ) )
                     => ( ( Y != pred_a_b_c_d )
                       => ( ! [X112: assertion_a_b_c_d] :
                              ( Y
                             != ( bounded_a_b_c_d @ X112 ) )
                         => ~ ! [X122: assertion_a_b_c_d] :
                                ( Y
                               != ( wildcard_a_b_c_d @ X122 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% assertion.exhaust
thf(fact_222_logic_Oframe__rule,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,C: set_Pr7868159745199425715_a_d_c,P2: assertion_a_b_c_d,Q: assertion_a_b_c_d,Delta: ( d > c ) > set_a,R4: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( valid_command_a_d_c @ Valid @ C )
       => ( ( safety7280469885071620222_a_d_c @ Plus @ Valid @ C )
         => ( ( frame_property_a_d_c @ Plus @ Valid @ C )
           => ( ( valid_8824771084768397689_b_c_d @ Plus @ Mult @ Valid @ P2 @ C @ Q @ Delta )
             => ( ( not_in_fv_a_b_c_d @ Plus @ Mult @ Valid @ R4 @ ( modified_a_d_c @ C ) )
               => ( valid_8824771084768397689_b_c_d @ Plus @ Mult @ Valid @ ( star_a_b_c_d @ P2 @ R4 ) @ C @ ( star_a_b_c_d @ Q @ R4 ) @ Delta ) ) ) ) ) ) ) ).

% logic.frame_rule
thf(fact_223_sat_Osimps_I8_J,axiom,
    ! [Sigma: a,S4: d > c,Delta: ( d > c ) > set_a,X3: d,A2: assertion_a_b_c_d] :
      ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( exists_d_a_b_c @ X3 @ A2 ) )
      = ( ? [V: c] : ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ ( fun_upd_d_c @ S4 @ X3 @ V ) @ Delta @ A2 ) ) ) ).

% sat.simps(8)
thf(fact_224_sat__forall,axiom,
    ! [Sigma: a,S4: d > c,X3: d,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
      ( ! [V2: c] : ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ ( fun_upd_d_c @ S4 @ X3 @ V2 ) @ Delta @ A2 )
     => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( forall_d_a_b_c @ X3 @ A2 ) ) ) ).

% sat_forall
thf(fact_225_assertion_Oinject_I9_J,axiom,
    ! [X912: d,X922: assertion_a_b_c_d,Y91: d,Y92: assertion_a_b_c_d] :
      ( ( ( forall_d_a_b_c @ X912 @ X922 )
        = ( forall_d_a_b_c @ Y91 @ Y92 ) )
      = ( ( X912 = Y91 )
        & ( X922 = Y92 ) ) ) ).

% assertion.inject(9)
thf(fact_226_assertion_Oinject_I8_J,axiom,
    ! [X812: d,X822: assertion_a_b_c_d,Y81: d,Y82: assertion_a_b_c_d] :
      ( ( ( exists_d_a_b_c @ X812 @ X822 )
        = ( exists_d_a_b_c @ Y81 @ Y82 ) )
      = ( ( X812 = Y81 )
        & ( X822 = Y82 ) ) ) ).

% assertion.inject(8)
thf(fact_227_not__Some__eq,axiom,
    ! [X3: option2362401199305441953_a_d_c] :
      ( ( ! [Y3: product_prod_a_d_c] :
            ( X3
           != ( some_P377817780860425132_a_d_c @ Y3 ) ) )
      = ( X3 = none_P1533316314068730928_a_d_c ) ) ).

% not_Some_eq
thf(fact_228_not__Some__eq,axiom,
    ! [X3: option_a] :
      ( ( ! [Y3: a] :
            ( X3
           != ( some_a @ Y3 ) ) )
      = ( X3 = none_a ) ) ).

% not_Some_eq
thf(fact_229_not__None__eq,axiom,
    ! [X3: option2362401199305441953_a_d_c] :
      ( ( X3 != none_P1533316314068730928_a_d_c )
      = ( ? [Y3: product_prod_a_d_c] :
            ( X3
            = ( some_P377817780860425132_a_d_c @ Y3 ) ) ) ) ).

% not_None_eq
thf(fact_230_not__None__eq,axiom,
    ! [X3: option_a] :
      ( ( X3 != none_a )
      = ( ? [Y3: a] :
            ( X3
            = ( some_a @ Y3 ) ) ) ) ).

% not_None_eq
thf(fact_231_sat_Osimps_I9_J,axiom,
    ! [Sigma: a,S4: d > c,Delta: ( d > c ) > set_a,X3: d,A2: assertion_a_b_c_d] :
      ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( forall_d_a_b_c @ X3 @ A2 ) )
      = ( ! [V: c] : ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ ( fun_upd_d_c @ S4 @ X3 @ V ) @ Delta @ A2 ) ) ) ).

% sat.simps(9)
thf(fact_232_assertion_Odistinct_I113_J,axiom,
    ! [X812: d,X822: assertion_a_b_c_d,X912: d,X922: assertion_a_b_c_d] :
      ( ( exists_d_a_b_c @ X812 @ X822 )
     != ( forall_d_a_b_c @ X912 @ X922 ) ) ).

% assertion.distinct(113)
thf(fact_233_combine__options__cases,axiom,
    ! [X3: option_a,P2: option_a > option2362401199305441953_a_d_c > $o,Y: option2362401199305441953_a_d_c] :
      ( ( ( X3 = none_a )
       => ( P2 @ X3 @ Y ) )
     => ( ( ( Y = none_P1533316314068730928_a_d_c )
         => ( P2 @ X3 @ Y ) )
       => ( ! [A4: a,B5: product_prod_a_d_c] :
              ( ( X3
                = ( some_a @ A4 ) )
             => ( ( Y
                  = ( some_P377817780860425132_a_d_c @ B5 ) )
               => ( P2 @ X3 @ Y ) ) )
         => ( P2 @ X3 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_234_combine__options__cases,axiom,
    ! [X3: option2362401199305441953_a_d_c,P2: option2362401199305441953_a_d_c > option_a > $o,Y: option_a] :
      ( ( ( X3 = none_P1533316314068730928_a_d_c )
       => ( P2 @ X3 @ Y ) )
     => ( ( ( Y = none_a )
         => ( P2 @ X3 @ Y ) )
       => ( ! [A4: product_prod_a_d_c,B5: a] :
              ( ( X3
                = ( some_P377817780860425132_a_d_c @ A4 ) )
             => ( ( Y
                  = ( some_a @ B5 ) )
               => ( P2 @ X3 @ Y ) ) )
         => ( P2 @ X3 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_235_combine__options__cases,axiom,
    ! [X3: option2362401199305441953_a_d_c,P2: option2362401199305441953_a_d_c > option2362401199305441953_a_d_c > $o,Y: option2362401199305441953_a_d_c] :
      ( ( ( X3 = none_P1533316314068730928_a_d_c )
       => ( P2 @ X3 @ Y ) )
     => ( ( ( Y = none_P1533316314068730928_a_d_c )
         => ( P2 @ X3 @ Y ) )
       => ( ! [A4: product_prod_a_d_c,B5: product_prod_a_d_c] :
              ( ( X3
                = ( some_P377817780860425132_a_d_c @ A4 ) )
             => ( ( Y
                  = ( some_P377817780860425132_a_d_c @ B5 ) )
               => ( P2 @ X3 @ Y ) ) )
         => ( P2 @ X3 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_236_combine__options__cases,axiom,
    ! [X3: option_a,P2: option_a > option_a > $o,Y: option_a] :
      ( ( ( X3 = none_a )
       => ( P2 @ X3 @ Y ) )
     => ( ( ( Y = none_a )
         => ( P2 @ X3 @ Y ) )
       => ( ! [A4: a,B5: a] :
              ( ( X3
                = ( some_a @ A4 ) )
             => ( ( Y
                  = ( some_a @ B5 ) )
               => ( P2 @ X3 @ Y ) ) )
         => ( P2 @ X3 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_237_split__option__all,axiom,
    ( ( ^ [P4: option2362401199305441953_a_d_c > $o] :
        ! [X5: option2362401199305441953_a_d_c] : ( P4 @ X5 ) )
    = ( ^ [P5: option2362401199305441953_a_d_c > $o] :
          ( ( P5 @ none_P1533316314068730928_a_d_c )
          & ! [X: product_prod_a_d_c] : ( P5 @ ( some_P377817780860425132_a_d_c @ X ) ) ) ) ) ).

% split_option_all
thf(fact_238_split__option__all,axiom,
    ( ( ^ [P4: option_a > $o] :
        ! [X5: option_a] : ( P4 @ X5 ) )
    = ( ^ [P5: option_a > $o] :
          ( ( P5 @ none_a )
          & ! [X: a] : ( P5 @ ( some_a @ X ) ) ) ) ) ).

% split_option_all
thf(fact_239_split__option__ex,axiom,
    ( ( ^ [P4: option2362401199305441953_a_d_c > $o] :
        ? [X5: option2362401199305441953_a_d_c] : ( P4 @ X5 ) )
    = ( ^ [P5: option2362401199305441953_a_d_c > $o] :
          ( ( P5 @ none_P1533316314068730928_a_d_c )
          | ? [X: product_prod_a_d_c] : ( P5 @ ( some_P377817780860425132_a_d_c @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_240_split__option__ex,axiom,
    ( ( ^ [P4: option_a > $o] :
        ? [X5: option_a] : ( P4 @ X5 ) )
    = ( ^ [P5: option_a > $o] :
          ( ( P5 @ none_a )
          | ? [X: a] : ( P5 @ ( some_a @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_241_option_Oexhaust,axiom,
    ! [Y: option2362401199305441953_a_d_c] :
      ( ( Y != none_P1533316314068730928_a_d_c )
     => ~ ! [X23: product_prod_a_d_c] :
            ( Y
           != ( some_P377817780860425132_a_d_c @ X23 ) ) ) ).

% option.exhaust
thf(fact_242_option_Oexhaust,axiom,
    ! [Y: option_a] :
      ( ( Y != none_a )
     => ~ ! [X23: a] :
            ( Y
           != ( some_a @ X23 ) ) ) ).

% option.exhaust
thf(fact_243_option_OdiscI,axiom,
    ! [Option: option2362401199305441953_a_d_c,X22: product_prod_a_d_c] :
      ( ( Option
        = ( some_P377817780860425132_a_d_c @ X22 ) )
     => ( Option != none_P1533316314068730928_a_d_c ) ) ).

% option.discI
thf(fact_244_option_OdiscI,axiom,
    ! [Option: option_a,X22: a] :
      ( ( Option
        = ( some_a @ X22 ) )
     => ( Option != none_a ) ) ).

% option.discI
thf(fact_245_option_Odistinct_I1_J,axiom,
    ! [X22: product_prod_a_d_c] :
      ( none_P1533316314068730928_a_d_c
     != ( some_P377817780860425132_a_d_c @ X22 ) ) ).

% option.distinct(1)
thf(fact_246_option_Odistinct_I1_J,axiom,
    ! [X22: a] :
      ( none_a
     != ( some_a @ X22 ) ) ).

% option.distinct(1)
thf(fact_247_logic_Ointuitionistic_Ocong,axiom,
    intuit4720955538653295669_b_d_c = intuit4720955538653295669_b_d_c ).

% logic.intuitionistic.cong
thf(fact_248_logic_Osat_Osimps_I9_J,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S4: d > c,Delta: ( d > c ) > set_a,X3: d,A2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( forall_d_a_b_c @ X3 @ A2 ) )
        = ( ! [V: c] : ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ ( fun_upd_d_c @ S4 @ X3 @ V ) @ Delta @ A2 ) ) ) ) ).

% logic.sat.simps(9)
thf(fact_249_logic_Osat__forall,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S4: d > c,X3: d,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ! [V2: c] : ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ ( fun_upd_d_c @ S4 @ X3 @ V2 ) @ Delta @ A2 )
       => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( forall_d_a_b_c @ X3 @ A2 ) ) ) ) ).

% logic.sat_forall
thf(fact_250_logic_Osat_Osimps_I8_J,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S4: d > c,Delta: ( d > c ) > set_a,X3: d,A2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( exists_d_a_b_c @ X3 @ A2 ) )
        = ( ? [V: c] : ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ ( fun_upd_d_c @ S4 @ X3 @ V ) @ Delta @ A2 ) ) ) ) ).

% logic.sat.simps(8)
thf(fact_251_assertion_Odistinct_I35_J,axiom,
    ! [X21: b,X222: assertion_a_b_c_d,X912: d,X922: assertion_a_b_c_d] :
      ( ( mult_b_a_c_d @ X21 @ X222 )
     != ( forall_d_a_b_c @ X912 @ X922 ) ) ).

% assertion.distinct(35)
thf(fact_252_assertion_Odistinct_I33_J,axiom,
    ! [X21: b,X222: assertion_a_b_c_d,X812: d,X822: assertion_a_b_c_d] :
      ( ( mult_b_a_c_d @ X21 @ X222 )
     != ( exists_d_a_b_c @ X812 @ X822 ) ) ).

% assertion.distinct(33)
thf(fact_253_assertion_Odistinct_I125_J,axiom,
    ! [X912: d,X922: assertion_a_b_c_d,X12: assertion_a_b_c_d] :
      ( ( forall_d_a_b_c @ X912 @ X922 )
     != ( wildcard_a_b_c_d @ X12 ) ) ).

% assertion.distinct(125)
thf(fact_254_assertion_Odistinct_I119_J,axiom,
    ! [X812: d,X822: assertion_a_b_c_d,X12: assertion_a_b_c_d] :
      ( ( exists_d_a_b_c @ X812 @ X822 )
     != ( wildcard_a_b_c_d @ X12 ) ) ).

% assertion.distinct(119)
thf(fact_255_assertion_Odistinct_I53_J,axiom,
    ! [X31: assertion_a_b_c_d,X32: assertion_a_b_c_d,X912: d,X922: assertion_a_b_c_d] :
      ( ( star_a_b_c_d @ X31 @ X32 )
     != ( forall_d_a_b_c @ X912 @ X922 ) ) ).

% assertion.distinct(53)
thf(fact_256_assertion_Odistinct_I83_J,axiom,
    ! [X51: assertion_a_b_c_d,X52: assertion_a_b_c_d,X912: d,X922: assertion_a_b_c_d] :
      ( ( or_a_b_c_d @ X51 @ X52 )
     != ( forall_d_a_b_c @ X912 @ X922 ) ) ).

% assertion.distinct(83)
thf(fact_257_assertion_Odistinct_I69_J,axiom,
    ! [X41: assertion_a_b_c_d,X42: assertion_a_b_c_d,X912: d,X922: assertion_a_b_c_d] :
      ( ( wand_a_b_c_d @ X41 @ X42 )
     != ( forall_d_a_b_c @ X912 @ X922 ) ) ).

% assertion.distinct(69)
thf(fact_258_assertion_Odistinct_I51_J,axiom,
    ! [X31: assertion_a_b_c_d,X32: assertion_a_b_c_d,X812: d,X822: assertion_a_b_c_d] :
      ( ( star_a_b_c_d @ X31 @ X32 )
     != ( exists_d_a_b_c @ X812 @ X822 ) ) ).

% assertion.distinct(51)
thf(fact_259_pre__logic_Ocompatible__def,axiom,
    ( pre_compatible_a
    = ( ^ [Plus2: a > a > option_a,A3: a,B3: a] :
          ( ( Plus2 @ A3 @ B3 )
         != none_a ) ) ) ).

% pre_logic.compatible_def
thf(fact_260_assertion_Odistinct_I81_J,axiom,
    ! [X51: assertion_a_b_c_d,X52: assertion_a_b_c_d,X812: d,X822: assertion_a_b_c_d] :
      ( ( or_a_b_c_d @ X51 @ X52 )
     != ( exists_d_a_b_c @ X812 @ X822 ) ) ).

% assertion.distinct(81)
thf(fact_261_assertion_Odistinct_I67_J,axiom,
    ! [X41: assertion_a_b_c_d,X42: assertion_a_b_c_d,X812: d,X822: assertion_a_b_c_d] :
      ( ( wand_a_b_c_d @ X41 @ X42 )
     != ( exists_d_a_b_c @ X812 @ X822 ) ) ).

% assertion.distinct(67)
thf(fact_262_assertion_Odistinct_I95_J,axiom,
    ! [X61: assertion_a_b_c_d,X62: assertion_a_b_c_d,X912: d,X922: assertion_a_b_c_d] :
      ( ( and_a_b_c_d @ X61 @ X62 )
     != ( forall_d_a_b_c @ X912 @ X922 ) ) ).

% assertion.distinct(95)
thf(fact_263_assertion_Odistinct_I123_J,axiom,
    ! [X912: d,X922: assertion_a_b_c_d,X11: assertion_a_b_c_d] :
      ( ( forall_d_a_b_c @ X912 @ X922 )
     != ( bounded_a_b_c_d @ X11 ) ) ).

% assertion.distinct(123)
thf(fact_264_assertion_Odistinct_I105_J,axiom,
    ! [X71: assertion_a_b_c_d,X72: assertion_a_b_c_d,X912: d,X922: assertion_a_b_c_d] :
      ( ( imp_a_b_c_d @ X71 @ X72 )
     != ( forall_d_a_b_c @ X912 @ X922 ) ) ).

% assertion.distinct(105)
thf(fact_265_assertion_Odistinct_I93_J,axiom,
    ! [X61: assertion_a_b_c_d,X62: assertion_a_b_c_d,X812: d,X822: assertion_a_b_c_d] :
      ( ( and_a_b_c_d @ X61 @ X62 )
     != ( exists_d_a_b_c @ X812 @ X822 ) ) ).

% assertion.distinct(93)
thf(fact_266_assertion_Odistinct_I117_J,axiom,
    ! [X812: d,X822: assertion_a_b_c_d,X11: assertion_a_b_c_d] :
      ( ( exists_d_a_b_c @ X812 @ X822 )
     != ( bounded_a_b_c_d @ X11 ) ) ).

% assertion.distinct(117)
thf(fact_267_assertion_Odistinct_I15_J,axiom,
    ! [X1: ( d > c ) > a > $o,X912: d,X922: assertion_a_b_c_d] :
      ( ( sem_d_c_a_b @ X1 )
     != ( forall_d_a_b_c @ X912 @ X922 ) ) ).

% assertion.distinct(15)
thf(fact_268_assertion_Odistinct_I103_J,axiom,
    ! [X71: assertion_a_b_c_d,X72: assertion_a_b_c_d,X812: d,X822: assertion_a_b_c_d] :
      ( ( imp_a_b_c_d @ X71 @ X72 )
     != ( exists_d_a_b_c @ X812 @ X822 ) ) ).

% assertion.distinct(103)
thf(fact_269_assertion_Odistinct_I13_J,axiom,
    ! [X1: ( d > c ) > a > $o,X812: d,X822: assertion_a_b_c_d] :
      ( ( sem_d_c_a_b @ X1 )
     != ( exists_d_a_b_c @ X812 @ X822 ) ) ).

% assertion.distinct(13)
thf(fact_270_assertion_Odistinct_I121_J,axiom,
    ! [X912: d,X922: assertion_a_b_c_d] :
      ( ( forall_d_a_b_c @ X912 @ X922 )
     != pred_a_b_c_d ) ).

% assertion.distinct(121)
thf(fact_271_assertion_Odistinct_I115_J,axiom,
    ! [X812: d,X822: assertion_a_b_c_d] :
      ( ( exists_d_a_b_c @ X812 @ X822 )
     != pred_a_b_c_d ) ).

% assertion.distinct(115)
thf(fact_272_fun__upd__apply,axiom,
    ( fun_upd_d_c
    = ( ^ [F: d > c,X: d,Y3: c,Z: d] : ( if_c @ ( Z = X ) @ Y3 @ ( F @ Z ) ) ) ) ).

% fun_upd_apply
thf(fact_273_fun__upd__triv,axiom,
    ! [F2: d > c,X3: d] :
      ( ( fun_upd_d_c @ F2 @ X3 @ ( F2 @ X3 ) )
      = F2 ) ).

% fun_upd_triv
thf(fact_274_fun__upd__upd,axiom,
    ! [F2: d > c,X3: d,Y: c,Z2: c] :
      ( ( fun_upd_d_c @ ( fun_upd_d_c @ F2 @ X3 @ Y ) @ X3 @ Z2 )
      = ( fun_upd_d_c @ F2 @ X3 @ Z2 ) ) ).

% fun_upd_upd
thf(fact_275_not__in__fv__mod,axiom,
    ! [A2: assertion_a_b_c_d,C: set_Pr7868159745199425715_a_d_c,Sigma: a,S4: d > c,Sigma6: a,S7: d > c,X3: a,Delta: ( d > c ) > set_a] :
      ( ( not_in_fv_a_b_c_d @ plus @ mult @ valid @ A2 @ ( modified_a_d_c @ C ) )
     => ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ Sigma @ S4 ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ Sigma6 @ S7 ) ) ) @ C )
       => ( ( sat_a_b_d_c @ plus @ mult @ valid @ X3 @ S4 @ Delta @ A2 )
          = ( sat_a_b_d_c @ plus @ mult @ valid @ X3 @ S7 @ Delta @ A2 ) ) ) ) ).

% not_in_fv_mod
thf(fact_276_safety__monotonicity__def,axiom,
    ! [C: set_Pr7868159745199425715_a_d_c] :
      ( ( safety7280469885071620222_a_d_c @ plus @ valid @ C )
      = ( ! [Sigma2: a,Sigma3: a,S: d > c] :
            ( ( ( valid @ Sigma3 )
              & ( pre_larger_a @ plus @ Sigma3 @ Sigma2 )
              & ( safe_a_d_c @ C @ ( product_Pair_a_d_c @ Sigma2 @ S ) ) )
           => ( safe_a_d_c @ C @ ( product_Pair_a_d_c @ Sigma3 @ S ) ) ) ) ) ).

% safety_monotonicity_def
thf(fact_277_fun__upd__idem__iff,axiom,
    ! [F2: d > c,X3: d,Y: c] :
      ( ( ( fun_upd_d_c @ F2 @ X3 @ Y )
        = F2 )
      = ( ( F2 @ X3 )
        = Y ) ) ).

% fun_upd_idem_iff
thf(fact_278_safe__def,axiom,
    ( safe_a_d_c
    = ( ^ [C2: set_Pr7868159745199425715_a_d_c,Sigma2: product_prod_a_d_c] :
          ~ ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ Sigma2 @ none_P1533316314068730928_a_d_c ) @ C2 ) ) ) ).

% safe_def
thf(fact_279_valid__command__def,axiom,
    ! [C: set_Pr7868159745199425715_a_d_c] :
      ( ( valid_command_a_d_c @ valid @ C )
      = ( ! [A3: a,B3: a,Sa: d > c,Sb: d > c] :
            ( ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ A3 @ Sa ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ B3 @ Sb ) ) ) @ C )
              & ( valid @ A3 ) )
           => ( valid @ B3 ) ) ) ) ).

% valid_command_def
thf(fact_280_frame__property__def,axiom,
    ! [C: set_Pr7868159745199425715_a_d_c] :
      ( ( frame_property_a_d_c @ plus @ valid @ C )
      = ( ! [Sigma2: a,Sigma_0: a,R3: a,Sigma3: a,S: d > c,S2: d > c] :
            ( ( ( valid @ Sigma2 )
              & ( valid @ Sigma3 )
              & ( safe_a_d_c @ C @ ( product_Pair_a_d_c @ Sigma_0 @ S ) )
              & ( ( some_a @ Sigma2 )
                = ( plus @ Sigma_0 @ R3 ) )
              & ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ Sigma2 @ S ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ Sigma3 @ S2 ) ) ) @ C ) )
           => ? [Sigma_02: a] :
                ( ( ( some_a @ Sigma3 )
                  = ( plus @ Sigma_02 @ R3 ) )
                & ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ Sigma_0 @ S ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ Sigma_02 @ S2 ) ) ) @ C ) ) ) ) ) ).

% frame_property_def
thf(fact_281_valid__hoare__triple__def,axiom,
    ! [P2: assertion_a_b_c_d,C: set_Pr7868159745199425715_a_d_c,Q: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( valid_8824771084768397689_b_c_d @ plus @ mult @ valid @ P2 @ C @ Q @ Delta )
      = ( ! [Sigma2: a,S: d > c] :
            ( ( ( valid @ Sigma2 )
              & ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma2 @ S @ Delta @ P2 ) )
           => ( ( safe_a_d_c @ C @ ( product_Pair_a_d_c @ Sigma2 @ S ) )
              & ! [Sigma3: a,S2: d > c] :
                  ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ Sigma2 @ S ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ Sigma3 @ S2 ) ) ) @ C )
                 => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma3 @ S2 @ Delta @ Q ) ) ) ) ) ) ).

% valid_hoare_triple_def
thf(fact_282_valid__hoare__tripleI,axiom,
    ! [Delta: ( d > c ) > set_a,P2: assertion_a_b_c_d,C: set_Pr7868159745199425715_a_d_c,Q: assertion_a_b_c_d] :
      ( ! [Sigma4: a,S5: d > c] :
          ( ( ( valid @ Sigma4 )
            & ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma4 @ S5 @ Delta @ P2 ) )
         => ( safe_a_d_c @ C @ ( product_Pair_a_d_c @ Sigma4 @ S5 ) ) )
     => ( ! [Sigma4: a,S5: d > c,Sigma5: a,S8: d > c] :
            ( ( ( valid @ Sigma4 )
              & ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma4 @ S5 @ Delta @ P2 ) )
           => ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ Sigma4 @ S5 ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ Sigma5 @ S8 ) ) ) @ C )
             => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma5 @ S8 @ Delta @ Q ) ) )
       => ( valid_8824771084768397689_b_c_d @ plus @ mult @ valid @ P2 @ C @ Q @ Delta ) ) ) ).

% valid_hoare_tripleI
thf(fact_283_logic_Osafe__def,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,C: set_Pr7868159745199425715_a_d_c,Sigma: product_prod_a_d_c] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( safe_a_d_c @ C @ Sigma )
        = ( ~ ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ Sigma @ none_P1533316314068730928_a_d_c ) @ C ) ) ) ) ).

% logic.safe_def
thf(fact_284_logic_Ovalid__command__def,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,C: set_Pr7868159745199425715_a_d_c] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( valid_command_a_d_c @ Valid @ C )
        = ( ! [A3: a,B3: a,Sa: d > c,Sb: d > c] :
              ( ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ A3 @ Sa ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ B3 @ Sb ) ) ) @ C )
                & ( Valid @ A3 ) )
             => ( Valid @ B3 ) ) ) ) ) ).

% logic.valid_command_def
thf(fact_285_logic_Ovalid__hoare__tripleI,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( d > c ) > set_a,P2: assertion_a_b_c_d,C: set_Pr7868159745199425715_a_d_c,Q: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ! [Sigma4: a,S5: d > c] :
            ( ( ( Valid @ Sigma4 )
              & ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma4 @ S5 @ Delta @ P2 ) )
           => ( safe_a_d_c @ C @ ( product_Pair_a_d_c @ Sigma4 @ S5 ) ) )
       => ( ! [Sigma4: a,S5: d > c,Sigma5: a,S8: d > c] :
              ( ( ( Valid @ Sigma4 )
                & ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma4 @ S5 @ Delta @ P2 ) )
             => ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ Sigma4 @ S5 ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ Sigma5 @ S8 ) ) ) @ C )
               => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma5 @ S8 @ Delta @ Q ) ) )
         => ( valid_8824771084768397689_b_c_d @ Plus @ Mult @ Valid @ P2 @ C @ Q @ Delta ) ) ) ) ).

% logic.valid_hoare_tripleI
thf(fact_286_logic_Ovalid__hoare__triple__def,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P2: assertion_a_b_c_d,C: set_Pr7868159745199425715_a_d_c,Q: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( valid_8824771084768397689_b_c_d @ Plus @ Mult @ Valid @ P2 @ C @ Q @ Delta )
        = ( ! [Sigma2: a,S: d > c] :
              ( ( ( Valid @ Sigma2 )
                & ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma2 @ S @ Delta @ P2 ) )
             => ( ( safe_a_d_c @ C @ ( product_Pair_a_d_c @ Sigma2 @ S ) )
                & ! [Sigma3: a,S2: d > c] :
                    ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ Sigma2 @ S ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ Sigma3 @ S2 ) ) ) @ C )
                   => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma3 @ S2 @ Delta @ Q ) ) ) ) ) ) ) ).

% logic.valid_hoare_triple_def
thf(fact_287_logic_Oframe__property__def,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,C: set_Pr7868159745199425715_a_d_c] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( frame_property_a_d_c @ Plus @ Valid @ C )
        = ( ! [Sigma2: a,Sigma_0: a,R3: a,Sigma3: a,S: d > c,S2: d > c] :
              ( ( ( Valid @ Sigma2 )
                & ( Valid @ Sigma3 )
                & ( safe_a_d_c @ C @ ( product_Pair_a_d_c @ Sigma_0 @ S ) )
                & ( ( some_a @ Sigma2 )
                  = ( Plus @ Sigma_0 @ R3 ) )
                & ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ Sigma2 @ S ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ Sigma3 @ S2 ) ) ) @ C ) )
             => ? [Sigma_02: a] :
                  ( ( ( some_a @ Sigma3 )
                    = ( Plus @ Sigma_02 @ R3 ) )
                  & ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ Sigma_0 @ S ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ Sigma_02 @ S2 ) ) ) @ C ) ) ) ) ) ) ).

% logic.frame_property_def
thf(fact_288_logic_Onot__in__fv__mod,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_c_d,C: set_Pr7868159745199425715_a_d_c,Sigma: a,S4: d > c,Sigma6: a,S7: d > c,X3: a,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( not_in_fv_a_b_c_d @ Plus @ Mult @ Valid @ A2 @ ( modified_a_d_c @ C ) )
       => ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ Sigma @ S4 ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ Sigma6 @ S7 ) ) ) @ C )
         => ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ X3 @ S4 @ Delta @ A2 )
            = ( sat_a_b_d_c @ Plus @ Mult @ Valid @ X3 @ S7 @ Delta @ A2 ) ) ) ) ) ).

% logic.not_in_fv_mod
thf(fact_289_logic_Osafety__monotonicity__def,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,C: set_Pr7868159745199425715_a_d_c] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( safety7280469885071620222_a_d_c @ Plus @ Valid @ C )
        = ( ! [Sigma2: a,Sigma3: a,S: d > c] :
              ( ( ( Valid @ Sigma3 )
                & ( pre_larger_a @ Plus @ Sigma3 @ Sigma2 )
                & ( safe_a_d_c @ C @ ( product_Pair_a_d_c @ Sigma2 @ S ) ) )
             => ( safe_a_d_c @ C @ ( product_Pair_a_d_c @ Sigma3 @ S ) ) ) ) ) ) ).

% logic.safety_monotonicity_def
thf(fact_290_fun__upd__def,axiom,
    ( fun_upd_d_c
    = ( ^ [F: d > c,A3: d,B3: c,X: d] : ( if_c @ ( X = A3 ) @ B3 @ ( F @ X ) ) ) ) ).

% fun_upd_def
thf(fact_291_fun__upd__eqD,axiom,
    ! [F2: d > c,X3: d,Y: c,G: d > c,Z2: c] :
      ( ( ( fun_upd_d_c @ F2 @ X3 @ Y )
        = ( fun_upd_d_c @ G @ X3 @ Z2 ) )
     => ( Y = Z2 ) ) ).

% fun_upd_eqD
thf(fact_292_fun__upd__idem,axiom,
    ! [F2: d > c,X3: d,Y: c] :
      ( ( ( F2 @ X3 )
        = Y )
     => ( ( fun_upd_d_c @ F2 @ X3 @ Y )
        = F2 ) ) ).

% fun_upd_idem
thf(fact_293_fun__upd__same,axiom,
    ! [F2: d > c,X3: d,Y: c] :
      ( ( fun_upd_d_c @ F2 @ X3 @ Y @ X3 )
      = Y ) ).

% fun_upd_same
thf(fact_294_fun__upd__other,axiom,
    ! [Z2: d,X3: d,F2: d > c,Y: c] :
      ( ( Z2 != X3 )
     => ( ( fun_upd_d_c @ F2 @ X3 @ Y @ Z2 )
        = ( F2 @ Z2 ) ) ) ).

% fun_upd_other
thf(fact_295_fun__upd__twist,axiom,
    ! [A: d,C: d,M: d > c,B: c,D: c] :
      ( ( A != C )
     => ( ( fun_upd_d_c @ ( fun_upd_d_c @ M @ A @ B ) @ C @ D )
        = ( fun_upd_d_c @ ( fun_upd_d_c @ M @ C @ D ) @ A @ B ) ) ) ).

% fun_upd_twist
thf(fact_296_prod_Oinject,axiom,
    ! [X1: product_prod_a_d_c,X22: option2362401199305441953_a_d_c,Y1: product_prod_a_d_c,Y2: option2362401199305441953_a_d_c] :
      ( ( ( produc5208860900648697099_a_d_c @ X1 @ X22 )
        = ( produc5208860900648697099_a_d_c @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_297_prod_Oinject,axiom,
    ! [X1: a,X22: d > c,Y1: a,Y2: d > c] :
      ( ( ( product_Pair_a_d_c @ X1 @ X22 )
        = ( product_Pair_a_d_c @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_298_old_Oprod_Oinject,axiom,
    ! [A: product_prod_a_d_c,B: option2362401199305441953_a_d_c,A5: product_prod_a_d_c,B4: option2362401199305441953_a_d_c] :
      ( ( ( produc5208860900648697099_a_d_c @ A @ B )
        = ( produc5208860900648697099_a_d_c @ A5 @ B4 ) )
      = ( ( A = A5 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_299_old_Oprod_Oinject,axiom,
    ! [A: a,B: d > c,A5: a,B4: d > c] :
      ( ( ( product_Pair_a_d_c @ A @ B )
        = ( product_Pair_a_d_c @ A5 @ B4 ) )
      = ( ( A = A5 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_300_old_Oprod_Oexhaust,axiom,
    ! [Y: produc6150846815813599699_a_d_c] :
      ~ ! [A4: product_prod_a_d_c,B5: option2362401199305441953_a_d_c] :
          ( Y
         != ( produc5208860900648697099_a_d_c @ A4 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_301_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_a_d_c] :
      ~ ! [A4: a,B5: d > c] :
          ( Y
         != ( product_Pair_a_d_c @ A4 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_302_Pair__inject,axiom,
    ! [A: product_prod_a_d_c,B: option2362401199305441953_a_d_c,A5: product_prod_a_d_c,B4: option2362401199305441953_a_d_c] :
      ( ( ( produc5208860900648697099_a_d_c @ A @ B )
        = ( produc5208860900648697099_a_d_c @ A5 @ B4 ) )
     => ~ ( ( A = A5 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_303_Pair__inject,axiom,
    ! [A: a,B: d > c,A5: a,B4: d > c] :
      ( ( ( product_Pair_a_d_c @ A @ B )
        = ( product_Pair_a_d_c @ A5 @ B4 ) )
     => ~ ( ( A = A5 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_304_prod__cases,axiom,
    ! [P2: produc6150846815813599699_a_d_c > $o,P: produc6150846815813599699_a_d_c] :
      ( ! [A4: product_prod_a_d_c,B5: option2362401199305441953_a_d_c] : ( P2 @ ( produc5208860900648697099_a_d_c @ A4 @ B5 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_305_prod__cases,axiom,
    ! [P2: product_prod_a_d_c > $o,P: product_prod_a_d_c] :
      ( ! [A4: a,B5: d > c] : ( P2 @ ( product_Pair_a_d_c @ A4 @ B5 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_306_surj__pair,axiom,
    ! [P: produc6150846815813599699_a_d_c] :
    ? [X2: product_prod_a_d_c,Y4: option2362401199305441953_a_d_c] :
      ( P
      = ( produc5208860900648697099_a_d_c @ X2 @ Y4 ) ) ).

% surj_pair
thf(fact_307_surj__pair,axiom,
    ! [P: product_prod_a_d_c] :
    ? [X2: a,Y4: d > c] :
      ( P
      = ( product_Pair_a_d_c @ X2 @ Y4 ) ) ).

% surj_pair
thf(fact_308_ssubst__Pair__rhs,axiom,
    ! [R2: product_prod_a_d_c,S4: option2362401199305441953_a_d_c,R4: set_Pr7868159745199425715_a_d_c,S7: option2362401199305441953_a_d_c] :
      ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ R2 @ S4 ) @ R4 )
     => ( ( S7 = S4 )
       => ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ R2 @ S7 ) @ R4 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_309_ssubst__Pair__rhs,axiom,
    ! [R2: a,S4: d > c,R4: set_Pr5506164808955685639_a_d_c,S7: d > c] :
      ( ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ R2 @ S4 ) @ R4 )
     => ( ( S7 = S4 )
       => ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ R2 @ S7 ) @ R4 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_310_option_Ocollapse,axiom,
    ! [Option: option2362401199305441953_a_d_c] :
      ( ( Option != none_P1533316314068730928_a_d_c )
     => ( ( some_P377817780860425132_a_d_c @ ( the_Pr6609529173119848253_a_d_c @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_311_option_Ocollapse,axiom,
    ! [Option: option_a] :
      ( ( Option != none_a )
     => ( ( some_a @ ( the_a @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_312_curryI,axiom,
    ! [F2: produc6150846815813599699_a_d_c > $o,A: product_prod_a_d_c,B: option2362401199305441953_a_d_c] :
      ( ( F2 @ ( produc5208860900648697099_a_d_c @ A @ B ) )
     => ( produc7912897043459099545_d_c_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_313_curryI,axiom,
    ! [F2: product_prod_a_d_c > $o,A: a,B: d > c] :
      ( ( F2 @ ( product_Pair_a_d_c @ A @ B ) )
     => ( produc8996622012541734719_d_c_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_314_option_Osel,axiom,
    ! [X22: product_prod_a_d_c] :
      ( ( the_Pr6609529173119848253_a_d_c @ ( some_P377817780860425132_a_d_c @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_315_option_Osel,axiom,
    ! [X22: a] :
      ( ( the_a @ ( some_a @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_316_option_Oexpand,axiom,
    ! [Option: option_a,Option2: option_a] :
      ( ( ( Option = none_a )
        = ( Option2 = none_a ) )
     => ( ( ( Option != none_a )
         => ( ( Option2 != none_a )
           => ( ( the_a @ Option )
              = ( the_a @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_317_curryD,axiom,
    ! [F2: produc6150846815813599699_a_d_c > $o,A: product_prod_a_d_c,B: option2362401199305441953_a_d_c] :
      ( ( produc7912897043459099545_d_c_o @ F2 @ A @ B )
     => ( F2 @ ( produc5208860900648697099_a_d_c @ A @ B ) ) ) ).

% curryD
thf(fact_318_curryD,axiom,
    ! [F2: product_prod_a_d_c > $o,A: a,B: d > c] :
      ( ( produc8996622012541734719_d_c_o @ F2 @ A @ B )
     => ( F2 @ ( product_Pair_a_d_c @ A @ B ) ) ) ).

% curryD
thf(fact_319_curryE,axiom,
    ! [F2: produc6150846815813599699_a_d_c > $o,A: product_prod_a_d_c,B: option2362401199305441953_a_d_c] :
      ( ( produc7912897043459099545_d_c_o @ F2 @ A @ B )
     => ( F2 @ ( produc5208860900648697099_a_d_c @ A @ B ) ) ) ).

% curryE
thf(fact_320_curryE,axiom,
    ! [F2: product_prod_a_d_c > $o,A: a,B: d > c] :
      ( ( produc8996622012541734719_d_c_o @ F2 @ A @ B )
     => ( F2 @ ( product_Pair_a_d_c @ A @ B ) ) ) ).

% curryE
thf(fact_321_option_Oexhaust__sel,axiom,
    ! [Option: option2362401199305441953_a_d_c] :
      ( ( Option != none_P1533316314068730928_a_d_c )
     => ( Option
        = ( some_P377817780860425132_a_d_c @ ( the_Pr6609529173119848253_a_d_c @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_322_option_Oexhaust__sel,axiom,
    ! [Option: option_a] :
      ( ( Option != none_a )
     => ( Option
        = ( some_a @ ( the_a @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_323_option_Osplit__sel__asm,axiom,
    ! [P2: $o > $o,F1: $o,F22: a > $o,Option: option_a] :
      ( ( P2 @ ( case_option_o_a @ F1 @ F22 @ Option ) )
      = ( ~ ( ( ( Option = none_a )
              & ~ ( P2 @ F1 ) )
            | ( ( Option
                = ( some_a @ ( the_a @ Option ) ) )
              & ~ ( P2 @ ( F22 @ ( the_a @ Option ) ) ) ) ) ) ) ).

% option.split_sel_asm
thf(fact_324_option_Osplit__sel,axiom,
    ! [P2: $o > $o,F1: $o,F22: a > $o,Option: option_a] :
      ( ( P2 @ ( case_option_o_a @ F1 @ F22 @ Option ) )
      = ( ( ( Option = none_a )
         => ( P2 @ F1 ) )
        & ( ( Option
            = ( some_a @ ( the_a @ Option ) ) )
         => ( P2 @ ( F22 @ ( the_a @ Option ) ) ) ) ) ) ).

% option.split_sel
thf(fact_325_map__add__upd__left,axiom,
    ! [M: a,E2: a > option_a,E1: a > option_a,U1: a] :
      ( ~ ( member_a @ M @ ( dom_a_a @ E2 ) )
     => ( ( map_add_a_a @ ( fun_upd_a_option_a @ E1 @ M @ ( some_a @ U1 ) ) @ E2 )
        = ( fun_upd_a_option_a @ ( map_add_a_a @ E1 @ E2 ) @ M @ ( some_a @ U1 ) ) ) ) ).

% map_add_upd_left
thf(fact_326_map__add__upd__left,axiom,
    ! [M: produc6150846815813599699_a_d_c,E2: produc6150846815813599699_a_d_c > option_a,E1: produc6150846815813599699_a_d_c > option_a,U1: a] :
      ( ~ ( member2052822272342364412_a_d_c @ M @ ( dom_Pr1451487535771488445_d_c_a @ E2 ) )
     => ( ( map_ad6461168807305607968_d_c_a @ ( fun_up4051926500471473070tion_a @ E1 @ M @ ( some_a @ U1 ) ) @ E2 )
        = ( fun_up4051926500471473070tion_a @ ( map_ad6461168807305607968_d_c_a @ E1 @ E2 ) @ M @ ( some_a @ U1 ) ) ) ) ).

% map_add_upd_left
thf(fact_327_map__add__upd__left,axiom,
    ! [M: option_a,E2: option_a > option_a,E1: option_a > option_a,U1: a] :
      ( ~ ( member_option_a @ M @ ( dom_option_a_a @ E2 ) )
     => ( ( map_add_option_a_a @ ( fun_up1079276522633388797tion_a @ E1 @ M @ ( some_a @ U1 ) ) @ E2 )
        = ( fun_up1079276522633388797tion_a @ ( map_add_option_a_a @ E1 @ E2 ) @ M @ ( some_a @ U1 ) ) ) ) ).

% map_add_upd_left
thf(fact_328_map__add__upd__left,axiom,
    ! [M: a,E2: a > option2362401199305441953_a_d_c,E1: a > option2362401199305441953_a_d_c,U1: product_prod_a_d_c] :
      ( ~ ( member_a @ M @ ( dom_a_2823175246211601021_a_d_c @ E2 ) )
     => ( ( map_ad1297432485515382362_a_d_c @ ( fun_up2590722781756628834_a_d_c @ E1 @ M @ ( some_P377817780860425132_a_d_c @ U1 ) ) @ E2 )
        = ( fun_up2590722781756628834_a_d_c @ ( map_ad1297432485515382362_a_d_c @ E1 @ E2 ) @ M @ ( some_P377817780860425132_a_d_c @ U1 ) ) ) ) ).

% map_add_upd_left
thf(fact_329_map__add__upd__left,axiom,
    ! [M: produc6150846815813599699_a_d_c,E2: produc6150846815813599699_a_d_c > option2362401199305441953_a_d_c,E1: produc6150846815813599699_a_d_c > option2362401199305441953_a_d_c,U1: product_prod_a_d_c] :
      ( ~ ( member2052822272342364412_a_d_c @ M @ ( dom_Pr4211630066744544850_a_d_c @ E2 ) )
     => ( ( map_ad6660179081546580527_a_d_c @ ( fun_up4469873178457468855_a_d_c @ E1 @ M @ ( some_P377817780860425132_a_d_c @ U1 ) ) @ E2 )
        = ( fun_up4469873178457468855_a_d_c @ ( map_ad6660179081546580527_a_d_c @ E1 @ E2 ) @ M @ ( some_P377817780860425132_a_d_c @ U1 ) ) ) ) ).

% map_add_upd_left
thf(fact_330_map__add__upd__left,axiom,
    ! [M: option_a,E2: option_a > option2362401199305441953_a_d_c,E1: option_a > option2362401199305441953_a_d_c,U1: product_prod_a_d_c] :
      ( ~ ( member_option_a @ M @ ( dom_op5450096724005979971_a_d_c @ E2 ) )
     => ( ( map_ad3059403310785688480_a_d_c @ ( fun_up1917685652830187432_a_d_c @ E1 @ M @ ( some_P377817780860425132_a_d_c @ U1 ) ) @ E2 )
        = ( fun_up1917685652830187432_a_d_c @ ( map_ad3059403310785688480_a_d_c @ E1 @ E2 ) @ M @ ( some_P377817780860425132_a_d_c @ U1 ) ) ) ) ).

% map_add_upd_left
thf(fact_331_option_Ocase__eq__if,axiom,
    ( case_option_o_a
    = ( ^ [F12: $o,F23: a > $o,Option3: option_a] :
          ( ( ( Option3 = none_a )
           => F12 )
          & ( ( Option3 != none_a )
           => ( F23 @ ( the_a @ Option3 ) ) ) ) ) ) ).

% option.case_eq_if
thf(fact_332_option_Oset__sel,axiom,
    ! [A: option4105738954736457113_a_d_c] :
      ( ( A != none_P7581502224897721140_a_d_c )
     => ( member2052822272342364412_a_d_c @ ( the_Pr308243380479556199_a_d_c @ A ) @ ( set_op2111387851716960097_a_d_c @ A ) ) ) ).

% option.set_sel
thf(fact_333_option_Oset__sel,axiom,
    ! [A: option_option_a] :
      ( ( A != none_option_a )
     => ( member_option_a @ ( the_option_a @ A ) @ ( set_option_option_a2 @ A ) ) ) ).

% option.set_sel
thf(fact_334_option_Oset__sel,axiom,
    ! [A: option_a] :
      ( ( A != none_a )
     => ( member_a @ ( the_a @ A ) @ ( set_option_a3 @ A ) ) ) ).

% option.set_sel
thf(fact_335_bind__eq__None__conv,axiom,
    ! [A: option_a,F2: a > option_a] :
      ( ( ( bind_a_a @ A @ F2 )
        = none_a )
      = ( ( A = none_a )
        | ( ( F2 @ ( the_a @ A ) )
          = none_a ) ) ) ).

% bind_eq_None_conv
thf(fact_336_elem__set,axiom,
    ! [X3: produc6150846815813599699_a_d_c,Xo: option4105738954736457113_a_d_c] :
      ( ( member2052822272342364412_a_d_c @ X3 @ ( set_op2111387851716960097_a_d_c @ Xo ) )
      = ( Xo
        = ( some_P8437211288446953144_a_d_c @ X3 ) ) ) ).

% elem_set
thf(fact_337_elem__set,axiom,
    ! [X3: option_a,Xo: option_option_a] :
      ( ( member_option_a @ X3 @ ( set_option_option_a2 @ Xo ) )
      = ( Xo
        = ( some_option_a @ X3 ) ) ) ).

% elem_set
thf(fact_338_elem__set,axiom,
    ! [X3: product_prod_a_d_c,Xo: option2362401199305441953_a_d_c] :
      ( ( member8140300833826434152_a_d_c @ X3 @ ( set_op6310949527382764995_a_d_c @ Xo ) )
      = ( Xo
        = ( some_P377817780860425132_a_d_c @ X3 ) ) ) ).

% elem_set
thf(fact_339_elem__set,axiom,
    ! [X3: a,Xo: option_a] :
      ( ( member_a @ X3 @ ( set_option_a3 @ Xo ) )
      = ( Xo
        = ( some_a @ X3 ) ) ) ).

% elem_set
thf(fact_340_bind__runit,axiom,
    ! [X3: option2362401199305441953_a_d_c] :
      ( ( bind_P7034654483831013730_a_d_c @ X3 @ some_P377817780860425132_a_d_c )
      = X3 ) ).

% bind_runit
thf(fact_341_bind__runit,axiom,
    ! [X3: option_a] :
      ( ( bind_a_a @ X3 @ some_a )
      = X3 ) ).

% bind_runit
thf(fact_342_fun__upd__None__if__notin__dom,axiom,
    ! [K: a,M: a > option_a] :
      ( ~ ( member_a @ K @ ( dom_a_a @ M ) )
     => ( ( fun_upd_a_option_a @ M @ K @ none_a )
        = M ) ) ).

% fun_upd_None_if_notin_dom
thf(fact_343_fun__upd__None__if__notin__dom,axiom,
    ! [K: produc6150846815813599699_a_d_c,M: produc6150846815813599699_a_d_c > option_a] :
      ( ~ ( member2052822272342364412_a_d_c @ K @ ( dom_Pr1451487535771488445_d_c_a @ M ) )
     => ( ( fun_up4051926500471473070tion_a @ M @ K @ none_a )
        = M ) ) ).

% fun_upd_None_if_notin_dom
thf(fact_344_fun__upd__None__if__notin__dom,axiom,
    ! [K: option_a,M: option_a > option_a] :
      ( ~ ( member_option_a @ K @ ( dom_option_a_a @ M ) )
     => ( ( fun_up1079276522633388797tion_a @ M @ K @ none_a )
        = M ) ) ).

% fun_upd_None_if_notin_dom
thf(fact_345_bind__option__cong,axiom,
    ! [X3: option_a,Y: option_a,F2: a > option_a,G: a > option_a] :
      ( ( X3 = Y )
     => ( ! [Z3: a] :
            ( ( member_a @ Z3 @ ( set_option_a3 @ Y ) )
           => ( ( F2 @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( bind_a_a @ X3 @ F2 )
          = ( bind_a_a @ Y @ G ) ) ) ) ).

% bind_option_cong
thf(fact_346_bind__option__cong__code,axiom,
    ! [X3: option_a,Y: option_a,F2: a > option_a] :
      ( ( X3 = Y )
     => ( ( bind_a_a @ X3 @ F2 )
        = ( bind_a_a @ Y @ F2 ) ) ) ).

% bind_option_cong_code
thf(fact_347_domD,axiom,
    ! [A: a,M: a > option_a] :
      ( ( member_a @ A @ ( dom_a_a @ M ) )
     => ? [B5: a] :
          ( ( M @ A )
          = ( some_a @ B5 ) ) ) ).

% domD
thf(fact_348_domD,axiom,
    ! [A: produc6150846815813599699_a_d_c,M: produc6150846815813599699_a_d_c > option_a] :
      ( ( member2052822272342364412_a_d_c @ A @ ( dom_Pr1451487535771488445_d_c_a @ M ) )
     => ? [B5: a] :
          ( ( M @ A )
          = ( some_a @ B5 ) ) ) ).

% domD
thf(fact_349_domD,axiom,
    ! [A: option_a,M: option_a > option_a] :
      ( ( member_option_a @ A @ ( dom_option_a_a @ M ) )
     => ? [B5: a] :
          ( ( M @ A )
          = ( some_a @ B5 ) ) ) ).

% domD
thf(fact_350_domD,axiom,
    ! [A: a,M: a > option2362401199305441953_a_d_c] :
      ( ( member_a @ A @ ( dom_a_2823175246211601021_a_d_c @ M ) )
     => ? [B5: product_prod_a_d_c] :
          ( ( M @ A )
          = ( some_P377817780860425132_a_d_c @ B5 ) ) ) ).

% domD
thf(fact_351_domD,axiom,
    ! [A: produc6150846815813599699_a_d_c,M: produc6150846815813599699_a_d_c > option2362401199305441953_a_d_c] :
      ( ( member2052822272342364412_a_d_c @ A @ ( dom_Pr4211630066744544850_a_d_c @ M ) )
     => ? [B5: product_prod_a_d_c] :
          ( ( M @ A )
          = ( some_P377817780860425132_a_d_c @ B5 ) ) ) ).

% domD
thf(fact_352_domD,axiom,
    ! [A: option_a,M: option_a > option2362401199305441953_a_d_c] :
      ( ( member_option_a @ A @ ( dom_op5450096724005979971_a_d_c @ M ) )
     => ? [B5: product_prod_a_d_c] :
          ( ( M @ A )
          = ( some_P377817780860425132_a_d_c @ B5 ) ) ) ).

% domD
thf(fact_353_domI,axiom,
    ! [M: a > option_a,A: a,B: a] :
      ( ( ( M @ A )
        = ( some_a @ B ) )
     => ( member_a @ A @ ( dom_a_a @ M ) ) ) ).

% domI
thf(fact_354_domI,axiom,
    ! [M: produc6150846815813599699_a_d_c > option_a,A: produc6150846815813599699_a_d_c,B: a] :
      ( ( ( M @ A )
        = ( some_a @ B ) )
     => ( member2052822272342364412_a_d_c @ A @ ( dom_Pr1451487535771488445_d_c_a @ M ) ) ) ).

% domI
thf(fact_355_domI,axiom,
    ! [M: option_a > option_a,A: option_a,B: a] :
      ( ( ( M @ A )
        = ( some_a @ B ) )
     => ( member_option_a @ A @ ( dom_option_a_a @ M ) ) ) ).

% domI
thf(fact_356_domI,axiom,
    ! [M: a > option2362401199305441953_a_d_c,A: a,B: product_prod_a_d_c] :
      ( ( ( M @ A )
        = ( some_P377817780860425132_a_d_c @ B ) )
     => ( member_a @ A @ ( dom_a_2823175246211601021_a_d_c @ M ) ) ) ).

% domI
thf(fact_357_domI,axiom,
    ! [M: produc6150846815813599699_a_d_c > option2362401199305441953_a_d_c,A: produc6150846815813599699_a_d_c,B: product_prod_a_d_c] :
      ( ( ( M @ A )
        = ( some_P377817780860425132_a_d_c @ B ) )
     => ( member2052822272342364412_a_d_c @ A @ ( dom_Pr4211630066744544850_a_d_c @ M ) ) ) ).

% domI
thf(fact_358_domI,axiom,
    ! [M: option_a > option2362401199305441953_a_d_c,A: option_a,B: product_prod_a_d_c] :
      ( ( ( M @ A )
        = ( some_P377817780860425132_a_d_c @ B ) )
     => ( member_option_a @ A @ ( dom_op5450096724005979971_a_d_c @ M ) ) ) ).

% domI
thf(fact_359_domIff,axiom,
    ! [A: a,M: a > option_a] :
      ( ( member_a @ A @ ( dom_a_a @ M ) )
      = ( ( M @ A )
       != none_a ) ) ).

% domIff
thf(fact_360_domIff,axiom,
    ! [A: produc6150846815813599699_a_d_c,M: produc6150846815813599699_a_d_c > option_a] :
      ( ( member2052822272342364412_a_d_c @ A @ ( dom_Pr1451487535771488445_d_c_a @ M ) )
      = ( ( M @ A )
       != none_a ) ) ).

% domIff
thf(fact_361_domIff,axiom,
    ! [A: option_a,M: option_a > option_a] :
      ( ( member_option_a @ A @ ( dom_option_a_a @ M ) )
      = ( ( M @ A )
       != none_a ) ) ).

% domIff
thf(fact_362_bind__eq__Some__conv,axiom,
    ! [F2: option2362401199305441953_a_d_c,G: product_prod_a_d_c > option_a,X3: a] :
      ( ( ( bind_P6705688431498269101_d_c_a @ F2 @ G )
        = ( some_a @ X3 ) )
      = ( ? [Y3: product_prod_a_d_c] :
            ( ( F2
              = ( some_P377817780860425132_a_d_c @ Y3 ) )
            & ( ( G @ Y3 )
              = ( some_a @ X3 ) ) ) ) ) ).

% bind_eq_Some_conv
thf(fact_363_bind__eq__Some__conv,axiom,
    ! [F2: option_a,G: a > option2362401199305441953_a_d_c,X3: product_prod_a_d_c] :
      ( ( ( bind_a3891568204378968849_a_d_c @ F2 @ G )
        = ( some_P377817780860425132_a_d_c @ X3 ) )
      = ( ? [Y3: a] :
            ( ( F2
              = ( some_a @ Y3 ) )
            & ( ( G @ Y3 )
              = ( some_P377817780860425132_a_d_c @ X3 ) ) ) ) ) ).

% bind_eq_Some_conv
thf(fact_364_bind__eq__Some__conv,axiom,
    ! [F2: option2362401199305441953_a_d_c,G: product_prod_a_d_c > option2362401199305441953_a_d_c,X3: product_prod_a_d_c] :
      ( ( ( bind_P7034654483831013730_a_d_c @ F2 @ G )
        = ( some_P377817780860425132_a_d_c @ X3 ) )
      = ( ? [Y3: product_prod_a_d_c] :
            ( ( F2
              = ( some_P377817780860425132_a_d_c @ Y3 ) )
            & ( ( G @ Y3 )
              = ( some_P377817780860425132_a_d_c @ X3 ) ) ) ) ) ).

% bind_eq_Some_conv
thf(fact_365_bind__eq__Some__conv,axiom,
    ! [F2: option_a,G: a > option_a,X3: a] :
      ( ( ( bind_a_a @ F2 @ G )
        = ( some_a @ X3 ) )
      = ( ? [Y3: a] :
            ( ( F2
              = ( some_a @ Y3 ) )
            & ( ( G @ Y3 )
              = ( some_a @ X3 ) ) ) ) ) ).

% bind_eq_Some_conv
thf(fact_366_Option_Obind__cong,axiom,
    ! [X3: option_a,Y: option_a,F2: a > option_a,G: a > option_a] :
      ( ( X3 = Y )
     => ( ! [A4: a] :
            ( ( Y
              = ( some_a @ A4 ) )
           => ( ( F2 @ A4 )
              = ( G @ A4 ) ) )
       => ( ( bind_a_a @ X3 @ F2 )
          = ( bind_a_a @ Y @ G ) ) ) ) ).

% Option.bind_cong
thf(fact_367_bind_Obind__lunit,axiom,
    ! [X3: a,F2: a > option_a] :
      ( ( bind_a_a @ ( some_a @ X3 ) @ F2 )
      = ( F2 @ X3 ) ) ).

% bind.bind_lunit
thf(fact_368_bind_Obind__lzero,axiom,
    ! [F2: a > option_a] :
      ( ( bind_a_a @ none_a @ F2 )
      = none_a ) ).

% bind.bind_lzero
thf(fact_369_ospec,axiom,
    ! [A2: option2362401199305441953_a_d_c,P2: product_prod_a_d_c > $o,X3: product_prod_a_d_c] :
      ( ! [X2: product_prod_a_d_c] :
          ( ( member8140300833826434152_a_d_c @ X2 @ ( set_op6310949527382764995_a_d_c @ A2 ) )
         => ( P2 @ X2 ) )
     => ( ( A2
          = ( some_P377817780860425132_a_d_c @ X3 ) )
       => ( P2 @ X3 ) ) ) ).

% ospec
thf(fact_370_ospec,axiom,
    ! [A2: option_a,P2: a > $o,X3: a] :
      ( ! [X2: a] :
          ( ( member_a @ X2 @ ( set_option_a3 @ A2 ) )
         => ( P2 @ X2 ) )
     => ( ( A2
          = ( some_a @ X3 ) )
       => ( P2 @ X3 ) ) ) ).

% ospec
thf(fact_371_option_Oset__intros,axiom,
    ! [X22: produc6150846815813599699_a_d_c] : ( member2052822272342364412_a_d_c @ X22 @ ( set_op2111387851716960097_a_d_c @ ( some_P8437211288446953144_a_d_c @ X22 ) ) ) ).

% option.set_intros
thf(fact_372_option_Oset__intros,axiom,
    ! [X22: option_a] : ( member_option_a @ X22 @ ( set_option_option_a2 @ ( some_option_a @ X22 ) ) ) ).

% option.set_intros
thf(fact_373_option_Oset__intros,axiom,
    ! [X22: product_prod_a_d_c] : ( member8140300833826434152_a_d_c @ X22 @ ( set_op6310949527382764995_a_d_c @ ( some_P377817780860425132_a_d_c @ X22 ) ) ) ).

% option.set_intros
thf(fact_374_option_Oset__intros,axiom,
    ! [X22: a] : ( member_a @ X22 @ ( set_option_a3 @ ( some_a @ X22 ) ) ) ).

% option.set_intros
thf(fact_375_option_Oset__cases,axiom,
    ! [E: produc6150846815813599699_a_d_c,A: option4105738954736457113_a_d_c] :
      ( ( member2052822272342364412_a_d_c @ E @ ( set_op2111387851716960097_a_d_c @ A ) )
     => ( A
        = ( some_P8437211288446953144_a_d_c @ E ) ) ) ).

% option.set_cases
thf(fact_376_option_Oset__cases,axiom,
    ! [E: option_a,A: option_option_a] :
      ( ( member_option_a @ E @ ( set_option_option_a2 @ A ) )
     => ( A
        = ( some_option_a @ E ) ) ) ).

% option.set_cases
thf(fact_377_option_Oset__cases,axiom,
    ! [E: product_prod_a_d_c,A: option2362401199305441953_a_d_c] :
      ( ( member8140300833826434152_a_d_c @ E @ ( set_op6310949527382764995_a_d_c @ A ) )
     => ( A
        = ( some_P377817780860425132_a_d_c @ E ) ) ) ).

% option.set_cases
thf(fact_378_option_Oset__cases,axiom,
    ! [E: a,A: option_a] :
      ( ( member_a @ E @ ( set_option_a3 @ A ) )
     => ( A
        = ( some_a @ E ) ) ) ).

% option.set_cases
thf(fact_379_option_Osimps_I5_J,axiom,
    ! [F1: $o,F22: a > $o,X22: a] :
      ( ( case_option_o_a @ F1 @ F22 @ ( some_a @ X22 ) )
      = ( F22 @ X22 ) ) ).

% option.simps(5)
thf(fact_380_option_Osimps_I4_J,axiom,
    ! [F1: $o,F22: a > $o] :
      ( ( case_option_o_a @ F1 @ F22 @ none_a )
      = F1 ) ).

% option.simps(4)
thf(fact_381_bind__split,axiom,
    ! [P2: option_a > $o,M: option2362401199305441953_a_d_c,F2: product_prod_a_d_c > option_a] :
      ( ( P2 @ ( bind_P6705688431498269101_d_c_a @ M @ F2 ) )
      = ( ( ( M = none_P1533316314068730928_a_d_c )
         => ( P2 @ none_a ) )
        & ! [V: product_prod_a_d_c] :
            ( ( M
              = ( some_P377817780860425132_a_d_c @ V ) )
           => ( P2 @ ( F2 @ V ) ) ) ) ) ).

% bind_split
thf(fact_382_bind__split,axiom,
    ! [P2: option_a > $o,M: option_a,F2: a > option_a] :
      ( ( P2 @ ( bind_a_a @ M @ F2 ) )
      = ( ( ( M = none_a )
         => ( P2 @ none_a ) )
        & ! [V: a] :
            ( ( M
              = ( some_a @ V ) )
           => ( P2 @ ( F2 @ V ) ) ) ) ) ).

% bind_split
thf(fact_383_bind__split__asm,axiom,
    ! [P2: option_a > $o,M: option2362401199305441953_a_d_c,F2: product_prod_a_d_c > option_a] :
      ( ( P2 @ ( bind_P6705688431498269101_d_c_a @ M @ F2 ) )
      = ( ~ ( ( ( M = none_P1533316314068730928_a_d_c )
              & ~ ( P2 @ none_a ) )
            | ? [X: product_prod_a_d_c] :
                ( ( M
                  = ( some_P377817780860425132_a_d_c @ X ) )
                & ~ ( P2 @ ( F2 @ X ) ) ) ) ) ) ).

% bind_split_asm
thf(fact_384_bind__split__asm,axiom,
    ! [P2: option_a > $o,M: option_a,F2: a > option_a] :
      ( ( P2 @ ( bind_a_a @ M @ F2 ) )
      = ( ~ ( ( ( M = none_a )
              & ~ ( P2 @ none_a ) )
            | ? [X: a] :
                ( ( M
                  = ( some_a @ X ) )
                & ~ ( P2 @ ( F2 @ X ) ) ) ) ) ) ).

% bind_split_asm
thf(fact_385_is__none__bind,axiom,
    ! [F2: option_a,G: a > option_a] :
      ( ( is_none_a @ ( bind_a_a @ F2 @ G ) )
      = ( ( is_none_a @ F2 )
        | ( is_none_a @ ( G @ ( the_a @ F2 ) ) ) ) ) ).

% is_none_bind
thf(fact_386_set__empty__eq,axiom,
    ! [Xo: option_option_a] :
      ( ( ( set_option_option_a2 @ Xo )
        = bot_bot_set_option_a )
      = ( Xo = none_option_a ) ) ).

% set_empty_eq
thf(fact_387_set__empty__eq,axiom,
    ! [Xo: option_a] :
      ( ( ( set_option_a3 @ Xo )
        = bot_bot_set_a )
      = ( Xo = none_a ) ) ).

% set_empty_eq
thf(fact_388_dom__eq__empty__conv,axiom,
    ! [F2: a > option_a] :
      ( ( ( dom_a_a @ F2 )
        = bot_bot_set_a )
      = ( F2
        = ( ^ [X: a] : none_a ) ) ) ).

% dom_eq_empty_conv
thf(fact_389_dom__eq__empty__conv,axiom,
    ! [F2: option_a > option_a] :
      ( ( ( dom_option_a_a @ F2 )
        = bot_bot_set_option_a )
      = ( F2
        = ( ^ [X: option_a] : none_a ) ) ) ).

% dom_eq_empty_conv
thf(fact_390_in__graphI,axiom,
    ! [M: product_prod_a_d_c > option5918686196580481265_a_d_c,K: product_prod_a_d_c,V3: option2362401199305441953_a_d_c] :
      ( ( ( M @ K )
        = ( some_o4835035200596650236_a_d_c @ V3 ) )
     => ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ K @ V3 ) @ ( graph_4360052036846219122_a_d_c @ M ) ) ) ).

% in_graphI
thf(fact_391_in__graphI,axiom,
    ! [M: a > option_d_c,K: a,V3: d > c] :
      ( ( ( M @ K )
        = ( some_d_c @ V3 ) )
     => ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ K @ V3 ) @ ( graph_a_d_c @ M ) ) ) ).

% in_graphI
thf(fact_392_is__none__code_I2_J,axiom,
    ! [X3: product_prod_a_d_c] :
      ~ ( is_non668464627475284981_a_d_c @ ( some_P377817780860425132_a_d_c @ X3 ) ) ).

% is_none_code(2)
thf(fact_393_is__none__code_I2_J,axiom,
    ! [X3: a] :
      ~ ( is_none_a @ ( some_a @ X3 ) ) ).

% is_none_code(2)
thf(fact_394_is__none__code_I1_J,axiom,
    is_none_a @ none_a ).

% is_none_code(1)
thf(fact_395_is__none__simps_I2_J,axiom,
    ! [X3: product_prod_a_d_c] :
      ~ ( is_non668464627475284981_a_d_c @ ( some_P377817780860425132_a_d_c @ X3 ) ) ).

% is_none_simps(2)
thf(fact_396_is__none__simps_I2_J,axiom,
    ! [X3: a] :
      ~ ( is_none_a @ ( some_a @ X3 ) ) ).

% is_none_simps(2)
thf(fact_397_is__none__simps_I1_J,axiom,
    is_none_a @ none_a ).

% is_none_simps(1)
thf(fact_398_Option_Ois__none__def,axiom,
    ( is_none_a
    = ( ^ [X: option_a] : ( X = none_a ) ) ) ).

% Option.is_none_def
thf(fact_399_option_Osimps_I14_J,axiom,
    ( ( set_option_option_a2 @ none_option_a )
    = bot_bot_set_option_a ) ).

% option.simps(14)
thf(fact_400_option_Osimps_I14_J,axiom,
    ( ( set_option_a3 @ none_a )
    = bot_bot_set_a ) ).

% option.simps(14)
thf(fact_401_in__graphD,axiom,
    ! [K: product_prod_a_d_c,V3: option2362401199305441953_a_d_c,M: product_prod_a_d_c > option5918686196580481265_a_d_c] :
      ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ K @ V3 ) @ ( graph_4360052036846219122_a_d_c @ M ) )
     => ( ( M @ K )
        = ( some_o4835035200596650236_a_d_c @ V3 ) ) ) ).

% in_graphD
thf(fact_402_in__graphD,axiom,
    ! [K: a,V3: d > c,M: a > option_d_c] :
      ( ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ K @ V3 ) @ ( graph_a_d_c @ M ) )
     => ( ( M @ K )
        = ( some_d_c @ V3 ) ) ) ).

% in_graphD
thf(fact_403_graph__map__upd,axiom,
    ! [M: product_prod_a_d_c > option5918686196580481265_a_d_c,K: product_prod_a_d_c,V3: option2362401199305441953_a_d_c] :
      ( ( graph_4360052036846219122_a_d_c @ ( fun_up7471186249482589761_a_d_c @ M @ K @ ( some_o4835035200596650236_a_d_c @ V3 ) ) )
      = ( insert4944988292672327587_a_d_c @ ( produc5208860900648697099_a_d_c @ K @ V3 ) @ ( graph_4360052036846219122_a_d_c @ ( fun_up7471186249482589761_a_d_c @ M @ K @ none_o7434091334857606016_a_d_c ) ) ) ) ).

% graph_map_upd
thf(fact_404_graph__map__upd,axiom,
    ! [M: a > option_d_c,K: a,V3: d > c] :
      ( ( graph_a_d_c @ ( fun_upd_a_option_d_c @ M @ K @ ( some_d_c @ V3 ) ) )
      = ( insert2041379952318371585_a_d_c @ ( product_Pair_a_d_c @ K @ V3 ) @ ( graph_a_d_c @ ( fun_upd_a_option_d_c @ M @ K @ none_d_c ) ) ) ) ).

% graph_map_upd
thf(fact_405_graph__restrictD_I2_J,axiom,
    ! [K: product_prod_a_d_c,V3: option2362401199305441953_a_d_c,M: product_prod_a_d_c > option5918686196580481265_a_d_c,A2: set_Pr5506164808955685639_a_d_c] :
      ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ K @ V3 ) @ ( graph_4360052036846219122_a_d_c @ ( restri115587479578113572_a_d_c @ M @ A2 ) ) )
     => ( ( M @ K )
        = ( some_o4835035200596650236_a_d_c @ V3 ) ) ) ).

% graph_restrictD(2)
thf(fact_406_graph__restrictD_I2_J,axiom,
    ! [K: a,V3: d > c,M: a > option_d_c,A2: set_a] :
      ( ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ K @ V3 ) @ ( graph_a_d_c @ ( restrict_map_a_d_c @ M @ A2 ) ) )
     => ( ( M @ K )
        = ( some_d_c @ V3 ) ) ) ).

% graph_restrictD(2)
thf(fact_407_option_Osimps_I15_J,axiom,
    ! [X22: product_prod_a_d_c] :
      ( ( set_op6310949527382764995_a_d_c @ ( some_P377817780860425132_a_d_c @ X22 ) )
      = ( insert2041379952318371585_a_d_c @ X22 @ bot_bo5281969120672942491_a_d_c ) ) ).

% option.simps(15)
thf(fact_408_option_Osimps_I15_J,axiom,
    ! [X22: option_a] :
      ( ( set_option_option_a2 @ ( some_option_a @ X22 ) )
      = ( insert_option_a @ X22 @ bot_bot_set_option_a ) ) ).

% option.simps(15)
thf(fact_409_option_Osimps_I15_J,axiom,
    ! [X22: a] :
      ( ( set_option_a3 @ ( some_a @ X22 ) )
      = ( insert_a @ X22 @ bot_bot_set_a ) ) ).

% option.simps(15)
thf(fact_410_restrict__out,axiom,
    ! [X3: a,A2: set_a,M: a > option_a] :
      ( ~ ( member_a @ X3 @ A2 )
     => ( ( restrict_map_a_a @ M @ A2 @ X3 )
        = none_a ) ) ).

% restrict_out
thf(fact_411_restrict__out,axiom,
    ! [X3: produc6150846815813599699_a_d_c,A2: set_Pr7868159745199425715_a_d_c,M: produc6150846815813599699_a_d_c > option_a] :
      ( ~ ( member2052822272342364412_a_d_c @ X3 @ A2 )
     => ( ( restri7722463564184762267_d_c_a @ M @ A2 @ X3 )
        = none_a ) ) ).

% restrict_out
thf(fact_412_restrict__out,axiom,
    ! [X3: option_a,A2: set_option_a,M: option_a > option_a] :
      ( ~ ( member_option_a @ X3 @ A2 )
     => ( ( restri3984065703976872170on_a_a @ M @ A2 @ X3 )
        = none_a ) ) ).

% restrict_out
thf(fact_413_restrict__map__to__empty,axiom,
    ! [M: a > option_a] :
      ( ( restrict_map_a_a @ M @ bot_bot_set_a )
      = ( ^ [X: a] : none_a ) ) ).

% restrict_map_to_empty
thf(fact_414_restrict__map__to__empty,axiom,
    ! [M: option_a > option_a] :
      ( ( restri3984065703976872170on_a_a @ M @ bot_bot_set_option_a )
      = ( ^ [X: option_a] : none_a ) ) ).

% restrict_map_to_empty
thf(fact_415_restrict__map__def,axiom,
    ( restrict_map_a_a
    = ( ^ [M2: a > option_a,A7: set_a,X: a] : ( if_option_a @ ( member_a @ X @ A7 ) @ ( M2 @ X ) @ none_a ) ) ) ).

% restrict_map_def
thf(fact_416_restrict__map__def,axiom,
    ( restri7722463564184762267_d_c_a
    = ( ^ [M2: produc6150846815813599699_a_d_c > option_a,A7: set_Pr7868159745199425715_a_d_c,X: produc6150846815813599699_a_d_c] : ( if_option_a @ ( member2052822272342364412_a_d_c @ X @ A7 ) @ ( M2 @ X ) @ none_a ) ) ) ).

% restrict_map_def
thf(fact_417_restrict__map__def,axiom,
    ( restri3984065703976872170on_a_a
    = ( ^ [M2: option_a > option_a,A7: set_option_a,X: option_a] : ( if_option_a @ ( member_option_a @ X @ A7 ) @ ( M2 @ X ) @ none_a ) ) ) ).

% restrict_map_def
thf(fact_418_insert__dom,axiom,
    ! [F2: a > option_a,X3: a,Y: a] :
      ( ( ( F2 @ X3 )
        = ( some_a @ Y ) )
     => ( ( insert_a @ X3 @ ( dom_a_a @ F2 ) )
        = ( dom_a_a @ F2 ) ) ) ).

% insert_dom
thf(fact_419_insert__dom,axiom,
    ! [F2: option_a > option_a,X3: option_a,Y: a] :
      ( ( ( F2 @ X3 )
        = ( some_a @ Y ) )
     => ( ( insert_option_a @ X3 @ ( dom_option_a_a @ F2 ) )
        = ( dom_option_a_a @ F2 ) ) ) ).

% insert_dom
thf(fact_420_insert__dom,axiom,
    ! [F2: a > option2362401199305441953_a_d_c,X3: a,Y: product_prod_a_d_c] :
      ( ( ( F2 @ X3 )
        = ( some_P377817780860425132_a_d_c @ Y ) )
     => ( ( insert_a @ X3 @ ( dom_a_2823175246211601021_a_d_c @ F2 ) )
        = ( dom_a_2823175246211601021_a_d_c @ F2 ) ) ) ).

% insert_dom
thf(fact_421_insert__dom,axiom,
    ! [F2: option_a > option2362401199305441953_a_d_c,X3: option_a,Y: product_prod_a_d_c] :
      ( ( ( F2 @ X3 )
        = ( some_P377817780860425132_a_d_c @ Y ) )
     => ( ( insert_option_a @ X3 @ ( dom_op5450096724005979971_a_d_c @ F2 ) )
        = ( dom_op5450096724005979971_a_d_c @ F2 ) ) ) ).

% insert_dom
thf(fact_422_graph__restrictD_I1_J,axiom,
    ! [K: product_prod_a_d_c,V3: option2362401199305441953_a_d_c,M: product_prod_a_d_c > option5918686196580481265_a_d_c,A2: set_Pr5506164808955685639_a_d_c] :
      ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ K @ V3 ) @ ( graph_4360052036846219122_a_d_c @ ( restri115587479578113572_a_d_c @ M @ A2 ) ) )
     => ( member8140300833826434152_a_d_c @ K @ A2 ) ) ).

% graph_restrictD(1)
thf(fact_423_graph__restrictD_I1_J,axiom,
    ! [K: a,V3: d > c,M: a > option_d_c,A2: set_a] :
      ( ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ K @ V3 ) @ ( graph_a_d_c @ ( restrict_map_a_d_c @ M @ A2 ) ) )
     => ( member_a @ K @ A2 ) ) ).

% graph_restrictD(1)
thf(fact_424_restrict__upd__same,axiom,
    ! [M: a > option_a,X3: a,Y: a] :
      ( ( restrict_map_a_a @ ( fun_upd_a_option_a @ M @ X3 @ ( some_a @ Y ) ) @ ( uminus_uminus_set_a @ ( insert_a @ X3 @ bot_bot_set_a ) ) )
      = ( restrict_map_a_a @ M @ ( uminus_uminus_set_a @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).

% restrict_upd_same
thf(fact_425_restrict__upd__same,axiom,
    ! [M: a > option2362401199305441953_a_d_c,X3: a,Y: product_prod_a_d_c] :
      ( ( restri70124959536540191_a_d_c @ ( fun_up2590722781756628834_a_d_c @ M @ X3 @ ( some_P377817780860425132_a_d_c @ Y ) ) @ ( uminus_uminus_set_a @ ( insert_a @ X3 @ bot_bot_set_a ) ) )
      = ( restri70124959536540191_a_d_c @ M @ ( uminus_uminus_set_a @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).

% restrict_upd_same
thf(fact_426_restrict__upd__same,axiom,
    ! [M: option_a > option_a,X3: option_a,Y: a] :
      ( ( restri3984065703976872170on_a_a @ ( fun_up1079276522633388797tion_a @ M @ X3 @ ( some_a @ Y ) ) @ ( uminus6205308855922866075tion_a @ ( insert_option_a @ X3 @ bot_bot_set_option_a ) ) )
      = ( restri3984065703976872170on_a_a @ M @ ( uminus6205308855922866075tion_a @ ( insert_option_a @ X3 @ bot_bot_set_option_a ) ) ) ) ).

% restrict_upd_same
thf(fact_427_restrict__upd__same,axiom,
    ! [M: option_a > option2362401199305441953_a_d_c,X3: option_a,Y: product_prod_a_d_c] :
      ( ( restri8381520612010263525_a_d_c @ ( fun_up1917685652830187432_a_d_c @ M @ X3 @ ( some_P377817780860425132_a_d_c @ Y ) ) @ ( uminus6205308855922866075tion_a @ ( insert_option_a @ X3 @ bot_bot_set_option_a ) ) )
      = ( restri8381520612010263525_a_d_c @ M @ ( uminus6205308855922866075tion_a @ ( insert_option_a @ X3 @ bot_bot_set_option_a ) ) ) ) ).

% restrict_upd_same
thf(fact_428_fun__upd__None__restrict,axiom,
    ! [X3: produc6150846815813599699_a_d_c,D2: set_Pr7868159745199425715_a_d_c,M: produc6150846815813599699_a_d_c > option_a] :
      ( ( ( member2052822272342364412_a_d_c @ X3 @ D2 )
       => ( ( fun_up4051926500471473070tion_a @ ( restri7722463564184762267_d_c_a @ M @ D2 ) @ X3 @ none_a )
          = ( restri7722463564184762267_d_c_a @ M @ ( minus_1640812273454246426_a_d_c @ D2 @ ( insert4944988292672327587_a_d_c @ X3 @ bot_bo1396921257525504799_a_d_c ) ) ) ) )
      & ( ~ ( member2052822272342364412_a_d_c @ X3 @ D2 )
       => ( ( fun_up4051926500471473070tion_a @ ( restri7722463564184762267_d_c_a @ M @ D2 ) @ X3 @ none_a )
          = ( restri7722463564184762267_d_c_a @ M @ D2 ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_429_fun__upd__None__restrict,axiom,
    ! [X3: a,D2: set_a,M: a > option_a] :
      ( ( ( member_a @ X3 @ D2 )
       => ( ( fun_upd_a_option_a @ ( restrict_map_a_a @ M @ D2 ) @ X3 @ none_a )
          = ( restrict_map_a_a @ M @ ( minus_minus_set_a @ D2 @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) )
      & ( ~ ( member_a @ X3 @ D2 )
       => ( ( fun_upd_a_option_a @ ( restrict_map_a_a @ M @ D2 ) @ X3 @ none_a )
          = ( restrict_map_a_a @ M @ D2 ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_430_fun__upd__None__restrict,axiom,
    ! [X3: option_a,D2: set_option_a,M: option_a > option_a] :
      ( ( ( member_option_a @ X3 @ D2 )
       => ( ( fun_up1079276522633388797tion_a @ ( restri3984065703976872170on_a_a @ M @ D2 ) @ X3 @ none_a )
          = ( restri3984065703976872170on_a_a @ M @ ( minus_1574173051537231627tion_a @ D2 @ ( insert_option_a @ X3 @ bot_bot_set_option_a ) ) ) ) )
      & ( ~ ( member_option_a @ X3 @ D2 )
       => ( ( fun_up1079276522633388797tion_a @ ( restri3984065703976872170on_a_a @ M @ D2 ) @ X3 @ none_a )
          = ( restri3984065703976872170on_a_a @ M @ D2 ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_431_dom__fun__upd,axiom,
    ! [Y: option_a,F2: a > option_a,X3: a] :
      ( ( ( Y = none_a )
       => ( ( dom_a_a @ ( fun_upd_a_option_a @ F2 @ X3 @ Y ) )
          = ( minus_minus_set_a @ ( dom_a_a @ F2 ) @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) )
      & ( ( Y != none_a )
       => ( ( dom_a_a @ ( fun_upd_a_option_a @ F2 @ X3 @ Y ) )
          = ( insert_a @ X3 @ ( dom_a_a @ F2 ) ) ) ) ) ).

% dom_fun_upd
thf(fact_432_dom__fun__upd,axiom,
    ! [Y: option_a,F2: option_a > option_a,X3: option_a] :
      ( ( ( Y = none_a )
       => ( ( dom_option_a_a @ ( fun_up1079276522633388797tion_a @ F2 @ X3 @ Y ) )
          = ( minus_1574173051537231627tion_a @ ( dom_option_a_a @ F2 ) @ ( insert_option_a @ X3 @ bot_bot_set_option_a ) ) ) )
      & ( ( Y != none_a )
       => ( ( dom_option_a_a @ ( fun_up1079276522633388797tion_a @ F2 @ X3 @ Y ) )
          = ( insert_option_a @ X3 @ ( dom_option_a_a @ F2 ) ) ) ) ) ).

% dom_fun_upd
thf(fact_433_dom__minus,axiom,
    ! [F2: a > option_a,X3: a,A2: set_a] :
      ( ( ( F2 @ X3 )
        = none_a )
     => ( ( minus_minus_set_a @ ( dom_a_a @ F2 ) @ ( insert_a @ X3 @ A2 ) )
        = ( minus_minus_set_a @ ( dom_a_a @ F2 ) @ A2 ) ) ) ).

% dom_minus
thf(fact_434_dom__minus,axiom,
    ! [F2: option_a > option_a,X3: option_a,A2: set_option_a] :
      ( ( ( F2 @ X3 )
        = none_a )
     => ( ( minus_1574173051537231627tion_a @ ( dom_option_a_a @ F2 ) @ ( insert_option_a @ X3 @ A2 ) )
        = ( minus_1574173051537231627tion_a @ ( dom_option_a_a @ F2 ) @ A2 ) ) ) ).

% dom_minus
thf(fact_435_restrict__complement__singleton__eq,axiom,
    ! [F2: a > option_a,X3: a] :
      ( ( restrict_map_a_a @ F2 @ ( uminus_uminus_set_a @ ( insert_a @ X3 @ bot_bot_set_a ) ) )
      = ( fun_upd_a_option_a @ F2 @ X3 @ none_a ) ) ).

% restrict_complement_singleton_eq
thf(fact_436_restrict__complement__singleton__eq,axiom,
    ! [F2: option_a > option_a,X3: option_a] :
      ( ( restri3984065703976872170on_a_a @ F2 @ ( uminus6205308855922866075tion_a @ ( insert_option_a @ X3 @ bot_bot_set_option_a ) ) )
      = ( fun_up1079276522633388797tion_a @ F2 @ X3 @ none_a ) ) ).

% restrict_complement_singleton_eq
thf(fact_437_ran__map__upd__Some,axiom,
    ! [M: a > option_a,X3: a,Y: a,Z2: a] :
      ( ( ( M @ X3 )
        = ( some_a @ Y ) )
     => ( ( inj_on_a_option_a @ M @ ( dom_a_a @ M ) )
       => ( ~ ( member_a @ Z2 @ ( ran_a_a @ M ) )
         => ( ( ran_a_a @ ( fun_upd_a_option_a @ M @ X3 @ ( some_a @ Z2 ) ) )
            = ( sup_sup_set_a @ ( minus_minus_set_a @ ( ran_a_a @ M ) @ ( insert_a @ Y @ bot_bot_set_a ) ) @ ( insert_a @ Z2 @ bot_bot_set_a ) ) ) ) ) ) ).

% ran_map_upd_Some
thf(fact_438_these__insert__Some,axiom,
    ! [X3: option_a,A2: set_option_option_a] :
      ( ( these_option_a @ ( insert605063979879581146tion_a @ ( some_option_a @ X3 ) @ A2 ) )
      = ( insert_option_a @ X3 @ ( these_option_a @ A2 ) ) ) ).

% these_insert_Some
thf(fact_439_these__insert__Some,axiom,
    ! [X3: product_prod_a_d_c,A2: set_op3089619689617166423_a_d_c] :
      ( ( these_4428306183215063892_a_d_c @ ( insert2578358782419847377_a_d_c @ ( some_P377817780860425132_a_d_c @ X3 ) @ A2 ) )
      = ( insert2041379952318371585_a_d_c @ X3 @ ( these_4428306183215063892_a_d_c @ A2 ) ) ) ).

% these_insert_Some
thf(fact_440_these__insert__Some,axiom,
    ! [X3: a,A2: set_option_a] :
      ( ( these_a @ ( insert_option_a @ ( some_a @ X3 ) @ A2 ) )
      = ( insert_a @ X3 @ ( these_a @ A2 ) ) ) ).

% these_insert_Some
thf(fact_441_Field__insert,axiom,
    ! [A: a,B: a,R2: set_Product_prod_a_a] :
      ( ( field_a @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 ) )
      = ( sup_sup_set_a @ ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( field_a @ R2 ) ) ) ).

% Field_insert
thf(fact_442_Field__insert,axiom,
    ! [A: option_a,B: option_a,R2: set_Pr7585778909603769095tion_a] :
      ( ( field_option_a @ ( insert1246254401036548087tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 ) )
      = ( sup_sup_set_option_a @ ( insert_option_a @ A @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( field_option_a @ R2 ) ) ) ).

% Field_insert
thf(fact_443_override__on__insert,axiom,
    ! [F2: d > c,G: d > c,X3: d,X6: set_d] :
      ( ( override_on_d_c @ F2 @ G @ ( insert_d @ X3 @ X6 ) )
      = ( fun_upd_d_c @ ( override_on_d_c @ F2 @ G @ X6 ) @ X3 @ ( G @ X3 ) ) ) ).

% override_on_insert
thf(fact_444_override__on__insert_H,axiom,
    ! [F2: d > c,G: d > c,X3: d,X6: set_d] :
      ( ( override_on_d_c @ F2 @ G @ ( insert_d @ X3 @ X6 ) )
      = ( override_on_d_c @ ( fun_upd_d_c @ F2 @ X3 @ ( G @ X3 ) ) @ G @ X6 ) ) ).

% override_on_insert'
thf(fact_445_inj__on__empty,axiom,
    ! [F2: a > option_a] : ( inj_on_a_option_a @ F2 @ bot_bot_set_a ) ).

% inj_on_empty
thf(fact_446_these__empty,axiom,
    ( ( these_option_a @ bot_bo4163488203964334806tion_a )
    = bot_bot_set_option_a ) ).

% these_empty
thf(fact_447_these__empty,axiom,
    ( ( these_a @ bot_bot_set_option_a )
    = bot_bot_set_a ) ).

% these_empty
thf(fact_448_inj__on__map__add__dom,axiom,
    ! [M: a > option_a,M3: a > option_a] :
      ( ( inj_on_a_option_a @ ( map_add_a_a @ M @ M3 ) @ ( dom_a_a @ M3 ) )
      = ( inj_on_a_option_a @ M3 @ ( dom_a_a @ M3 ) ) ) ).

% inj_on_map_add_dom
thf(fact_449_these__insert__None,axiom,
    ! [A2: set_option_a] :
      ( ( these_a @ ( insert_option_a @ none_a @ A2 ) )
      = ( these_a @ A2 ) ) ).

% these_insert_None
thf(fact_450_inj__on__diff,axiom,
    ! [F2: a > option_a,A2: set_a,B2: set_a] :
      ( ( inj_on_a_option_a @ F2 @ A2 )
     => ( inj_on_a_option_a @ F2 @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).

% inj_on_diff
thf(fact_451_inj__Some,axiom,
    ! [A2: set_Pr5506164808955685639_a_d_c] : ( inj_on4952834208159004389_a_d_c @ some_P377817780860425132_a_d_c @ A2 ) ).

% inj_Some
thf(fact_452_inj__Some,axiom,
    ! [A2: set_a] : ( inj_on_a_option_a @ some_a @ A2 ) ).

% inj_Some
thf(fact_453_FieldI1,axiom,
    ! [I: a,J: a,R4: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R4 )
     => ( member_a @ I @ ( field_a @ R4 ) ) ) ).

% FieldI1
thf(fact_454_FieldI1,axiom,
    ! [I: produc6150846815813599699_a_d_c,J: produc6150846815813599699_a_d_c,R4: set_Pr8390707888732215751_a_d_c] :
      ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ I @ J ) @ R4 )
     => ( member2052822272342364412_a_d_c @ I @ ( field_6537103990374211872_a_d_c @ R4 ) ) ) ).

% FieldI1
thf(fact_455_FieldI1,axiom,
    ! [I: option_a,J: option_a,R4: set_Pr7585778909603769095tion_a] :
      ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ I @ J ) @ R4 )
     => ( member_option_a @ I @ ( field_option_a @ R4 ) ) ) ).

% FieldI1
thf(fact_456_FieldI2,axiom,
    ! [I: a,J: a,R4: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R4 )
     => ( member_a @ J @ ( field_a @ R4 ) ) ) ).

% FieldI2
thf(fact_457_FieldI2,axiom,
    ! [I: produc6150846815813599699_a_d_c,J: produc6150846815813599699_a_d_c,R4: set_Pr8390707888732215751_a_d_c] :
      ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ I @ J ) @ R4 )
     => ( member2052822272342364412_a_d_c @ J @ ( field_6537103990374211872_a_d_c @ R4 ) ) ) ).

% FieldI2
thf(fact_458_FieldI2,axiom,
    ! [I: option_a,J: option_a,R4: set_Pr7585778909603769095tion_a] :
      ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ I @ J ) @ R4 )
     => ( member_option_a @ J @ ( field_option_a @ R4 ) ) ) ).

% FieldI2
thf(fact_459_inj__onD,axiom,
    ! [F2: a > option_a,A2: set_a,X3: a,Y: a] :
      ( ( inj_on_a_option_a @ F2 @ A2 )
     => ( ( ( F2 @ X3 )
          = ( F2 @ Y ) )
       => ( ( member_a @ X3 @ A2 )
         => ( ( member_a @ Y @ A2 )
           => ( X3 = Y ) ) ) ) ) ).

% inj_onD
thf(fact_460_inj__onI,axiom,
    ! [A2: set_a,F2: a > option_a] :
      ( ! [X2: a,Y4: a] :
          ( ( member_a @ X2 @ A2 )
         => ( ( member_a @ Y4 @ A2 )
           => ( ( ( F2 @ X2 )
                = ( F2 @ Y4 ) )
             => ( X2 = Y4 ) ) ) )
     => ( inj_on_a_option_a @ F2 @ A2 ) ) ).

% inj_onI
thf(fact_461_inj__on__def,axiom,
    ( inj_on_a_option_a
    = ( ^ [F: a > option_a,A7: set_a] :
        ! [X: a] :
          ( ( member_a @ X @ A7 )
         => ! [Y3: a] :
              ( ( member_a @ Y3 @ A7 )
             => ( ( ( F @ X )
                  = ( F @ Y3 ) )
               => ( X = Y3 ) ) ) ) ) ) ).

% inj_on_def
thf(fact_462_inj__on__cong,axiom,
    ! [A2: set_a,F2: a > option_a,G: a > option_a] :
      ( ! [A4: a] :
          ( ( member_a @ A4 @ A2 )
         => ( ( F2 @ A4 )
            = ( G @ A4 ) ) )
     => ( ( inj_on_a_option_a @ F2 @ A2 )
        = ( inj_on_a_option_a @ G @ A2 ) ) ) ).

% inj_on_cong
thf(fact_463_inj__on__eq__iff,axiom,
    ! [F2: a > option_a,A2: set_a,X3: a,Y: a] :
      ( ( inj_on_a_option_a @ F2 @ A2 )
     => ( ( member_a @ X3 @ A2 )
       => ( ( member_a @ Y @ A2 )
         => ( ( ( F2 @ X3 )
              = ( F2 @ Y ) )
            = ( X3 = Y ) ) ) ) ) ).

% inj_on_eq_iff
thf(fact_464_inj__on__contraD,axiom,
    ! [F2: a > option_a,A2: set_a,X3: a,Y: a] :
      ( ( inj_on_a_option_a @ F2 @ A2 )
     => ( ( X3 != Y )
       => ( ( member_a @ X3 @ A2 )
         => ( ( member_a @ Y @ A2 )
           => ( ( F2 @ X3 )
             != ( F2 @ Y ) ) ) ) ) ) ).

% inj_on_contraD
thf(fact_465_inj__on__inverseI,axiom,
    ! [A2: set_a,G: option_a > a,F2: a > option_a] :
      ( ! [X2: a] :
          ( ( member_a @ X2 @ A2 )
         => ( ( G @ ( F2 @ X2 ) )
            = X2 ) )
     => ( inj_on_a_option_a @ F2 @ A2 ) ) ).

% inj_on_inverseI
thf(fact_466_inj__on__Int,axiom,
    ! [F2: a > option_a,A2: set_a,B2: set_a] :
      ( ( ( inj_on_a_option_a @ F2 @ A2 )
        | ( inj_on_a_option_a @ F2 @ B2 ) )
     => ( inj_on_a_option_a @ F2 @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ).

% inj_on_Int
thf(fact_467_in__these__eq,axiom,
    ! [X3: produc6150846815813599699_a_d_c,A2: set_op4102275208527013369_a_d_c] :
      ( ( member2052822272342364412_a_d_c @ X3 @ ( these_6727699095634540816_a_d_c @ A2 ) )
      = ( member4494181189314409282_a_d_c @ ( some_P8437211288446953144_a_d_c @ X3 ) @ A2 ) ) ).

% in_these_eq
thf(fact_468_in__these__eq,axiom,
    ! [X3: option_a,A2: set_option_option_a] :
      ( ( member_option_a @ X3 @ ( these_option_a @ A2 ) )
      = ( member5113800082084363315tion_a @ ( some_option_a @ X3 ) @ A2 ) ) ).

% in_these_eq
thf(fact_469_in__these__eq,axiom,
    ! [X3: product_prod_a_d_c,A2: set_op3089619689617166423_a_d_c] :
      ( ( member8140300833826434152_a_d_c @ X3 @ ( these_4428306183215063892_a_d_c @ A2 ) )
      = ( member1874962706598213560_a_d_c @ ( some_P377817780860425132_a_d_c @ X3 ) @ A2 ) ) ).

% in_these_eq
thf(fact_470_in__these__eq,axiom,
    ! [X3: a,A2: set_option_a] :
      ( ( member_a @ X3 @ ( these_a @ A2 ) )
      = ( member_option_a @ ( some_a @ X3 ) @ A2 ) ) ).

% in_these_eq
thf(fact_471_these__empty__eq,axiom,
    ! [B2: set_option_option_a] :
      ( ( ( these_option_a @ B2 )
        = bot_bot_set_option_a )
      = ( ( B2 = bot_bo4163488203964334806tion_a )
        | ( B2
          = ( insert605063979879581146tion_a @ none_option_a @ bot_bo4163488203964334806tion_a ) ) ) ) ).

% these_empty_eq
thf(fact_472_these__empty__eq,axiom,
    ! [B2: set_option_a] :
      ( ( ( these_a @ B2 )
        = bot_bot_set_a )
      = ( ( B2 = bot_bot_set_option_a )
        | ( B2
          = ( insert_option_a @ none_a @ bot_bot_set_option_a ) ) ) ) ).

% these_empty_eq
thf(fact_473_these__not__empty__eq,axiom,
    ! [B2: set_option_option_a] :
      ( ( ( these_option_a @ B2 )
       != bot_bot_set_option_a )
      = ( ( B2 != bot_bo4163488203964334806tion_a )
        & ( B2
         != ( insert605063979879581146tion_a @ none_option_a @ bot_bo4163488203964334806tion_a ) ) ) ) ).

% these_not_empty_eq
thf(fact_474_these__not__empty__eq,axiom,
    ! [B2: set_option_a] :
      ( ( ( these_a @ B2 )
       != bot_bot_set_a )
      = ( ( B2 != bot_bot_set_option_a )
        & ( B2
         != ( insert_option_a @ none_a @ bot_bot_set_option_a ) ) ) ) ).

% these_not_empty_eq
thf(fact_475_inj__on__Un,axiom,
    ! [F2: option_a > a,A2: set_option_a,B2: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ ( sup_sup_set_option_a @ A2 @ B2 ) )
      = ( ( inj_on_option_a_a @ F2 @ A2 )
        & ( inj_on_option_a_a @ F2 @ B2 )
        & ( ( inf_inf_set_a @ ( image_option_a_a2 @ F2 @ ( minus_1574173051537231627tion_a @ A2 @ B2 ) ) @ ( image_option_a_a2 @ F2 @ ( minus_1574173051537231627tion_a @ B2 @ A2 ) ) )
          = bot_bot_set_a ) ) ) ).

% inj_on_Un
thf(fact_476_inj__on__Un,axiom,
    ! [F2: a > option_a,A2: set_a,B2: set_a] :
      ( ( inj_on_a_option_a @ F2 @ ( sup_sup_set_a @ A2 @ B2 ) )
      = ( ( inj_on_a_option_a @ F2 @ A2 )
        & ( inj_on_a_option_a @ F2 @ B2 )
        & ( ( inf_inf_set_option_a @ ( image_a_option_a2 @ F2 @ ( minus_minus_set_a @ A2 @ B2 ) ) @ ( image_a_option_a2 @ F2 @ ( minus_minus_set_a @ B2 @ A2 ) ) )
          = bot_bot_set_option_a ) ) ) ).

% inj_on_Un
thf(fact_477_inj__on__insert,axiom,
    ! [F2: a > a,A: a,A2: set_a] :
      ( ( inj_on_a_a @ F2 @ ( insert_a @ A @ A2 ) )
      = ( ( inj_on_a_a @ F2 @ A2 )
        & ~ ( member_a @ ( F2 @ A ) @ ( image_a_a2 @ F2 @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_478_inj__on__insert,axiom,
    ! [F2: a > produc6150846815813599699_a_d_c,A: a,A2: set_a] :
      ( ( inj_on3709328703910593990_a_d_c @ F2 @ ( insert_a @ A @ A2 ) )
      = ( ( inj_on3709328703910593990_a_d_c @ F2 @ A2 )
        & ~ ( member2052822272342364412_a_d_c @ ( F2 @ A ) @ ( image_6884406963069720154_a_d_c @ F2 @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_479_inj__on__insert,axiom,
    ! [F2: a > option_a,A: a,A2: set_a] :
      ( ( inj_on_a_option_a @ F2 @ ( insert_a @ A @ A2 ) )
      = ( ( inj_on_a_option_a @ F2 @ A2 )
        & ~ ( member_option_a @ ( F2 @ A ) @ ( image_a_option_a2 @ F2 @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_480_inj__on__insert,axiom,
    ! [F2: option_a > a,A: option_a,A2: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ ( insert_option_a @ A @ A2 ) )
      = ( ( inj_on_option_a_a @ F2 @ A2 )
        & ~ ( member_a @ ( F2 @ A ) @ ( image_option_a_a2 @ F2 @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_481_inj__on__insert,axiom,
    ! [F2: option_a > produc6150846815813599699_a_d_c,A: option_a,A2: set_option_a] :
      ( ( inj_on7085348629561674240_a_d_c @ F2 @ ( insert_option_a @ A @ A2 ) )
      = ( ( inj_on7085348629561674240_a_d_c @ F2 @ A2 )
        & ~ ( member2052822272342364412_a_d_c @ ( F2 @ A ) @ ( image_191531556693044372_a_d_c @ F2 @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_482_inj__on__insert,axiom,
    ! [F2: option_a > option_a,A: option_a,A2: set_option_a] :
      ( ( inj_on8559383841115902449tion_a @ F2 @ ( insert_option_a @ A @ A2 ) )
      = ( ( inj_on8559383841115902449tion_a @ F2 @ A2 )
        & ~ ( member_option_a @ ( F2 @ A ) @ ( image_7439109396645324421tion_a @ F2 @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_483_Range__insert,axiom,
    ! [A: product_prod_a_d_c,B: option2362401199305441953_a_d_c,R2: set_Pr7868159745199425715_a_d_c] :
      ( ( range_8197308513517232617_a_d_c @ ( insert4944988292672327587_a_d_c @ ( produc5208860900648697099_a_d_c @ A @ B ) @ R2 ) )
      = ( insert2578358782419847377_a_d_c @ B @ ( range_8197308513517232617_a_d_c @ R2 ) ) ) ).

% Range_insert
thf(fact_484_Range__insert,axiom,
    ! [A: a,B: d > c,R2: set_Pr5506164808955685639_a_d_c] :
      ( ( range_a_d_c @ ( insert2041379952318371585_a_d_c @ ( product_Pair_a_d_c @ A @ B ) @ R2 ) )
      = ( insert_d_c @ B @ ( range_a_d_c @ R2 ) ) ) ).

% Range_insert
thf(fact_485_refl__on__singleton,axiom,
    ! [X3: a] : ( refl_on_a @ ( insert_a @ X3 @ bot_bot_set_a ) @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ X3 @ X3 ) @ bot_bo3357376287454694259od_a_a ) ) ).

% refl_on_singleton
thf(fact_486_refl__on__singleton,axiom,
    ! [X3: option_a] : ( refl_on_option_a @ ( insert_option_a @ X3 @ bot_bot_set_option_a ) @ ( insert1246254401036548087tion_a @ ( produc9011544418120257559tion_a @ X3 @ X3 ) @ bot_bo235252021745139059tion_a ) ) ).

% refl_on_singleton
thf(fact_487_Domain__insert,axiom,
    ! [A: product_prod_a_d_c,B: option2362401199305441953_a_d_c,R2: set_Pr7868159745199425715_a_d_c] :
      ( ( domain4774513116958948480_a_d_c @ ( insert4944988292672327587_a_d_c @ ( produc5208860900648697099_a_d_c @ A @ B ) @ R2 ) )
      = ( insert2041379952318371585_a_d_c @ A @ ( domain4774513116958948480_a_d_c @ R2 ) ) ) ).

% Domain_insert
thf(fact_488_Domain__insert,axiom,
    ! [A: a,B: d > c,R2: set_Pr5506164808955685639_a_d_c] :
      ( ( domain_a_d_c @ ( insert2041379952318371585_a_d_c @ ( product_Pair_a_d_c @ A @ B ) @ R2 ) )
      = ( insert_a @ A @ ( domain_a_d_c @ R2 ) ) ) ).

% Domain_insert
thf(fact_489_image__map__upd,axiom,
    ! [X3: a,A2: set_a,M: a > option_a,Y: a] :
      ( ~ ( member_a @ X3 @ A2 )
     => ( ( image_a_option_a2 @ ( fun_upd_a_option_a @ M @ X3 @ ( some_a @ Y ) ) @ A2 )
        = ( image_a_option_a2 @ M @ A2 ) ) ) ).

% image_map_upd
thf(fact_490_image__map__upd,axiom,
    ! [X3: produc6150846815813599699_a_d_c,A2: set_Pr7868159745199425715_a_d_c,M: produc6150846815813599699_a_d_c > option_a,Y: a] :
      ( ~ ( member2052822272342364412_a_d_c @ X3 @ A2 )
     => ( ( image_5971021413422854966tion_a @ ( fun_up4051926500471473070tion_a @ M @ X3 @ ( some_a @ Y ) ) @ A2 )
        = ( image_5971021413422854966tion_a @ M @ A2 ) ) ) ).

% image_map_upd
thf(fact_491_image__map__upd,axiom,
    ! [X3: option_a,A2: set_option_a,M: option_a > option_a,Y: a] :
      ( ~ ( member_option_a @ X3 @ A2 )
     => ( ( image_7439109396645324421tion_a @ ( fun_up1079276522633388797tion_a @ M @ X3 @ ( some_a @ Y ) ) @ A2 )
        = ( image_7439109396645324421tion_a @ M @ A2 ) ) ) ).

% image_map_upd
thf(fact_492_image__map__upd,axiom,
    ! [X3: a,A2: set_a,M: a > option2362401199305441953_a_d_c,Y: product_prod_a_d_c] :
      ( ~ ( member_a @ X3 @ A2 )
     => ( ( image_8958380359779546586_a_d_c @ ( fun_up2590722781756628834_a_d_c @ M @ X3 @ ( some_P377817780860425132_a_d_c @ Y ) ) @ A2 )
        = ( image_8958380359779546586_a_d_c @ M @ A2 ) ) ) ).

% image_map_upd
thf(fact_493_image__map__upd,axiom,
    ! [X3: produc6150846815813599699_a_d_c,A2: set_Pr7868159745199425715_a_d_c,M: produc6150846815813599699_a_d_c > option2362401199305441953_a_d_c,Y: product_prod_a_d_c] :
      ( ~ ( member2052822272342364412_a_d_c @ X3 @ A2 )
     => ( ( image_1986814276350215727_a_d_c @ ( fun_up4469873178457468855_a_d_c @ M @ X3 @ ( some_P377817780860425132_a_d_c @ Y ) ) @ A2 )
        = ( image_1986814276350215727_a_d_c @ M @ A2 ) ) ) ).

% image_map_upd
thf(fact_494_image__map__upd,axiom,
    ! [X3: option_a,A2: set_option_a,M: option_a > option2362401199305441953_a_d_c,Y: product_prod_a_d_c] :
      ( ~ ( member_option_a @ X3 @ A2 )
     => ( ( image_8868864280970703392_a_d_c @ ( fun_up1917685652830187432_a_d_c @ M @ X3 @ ( some_P377817780860425132_a_d_c @ Y ) ) @ A2 )
        = ( image_8868864280970703392_a_d_c @ M @ A2 ) ) ) ).

% image_map_upd
thf(fact_495_these__image__Some__eq,axiom,
    ! [A2: set_Pr5506164808955685639_a_d_c] :
      ( ( these_4428306183215063892_a_d_c @ ( image_2856777003974851449_a_d_c @ some_P377817780860425132_a_d_c @ A2 ) )
      = A2 ) ).

% these_image_Some_eq
thf(fact_496_these__image__Some__eq,axiom,
    ! [A2: set_a] :
      ( ( these_a @ ( image_a_option_a2 @ some_a @ A2 ) )
      = A2 ) ).

% these_image_Some_eq
thf(fact_497_inj__on__image__iff,axiom,
    ! [A2: set_a,G: a > option_a,F2: a > a] :
      ( ! [X2: a] :
          ( ( member_a @ X2 @ A2 )
         => ! [Xa: a] :
              ( ( member_a @ Xa @ A2 )
             => ( ( ( G @ ( F2 @ X2 ) )
                  = ( G @ ( F2 @ Xa ) ) )
                = ( ( G @ X2 )
                  = ( G @ Xa ) ) ) ) )
     => ( ( inj_on_a_a @ F2 @ A2 )
       => ( ( inj_on_a_option_a @ G @ ( image_a_a2 @ F2 @ A2 ) )
          = ( inj_on_a_option_a @ G @ A2 ) ) ) ) ).

% inj_on_image_iff
thf(fact_498_refl__onD2,axiom,
    ! [A2: set_a,R2: set_Product_prod_a_a,X3: a,Y: a] :
      ( ( refl_on_a @ A2 @ R2 )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R2 )
       => ( member_a @ Y @ A2 ) ) ) ).

% refl_onD2
thf(fact_499_refl__onD2,axiom,
    ! [A2: set_Pr7868159745199425715_a_d_c,R2: set_Pr8390707888732215751_a_d_c,X3: produc6150846815813599699_a_d_c,Y: produc6150846815813599699_a_d_c] :
      ( ( refl_o1599096589449526226_a_d_c @ A2 @ R2 )
     => ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ X3 @ Y ) @ R2 )
       => ( member2052822272342364412_a_d_c @ Y @ A2 ) ) ) ).

% refl_onD2
thf(fact_500_refl__onD2,axiom,
    ! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,X3: option_a,Y: option_a] :
      ( ( refl_on_option_a @ A2 @ R2 )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y ) @ R2 )
       => ( member_option_a @ Y @ A2 ) ) ) ).

% refl_onD2
thf(fact_501_refl__onD1,axiom,
    ! [A2: set_a,R2: set_Product_prod_a_a,X3: a,Y: a] :
      ( ( refl_on_a @ A2 @ R2 )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R2 )
       => ( member_a @ X3 @ A2 ) ) ) ).

% refl_onD1
thf(fact_502_refl__onD1,axiom,
    ! [A2: set_Pr7868159745199425715_a_d_c,R2: set_Pr8390707888732215751_a_d_c,X3: produc6150846815813599699_a_d_c,Y: produc6150846815813599699_a_d_c] :
      ( ( refl_o1599096589449526226_a_d_c @ A2 @ R2 )
     => ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ X3 @ Y ) @ R2 )
       => ( member2052822272342364412_a_d_c @ X3 @ A2 ) ) ) ).

% refl_onD1
thf(fact_503_refl__onD1,axiom,
    ! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,X3: option_a,Y: option_a] :
      ( ( refl_on_option_a @ A2 @ R2 )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y ) @ R2 )
       => ( member_option_a @ X3 @ A2 ) ) ) ).

% refl_onD1
thf(fact_504_refl__onD,axiom,
    ! [A2: set_a,R2: set_Product_prod_a_a,A: a] :
      ( ( refl_on_a @ A2 @ R2 )
     => ( ( member_a @ A @ A2 )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A ) @ R2 ) ) ) ).

% refl_onD
thf(fact_505_refl__onD,axiom,
    ! [A2: set_Pr7868159745199425715_a_d_c,R2: set_Pr8390707888732215751_a_d_c,A: produc6150846815813599699_a_d_c] :
      ( ( refl_o1599096589449526226_a_d_c @ A2 @ R2 )
     => ( ( member2052822272342364412_a_d_c @ A @ A2 )
       => ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ A @ A ) @ R2 ) ) ) ).

% refl_onD
thf(fact_506_refl__onD,axiom,
    ! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,A: option_a] :
      ( ( refl_on_option_a @ A2 @ R2 )
     => ( ( member_option_a @ A @ A2 )
       => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ A ) @ R2 ) ) ) ).

% refl_onD
thf(fact_507_Domain__iff,axiom,
    ! [A: product_prod_a_d_c,R2: set_Pr7868159745199425715_a_d_c] :
      ( ( member8140300833826434152_a_d_c @ A @ ( domain4774513116958948480_a_d_c @ R2 ) )
      = ( ? [Y3: option2362401199305441953_a_d_c] : ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ A @ Y3 ) @ R2 ) ) ) ).

% Domain_iff
thf(fact_508_Domain__iff,axiom,
    ! [A: a,R2: set_Pr5506164808955685639_a_d_c] :
      ( ( member_a @ A @ ( domain_a_d_c @ R2 ) )
      = ( ? [Y3: d > c] : ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ A @ Y3 ) @ R2 ) ) ) ).

% Domain_iff
thf(fact_509_DomainE,axiom,
    ! [A: product_prod_a_d_c,R2: set_Pr7868159745199425715_a_d_c] :
      ( ( member8140300833826434152_a_d_c @ A @ ( domain4774513116958948480_a_d_c @ R2 ) )
     => ~ ! [B5: option2362401199305441953_a_d_c] :
            ~ ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ A @ B5 ) @ R2 ) ) ).

% DomainE
thf(fact_510_DomainE,axiom,
    ! [A: a,R2: set_Pr5506164808955685639_a_d_c] :
      ( ( member_a @ A @ ( domain_a_d_c @ R2 ) )
     => ~ ! [B5: d > c] :
            ~ ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ A @ B5 ) @ R2 ) ) ).

% DomainE
thf(fact_511_Domain_ODomainI,axiom,
    ! [A: product_prod_a_d_c,B: option2362401199305441953_a_d_c,R2: set_Pr7868159745199425715_a_d_c] :
      ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ A @ B ) @ R2 )
     => ( member8140300833826434152_a_d_c @ A @ ( domain4774513116958948480_a_d_c @ R2 ) ) ) ).

% Domain.DomainI
thf(fact_512_Domain_ODomainI,axiom,
    ! [A: a,B: d > c,R2: set_Pr5506164808955685639_a_d_c] :
      ( ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ A @ B ) @ R2 )
     => ( member_a @ A @ ( domain_a_d_c @ R2 ) ) ) ).

% Domain.DomainI
thf(fact_513_Domain_Osimps,axiom,
    ! [A: product_prod_a_d_c,R2: set_Pr7868159745199425715_a_d_c] :
      ( ( member8140300833826434152_a_d_c @ A @ ( domain4774513116958948480_a_d_c @ R2 ) )
      = ( ? [A3: product_prod_a_d_c,B3: option2362401199305441953_a_d_c] :
            ( ( A = A3 )
            & ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ A3 @ B3 ) @ R2 ) ) ) ) ).

% Domain.simps
thf(fact_514_Domain_Osimps,axiom,
    ! [A: a,R2: set_Pr5506164808955685639_a_d_c] :
      ( ( member_a @ A @ ( domain_a_d_c @ R2 ) )
      = ( ? [A3: a,B3: d > c] :
            ( ( A = A3 )
            & ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ A3 @ B3 ) @ R2 ) ) ) ) ).

% Domain.simps
thf(fact_515_Domain_Ocases,axiom,
    ! [A: product_prod_a_d_c,R2: set_Pr7868159745199425715_a_d_c] :
      ( ( member8140300833826434152_a_d_c @ A @ ( domain4774513116958948480_a_d_c @ R2 ) )
     => ~ ! [B5: option2362401199305441953_a_d_c] :
            ~ ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ A @ B5 ) @ R2 ) ) ).

% Domain.cases
thf(fact_516_Domain_Ocases,axiom,
    ! [A: a,R2: set_Pr5506164808955685639_a_d_c] :
      ( ( member_a @ A @ ( domain_a_d_c @ R2 ) )
     => ~ ! [B5: d > c] :
            ~ ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ A @ B5 ) @ R2 ) ) ).

% Domain.cases
thf(fact_517_Range__iff,axiom,
    ! [A: option2362401199305441953_a_d_c,R2: set_Pr7868159745199425715_a_d_c] :
      ( ( member1874962706598213560_a_d_c @ A @ ( range_8197308513517232617_a_d_c @ R2 ) )
      = ( ? [Y3: product_prod_a_d_c] : ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ Y3 @ A ) @ R2 ) ) ) ).

% Range_iff
thf(fact_518_Range__iff,axiom,
    ! [A: d > c,R2: set_Pr5506164808955685639_a_d_c] :
      ( ( member_d_c @ A @ ( range_a_d_c @ R2 ) )
      = ( ? [Y3: a] : ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ Y3 @ A ) @ R2 ) ) ) ).

% Range_iff
thf(fact_519_RangeE,axiom,
    ! [B: option2362401199305441953_a_d_c,R2: set_Pr7868159745199425715_a_d_c] :
      ( ( member1874962706598213560_a_d_c @ B @ ( range_8197308513517232617_a_d_c @ R2 ) )
     => ~ ! [A4: product_prod_a_d_c] :
            ~ ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ A4 @ B ) @ R2 ) ) ).

% RangeE
thf(fact_520_RangeE,axiom,
    ! [B: d > c,R2: set_Pr5506164808955685639_a_d_c] :
      ( ( member_d_c @ B @ ( range_a_d_c @ R2 ) )
     => ~ ! [A4: a] :
            ~ ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ A4 @ B ) @ R2 ) ) ).

% RangeE
thf(fact_521_Range_Ointros,axiom,
    ! [A: product_prod_a_d_c,B: option2362401199305441953_a_d_c,R2: set_Pr7868159745199425715_a_d_c] :
      ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ A @ B ) @ R2 )
     => ( member1874962706598213560_a_d_c @ B @ ( range_8197308513517232617_a_d_c @ R2 ) ) ) ).

% Range.intros
thf(fact_522_Range_Ointros,axiom,
    ! [A: a,B: d > c,R2: set_Pr5506164808955685639_a_d_c] :
      ( ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ A @ B ) @ R2 )
     => ( member_d_c @ B @ ( range_a_d_c @ R2 ) ) ) ).

% Range.intros
thf(fact_523_Range_Osimps,axiom,
    ! [A: option2362401199305441953_a_d_c,R2: set_Pr7868159745199425715_a_d_c] :
      ( ( member1874962706598213560_a_d_c @ A @ ( range_8197308513517232617_a_d_c @ R2 ) )
      = ( ? [A3: product_prod_a_d_c,B3: option2362401199305441953_a_d_c] :
            ( ( A = B3 )
            & ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ A3 @ B3 ) @ R2 ) ) ) ) ).

% Range.simps
thf(fact_524_Range_Osimps,axiom,
    ! [A: d > c,R2: set_Pr5506164808955685639_a_d_c] :
      ( ( member_d_c @ A @ ( range_a_d_c @ R2 ) )
      = ( ? [A3: a,B3: d > c] :
            ( ( A = B3 )
            & ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ A3 @ B3 ) @ R2 ) ) ) ) ).

% Range.simps
thf(fact_525_Range_Ocases,axiom,
    ! [A: option2362401199305441953_a_d_c,R2: set_Pr7868159745199425715_a_d_c] :
      ( ( member1874962706598213560_a_d_c @ A @ ( range_8197308513517232617_a_d_c @ R2 ) )
     => ~ ! [A4: product_prod_a_d_c] :
            ~ ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ A4 @ A ) @ R2 ) ) ).

% Range.cases
thf(fact_526_Range_Ocases,axiom,
    ! [A: d > c,R2: set_Pr5506164808955685639_a_d_c] :
      ( ( member_d_c @ A @ ( range_a_d_c @ R2 ) )
     => ~ ! [A4: a] :
            ~ ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ A4 @ A ) @ R2 ) ) ).

% Range.cases
thf(fact_527_inj__img__insertE,axiom,
    ! [F2: a > a,A2: set_a,X3: a,B2: set_a] :
      ( ( inj_on_a_a @ F2 @ A2 )
     => ( ~ ( member_a @ X3 @ B2 )
       => ( ( ( insert_a @ X3 @ B2 )
            = ( image_a_a2 @ F2 @ A2 ) )
         => ~ ! [X7: a,A8: set_a] :
                ( ~ ( member_a @ X7 @ A8 )
               => ( ( A2
                    = ( insert_a @ X7 @ A8 ) )
                 => ( ( X3
                      = ( F2 @ X7 ) )
                   => ( B2
                     != ( image_a_a2 @ F2 @ A8 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_528_inj__img__insertE,axiom,
    ! [F2: produc6150846815813599699_a_d_c > a,A2: set_Pr7868159745199425715_a_d_c,X3: a,B2: set_a] :
      ( ( inj_on6701986109746958172_d_c_a @ F2 @ A2 )
     => ( ~ ( member_a @ X3 @ B2 )
       => ( ( ( insert_a @ X3 @ B2 )
            = ( image_653692332051308528_d_c_a @ F2 @ A2 ) )
         => ~ ! [X7: produc6150846815813599699_a_d_c,A8: set_Pr7868159745199425715_a_d_c] :
                ( ~ ( member2052822272342364412_a_d_c @ X7 @ A8 )
               => ( ( A2
                    = ( insert4944988292672327587_a_d_c @ X7 @ A8 ) )
                 => ( ( X3
                      = ( F2 @ X7 ) )
                   => ( B2
                     != ( image_653692332051308528_d_c_a @ F2 @ A8 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_529_inj__img__insertE,axiom,
    ! [F2: option_a > a,A2: set_option_a,X3: a,B2: set_a] :
      ( ( inj_on_option_a_a @ F2 @ A2 )
     => ( ~ ( member_a @ X3 @ B2 )
       => ( ( ( insert_a @ X3 @ B2 )
            = ( image_option_a_a2 @ F2 @ A2 ) )
         => ~ ! [X7: option_a,A8: set_option_a] :
                ( ~ ( member_option_a @ X7 @ A8 )
               => ( ( A2
                    = ( insert_option_a @ X7 @ A8 ) )
                 => ( ( X3
                      = ( F2 @ X7 ) )
                   => ( B2
                     != ( image_option_a_a2 @ F2 @ A8 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_530_inj__img__insertE,axiom,
    ! [F2: a > produc6150846815813599699_a_d_c,A2: set_a,X3: produc6150846815813599699_a_d_c,B2: set_Pr7868159745199425715_a_d_c] :
      ( ( inj_on3709328703910593990_a_d_c @ F2 @ A2 )
     => ( ~ ( member2052822272342364412_a_d_c @ X3 @ B2 )
       => ( ( ( insert4944988292672327587_a_d_c @ X3 @ B2 )
            = ( image_6884406963069720154_a_d_c @ F2 @ A2 ) )
         => ~ ! [X7: a,A8: set_a] :
                ( ~ ( member_a @ X7 @ A8 )
               => ( ( A2
                    = ( insert_a @ X7 @ A8 ) )
                 => ( ( X3
                      = ( F2 @ X7 ) )
                   => ( B2
                     != ( image_6884406963069720154_a_d_c @ F2 @ A8 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_531_inj__img__insertE,axiom,
    ! [F2: produc6150846815813599699_a_d_c > produc6150846815813599699_a_d_c,A2: set_Pr7868159745199425715_a_d_c,X3: produc6150846815813599699_a_d_c,B2: set_Pr7868159745199425715_a_d_c] :
      ( ( inj_on5053348487737048625_a_d_c @ F2 @ A2 )
     => ( ~ ( member2052822272342364412_a_d_c @ X3 @ B2 )
       => ( ( ( insert4944988292672327587_a_d_c @ X3 @ B2 )
            = ( image_548751279113285317_a_d_c @ F2 @ A2 ) )
         => ~ ! [X7: produc6150846815813599699_a_d_c,A8: set_Pr7868159745199425715_a_d_c] :
                ( ~ ( member2052822272342364412_a_d_c @ X7 @ A8 )
               => ( ( A2
                    = ( insert4944988292672327587_a_d_c @ X7 @ A8 ) )
                 => ( ( X3
                      = ( F2 @ X7 ) )
                   => ( B2
                     != ( image_548751279113285317_a_d_c @ F2 @ A8 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_532_inj__img__insertE,axiom,
    ! [F2: option_a > produc6150846815813599699_a_d_c,A2: set_option_a,X3: produc6150846815813599699_a_d_c,B2: set_Pr7868159745199425715_a_d_c] :
      ( ( inj_on7085348629561674240_a_d_c @ F2 @ A2 )
     => ( ~ ( member2052822272342364412_a_d_c @ X3 @ B2 )
       => ( ( ( insert4944988292672327587_a_d_c @ X3 @ B2 )
            = ( image_191531556693044372_a_d_c @ F2 @ A2 ) )
         => ~ ! [X7: option_a,A8: set_option_a] :
                ( ~ ( member_option_a @ X7 @ A8 )
               => ( ( A2
                    = ( insert_option_a @ X7 @ A8 ) )
                 => ( ( X3
                      = ( F2 @ X7 ) )
                   => ( B2
                     != ( image_191531556693044372_a_d_c @ F2 @ A8 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_533_inj__img__insertE,axiom,
    ! [F2: a > option_a,A2: set_a,X3: option_a,B2: set_option_a] :
      ( ( inj_on_a_option_a @ F2 @ A2 )
     => ( ~ ( member_option_a @ X3 @ B2 )
       => ( ( ( insert_option_a @ X3 @ B2 )
            = ( image_a_option_a2 @ F2 @ A2 ) )
         => ~ ! [X7: a,A8: set_a] :
                ( ~ ( member_a @ X7 @ A8 )
               => ( ( A2
                    = ( insert_a @ X7 @ A8 ) )
                 => ( ( X3
                      = ( F2 @ X7 ) )
                   => ( B2
                     != ( image_a_option_a2 @ F2 @ A8 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_534_inj__img__insertE,axiom,
    ! [F2: produc6150846815813599699_a_d_c > option_a,A2: set_Pr7868159745199425715_a_d_c,X3: option_a,B2: set_option_a] :
      ( ( inj_on3641466449436709026tion_a @ F2 @ A2 )
     => ( ~ ( member_option_a @ X3 @ B2 )
       => ( ( ( insert_option_a @ X3 @ B2 )
            = ( image_5971021413422854966tion_a @ F2 @ A2 ) )
         => ~ ! [X7: produc6150846815813599699_a_d_c,A8: set_Pr7868159745199425715_a_d_c] :
                ( ~ ( member2052822272342364412_a_d_c @ X7 @ A8 )
               => ( ( A2
                    = ( insert4944988292672327587_a_d_c @ X7 @ A8 ) )
                 => ( ( X3
                      = ( F2 @ X7 ) )
                   => ( B2
                     != ( image_5971021413422854966tion_a @ F2 @ A8 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_535_inj__img__insertE,axiom,
    ! [F2: option_a > option_a,A2: set_option_a,X3: option_a,B2: set_option_a] :
      ( ( inj_on8559383841115902449tion_a @ F2 @ A2 )
     => ( ~ ( member_option_a @ X3 @ B2 )
       => ( ( ( insert_option_a @ X3 @ B2 )
            = ( image_7439109396645324421tion_a @ F2 @ A2 ) )
         => ~ ! [X7: option_a,A8: set_option_a] :
                ( ~ ( member_option_a @ X7 @ A8 )
               => ( ( A2
                    = ( insert_option_a @ X7 @ A8 ) )
                 => ( ( X3
                      = ( F2 @ X7 ) )
                   => ( B2
                     != ( image_7439109396645324421tion_a @ F2 @ A8 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_536_inj__on__Un__image__eq__iff,axiom,
    ! [F2: option_a > a,A2: set_option_a,B2: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ ( sup_sup_set_option_a @ A2 @ B2 ) )
     => ( ( ( image_option_a_a2 @ F2 @ A2 )
          = ( image_option_a_a2 @ F2 @ B2 ) )
        = ( A2 = B2 ) ) ) ).

% inj_on_Un_image_eq_iff
thf(fact_537_inj__on__Un__image__eq__iff,axiom,
    ! [F2: a > option_a,A2: set_a,B2: set_a] :
      ( ( inj_on_a_option_a @ F2 @ ( sup_sup_set_a @ A2 @ B2 ) )
     => ( ( ( image_a_option_a2 @ F2 @ A2 )
          = ( image_a_option_a2 @ F2 @ B2 ) )
        = ( A2 = B2 ) ) ) ).

% inj_on_Un_image_eq_iff
thf(fact_538_None__notin__image__Some,axiom,
    ! [A2: set_Pr5506164808955685639_a_d_c] :
      ~ ( member1874962706598213560_a_d_c @ none_P1533316314068730928_a_d_c @ ( image_2856777003974851449_a_d_c @ some_P377817780860425132_a_d_c @ A2 ) ) ).

% None_notin_image_Some
thf(fact_539_None__notin__image__Some,axiom,
    ! [A2: set_a] :
      ~ ( member_option_a @ none_a @ ( image_a_option_a2 @ some_a @ A2 ) ) ).

% None_notin_image_Some
thf(fact_540_inj__on__fun__updI,axiom,
    ! [F2: option_a > a,A2: set_option_a,Y: a,X3: option_a] :
      ( ( inj_on_option_a_a @ F2 @ A2 )
     => ( ~ ( member_a @ Y @ ( image_option_a_a2 @ F2 @ A2 ) )
       => ( inj_on_option_a_a @ ( fun_upd_option_a_a @ F2 @ X3 @ Y ) @ A2 ) ) ) ).

% inj_on_fun_updI
thf(fact_541_inj__on__fun__updI,axiom,
    ! [F2: a > option_a,A2: set_a,Y: option_a,X3: a] :
      ( ( inj_on_a_option_a @ F2 @ A2 )
     => ( ~ ( member_option_a @ Y @ ( image_a_option_a2 @ F2 @ A2 ) )
       => ( inj_on_a_option_a @ ( fun_upd_a_option_a @ F2 @ X3 @ Y ) @ A2 ) ) ) ).

% inj_on_fun_updI
thf(fact_542_inj__on__fun__updI,axiom,
    ! [F2: d > c,A2: set_d,Y: c,X3: d] :
      ( ( inj_on_d_c @ F2 @ A2 )
     => ( ~ ( member_c @ Y @ ( image_d_c @ F2 @ A2 ) )
       => ( inj_on_d_c @ ( fun_upd_d_c @ F2 @ X3 @ Y ) @ A2 ) ) ) ).

% inj_on_fun_updI
thf(fact_543_fun__upd__image,axiom,
    ! [X3: produc6150846815813599699_a_d_c,A2: set_Pr7868159745199425715_a_d_c,F2: produc6150846815813599699_a_d_c > a,Y: a] :
      ( ( ( member2052822272342364412_a_d_c @ X3 @ A2 )
       => ( ( image_653692332051308528_d_c_a @ ( fun_up3467885399696393320_d_c_a @ F2 @ X3 @ Y ) @ A2 )
          = ( insert_a @ Y @ ( image_653692332051308528_d_c_a @ F2 @ ( minus_1640812273454246426_a_d_c @ A2 @ ( insert4944988292672327587_a_d_c @ X3 @ bot_bo1396921257525504799_a_d_c ) ) ) ) ) )
      & ( ~ ( member2052822272342364412_a_d_c @ X3 @ A2 )
       => ( ( image_653692332051308528_d_c_a @ ( fun_up3467885399696393320_d_c_a @ F2 @ X3 @ Y ) @ A2 )
          = ( image_653692332051308528_d_c_a @ F2 @ A2 ) ) ) ) ).

% fun_upd_image
thf(fact_544_fun__upd__image,axiom,
    ! [X3: produc6150846815813599699_a_d_c,A2: set_Pr7868159745199425715_a_d_c,F2: produc6150846815813599699_a_d_c > option_a,Y: option_a] :
      ( ( ( member2052822272342364412_a_d_c @ X3 @ A2 )
       => ( ( image_5971021413422854966tion_a @ ( fun_up4051926500471473070tion_a @ F2 @ X3 @ Y ) @ A2 )
          = ( insert_option_a @ Y @ ( image_5971021413422854966tion_a @ F2 @ ( minus_1640812273454246426_a_d_c @ A2 @ ( insert4944988292672327587_a_d_c @ X3 @ bot_bo1396921257525504799_a_d_c ) ) ) ) ) )
      & ( ~ ( member2052822272342364412_a_d_c @ X3 @ A2 )
       => ( ( image_5971021413422854966tion_a @ ( fun_up4051926500471473070tion_a @ F2 @ X3 @ Y ) @ A2 )
          = ( image_5971021413422854966tion_a @ F2 @ A2 ) ) ) ) ).

% fun_upd_image
thf(fact_545_fun__upd__image,axiom,
    ! [X3: d,A2: set_d,F2: d > c,Y: c] :
      ( ( ( member_d @ X3 @ A2 )
       => ( ( image_d_c @ ( fun_upd_d_c @ F2 @ X3 @ Y ) @ A2 )
          = ( insert_c @ Y @ ( image_d_c @ F2 @ ( minus_minus_set_d @ A2 @ ( insert_d @ X3 @ bot_bot_set_d ) ) ) ) ) )
      & ( ~ ( member_d @ X3 @ A2 )
       => ( ( image_d_c @ ( fun_upd_d_c @ F2 @ X3 @ Y ) @ A2 )
          = ( image_d_c @ F2 @ A2 ) ) ) ) ).

% fun_upd_image
thf(fact_546_fun__upd__image,axiom,
    ! [X3: a,A2: set_a,F2: a > a,Y: a] :
      ( ( ( member_a @ X3 @ A2 )
       => ( ( image_a_a2 @ ( fun_upd_a_a @ F2 @ X3 @ Y ) @ A2 )
          = ( insert_a @ Y @ ( image_a_a2 @ F2 @ ( minus_minus_set_a @ A2 @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) )
      & ( ~ ( member_a @ X3 @ A2 )
       => ( ( image_a_a2 @ ( fun_upd_a_a @ F2 @ X3 @ Y ) @ A2 )
          = ( image_a_a2 @ F2 @ A2 ) ) ) ) ).

% fun_upd_image
thf(fact_547_fun__upd__image,axiom,
    ! [X3: a,A2: set_a,F2: a > option_a,Y: option_a] :
      ( ( ( member_a @ X3 @ A2 )
       => ( ( image_a_option_a2 @ ( fun_upd_a_option_a @ F2 @ X3 @ Y ) @ A2 )
          = ( insert_option_a @ Y @ ( image_a_option_a2 @ F2 @ ( minus_minus_set_a @ A2 @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) )
      & ( ~ ( member_a @ X3 @ A2 )
       => ( ( image_a_option_a2 @ ( fun_upd_a_option_a @ F2 @ X3 @ Y ) @ A2 )
          = ( image_a_option_a2 @ F2 @ A2 ) ) ) ) ).

% fun_upd_image
thf(fact_548_fun__upd__image,axiom,
    ! [X3: option_a,A2: set_option_a,F2: option_a > a,Y: a] :
      ( ( ( member_option_a @ X3 @ A2 )
       => ( ( image_option_a_a2 @ ( fun_upd_option_a_a @ F2 @ X3 @ Y ) @ A2 )
          = ( insert_a @ Y @ ( image_option_a_a2 @ F2 @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ X3 @ bot_bot_set_option_a ) ) ) ) ) )
      & ( ~ ( member_option_a @ X3 @ A2 )
       => ( ( image_option_a_a2 @ ( fun_upd_option_a_a @ F2 @ X3 @ Y ) @ A2 )
          = ( image_option_a_a2 @ F2 @ A2 ) ) ) ) ).

% fun_upd_image
thf(fact_549_fun__upd__image,axiom,
    ! [X3: option_a,A2: set_option_a,F2: option_a > option_a,Y: option_a] :
      ( ( ( member_option_a @ X3 @ A2 )
       => ( ( image_7439109396645324421tion_a @ ( fun_up1079276522633388797tion_a @ F2 @ X3 @ Y ) @ A2 )
          = ( insert_option_a @ Y @ ( image_7439109396645324421tion_a @ F2 @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ X3 @ bot_bot_set_option_a ) ) ) ) ) )
      & ( ~ ( member_option_a @ X3 @ A2 )
       => ( ( image_7439109396645324421tion_a @ ( fun_up1079276522633388797tion_a @ F2 @ X3 @ Y ) @ A2 )
          = ( image_7439109396645324421tion_a @ F2 @ A2 ) ) ) ) ).

% fun_upd_image
thf(fact_550_refl__on__domain,axiom,
    ! [A2: set_a,R2: set_Product_prod_a_a,A: a,B: a] :
      ( ( refl_on_a @ A2 @ R2 )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
       => ( ( member_a @ A @ A2 )
          & ( member_a @ B @ A2 ) ) ) ) ).

% refl_on_domain
thf(fact_551_refl__on__domain,axiom,
    ! [A2: set_Pr7868159745199425715_a_d_c,R2: set_Pr8390707888732215751_a_d_c,A: produc6150846815813599699_a_d_c,B: produc6150846815813599699_a_d_c] :
      ( ( refl_o1599096589449526226_a_d_c @ A2 @ R2 )
     => ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ A @ B ) @ R2 )
       => ( ( member2052822272342364412_a_d_c @ A @ A2 )
          & ( member2052822272342364412_a_d_c @ B @ A2 ) ) ) ) ).

% refl_on_domain
thf(fact_552_refl__on__domain,axiom,
    ! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
      ( ( refl_on_option_a @ A2 @ R2 )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 )
       => ( ( member_option_a @ A @ A2 )
          & ( member_option_a @ B @ A2 ) ) ) ) ).

% refl_on_domain
thf(fact_553_linear__order__on__singleton,axiom,
    ! [X3: a] : ( order_8768733634509060147r_on_a @ ( insert_a @ X3 @ bot_bot_set_a ) @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ X3 @ X3 ) @ bot_bo3357376287454694259od_a_a ) ) ).

% linear_order_on_singleton
thf(fact_554_linear__order__on__singleton,axiom,
    ! [X3: option_a] : ( order_7850372301378808569tion_a @ ( insert_option_a @ X3 @ bot_bot_set_option_a ) @ ( insert1246254401036548087tion_a @ ( produc9011544418120257559tion_a @ X3 @ X3 ) @ bot_bo235252021745139059tion_a ) ) ).

% linear_order_on_singleton
thf(fact_555_the__inv__into__onto,axiom,
    ! [F2: option_a > a,A2: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ A2 )
     => ( ( image_a_option_a2 @ ( the_in1757154643552616557on_a_a @ A2 @ F2 ) @ ( image_option_a_a2 @ F2 @ A2 ) )
        = A2 ) ) ).

% the_inv_into_onto
thf(fact_556_the__inv__into__onto,axiom,
    ! [F2: a > option_a,A2: set_a] :
      ( ( inj_on_a_option_a @ F2 @ A2 )
     => ( ( image_option_a_a2 @ ( the_in8758012798868597241tion_a @ A2 @ F2 ) @ ( image_a_option_a2 @ F2 @ A2 ) )
        = A2 ) ) ).

% the_inv_into_onto
thf(fact_557_image__set__diff,axiom,
    ! [F2: option_a > a,A2: set_option_a,B2: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ top_top_set_option_a )
     => ( ( image_option_a_a2 @ F2 @ ( minus_1574173051537231627tion_a @ A2 @ B2 ) )
        = ( minus_minus_set_a @ ( image_option_a_a2 @ F2 @ A2 ) @ ( image_option_a_a2 @ F2 @ B2 ) ) ) ) ).

% image_set_diff
thf(fact_558_image__set__diff,axiom,
    ! [F2: a > option_a,A2: set_a,B2: set_a] :
      ( ( inj_on_a_option_a @ F2 @ top_top_set_a )
     => ( ( image_a_option_a2 @ F2 @ ( minus_minus_set_a @ A2 @ B2 ) )
        = ( minus_1574173051537231627tion_a @ ( image_a_option_a2 @ F2 @ A2 ) @ ( image_a_option_a2 @ F2 @ B2 ) ) ) ) ).

% image_set_diff
thf(fact_559_the__inv__f__f,axiom,
    ! [F2: a > option_a,X3: a] :
      ( ( inj_on_a_option_a @ F2 @ top_top_set_a )
     => ( ( the_in8758012798868597241tion_a @ top_top_set_a @ F2 @ ( F2 @ X3 ) )
        = X3 ) ) ).

% the_inv_f_f
thf(fact_560_surj__def,axiom,
    ! [F2: option_a > option_a] :
      ( ( ( image_7439109396645324421tion_a @ F2 @ top_top_set_option_a )
        = top_top_set_option_a )
      = ( ! [Y3: option_a] :
          ? [X: option_a] :
            ( Y3
            = ( F2 @ X ) ) ) ) ).

% surj_def
thf(fact_561_surj__def,axiom,
    ! [F2: option_a > a] :
      ( ( ( image_option_a_a2 @ F2 @ top_top_set_option_a )
        = top_top_set_a )
      = ( ! [Y3: a] :
          ? [X: option_a] :
            ( Y3
            = ( F2 @ X ) ) ) ) ).

% surj_def
thf(fact_562_surj__def,axiom,
    ! [F2: a > option_a] :
      ( ( ( image_a_option_a2 @ F2 @ top_top_set_a )
        = top_top_set_option_a )
      = ( ! [Y3: option_a] :
          ? [X: a] :
            ( Y3
            = ( F2 @ X ) ) ) ) ).

% surj_def
thf(fact_563_surj__def,axiom,
    ! [F2: a > a] :
      ( ( ( image_a_a2 @ F2 @ top_top_set_a )
        = top_top_set_a )
      = ( ! [Y3: a] :
          ? [X: a] :
            ( Y3
            = ( F2 @ X ) ) ) ) ).

% surj_def
thf(fact_564_surjI,axiom,
    ! [G: option_a > option_a,F2: option_a > option_a] :
      ( ! [X2: option_a] :
          ( ( G @ ( F2 @ X2 ) )
          = X2 )
     => ( ( image_7439109396645324421tion_a @ G @ top_top_set_option_a )
        = top_top_set_option_a ) ) ).

% surjI
thf(fact_565_surjI,axiom,
    ! [G: option_a > a,F2: a > option_a] :
      ( ! [X2: a] :
          ( ( G @ ( F2 @ X2 ) )
          = X2 )
     => ( ( image_option_a_a2 @ G @ top_top_set_option_a )
        = top_top_set_a ) ) ).

% surjI
thf(fact_566_surjI,axiom,
    ! [G: a > option_a,F2: option_a > a] :
      ( ! [X2: option_a] :
          ( ( G @ ( F2 @ X2 ) )
          = X2 )
     => ( ( image_a_option_a2 @ G @ top_top_set_a )
        = top_top_set_option_a ) ) ).

% surjI
thf(fact_567_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_568_surjE,axiom,
    ! [F2: option_a > option_a,Y: option_a] :
      ( ( ( image_7439109396645324421tion_a @ F2 @ top_top_set_option_a )
        = top_top_set_option_a )
     => ~ ! [X2: option_a] :
            ( Y
           != ( F2 @ X2 ) ) ) ).

% surjE
thf(fact_569_surjE,axiom,
    ! [F2: option_a > a,Y: a] :
      ( ( ( image_option_a_a2 @ F2 @ top_top_set_option_a )
        = top_top_set_a )
     => ~ ! [X2: option_a] :
            ( Y
           != ( F2 @ X2 ) ) ) ).

% surjE
thf(fact_570_surjE,axiom,
    ! [F2: a > option_a,Y: option_a] :
      ( ( ( image_a_option_a2 @ F2 @ top_top_set_a )
        = top_top_set_option_a )
     => ~ ! [X2: a] :
            ( Y
           != ( F2 @ X2 ) ) ) ).

% surjE
thf(fact_571_surjE,axiom,
    ! [F2: a > a,Y: a] :
      ( ( ( image_a_a2 @ F2 @ top_top_set_a )
        = top_top_set_a )
     => ~ ! [X2: a] :
            ( Y
           != ( F2 @ X2 ) ) ) ).

% surjE
thf(fact_572_surjD,axiom,
    ! [F2: option_a > option_a,Y: option_a] :
      ( ( ( image_7439109396645324421tion_a @ F2 @ top_top_set_option_a )
        = top_top_set_option_a )
     => ? [X2: option_a] :
          ( Y
          = ( F2 @ X2 ) ) ) ).

% surjD
thf(fact_573_surjD,axiom,
    ! [F2: option_a > a,Y: a] :
      ( ( ( image_option_a_a2 @ F2 @ top_top_set_option_a )
        = top_top_set_a )
     => ? [X2: option_a] :
          ( Y
          = ( F2 @ X2 ) ) ) ).

% surjD
thf(fact_574_surjD,axiom,
    ! [F2: a > option_a,Y: option_a] :
      ( ( ( image_a_option_a2 @ F2 @ top_top_set_a )
        = top_top_set_option_a )
     => ? [X2: a] :
          ( Y
          = ( F2 @ X2 ) ) ) ).

% surjD
thf(fact_575_surjD,axiom,
    ! [F2: a > a,Y: a] :
      ( ( ( image_a_a2 @ F2 @ top_top_set_a )
        = top_top_set_a )
     => ? [X2: a] :
          ( Y
          = ( F2 @ X2 ) ) ) ).

% surjD
thf(fact_576_injD,axiom,
    ! [F2: a > option_a,X3: a,Y: a] :
      ( ( inj_on_a_option_a @ F2 @ top_top_set_a )
     => ( ( ( F2 @ X3 )
          = ( F2 @ Y ) )
       => ( X3 = Y ) ) ) ).

% injD
thf(fact_577_injI,axiom,
    ! [F2: a > option_a] :
      ( ! [X2: a,Y4: a] :
          ( ( ( F2 @ X2 )
            = ( F2 @ Y4 ) )
         => ( X2 = Y4 ) )
     => ( inj_on_a_option_a @ F2 @ top_top_set_a ) ) ).

% injI
thf(fact_578_inj__eq,axiom,
    ! [F2: a > option_a,X3: a,Y: a] :
      ( ( inj_on_a_option_a @ F2 @ top_top_set_a )
     => ( ( ( F2 @ X3 )
          = ( F2 @ Y ) )
        = ( X3 = Y ) ) ) ).

% inj_eq
thf(fact_579_inj__def,axiom,
    ! [F2: a > option_a] :
      ( ( inj_on_a_option_a @ F2 @ top_top_set_a )
      = ( ! [X: a,Y3: a] :
            ( ( ( F2 @ X )
              = ( F2 @ Y3 ) )
           => ( X = Y3 ) ) ) ) ).

% inj_def
thf(fact_580_the__inv__into__f__f,axiom,
    ! [F2: a > option_a,A2: set_a,X3: a] :
      ( ( inj_on_a_option_a @ F2 @ A2 )
     => ( ( member_a @ X3 @ A2 )
       => ( ( the_in8758012798868597241tion_a @ A2 @ F2 @ ( F2 @ X3 ) )
          = X3 ) ) ) ).

% the_inv_into_f_f
thf(fact_581_the__inv__into__f__eq,axiom,
    ! [F2: a > option_a,A2: set_a,X3: a,Y: option_a] :
      ( ( inj_on_a_option_a @ F2 @ A2 )
     => ( ( ( F2 @ X3 )
          = Y )
       => ( ( member_a @ X3 @ A2 )
         => ( ( the_in8758012798868597241tion_a @ A2 @ F2 @ Y )
            = X3 ) ) ) ) ).

% the_inv_into_f_eq
thf(fact_582_UNIV__option__conv,axiom,
    ( top_to2552133556407834375_a_d_c
    = ( insert2578358782419847377_a_d_c @ none_P1533316314068730928_a_d_c @ ( image_2856777003974851449_a_d_c @ some_P377817780860425132_a_d_c @ top_to1491896916071365047_a_d_c ) ) ) ).

% UNIV_option_conv
thf(fact_583_UNIV__option__conv,axiom,
    ( top_to1659475022456381882tion_a
    = ( insert605063979879581146tion_a @ none_option_a @ ( image_2132136900116418507tion_a @ some_option_a @ top_top_set_option_a ) ) ) ).

% UNIV_option_conv
thf(fact_584_UNIV__option__conv,axiom,
    ( top_top_set_option_a
    = ( insert_option_a @ none_a @ ( image_a_option_a2 @ some_a @ top_top_set_a ) ) ) ).

% UNIV_option_conv
thf(fact_585_inj__image__mem__iff,axiom,
    ! [F2: produc6150846815813599699_a_d_c > a,A: produc6150846815813599699_a_d_c,A2: set_Pr7868159745199425715_a_d_c] :
      ( ( inj_on6701986109746958172_d_c_a @ F2 @ top_to140874604318701059_a_d_c )
     => ( ( member_a @ ( F2 @ A ) @ ( image_653692332051308528_d_c_a @ F2 @ A2 ) )
        = ( member2052822272342364412_a_d_c @ A @ A2 ) ) ) ).

% inj_image_mem_iff
thf(fact_586_inj__image__mem__iff,axiom,
    ! [F2: produc6150846815813599699_a_d_c > produc6150846815813599699_a_d_c,A: produc6150846815813599699_a_d_c,A2: set_Pr7868159745199425715_a_d_c] :
      ( ( inj_on5053348487737048625_a_d_c @ F2 @ top_to140874604318701059_a_d_c )
     => ( ( member2052822272342364412_a_d_c @ ( F2 @ A ) @ ( image_548751279113285317_a_d_c @ F2 @ A2 ) )
        = ( member2052822272342364412_a_d_c @ A @ A2 ) ) ) ).

% inj_image_mem_iff
thf(fact_587_inj__image__mem__iff,axiom,
    ! [F2: produc6150846815813599699_a_d_c > option_a,A: produc6150846815813599699_a_d_c,A2: set_Pr7868159745199425715_a_d_c] :
      ( ( inj_on3641466449436709026tion_a @ F2 @ top_to140874604318701059_a_d_c )
     => ( ( member_option_a @ ( F2 @ A ) @ ( image_5971021413422854966tion_a @ F2 @ A2 ) )
        = ( member2052822272342364412_a_d_c @ A @ A2 ) ) ) ).

% inj_image_mem_iff
thf(fact_588_inj__image__mem__iff,axiom,
    ! [F2: option_a > a,A: option_a,A2: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ top_top_set_option_a )
     => ( ( member_a @ ( F2 @ A ) @ ( image_option_a_a2 @ F2 @ A2 ) )
        = ( member_option_a @ A @ A2 ) ) ) ).

% inj_image_mem_iff
thf(fact_589_inj__image__mem__iff,axiom,
    ! [F2: option_a > produc6150846815813599699_a_d_c,A: option_a,A2: set_option_a] :
      ( ( inj_on7085348629561674240_a_d_c @ F2 @ top_top_set_option_a )
     => ( ( member2052822272342364412_a_d_c @ ( F2 @ A ) @ ( image_191531556693044372_a_d_c @ F2 @ A2 ) )
        = ( member_option_a @ A @ A2 ) ) ) ).

% inj_image_mem_iff
thf(fact_590_inj__image__mem__iff,axiom,
    ! [F2: option_a > option_a,A: option_a,A2: set_option_a] :
      ( ( inj_on8559383841115902449tion_a @ F2 @ top_top_set_option_a )
     => ( ( member_option_a @ ( F2 @ A ) @ ( image_7439109396645324421tion_a @ F2 @ A2 ) )
        = ( member_option_a @ A @ A2 ) ) ) ).

% inj_image_mem_iff
thf(fact_591_inj__image__mem__iff,axiom,
    ! [F2: a > a,A: a,A2: set_a] :
      ( ( inj_on_a_a @ F2 @ top_top_set_a )
     => ( ( member_a @ ( F2 @ A ) @ ( image_a_a2 @ F2 @ A2 ) )
        = ( member_a @ A @ A2 ) ) ) ).

% inj_image_mem_iff
thf(fact_592_inj__image__mem__iff,axiom,
    ! [F2: a > produc6150846815813599699_a_d_c,A: a,A2: set_a] :
      ( ( inj_on3709328703910593990_a_d_c @ F2 @ top_top_set_a )
     => ( ( member2052822272342364412_a_d_c @ ( F2 @ A ) @ ( image_6884406963069720154_a_d_c @ F2 @ A2 ) )
        = ( member_a @ A @ A2 ) ) ) ).

% inj_image_mem_iff
thf(fact_593_inj__image__mem__iff,axiom,
    ! [F2: a > option_a,A: a,A2: set_a] :
      ( ( inj_on_a_option_a @ F2 @ top_top_set_a )
     => ( ( member_option_a @ ( F2 @ A ) @ ( image_a_option_a2 @ F2 @ A2 ) )
        = ( member_a @ A @ A2 ) ) ) ).

% inj_image_mem_iff
thf(fact_594_inj__image__eq__iff,axiom,
    ! [F2: option_a > a,A2: set_option_a,B2: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ top_top_set_option_a )
     => ( ( ( image_option_a_a2 @ F2 @ A2 )
          = ( image_option_a_a2 @ F2 @ B2 ) )
        = ( A2 = B2 ) ) ) ).

% inj_image_eq_iff
thf(fact_595_inj__image__eq__iff,axiom,
    ! [F2: a > option_a,A2: set_a,B2: set_a] :
      ( ( inj_on_a_option_a @ F2 @ top_top_set_a )
     => ( ( ( image_a_option_a2 @ F2 @ A2 )
          = ( image_a_option_a2 @ F2 @ B2 ) )
        = ( A2 = B2 ) ) ) ).

% inj_image_eq_iff
thf(fact_596_range__ex1__eq,axiom,
    ! [F2: option_a > a,B: a] :
      ( ( inj_on_option_a_a @ F2 @ top_top_set_option_a )
     => ( ( member_a @ B @ ( image_option_a_a2 @ F2 @ top_top_set_option_a ) )
        = ( ? [X: option_a] :
              ( ( B
                = ( F2 @ X ) )
              & ! [Y3: option_a] :
                  ( ( B
                    = ( F2 @ Y3 ) )
                 => ( Y3 = X ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_597_range__ex1__eq,axiom,
    ! [F2: option_a > produc6150846815813599699_a_d_c,B: produc6150846815813599699_a_d_c] :
      ( ( inj_on7085348629561674240_a_d_c @ F2 @ top_top_set_option_a )
     => ( ( member2052822272342364412_a_d_c @ B @ ( image_191531556693044372_a_d_c @ F2 @ top_top_set_option_a ) )
        = ( ? [X: option_a] :
              ( ( B
                = ( F2 @ X ) )
              & ! [Y3: option_a] :
                  ( ( B
                    = ( F2 @ Y3 ) )
                 => ( Y3 = X ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_598_range__ex1__eq,axiom,
    ! [F2: option_a > option_a,B: option_a] :
      ( ( inj_on8559383841115902449tion_a @ F2 @ top_top_set_option_a )
     => ( ( member_option_a @ B @ ( image_7439109396645324421tion_a @ F2 @ top_top_set_option_a ) )
        = ( ? [X: option_a] :
              ( ( B
                = ( F2 @ X ) )
              & ! [Y3: option_a] :
                  ( ( B
                    = ( F2 @ Y3 ) )
                 => ( Y3 = X ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_599_range__ex1__eq,axiom,
    ! [F2: a > a,B: a] :
      ( ( inj_on_a_a @ F2 @ top_top_set_a )
     => ( ( member_a @ B @ ( image_a_a2 @ F2 @ top_top_set_a ) )
        = ( ? [X: a] :
              ( ( B
                = ( F2 @ X ) )
              & ! [Y3: a] :
                  ( ( B
                    = ( F2 @ Y3 ) )
                 => ( Y3 = X ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_600_range__ex1__eq,axiom,
    ! [F2: a > produc6150846815813599699_a_d_c,B: produc6150846815813599699_a_d_c] :
      ( ( inj_on3709328703910593990_a_d_c @ F2 @ top_top_set_a )
     => ( ( member2052822272342364412_a_d_c @ B @ ( image_6884406963069720154_a_d_c @ F2 @ top_top_set_a ) )
        = ( ? [X: a] :
              ( ( B
                = ( F2 @ X ) )
              & ! [Y3: a] :
                  ( ( B
                    = ( F2 @ Y3 ) )
                 => ( Y3 = X ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_601_range__ex1__eq,axiom,
    ! [F2: a > option_a,B: option_a] :
      ( ( inj_on_a_option_a @ F2 @ top_top_set_a )
     => ( ( member_option_a @ B @ ( image_a_option_a2 @ F2 @ top_top_set_a ) )
        = ( ? [X: a] :
              ( ( B
                = ( F2 @ X ) )
              & ! [Y3: a] :
                  ( ( B
                    = ( F2 @ Y3 ) )
                 => ( Y3 = X ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_602_notin__range__Some,axiom,
    ! [X3: option2362401199305441953_a_d_c] :
      ( ( ~ ( member1874962706598213560_a_d_c @ X3 @ ( image_2856777003974851449_a_d_c @ some_P377817780860425132_a_d_c @ top_to1491896916071365047_a_d_c ) ) )
      = ( X3 = none_P1533316314068730928_a_d_c ) ) ).

% notin_range_Some
thf(fact_603_notin__range__Some,axiom,
    ! [X3: option_option_a] :
      ( ( ~ ( member5113800082084363315tion_a @ X3 @ ( image_2132136900116418507tion_a @ some_option_a @ top_top_set_option_a ) ) )
      = ( X3 = none_option_a ) ) ).

% notin_range_Some
thf(fact_604_notin__range__Some,axiom,
    ! [X3: option_a] :
      ( ( ~ ( member_option_a @ X3 @ ( image_a_option_a2 @ some_a @ top_top_set_a ) ) )
      = ( X3 = none_a ) ) ).

% notin_range_Some
thf(fact_605_reflI,axiom,
    ! [R2: set_Pr7585778909603769095tion_a] :
      ( ! [X2: option_a] : ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ X2 ) @ R2 )
     => ( refl_on_option_a @ top_top_set_option_a @ R2 ) ) ).

% reflI
thf(fact_606_reflI,axiom,
    ! [R2: set_Product_prod_a_a] :
      ( ! [X2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ X2 ) @ R2 )
     => ( refl_on_a @ top_top_set_a @ R2 ) ) ).

% reflI
thf(fact_607_reflD,axiom,
    ! [R2: set_Pr7585778909603769095tion_a,A: option_a] :
      ( ( refl_on_option_a @ top_top_set_option_a @ R2 )
     => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ A ) @ R2 ) ) ).

% reflD
thf(fact_608_reflD,axiom,
    ! [R2: set_Product_prod_a_a,A: a] :
      ( ( refl_on_a @ top_top_set_a @ R2 )
     => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A ) @ R2 ) ) ).

% reflD
thf(fact_609_inj__on__the__inv__into,axiom,
    ! [F2: option_a > a,A2: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ A2 )
     => ( inj_on_a_option_a @ ( the_in1757154643552616557on_a_a @ A2 @ F2 ) @ ( image_option_a_a2 @ F2 @ A2 ) ) ) ).

% inj_on_the_inv_into
thf(fact_610_inj__on__the__inv__into,axiom,
    ! [F2: a > option_a,A2: set_a] :
      ( ( inj_on_a_option_a @ F2 @ A2 )
     => ( inj_on_option_a_a @ ( the_in8758012798868597241tion_a @ A2 @ F2 ) @ ( image_a_option_a2 @ F2 @ A2 ) ) ) ).

% inj_on_the_inv_into
thf(fact_611_f__the__inv__into__f,axiom,
    ! [F2: option_a > a,A2: set_option_a,Y: a] :
      ( ( inj_on_option_a_a @ F2 @ A2 )
     => ( ( member_a @ Y @ ( image_option_a_a2 @ F2 @ A2 ) )
       => ( ( F2 @ ( the_in1757154643552616557on_a_a @ A2 @ F2 @ Y ) )
          = Y ) ) ) ).

% f_the_inv_into_f
thf(fact_612_f__the__inv__into__f,axiom,
    ! [F2: a > option_a,A2: set_a,Y: option_a] :
      ( ( inj_on_a_option_a @ F2 @ A2 )
     => ( ( member_option_a @ Y @ ( image_a_option_a2 @ F2 @ A2 ) )
       => ( ( F2 @ ( the_in8758012798868597241tion_a @ A2 @ F2 @ Y ) )
          = Y ) ) ) ).

% f_the_inv_into_f
thf(fact_613_image__Int,axiom,
    ! [F2: option_a > a,A2: set_option_a,B2: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ top_top_set_option_a )
     => ( ( image_option_a_a2 @ F2 @ ( inf_inf_set_option_a @ A2 @ B2 ) )
        = ( inf_inf_set_a @ ( image_option_a_a2 @ F2 @ A2 ) @ ( image_option_a_a2 @ F2 @ B2 ) ) ) ) ).

% image_Int
thf(fact_614_image__Int,axiom,
    ! [F2: a > option_a,A2: set_a,B2: set_a] :
      ( ( inj_on_a_option_a @ F2 @ top_top_set_a )
     => ( ( image_a_option_a2 @ F2 @ ( inf_inf_set_a @ A2 @ B2 ) )
        = ( inf_inf_set_option_a @ ( image_a_option_a2 @ F2 @ A2 ) @ ( image_a_option_a2 @ F2 @ B2 ) ) ) ) ).

% image_Int
thf(fact_615_finite__range__updI,axiom,
    ! [F2: option_a > option_a,A: option_a,B: a] :
      ( ( finite1674126218327898605tion_a @ ( image_7439109396645324421tion_a @ F2 @ top_top_set_option_a ) )
     => ( finite1674126218327898605tion_a @ ( image_7439109396645324421tion_a @ ( fun_up1079276522633388797tion_a @ F2 @ A @ ( some_a @ B ) ) @ top_top_set_option_a ) ) ) ).

% finite_range_updI
thf(fact_616_finite__range__updI,axiom,
    ! [F2: option_a > option2362401199305441953_a_d_c,A: option_a,B: product_prod_a_d_c] :
      ( ( finite8199294546587094200_a_d_c @ ( image_8868864280970703392_a_d_c @ F2 @ top_top_set_option_a ) )
     => ( finite8199294546587094200_a_d_c @ ( image_8868864280970703392_a_d_c @ ( fun_up1917685652830187432_a_d_c @ F2 @ A @ ( some_P377817780860425132_a_d_c @ B ) ) @ top_top_set_option_a ) ) ) ).

% finite_range_updI
thf(fact_617_finite__range__updI,axiom,
    ! [F2: a > option_a,A: a,B: a] :
      ( ( finite1674126218327898605tion_a @ ( image_a_option_a2 @ F2 @ top_top_set_a ) )
     => ( finite1674126218327898605tion_a @ ( image_a_option_a2 @ ( fun_upd_a_option_a @ F2 @ A @ ( some_a @ B ) ) @ top_top_set_a ) ) ) ).

% finite_range_updI
thf(fact_618_finite__range__updI,axiom,
    ! [F2: a > option2362401199305441953_a_d_c,A: a,B: product_prod_a_d_c] :
      ( ( finite8199294546587094200_a_d_c @ ( image_8958380359779546586_a_d_c @ F2 @ top_top_set_a ) )
     => ( finite8199294546587094200_a_d_c @ ( image_8958380359779546586_a_d_c @ ( fun_up2590722781756628834_a_d_c @ F2 @ A @ ( some_P377817780860425132_a_d_c @ B ) ) @ top_top_set_a ) ) ) ).

% finite_range_updI
thf(fact_619_inj__image__Compl__subset,axiom,
    ! [F2: option_a > a,A2: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ top_top_set_option_a )
     => ( ord_less_eq_set_a @ ( image_option_a_a2 @ F2 @ ( uminus6205308855922866075tion_a @ A2 ) ) @ ( uminus_uminus_set_a @ ( image_option_a_a2 @ F2 @ A2 ) ) ) ) ).

% inj_image_Compl_subset
thf(fact_620_inj__image__Compl__subset,axiom,
    ! [F2: a > option_a,A2: set_a] :
      ( ( inj_on_a_option_a @ F2 @ top_top_set_a )
     => ( ord_le1955136853071979460tion_a @ ( image_a_option_a2 @ F2 @ ( uminus_uminus_set_a @ A2 ) ) @ ( uminus6205308855922866075tion_a @ ( image_a_option_a2 @ F2 @ A2 ) ) ) ) ).

% inj_image_Compl_subset
thf(fact_621_trans__singleton,axiom,
    ! [A: option_a] : ( trans_on_option_a @ top_top_set_option_a @ ( insert1246254401036548087tion_a @ ( produc9011544418120257559tion_a @ A @ A ) @ bot_bo235252021745139059tion_a ) ) ).

% trans_singleton
thf(fact_622_trans__singleton,axiom,
    ! [A: a] : ( trans_on_a @ top_top_set_a @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ A @ A ) @ bot_bo3357376287454694259od_a_a ) ) ).

% trans_singleton
thf(fact_623_Linear__order__in__diff__Id,axiom,
    ! [R2: set_Product_prod_a_a,A: a,B: a] :
      ( ( order_8768733634509060147r_on_a @ ( field_a @ R2 ) @ R2 )
     => ( ( member_a @ A @ ( field_a @ R2 ) )
       => ( ( member_a @ B @ ( field_a @ R2 ) )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
            = ( ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ A ) @ ( minus_6817036919807184750od_a_a @ R2 @ id_a2 ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_624_Linear__order__in__diff__Id,axiom,
    ! [R2: set_Pr8390707888732215751_a_d_c,A: produc6150846815813599699_a_d_c,B: produc6150846815813599699_a_d_c] :
      ( ( order_2462247862883012360_a_d_c @ ( field_6537103990374211872_a_d_c @ R2 ) @ R2 )
     => ( ( member2052822272342364412_a_d_c @ A @ ( field_6537103990374211872_a_d_c @ R2 ) )
       => ( ( member2052822272342364412_a_d_c @ B @ ( field_6537103990374211872_a_d_c @ R2 ) )
         => ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ A @ B ) @ R2 )
            = ( ~ ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ B @ A ) @ ( minus_1498755887985271854_a_d_c @ R2 @ id_Pro269462521913453359_a_d_c ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_625_Linear__order__in__diff__Id,axiom,
    ! [R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
      ( ( order_7850372301378808569tion_a @ ( field_option_a @ R2 ) @ R2 )
     => ( ( member_option_a @ A @ ( field_option_a @ R2 ) )
       => ( ( member_option_a @ B @ ( field_option_a @ R2 ) )
         => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 )
            = ( ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ A ) @ ( minus_6512073291116468334tion_a @ R2 @ id_option_a2 ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_626_finite__option__UNIV,axiom,
    ( ( finite8114217219359860531tion_a @ top_to1659475022456381882tion_a )
    = ( finite1674126218327898605tion_a @ top_top_set_option_a ) ) ).

% finite_option_UNIV
thf(fact_627_finite__option__UNIV,axiom,
    ( ( finite1674126218327898605tion_a @ top_top_set_option_a )
    = ( finite_finite_a @ top_top_set_a ) ) ).

% finite_option_UNIV
thf(fact_628_subrelI,axiom,
    ! [R2: set_Pr7868159745199425715_a_d_c,S4: set_Pr7868159745199425715_a_d_c] :
      ( ! [X2: product_prod_a_d_c,Y4: option2362401199305441953_a_d_c] :
          ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ X2 @ Y4 ) @ R2 )
         => ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ X2 @ Y4 ) @ S4 ) )
     => ( ord_le3761643085089054803_a_d_c @ R2 @ S4 ) ) ).

% subrelI
thf(fact_629_subrelI,axiom,
    ! [R2: set_Pr5506164808955685639_a_d_c,S4: set_Pr5506164808955685639_a_d_c] :
      ( ! [X2: a,Y4: d > c] :
          ( ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ X2 @ Y4 ) @ R2 )
         => ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ X2 @ Y4 ) @ S4 ) )
     => ( ord_le1047748603589151591_a_d_c @ R2 @ S4 ) ) ).

% subrelI
thf(fact_630_trans__onD,axiom,
    ! [A2: set_a,R2: set_Product_prod_a_a,X3: a,Y: a,Z2: a] :
      ( ( trans_on_a @ A2 @ R2 )
     => ( ( member_a @ X3 @ A2 )
       => ( ( member_a @ Y @ A2 )
         => ( ( member_a @ Z2 @ A2 )
           => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R2 )
             => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z2 ) @ R2 )
               => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Z2 ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_631_trans__onD,axiom,
    ! [A2: set_Pr7868159745199425715_a_d_c,R2: set_Pr8390707888732215751_a_d_c,X3: produc6150846815813599699_a_d_c,Y: produc6150846815813599699_a_d_c,Z2: produc6150846815813599699_a_d_c] :
      ( ( trans_3980220593905352563_a_d_c @ A2 @ R2 )
     => ( ( member2052822272342364412_a_d_c @ X3 @ A2 )
       => ( ( member2052822272342364412_a_d_c @ Y @ A2 )
         => ( ( member2052822272342364412_a_d_c @ Z2 @ A2 )
           => ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ X3 @ Y ) @ R2 )
             => ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ Y @ Z2 ) @ R2 )
               => ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ X3 @ Z2 ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_632_trans__onD,axiom,
    ! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,X3: option_a,Y: option_a,Z2: option_a] :
      ( ( trans_on_option_a @ A2 @ R2 )
     => ( ( member_option_a @ X3 @ A2 )
       => ( ( member_option_a @ Y @ A2 )
         => ( ( member_option_a @ Z2 @ A2 )
           => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y ) @ R2 )
             => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y @ Z2 ) @ R2 )
               => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Z2 ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_633_trans__onI,axiom,
    ! [A2: set_a,R2: set_Product_prod_a_a] :
      ( ! [X2: a,Y4: a,Z3: a] :
          ( ( member_a @ X2 @ A2 )
         => ( ( member_a @ Y4 @ A2 )
           => ( ( member_a @ Z3 @ A2 )
             => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y4 ) @ R2 )
               => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ Z3 ) @ R2 )
                 => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Z3 ) @ R2 ) ) ) ) ) )
     => ( trans_on_a @ A2 @ R2 ) ) ).

% trans_onI
thf(fact_634_trans__onI,axiom,
    ! [A2: set_Pr7868159745199425715_a_d_c,R2: set_Pr8390707888732215751_a_d_c] :
      ( ! [X2: produc6150846815813599699_a_d_c,Y4: produc6150846815813599699_a_d_c,Z3: produc6150846815813599699_a_d_c] :
          ( ( member2052822272342364412_a_d_c @ X2 @ A2 )
         => ( ( member2052822272342364412_a_d_c @ Y4 @ A2 )
           => ( ( member2052822272342364412_a_d_c @ Z3 @ A2 )
             => ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ X2 @ Y4 ) @ R2 )
               => ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ Y4 @ Z3 ) @ R2 )
                 => ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ X2 @ Z3 ) @ R2 ) ) ) ) ) )
     => ( trans_3980220593905352563_a_d_c @ A2 @ R2 ) ) ).

% trans_onI
thf(fact_635_trans__onI,axiom,
    ! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a] :
      ( ! [X2: option_a,Y4: option_a,Z3: option_a] :
          ( ( member_option_a @ X2 @ A2 )
         => ( ( member_option_a @ Y4 @ A2 )
           => ( ( member_option_a @ Z3 @ A2 )
             => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y4 ) @ R2 )
               => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y4 @ Z3 ) @ R2 )
                 => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Z3 ) @ R2 ) ) ) ) ) )
     => ( trans_on_option_a @ A2 @ R2 ) ) ).

% trans_onI
thf(fact_636_inj__on__subset,axiom,
    ! [F2: a > option_a,A2: set_a,B2: set_a] :
      ( ( inj_on_a_option_a @ F2 @ A2 )
     => ( ( ord_less_eq_set_a @ B2 @ A2 )
       => ( inj_on_a_option_a @ F2 @ B2 ) ) ) ).

% inj_on_subset
thf(fact_637_subset__inj__on,axiom,
    ! [F2: a > option_a,B2: set_a,A2: set_a] :
      ( ( inj_on_a_option_a @ F2 @ B2 )
     => ( ( ord_less_eq_set_a @ A2 @ B2 )
       => ( inj_on_a_option_a @ F2 @ A2 ) ) ) ).

% subset_inj_on
thf(fact_638_finite__range__Some,axiom,
    ( ( finite8199294546587094200_a_d_c @ ( image_2856777003974851449_a_d_c @ some_P377817780860425132_a_d_c @ top_to1491896916071365047_a_d_c ) )
    = ( finite5780817786756972904_a_d_c @ top_to1491896916071365047_a_d_c ) ) ).

% finite_range_Some
thf(fact_639_finite__range__Some,axiom,
    ( ( finite8114217219359860531tion_a @ ( image_2132136900116418507tion_a @ some_option_a @ top_top_set_option_a ) )
    = ( finite1674126218327898605tion_a @ top_top_set_option_a ) ) ).

% finite_range_Some
thf(fact_640_finite__range__Some,axiom,
    ( ( finite1674126218327898605tion_a @ ( image_a_option_a2 @ some_a @ top_top_set_a ) )
    = ( finite_finite_a @ top_top_set_a ) ) ).

% finite_range_Some
thf(fact_641_transD,axiom,
    ! [R2: set_Pr7585778909603769095tion_a,X3: option_a,Y: option_a,Z2: option_a] :
      ( ( trans_on_option_a @ top_top_set_option_a @ R2 )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y ) @ R2 )
       => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y @ Z2 ) @ R2 )
         => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Z2 ) @ R2 ) ) ) ) ).

% transD
thf(fact_642_transD,axiom,
    ! [R2: set_Product_prod_a_a,X3: a,Y: a,Z2: a] :
      ( ( trans_on_a @ top_top_set_a @ R2 )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R2 )
       => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z2 ) @ R2 )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Z2 ) @ R2 ) ) ) ) ).

% transD
thf(fact_643_transE,axiom,
    ! [R2: set_Pr7585778909603769095tion_a,X3: option_a,Y: option_a,Z2: option_a] :
      ( ( trans_on_option_a @ top_top_set_option_a @ R2 )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y ) @ R2 )
       => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y @ Z2 ) @ R2 )
         => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Z2 ) @ R2 ) ) ) ) ).

% transE
thf(fact_644_transE,axiom,
    ! [R2: set_Product_prod_a_a,X3: a,Y: a,Z2: a] :
      ( ( trans_on_a @ top_top_set_a @ R2 )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R2 )
       => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z2 ) @ R2 )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Z2 ) @ R2 ) ) ) ) ).

% transE
thf(fact_645_transI,axiom,
    ! [R2: set_Pr7585778909603769095tion_a] :
      ( ! [X2: option_a,Y4: option_a,Z3: option_a] :
          ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y4 ) @ R2 )
         => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y4 @ Z3 ) @ R2 )
           => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Z3 ) @ R2 ) ) )
     => ( trans_on_option_a @ top_top_set_option_a @ R2 ) ) ).

% transI
thf(fact_646_transI,axiom,
    ! [R2: set_Product_prod_a_a] :
      ( ! [X2: a,Y4: a,Z3: a] :
          ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y4 ) @ R2 )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ Z3 ) @ R2 )
           => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Z3 ) @ R2 ) ) )
     => ( trans_on_a @ top_top_set_a @ R2 ) ) ).

% transI
thf(fact_647_inj__on__image__eq__iff,axiom,
    ! [F2: option_a > a,C3: set_option_a,A2: set_option_a,B2: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ C3 )
     => ( ( ord_le1955136853071979460tion_a @ A2 @ C3 )
       => ( ( ord_le1955136853071979460tion_a @ B2 @ C3 )
         => ( ( ( image_option_a_a2 @ F2 @ A2 )
              = ( image_option_a_a2 @ F2 @ B2 ) )
            = ( A2 = B2 ) ) ) ) ) ).

% inj_on_image_eq_iff
thf(fact_648_inj__on__image__eq__iff,axiom,
    ! [F2: a > option_a,C3: set_a,A2: set_a,B2: set_a] :
      ( ( inj_on_a_option_a @ F2 @ C3 )
     => ( ( ord_less_eq_set_a @ A2 @ C3 )
       => ( ( ord_less_eq_set_a @ B2 @ C3 )
         => ( ( ( image_a_option_a2 @ F2 @ A2 )
              = ( image_a_option_a2 @ F2 @ B2 ) )
            = ( A2 = B2 ) ) ) ) ) ).

% inj_on_image_eq_iff
thf(fact_649_inj__on__image__mem__iff,axiom,
    ! [F2: a > a,B2: set_a,A: a,A2: set_a] :
      ( ( inj_on_a_a @ F2 @ B2 )
     => ( ( member_a @ A @ B2 )
       => ( ( ord_less_eq_set_a @ A2 @ B2 )
         => ( ( member_a @ ( F2 @ A ) @ ( image_a_a2 @ F2 @ A2 ) )
            = ( member_a @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_650_inj__on__image__mem__iff,axiom,
    ! [F2: a > produc6150846815813599699_a_d_c,B2: set_a,A: a,A2: set_a] :
      ( ( inj_on3709328703910593990_a_d_c @ F2 @ B2 )
     => ( ( member_a @ A @ B2 )
       => ( ( ord_less_eq_set_a @ A2 @ B2 )
         => ( ( member2052822272342364412_a_d_c @ ( F2 @ A ) @ ( image_6884406963069720154_a_d_c @ F2 @ A2 ) )
            = ( member_a @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_651_inj__on__image__mem__iff,axiom,
    ! [F2: a > option_a,B2: set_a,A: a,A2: set_a] :
      ( ( inj_on_a_option_a @ F2 @ B2 )
     => ( ( member_a @ A @ B2 )
       => ( ( ord_less_eq_set_a @ A2 @ B2 )
         => ( ( member_option_a @ ( F2 @ A ) @ ( image_a_option_a2 @ F2 @ A2 ) )
            = ( member_a @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_652_inj__on__image__mem__iff,axiom,
    ! [F2: produc6150846815813599699_a_d_c > a,B2: set_Pr7868159745199425715_a_d_c,A: produc6150846815813599699_a_d_c,A2: set_Pr7868159745199425715_a_d_c] :
      ( ( inj_on6701986109746958172_d_c_a @ F2 @ B2 )
     => ( ( member2052822272342364412_a_d_c @ A @ B2 )
       => ( ( ord_le3761643085089054803_a_d_c @ A2 @ B2 )
         => ( ( member_a @ ( F2 @ A ) @ ( image_653692332051308528_d_c_a @ F2 @ A2 ) )
            = ( member2052822272342364412_a_d_c @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_653_inj__on__image__mem__iff,axiom,
    ! [F2: produc6150846815813599699_a_d_c > produc6150846815813599699_a_d_c,B2: set_Pr7868159745199425715_a_d_c,A: produc6150846815813599699_a_d_c,A2: set_Pr7868159745199425715_a_d_c] :
      ( ( inj_on5053348487737048625_a_d_c @ F2 @ B2 )
     => ( ( member2052822272342364412_a_d_c @ A @ B2 )
       => ( ( ord_le3761643085089054803_a_d_c @ A2 @ B2 )
         => ( ( member2052822272342364412_a_d_c @ ( F2 @ A ) @ ( image_548751279113285317_a_d_c @ F2 @ A2 ) )
            = ( member2052822272342364412_a_d_c @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_654_inj__on__image__mem__iff,axiom,
    ! [F2: produc6150846815813599699_a_d_c > option_a,B2: set_Pr7868159745199425715_a_d_c,A: produc6150846815813599699_a_d_c,A2: set_Pr7868159745199425715_a_d_c] :
      ( ( inj_on3641466449436709026tion_a @ F2 @ B2 )
     => ( ( member2052822272342364412_a_d_c @ A @ B2 )
       => ( ( ord_le3761643085089054803_a_d_c @ A2 @ B2 )
         => ( ( member_option_a @ ( F2 @ A ) @ ( image_5971021413422854966tion_a @ F2 @ A2 ) )
            = ( member2052822272342364412_a_d_c @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_655_inj__on__image__mem__iff,axiom,
    ! [F2: option_a > a,B2: set_option_a,A: option_a,A2: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ B2 )
     => ( ( member_option_a @ A @ B2 )
       => ( ( ord_le1955136853071979460tion_a @ A2 @ B2 )
         => ( ( member_a @ ( F2 @ A ) @ ( image_option_a_a2 @ F2 @ A2 ) )
            = ( member_option_a @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_656_inj__on__image__mem__iff,axiom,
    ! [F2: option_a > produc6150846815813599699_a_d_c,B2: set_option_a,A: option_a,A2: set_option_a] :
      ( ( inj_on7085348629561674240_a_d_c @ F2 @ B2 )
     => ( ( member_option_a @ A @ B2 )
       => ( ( ord_le1955136853071979460tion_a @ A2 @ B2 )
         => ( ( member2052822272342364412_a_d_c @ ( F2 @ A ) @ ( image_191531556693044372_a_d_c @ F2 @ A2 ) )
            = ( member_option_a @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_657_inj__on__image__mem__iff,axiom,
    ! [F2: option_a > option_a,B2: set_option_a,A: option_a,A2: set_option_a] :
      ( ( inj_on8559383841115902449tion_a @ F2 @ B2 )
     => ( ( member_option_a @ A @ B2 )
       => ( ( ord_le1955136853071979460tion_a @ A2 @ B2 )
         => ( ( member_option_a @ ( F2 @ A ) @ ( image_7439109396645324421tion_a @ F2 @ A2 ) )
            = ( member_option_a @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_658_finite__ran,axiom,
    ! [P: option_a > option_option_a] :
      ( ( finite1674126218327898605tion_a @ ( dom_op4724496951392727122tion_a @ P ) )
     => ( finite1674126218327898605tion_a @ ( ran_op6317565877353657455tion_a @ P ) ) ) ).

% finite_ran
thf(fact_659_finite__ran,axiom,
    ! [P: option_a > option_a] :
      ( ( finite1674126218327898605tion_a @ ( dom_option_a_a @ P ) )
     => ( finite_finite_a @ ( ran_option_a_a @ P ) ) ) ).

% finite_ran
thf(fact_660_finite__ran,axiom,
    ! [P: a > option_option_a] :
      ( ( finite_finite_a @ ( dom_a_option_a @ P ) )
     => ( finite1674126218327898605tion_a @ ( ran_a_option_a @ P ) ) ) ).

% finite_ran
thf(fact_661_finite__ran,axiom,
    ! [P: a > option_a] :
      ( ( finite_finite_a @ ( dom_a_a @ P ) )
     => ( finite_finite_a @ ( ran_a_a @ P ) ) ) ).

% finite_ran
thf(fact_662_inj__image__subset__iff,axiom,
    ! [F2: option_a > a,A2: set_option_a,B2: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ top_top_set_option_a )
     => ( ( ord_less_eq_set_a @ ( image_option_a_a2 @ F2 @ A2 ) @ ( image_option_a_a2 @ F2 @ B2 ) )
        = ( ord_le1955136853071979460tion_a @ A2 @ B2 ) ) ) ).

% inj_image_subset_iff
thf(fact_663_inj__image__subset__iff,axiom,
    ! [F2: a > option_a,A2: set_a,B2: set_a] :
      ( ( inj_on_a_option_a @ F2 @ top_top_set_a )
     => ( ( ord_le1955136853071979460tion_a @ ( image_a_option_a2 @ F2 @ A2 ) @ ( image_a_option_a2 @ F2 @ B2 ) )
        = ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).

% inj_image_subset_iff
thf(fact_664_inj__on__image__Int,axiom,
    ! [F2: option_a > a,C3: set_option_a,A2: set_option_a,B2: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ C3 )
     => ( ( ord_le1955136853071979460tion_a @ A2 @ C3 )
       => ( ( ord_le1955136853071979460tion_a @ B2 @ C3 )
         => ( ( image_option_a_a2 @ F2 @ ( inf_inf_set_option_a @ A2 @ B2 ) )
            = ( inf_inf_set_a @ ( image_option_a_a2 @ F2 @ A2 ) @ ( image_option_a_a2 @ F2 @ B2 ) ) ) ) ) ) ).

% inj_on_image_Int
thf(fact_665_inj__on__image__Int,axiom,
    ! [F2: a > option_a,C3: set_a,A2: set_a,B2: set_a] :
      ( ( inj_on_a_option_a @ F2 @ C3 )
     => ( ( ord_less_eq_set_a @ A2 @ C3 )
       => ( ( ord_less_eq_set_a @ B2 @ C3 )
         => ( ( image_a_option_a2 @ F2 @ ( inf_inf_set_a @ A2 @ B2 ) )
            = ( inf_inf_set_option_a @ ( image_a_option_a2 @ F2 @ A2 ) @ ( image_a_option_a2 @ F2 @ B2 ) ) ) ) ) ) ).

% inj_on_image_Int
thf(fact_666_inj__on__image__set__diff,axiom,
    ! [F2: option_a > a,C3: set_option_a,A2: set_option_a,B2: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ C3 )
     => ( ( ord_le1955136853071979460tion_a @ ( minus_1574173051537231627tion_a @ A2 @ B2 ) @ C3 )
       => ( ( ord_le1955136853071979460tion_a @ B2 @ C3 )
         => ( ( image_option_a_a2 @ F2 @ ( minus_1574173051537231627tion_a @ A2 @ B2 ) )
            = ( minus_minus_set_a @ ( image_option_a_a2 @ F2 @ A2 ) @ ( image_option_a_a2 @ F2 @ B2 ) ) ) ) ) ) ).

% inj_on_image_set_diff
thf(fact_667_inj__on__image__set__diff,axiom,
    ! [F2: a > option_a,C3: set_a,A2: set_a,B2: set_a] :
      ( ( inj_on_a_option_a @ F2 @ C3 )
     => ( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ C3 )
       => ( ( ord_less_eq_set_a @ B2 @ C3 )
         => ( ( image_a_option_a2 @ F2 @ ( minus_minus_set_a @ A2 @ B2 ) )
            = ( minus_1574173051537231627tion_a @ ( image_a_option_a2 @ F2 @ A2 ) @ ( image_a_option_a2 @ F2 @ B2 ) ) ) ) ) ) ).

% inj_on_image_set_diff
thf(fact_668_finite__map__freshness,axiom,
    ! [F2: option_a > option_a] :
      ( ( finite1674126218327898605tion_a @ ( dom_option_a_a @ F2 ) )
     => ( ~ ( finite1674126218327898605tion_a @ top_top_set_option_a )
       => ? [X2: option_a] :
            ( ( F2 @ X2 )
            = none_a ) ) ) ).

% finite_map_freshness
thf(fact_669_finite__map__freshness,axiom,
    ! [F2: a > option_a] :
      ( ( finite_finite_a @ ( dom_a_a @ F2 ) )
     => ( ~ ( finite_finite_a @ top_top_set_a )
       => ? [X2: a] :
            ( ( F2 @ X2 )
            = none_a ) ) ) ).

% finite_map_freshness
thf(fact_670_surj__Compl__image__subset,axiom,
    ! [F2: option_a > option_a,A2: set_option_a] :
      ( ( ( image_7439109396645324421tion_a @ F2 @ top_top_set_option_a )
        = top_top_set_option_a )
     => ( ord_le1955136853071979460tion_a @ ( uminus6205308855922866075tion_a @ ( image_7439109396645324421tion_a @ F2 @ A2 ) ) @ ( image_7439109396645324421tion_a @ F2 @ ( uminus6205308855922866075tion_a @ A2 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_671_surj__Compl__image__subset,axiom,
    ! [F2: option_a > a,A2: set_option_a] :
      ( ( ( image_option_a_a2 @ F2 @ top_top_set_option_a )
        = top_top_set_a )
     => ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ ( image_option_a_a2 @ F2 @ A2 ) ) @ ( image_option_a_a2 @ F2 @ ( uminus6205308855922866075tion_a @ A2 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_672_surj__Compl__image__subset,axiom,
    ! [F2: a > option_a,A2: set_a] :
      ( ( ( image_a_option_a2 @ F2 @ top_top_set_a )
        = top_top_set_option_a )
     => ( ord_le1955136853071979460tion_a @ ( uminus6205308855922866075tion_a @ ( image_a_option_a2 @ F2 @ A2 ) ) @ ( image_a_option_a2 @ F2 @ ( uminus_uminus_set_a @ A2 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_673_surj__Compl__image__subset,axiom,
    ! [F2: a > a,A2: 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 @ A2 ) ) @ ( image_a_a2 @ F2 @ ( uminus_uminus_set_a @ A2 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_674_the__inv__into__into,axiom,
    ! [F2: a > a,A2: set_a,X3: a,B2: set_a] :
      ( ( inj_on_a_a @ F2 @ A2 )
     => ( ( member_a @ X3 @ ( image_a_a2 @ F2 @ A2 ) )
       => ( ( ord_less_eq_set_a @ A2 @ B2 )
         => ( member_a @ ( the_inv_into_a_a @ A2 @ F2 @ X3 ) @ B2 ) ) ) ) ).

% the_inv_into_into
thf(fact_675_the__inv__into__into,axiom,
    ! [F2: produc6150846815813599699_a_d_c > a,A2: set_Pr7868159745199425715_a_d_c,X3: a,B2: set_Pr7868159745199425715_a_d_c] :
      ( ( inj_on6701986109746958172_d_c_a @ F2 @ A2 )
     => ( ( member_a @ X3 @ ( image_653692332051308528_d_c_a @ F2 @ A2 ) )
       => ( ( ord_le3761643085089054803_a_d_c @ A2 @ B2 )
         => ( member2052822272342364412_a_d_c @ ( the_in2380739839789275038_d_c_a @ A2 @ F2 @ X3 ) @ B2 ) ) ) ) ).

% the_inv_into_into
thf(fact_676_the__inv__into__into,axiom,
    ! [F2: option_a > a,A2: set_option_a,X3: a,B2: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ A2 )
     => ( ( member_a @ X3 @ ( image_option_a_a2 @ F2 @ A2 ) )
       => ( ( ord_le1955136853071979460tion_a @ A2 @ B2 )
         => ( member_option_a @ ( the_in1757154643552616557on_a_a @ A2 @ F2 @ X3 ) @ B2 ) ) ) ) ).

% the_inv_into_into
thf(fact_677_the__inv__into__into,axiom,
    ! [F2: a > produc6150846815813599699_a_d_c,A2: set_a,X3: produc6150846815813599699_a_d_c,B2: set_a] :
      ( ( inj_on3709328703910593990_a_d_c @ F2 @ A2 )
     => ( ( member2052822272342364412_a_d_c @ X3 @ ( image_6884406963069720154_a_d_c @ F2 @ A2 ) )
       => ( ( ord_less_eq_set_a @ A2 @ B2 )
         => ( member_a @ ( the_in8611454470807686664_a_d_c @ A2 @ F2 @ X3 ) @ B2 ) ) ) ) ).

% the_inv_into_into
thf(fact_678_the__inv__into__into,axiom,
    ! [F2: produc6150846815813599699_a_d_c > produc6150846815813599699_a_d_c,A2: set_Pr7868159745199425715_a_d_c,X3: produc6150846815813599699_a_d_c,B2: set_Pr7868159745199425715_a_d_c] :
      ( ( inj_on5053348487737048625_a_d_c @ F2 @ A2 )
     => ( ( member2052822272342364412_a_d_c @ X3 @ ( image_548751279113285317_a_d_c @ F2 @ A2 ) )
       => ( ( ord_le3761643085089054803_a_d_c @ A2 @ B2 )
         => ( member2052822272342364412_a_d_c @ ( the_in6963316073763403379_a_d_c @ A2 @ F2 @ X3 ) @ B2 ) ) ) ) ).

% the_inv_into_into
thf(fact_679_the__inv__into__into,axiom,
    ! [F2: option_a > produc6150846815813599699_a_d_c,A2: set_option_a,X3: produc6150846815813599699_a_d_c,B2: set_option_a] :
      ( ( inj_on7085348629561674240_a_d_c @ F2 @ A2 )
     => ( ( member2052822272342364412_a_d_c @ X3 @ ( image_191531556693044372_a_d_c @ F2 @ A2 ) )
       => ( ( ord_le1955136853071979460tion_a @ A2 @ B2 )
         => ( member_option_a @ ( the_in5027906823813023554_a_d_c @ A2 @ F2 @ X3 ) @ B2 ) ) ) ) ).

% the_inv_into_into
thf(fact_680_the__inv__into__into,axiom,
    ! [F2: a > option_a,A2: set_a,X3: option_a,B2: set_a] :
      ( ( inj_on_a_option_a @ F2 @ A2 )
     => ( ( member_option_a @ X3 @ ( image_a_option_a2 @ F2 @ A2 ) )
       => ( ( ord_less_eq_set_a @ A2 @ B2 )
         => ( member_a @ ( the_in8758012798868597241tion_a @ A2 @ F2 @ X3 ) @ B2 ) ) ) ) ).

% the_inv_into_into
thf(fact_681_the__inv__into__into,axiom,
    ! [F2: produc6150846815813599699_a_d_c > option_a,A2: set_Pr7868159745199425715_a_d_c,X3: option_a,B2: set_Pr7868159745199425715_a_d_c] :
      ( ( inj_on3641466449436709026tion_a @ F2 @ A2 )
     => ( ( member_option_a @ X3 @ ( image_5971021413422854966tion_a @ F2 @ A2 ) )
       => ( ( ord_le3761643085089054803_a_d_c @ A2 @ B2 )
         => ( member2052822272342364412_a_d_c @ ( the_in1584024643688058340tion_a @ A2 @ F2 @ X3 ) @ B2 ) ) ) ) ).

% the_inv_into_into
thf(fact_682_the__inv__into__into,axiom,
    ! [F2: option_a > option_a,A2: set_option_a,X3: option_a,B2: set_option_a] :
      ( ( inj_on8559383841115902449tion_a @ F2 @ A2 )
     => ( ( member_option_a @ X3 @ ( image_7439109396645324421tion_a @ F2 @ A2 ) )
       => ( ( ord_le1955136853071979460tion_a @ A2 @ B2 )
         => ( member_option_a @ ( the_in2538339130118444083tion_a @ A2 @ F2 @ X3 ) @ B2 ) ) ) ) ).

% the_inv_into_into
thf(fact_683_Linear__order__wf__diff__Id,axiom,
    ! [R2: set_Product_prod_a_a] :
      ( ( order_8768733634509060147r_on_a @ ( field_a @ R2 ) @ R2 )
     => ( ( wf_a @ ( minus_6817036919807184750od_a_a @ R2 @ id_a2 ) )
        = ( ! [A7: set_a] :
              ( ( ord_less_eq_set_a @ A7 @ ( field_a @ R2 ) )
             => ( ( A7 != bot_bot_set_a )
               => ? [X: a] :
                    ( ( member_a @ X @ A7 )
                    & ! [Y3: a] :
                        ( ( member_a @ Y3 @ A7 )
                       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_684_Linear__order__wf__diff__Id,axiom,
    ! [R2: set_Pr7585778909603769095tion_a] :
      ( ( order_7850372301378808569tion_a @ ( field_option_a @ R2 ) @ R2 )
     => ( ( wf_option_a @ ( minus_6512073291116468334tion_a @ R2 @ id_option_a2 ) )
        = ( ! [A7: set_option_a] :
              ( ( ord_le1955136853071979460tion_a @ A7 @ ( field_option_a @ R2 ) )
             => ( ( A7 != bot_bot_set_option_a )
               => ? [X: option_a] :
                    ( ( member_option_a @ X @ A7 )
                    & ! [Y3: option_a] :
                        ( ( member_option_a @ Y3 @ A7 )
                       => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_685_total__onI,axiom,
    ! [A2: set_a,R2: set_Product_prod_a_a] :
      ( ! [X2: a,Y4: a] :
          ( ( member_a @ X2 @ A2 )
         => ( ( member_a @ Y4 @ A2 )
           => ( ( X2 != Y4 )
             => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y4 ) @ R2 )
                | ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X2 ) @ R2 ) ) ) ) )
     => ( total_on_a @ A2 @ R2 ) ) ).

% total_onI
thf(fact_686_total__onI,axiom,
    ! [A2: set_Pr7868159745199425715_a_d_c,R2: set_Pr8390707888732215751_a_d_c] :
      ( ! [X2: produc6150846815813599699_a_d_c,Y4: produc6150846815813599699_a_d_c] :
          ( ( member2052822272342364412_a_d_c @ X2 @ A2 )
         => ( ( member2052822272342364412_a_d_c @ Y4 @ A2 )
           => ( ( X2 != Y4 )
             => ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ X2 @ Y4 ) @ R2 )
                | ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ Y4 @ X2 ) @ R2 ) ) ) ) )
     => ( total_8418979534536575319_a_d_c @ A2 @ R2 ) ) ).

% total_onI
thf(fact_687_total__onI,axiom,
    ! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a] :
      ( ! [X2: option_a,Y4: option_a] :
          ( ( member_option_a @ X2 @ A2 )
         => ( ( member_option_a @ Y4 @ A2 )
           => ( ( X2 != Y4 )
             => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y4 ) @ R2 )
                | ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y4 @ X2 ) @ R2 ) ) ) ) )
     => ( total_on_option_a @ A2 @ R2 ) ) ).

% total_onI
thf(fact_688_totalI,axiom,
    ! [R2: set_Pr7585778909603769095tion_a] :
      ( ! [X2: option_a,Y4: option_a] :
          ( ( X2 != Y4 )
         => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y4 ) @ R2 )
            | ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y4 @ X2 ) @ R2 ) ) )
     => ( total_on_option_a @ top_top_set_option_a @ R2 ) ) ).

% totalI
thf(fact_689_totalI,axiom,
    ! [R2: set_Product_prod_a_a] :
      ( ! [X2: a,Y4: a] :
          ( ( X2 != Y4 )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y4 ) @ R2 )
            | ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X2 ) @ R2 ) ) )
     => ( total_on_a @ top_top_set_a @ R2 ) ) ).

% totalI
thf(fact_690_wf__eq__minimal2,axiom,
    ( wf_a
    = ( ^ [R3: set_Product_prod_a_a] :
        ! [A7: set_a] :
          ( ( ( ord_less_eq_set_a @ A7 @ ( field_a @ R3 ) )
            & ( A7 != bot_bot_set_a ) )
         => ? [X: a] :
              ( ( member_a @ X @ A7 )
              & ! [Y3: a] :
                  ( ( member_a @ Y3 @ A7 )
                 => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X ) @ R3 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_691_wf__eq__minimal2,axiom,
    ( wf_option_a
    = ( ^ [R3: set_Pr7585778909603769095tion_a] :
        ! [A7: set_option_a] :
          ( ( ( ord_le1955136853071979460tion_a @ A7 @ ( field_option_a @ R3 ) )
            & ( A7 != bot_bot_set_option_a ) )
         => ? [X: option_a] :
              ( ( member_option_a @ X @ A7 )
              & ! [Y3: option_a] :
                  ( ( member_option_a @ Y3 @ A7 )
                 => ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y3 @ X ) @ R3 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_692_wfE__min_H,axiom,
    ! [R4: set_Pr8390707888732215751_a_d_c,Q: set_Pr7868159745199425715_a_d_c] :
      ( ( wf_Pro1713206634478224572_a_d_c @ R4 )
     => ( ( Q != bot_bo1396921257525504799_a_d_c )
       => ~ ! [Z3: produc6150846815813599699_a_d_c] :
              ( ( member2052822272342364412_a_d_c @ Z3 @ Q )
             => ~ ! [Y5: produc6150846815813599699_a_d_c] :
                    ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ Y5 @ Z3 ) @ R4 )
                   => ~ ( member2052822272342364412_a_d_c @ Y5 @ Q ) ) ) ) ) ).

% wfE_min'
thf(fact_693_wfE__min_H,axiom,
    ! [R4: set_Product_prod_a_a,Q: set_a] :
      ( ( wf_a @ R4 )
     => ( ( Q != bot_bot_set_a )
       => ~ ! [Z3: a] :
              ( ( member_a @ Z3 @ Q )
             => ~ ! [Y5: a] :
                    ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ Z3 ) @ R4 )
                   => ~ ( member_a @ Y5 @ Q ) ) ) ) ) ).

% wfE_min'
thf(fact_694_wfE__min_H,axiom,
    ! [R4: set_Pr7585778909603769095tion_a,Q: set_option_a] :
      ( ( wf_option_a @ R4 )
     => ( ( Q != bot_bot_set_option_a )
       => ~ ! [Z3: option_a] :
              ( ( member_option_a @ Z3 @ Q )
             => ~ ! [Y5: option_a] :
                    ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y5 @ Z3 ) @ R4 )
                   => ~ ( member_option_a @ Y5 @ Q ) ) ) ) ) ).

% wfE_min'
thf(fact_695_wfE__min,axiom,
    ! [R4: set_Product_prod_a_a,X3: a,Q: set_a] :
      ( ( wf_a @ R4 )
     => ( ( member_a @ X3 @ Q )
       => ~ ! [Z3: a] :
              ( ( member_a @ Z3 @ Q )
             => ~ ! [Y5: a] :
                    ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ Z3 ) @ R4 )
                   => ~ ( member_a @ Y5 @ Q ) ) ) ) ) ).

% wfE_min
thf(fact_696_wfE__min,axiom,
    ! [R4: set_Pr8390707888732215751_a_d_c,X3: produc6150846815813599699_a_d_c,Q: set_Pr7868159745199425715_a_d_c] :
      ( ( wf_Pro1713206634478224572_a_d_c @ R4 )
     => ( ( member2052822272342364412_a_d_c @ X3 @ Q )
       => ~ ! [Z3: produc6150846815813599699_a_d_c] :
              ( ( member2052822272342364412_a_d_c @ Z3 @ Q )
             => ~ ! [Y5: produc6150846815813599699_a_d_c] :
                    ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ Y5 @ Z3 ) @ R4 )
                   => ~ ( member2052822272342364412_a_d_c @ Y5 @ Q ) ) ) ) ) ).

% wfE_min
thf(fact_697_wfE__min,axiom,
    ! [R4: set_Pr7585778909603769095tion_a,X3: option_a,Q: set_option_a] :
      ( ( wf_option_a @ R4 )
     => ( ( member_option_a @ X3 @ Q )
       => ~ ! [Z3: option_a] :
              ( ( member_option_a @ Z3 @ Q )
             => ~ ! [Y5: option_a] :
                    ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y5 @ Z3 ) @ R4 )
                   => ~ ( member_option_a @ Y5 @ Q ) ) ) ) ) ).

% wfE_min
thf(fact_698_wfI__min,axiom,
    ! [R4: set_Product_prod_a_a] :
      ( ! [X2: a,Q4: set_a] :
          ( ( member_a @ X2 @ Q4 )
         => ? [Xa2: a] :
              ( ( member_a @ Xa2 @ Q4 )
              & ! [Y4: a] :
                  ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ Xa2 ) @ R4 )
                 => ~ ( member_a @ Y4 @ Q4 ) ) ) )
     => ( wf_a @ R4 ) ) ).

% wfI_min
thf(fact_699_wfI__min,axiom,
    ! [R4: set_Pr8390707888732215751_a_d_c] :
      ( ! [X2: produc6150846815813599699_a_d_c,Q4: set_Pr7868159745199425715_a_d_c] :
          ( ( member2052822272342364412_a_d_c @ X2 @ Q4 )
         => ? [Xa2: produc6150846815813599699_a_d_c] :
              ( ( member2052822272342364412_a_d_c @ Xa2 @ Q4 )
              & ! [Y4: produc6150846815813599699_a_d_c] :
                  ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ Y4 @ Xa2 ) @ R4 )
                 => ~ ( member2052822272342364412_a_d_c @ Y4 @ Q4 ) ) ) )
     => ( wf_Pro1713206634478224572_a_d_c @ R4 ) ) ).

% wfI_min
thf(fact_700_wfI__min,axiom,
    ! [R4: set_Pr7585778909603769095tion_a] :
      ( ! [X2: option_a,Q4: set_option_a] :
          ( ( member_option_a @ X2 @ Q4 )
         => ? [Xa2: option_a] :
              ( ( member_option_a @ Xa2 @ Q4 )
              & ! [Y4: option_a] :
                  ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y4 @ Xa2 ) @ R4 )
                 => ~ ( member_option_a @ Y4 @ Q4 ) ) ) )
     => ( wf_option_a @ R4 ) ) ).

% wfI_min
thf(fact_701_wf__eq__minimal,axiom,
    ( wf_a
    = ( ^ [R3: set_Product_prod_a_a] :
        ! [Q5: set_a] :
          ( ? [X: a] : ( member_a @ X @ Q5 )
         => ? [X: a] :
              ( ( member_a @ X @ Q5 )
              & ! [Y3: a] :
                  ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X ) @ R3 )
                 => ~ ( member_a @ Y3 @ Q5 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_702_wf__eq__minimal,axiom,
    ( wf_Pro1713206634478224572_a_d_c
    = ( ^ [R3: set_Pr8390707888732215751_a_d_c] :
        ! [Q5: set_Pr7868159745199425715_a_d_c] :
          ( ? [X: produc6150846815813599699_a_d_c] : ( member2052822272342364412_a_d_c @ X @ Q5 )
         => ? [X: produc6150846815813599699_a_d_c] :
              ( ( member2052822272342364412_a_d_c @ X @ Q5 )
              & ! [Y3: produc6150846815813599699_a_d_c] :
                  ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ Y3 @ X ) @ R3 )
                 => ~ ( member2052822272342364412_a_d_c @ Y3 @ Q5 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_703_wf__eq__minimal,axiom,
    ( wf_option_a
    = ( ^ [R3: set_Pr7585778909603769095tion_a] :
        ! [Q5: set_option_a] :
          ( ? [X: option_a] : ( member_option_a @ X @ Q5 )
         => ? [X: option_a] :
              ( ( member_option_a @ X @ Q5 )
              & ! [Y3: option_a] :
                  ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y3 @ X ) @ R3 )
                 => ~ ( member_option_a @ Y3 @ Q5 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_704_wo__rel_Ocases__Total3,axiom,
    ! [R2: set_Product_prod_a_a,A: a,B: a,Phi: a > a > $o] :
      ( ( bNF_We1162827675446709994_rel_a @ R2 )
     => ( ( ord_less_eq_set_a @ ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( field_a @ R2 ) )
       => ( ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( minus_6817036919807184750od_a_a @ R2 @ id_a2 ) )
              | ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ A ) @ ( minus_6817036919807184750od_a_a @ R2 @ id_a2 ) ) )
           => ( Phi @ A @ B ) )
         => ( ( ( A = B )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_705_wo__rel_Ocases__Total3,axiom,
    ! [R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a,Phi: option_a > option_a > $o] :
      ( ( bNF_We8432232079604507440tion_a @ R2 )
     => ( ( ord_le1955136853071979460tion_a @ ( insert_option_a @ A @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( field_option_a @ R2 ) )
       => ( ( ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ ( minus_6512073291116468334tion_a @ R2 @ id_option_a2 ) )
              | ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ A ) @ ( minus_6512073291116468334tion_a @ R2 @ id_option_a2 ) ) )
           => ( Phi @ A @ B ) )
         => ( ( ( A = B )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_706_finite__range__map__of__map__add,axiom,
    ! [F2: option_a > option_a,L: list_P6260409590414597735on_a_a] :
      ( ( finite1674126218327898605tion_a @ ( image_7439109396645324421tion_a @ F2 @ top_top_set_option_a ) )
     => ( finite1674126218327898605tion_a @ ( image_7439109396645324421tion_a @ ( map_add_option_a_a @ F2 @ ( map_of_option_a_a @ L ) ) @ top_top_set_option_a ) ) ) ).

% finite_range_map_of_map_add
thf(fact_707_finite__range__map__of__map__add,axiom,
    ! [F2: a > option_a,L: list_P1396940483166286381od_a_a] :
      ( ( finite1674126218327898605tion_a @ ( image_a_option_a2 @ F2 @ top_top_set_a ) )
     => ( finite1674126218327898605tion_a @ ( image_a_option_a2 @ ( map_add_a_a @ F2 @ ( map_of_a_a @ L ) ) @ top_top_set_a ) ) ) ).

% finite_range_map_of_map_add
thf(fact_708_under__incr,axiom,
    ! [R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
      ( ( trans_on_option_a @ top_top_set_option_a @ R2 )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 )
       => ( ord_le1955136853071979460tion_a @ ( order_under_option_a @ R2 @ A ) @ ( order_under_option_a @ R2 @ B ) ) ) ) ).

% under_incr
thf(fact_709_under__incr,axiom,
    ! [R2: set_Product_prod_a_a,A: a,B: a] :
      ( ( trans_on_a @ top_top_set_a @ R2 )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
       => ( ord_less_eq_set_a @ ( order_under_a @ R2 @ A ) @ ( order_under_a @ R2 @ B ) ) ) ) ).

% under_incr
thf(fact_710_wo__rel_Oin__notinI,axiom,
    ! [R2: set_Product_prod_a_a,J: a,I: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R2 )
     => ( ( ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ J @ I ) @ R2 )
          | ( J = I ) )
       => ( ( member_a @ I @ ( field_a @ R2 ) )
         => ( ( member_a @ J @ ( field_a @ R2 ) )
           => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R2 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_711_wo__rel_Oin__notinI,axiom,
    ! [R2: set_Pr8390707888732215751_a_d_c,J: produc6150846815813599699_a_d_c,I: produc6150846815813599699_a_d_c] :
      ( ( bNF_We920288868220767679_a_d_c @ R2 )
     => ( ( ~ ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ J @ I ) @ R2 )
          | ( J = I ) )
       => ( ( member2052822272342364412_a_d_c @ I @ ( field_6537103990374211872_a_d_c @ R2 ) )
         => ( ( member2052822272342364412_a_d_c @ J @ ( field_6537103990374211872_a_d_c @ R2 ) )
           => ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ I @ J ) @ R2 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_712_wo__rel_Oin__notinI,axiom,
    ! [R2: set_Pr7585778909603769095tion_a,J: option_a,I: option_a] :
      ( ( bNF_We8432232079604507440tion_a @ R2 )
     => ( ( ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ J @ I ) @ R2 )
          | ( J = I ) )
       => ( ( member_option_a @ I @ ( field_option_a @ R2 ) )
         => ( ( member_option_a @ J @ ( field_option_a @ R2 ) )
           => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ I @ J ) @ R2 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_713_well__order__induct__imp,axiom,
    ! [R2: set_Product_prod_a_a,P2: a > $o,A: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R2 )
     => ( ! [X2: a] :
            ( ! [Y5: a] :
                ( ( ( Y5 != X2 )
                  & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ X2 ) @ R2 ) )
               => ( ( member_a @ Y5 @ ( field_a @ R2 ) )
                 => ( P2 @ Y5 ) ) )
           => ( ( member_a @ X2 @ ( field_a @ R2 ) )
             => ( P2 @ X2 ) ) )
       => ( ( member_a @ A @ ( field_a @ R2 ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_714_well__order__induct__imp,axiom,
    ! [R2: set_Pr8390707888732215751_a_d_c,P2: produc6150846815813599699_a_d_c > $o,A: produc6150846815813599699_a_d_c] :
      ( ( bNF_We920288868220767679_a_d_c @ R2 )
     => ( ! [X2: produc6150846815813599699_a_d_c] :
            ( ! [Y5: produc6150846815813599699_a_d_c] :
                ( ( ( Y5 != X2 )
                  & ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ Y5 @ X2 ) @ R2 ) )
               => ( ( member2052822272342364412_a_d_c @ Y5 @ ( field_6537103990374211872_a_d_c @ R2 ) )
                 => ( P2 @ Y5 ) ) )
           => ( ( member2052822272342364412_a_d_c @ X2 @ ( field_6537103990374211872_a_d_c @ R2 ) )
             => ( P2 @ X2 ) ) )
       => ( ( member2052822272342364412_a_d_c @ A @ ( field_6537103990374211872_a_d_c @ R2 ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_715_well__order__induct__imp,axiom,
    ! [R2: set_Pr7585778909603769095tion_a,P2: option_a > $o,A: option_a] :
      ( ( bNF_We8432232079604507440tion_a @ R2 )
     => ( ! [X2: option_a] :
            ( ! [Y5: option_a] :
                ( ( ( Y5 != X2 )
                  & ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y5 @ X2 ) @ R2 ) )
               => ( ( member_option_a @ Y5 @ ( field_option_a @ R2 ) )
                 => ( P2 @ Y5 ) ) )
           => ( ( member_option_a @ X2 @ ( field_option_a @ R2 ) )
             => ( P2 @ X2 ) ) )
       => ( ( member_option_a @ A @ ( field_option_a @ R2 ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_716_wo__rel_Ocases__Total,axiom,
    ! [R2: set_Product_prod_a_a,A: a,B: a,Phi: a > a > $o] :
      ( ( bNF_We1162827675446709994_rel_a @ R2 )
     => ( ( ord_less_eq_set_a @ ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( field_a @ R2 ) )
       => ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
           => ( Phi @ A @ B ) )
         => ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ A ) @ R2 )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_717_wo__rel_Ocases__Total,axiom,
    ! [R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a,Phi: option_a > option_a > $o] :
      ( ( bNF_We8432232079604507440tion_a @ R2 )
     => ( ( ord_le1955136853071979460tion_a @ ( insert_option_a @ A @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( field_option_a @ R2 ) )
       => ( ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 )
           => ( Phi @ A @ B ) )
         => ( ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ A ) @ R2 )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_718_finite__range__map__of,axiom,
    ! [Xys: list_P6260409590414597735on_a_a] : ( finite1674126218327898605tion_a @ ( image_7439109396645324421tion_a @ ( map_of_option_a_a @ Xys ) @ top_top_set_option_a ) ) ).

% finite_range_map_of
thf(fact_719_finite__range__map__of,axiom,
    ! [Xys: list_P1396940483166286381od_a_a] : ( finite1674126218327898605tion_a @ ( image_a_option_a2 @ ( map_of_a_a @ Xys ) @ top_top_set_a ) ) ).

% finite_range_map_of
thf(fact_720_wo__rel_Oequals__minim,axiom,
    ! [R2: set_Product_prod_a_a,B2: set_a,A: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R2 )
     => ( ( ord_less_eq_set_a @ B2 @ ( field_a @ R2 ) )
       => ( ( member_a @ A @ B2 )
         => ( ! [B5: a] :
                ( ( member_a @ B5 @ B2 )
               => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B5 ) @ R2 ) )
           => ( A
              = ( bNF_We5615626441682584778inim_a @ R2 @ B2 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_721_wo__rel_Oequals__minim,axiom,
    ! [R2: set_Pr8390707888732215751_a_d_c,B2: set_Pr7868159745199425715_a_d_c,A: produc6150846815813599699_a_d_c] :
      ( ( bNF_We920288868220767679_a_d_c @ R2 )
     => ( ( ord_le3761643085089054803_a_d_c @ B2 @ ( field_6537103990374211872_a_d_c @ R2 ) )
       => ( ( member2052822272342364412_a_d_c @ A @ B2 )
         => ( ! [B5: produc6150846815813599699_a_d_c] :
                ( ( member2052822272342364412_a_d_c @ B5 @ B2 )
               => ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ A @ B5 ) @ R2 ) )
           => ( A
              = ( bNF_We821194747749729183_a_d_c @ R2 @ B2 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_722_wo__rel_Oequals__minim,axiom,
    ! [R2: set_Pr7585778909603769095tion_a,B2: set_option_a,A: option_a] :
      ( ( bNF_We8432232079604507440tion_a @ R2 )
     => ( ( ord_le1955136853071979460tion_a @ B2 @ ( field_option_a @ R2 ) )
       => ( ( member_option_a @ A @ B2 )
         => ( ! [B5: option_a] :
                ( ( member_option_a @ B5 @ B2 )
               => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B5 ) @ R2 ) )
           => ( A
              = ( bNF_We6579146059749918992tion_a @ R2 @ B2 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_723_wo__rel_Ominim__least,axiom,
    ! [R2: set_Product_prod_a_a,B2: set_a,B: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R2 )
     => ( ( ord_less_eq_set_a @ B2 @ ( field_a @ R2 ) )
       => ( ( member_a @ B @ B2 )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( bNF_We5615626441682584778inim_a @ R2 @ B2 ) @ B ) @ R2 ) ) ) ) ).

% wo_rel.minim_least
thf(fact_724_wo__rel_Ominim__least,axiom,
    ! [R2: set_Pr8390707888732215751_a_d_c,B2: set_Pr7868159745199425715_a_d_c,B: produc6150846815813599699_a_d_c] :
      ( ( bNF_We920288868220767679_a_d_c @ R2 )
     => ( ( ord_le3761643085089054803_a_d_c @ B2 @ ( field_6537103990374211872_a_d_c @ R2 ) )
       => ( ( member2052822272342364412_a_d_c @ B @ B2 )
         => ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ ( bNF_We821194747749729183_a_d_c @ R2 @ B2 ) @ B ) @ R2 ) ) ) ) ).

% wo_rel.minim_least
thf(fact_725_wo__rel_Ominim__least,axiom,
    ! [R2: set_Pr7585778909603769095tion_a,B2: set_option_a,B: option_a] :
      ( ( bNF_We8432232079604507440tion_a @ R2 )
     => ( ( ord_le1955136853071979460tion_a @ B2 @ ( field_option_a @ R2 ) )
       => ( ( member_option_a @ B @ B2 )
         => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ ( bNF_We6579146059749918992tion_a @ R2 @ B2 ) @ B ) @ R2 ) ) ) ) ).

% wo_rel.minim_least
thf(fact_726_wo__rel_Omax2__greater__among,axiom,
    ! [R2: set_Pr8390707888732215751_a_d_c,A: produc6150846815813599699_a_d_c,B: produc6150846815813599699_a_d_c] :
      ( ( bNF_We920288868220767679_a_d_c @ R2 )
     => ( ( member2052822272342364412_a_d_c @ A @ ( field_6537103990374211872_a_d_c @ R2 ) )
       => ( ( member2052822272342364412_a_d_c @ B @ ( field_6537103990374211872_a_d_c @ R2 ) )
         => ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ A @ ( bNF_We4538586881274025761_a_d_c @ R2 @ A @ B ) ) @ R2 )
            & ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ B @ ( bNF_We4538586881274025761_a_d_c @ R2 @ A @ B ) ) @ R2 )
            & ( member2052822272342364412_a_d_c @ ( bNF_We4538586881274025761_a_d_c @ R2 @ A @ B ) @ ( insert4944988292672327587_a_d_c @ A @ ( insert4944988292672327587_a_d_c @ B @ bot_bo1396921257525504799_a_d_c ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_727_wo__rel_Omax2__greater__among,axiom,
    ! [R2: set_Product_prod_a_a,A: a,B: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R2 )
     => ( ( member_a @ A @ ( field_a @ R2 ) )
       => ( ( member_a @ B @ ( field_a @ R2 ) )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ ( bNF_We3763454674811381836max2_a @ R2 @ A @ B ) ) @ R2 )
            & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ ( bNF_We3763454674811381836max2_a @ R2 @ A @ B ) ) @ R2 )
            & ( member_a @ ( bNF_We3763454674811381836max2_a @ R2 @ A @ B ) @ ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_728_wo__rel_Omax2__greater__among,axiom,
    ! [R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
      ( ( bNF_We8432232079604507440tion_a @ R2 )
     => ( ( member_option_a @ A @ ( field_option_a @ R2 ) )
       => ( ( member_option_a @ B @ ( field_option_a @ R2 ) )
         => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ ( bNF_We4567742444881707410tion_a @ R2 @ A @ B ) ) @ R2 )
            & ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ ( bNF_We4567742444881707410tion_a @ R2 @ A @ B ) ) @ R2 )
            & ( member_option_a @ ( bNF_We4567742444881707410tion_a @ R2 @ A @ B ) @ ( insert_option_a @ A @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_729_Linear__order__Well__order__iff,axiom,
    ! [R2: set_Product_prod_a_a] :
      ( ( order_8768733634509060147r_on_a @ ( field_a @ R2 ) @ R2 )
     => ( ( order_6972113574731384220r_on_a @ ( field_a @ R2 ) @ R2 )
        = ( ! [A7: set_a] :
              ( ( ord_less_eq_set_a @ A7 @ ( field_a @ R2 ) )
             => ( ( A7 != bot_bot_set_a )
               => ? [X: a] :
                    ( ( member_a @ X @ A7 )
                    & ! [Y3: a] :
                        ( ( member_a @ Y3 @ A7 )
                       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_730_Linear__order__Well__order__iff,axiom,
    ! [R2: set_Pr7585778909603769095tion_a] :
      ( ( order_7850372301378808569tion_a @ ( field_option_a @ R2 ) @ R2 )
     => ( ( order_4821795997958563554tion_a @ ( field_option_a @ R2 ) @ R2 )
        = ( ! [A7: set_option_a] :
              ( ( ord_le1955136853071979460tion_a @ A7 @ ( field_option_a @ R2 ) )
             => ( ( A7 != bot_bot_set_option_a )
               => ? [X: option_a] :
                    ( ( member_option_a @ X @ A7 )
                    & ! [Y3: option_a] :
                        ( ( member_option_a @ Y3 @ A7 )
                       => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_731_well__order__on__domain,axiom,
    ! [A2: set_a,R2: set_Product_prod_a_a,A: a,B: a] :
      ( ( order_6972113574731384220r_on_a @ A2 @ R2 )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
       => ( ( member_a @ A @ A2 )
          & ( member_a @ B @ A2 ) ) ) ) ).

% well_order_on_domain
thf(fact_732_well__order__on__domain,axiom,
    ! [A2: set_Pr7868159745199425715_a_d_c,R2: set_Pr8390707888732215751_a_d_c,A: produc6150846815813599699_a_d_c,B: produc6150846815813599699_a_d_c] :
      ( ( order_1117645225797497969_a_d_c @ A2 @ R2 )
     => ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ A @ B ) @ R2 )
       => ( ( member2052822272342364412_a_d_c @ A @ A2 )
          & ( member2052822272342364412_a_d_c @ B @ A2 ) ) ) ) ).

% well_order_on_domain
thf(fact_733_well__order__on__domain,axiom,
    ! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
      ( ( order_4821795997958563554tion_a @ A2 @ R2 )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 )
       => ( ( member_option_a @ A @ A2 )
          & ( member_option_a @ B @ A2 ) ) ) ) ).

% well_order_on_domain
thf(fact_734_wo__rel_Omax2__equals1,axiom,
    ! [R2: set_Product_prod_a_a,A: a,B: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R2 )
     => ( ( member_a @ A @ ( field_a @ R2 ) )
       => ( ( member_a @ B @ ( field_a @ R2 ) )
         => ( ( ( bNF_We3763454674811381836max2_a @ R2 @ A @ B )
              = A )
            = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ A ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_735_wo__rel_Omax2__equals1,axiom,
    ! [R2: set_Pr8390707888732215751_a_d_c,A: produc6150846815813599699_a_d_c,B: produc6150846815813599699_a_d_c] :
      ( ( bNF_We920288868220767679_a_d_c @ R2 )
     => ( ( member2052822272342364412_a_d_c @ A @ ( field_6537103990374211872_a_d_c @ R2 ) )
       => ( ( member2052822272342364412_a_d_c @ B @ ( field_6537103990374211872_a_d_c @ R2 ) )
         => ( ( ( bNF_We4538586881274025761_a_d_c @ R2 @ A @ B )
              = A )
            = ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ B @ A ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_736_wo__rel_Omax2__equals1,axiom,
    ! [R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
      ( ( bNF_We8432232079604507440tion_a @ R2 )
     => ( ( member_option_a @ A @ ( field_option_a @ R2 ) )
       => ( ( member_option_a @ B @ ( field_option_a @ R2 ) )
         => ( ( ( bNF_We4567742444881707410tion_a @ R2 @ A @ B )
              = A )
            = ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ A ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_737_wo__rel_Omax2__equals2,axiom,
    ! [R2: set_Product_prod_a_a,A: a,B: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R2 )
     => ( ( member_a @ A @ ( field_a @ R2 ) )
       => ( ( member_a @ B @ ( field_a @ R2 ) )
         => ( ( ( bNF_We3763454674811381836max2_a @ R2 @ A @ B )
              = B )
            = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_738_wo__rel_Omax2__equals2,axiom,
    ! [R2: set_Pr8390707888732215751_a_d_c,A: produc6150846815813599699_a_d_c,B: produc6150846815813599699_a_d_c] :
      ( ( bNF_We920288868220767679_a_d_c @ R2 )
     => ( ( member2052822272342364412_a_d_c @ A @ ( field_6537103990374211872_a_d_c @ R2 ) )
       => ( ( member2052822272342364412_a_d_c @ B @ ( field_6537103990374211872_a_d_c @ R2 ) )
         => ( ( ( bNF_We4538586881274025761_a_d_c @ R2 @ A @ B )
              = B )
            = ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ A @ B ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_739_wo__rel_Omax2__equals2,axiom,
    ! [R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
      ( ( bNF_We8432232079604507440tion_a @ R2 )
     => ( ( member_option_a @ A @ ( field_option_a @ R2 ) )
       => ( ( member_option_a @ B @ ( field_option_a @ R2 ) )
         => ( ( ( bNF_We4567742444881707410tion_a @ R2 @ A @ B )
              = B )
            = ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_740_wo__rel_Omax2__greater,axiom,
    ! [R2: set_Product_prod_a_a,A: a,B: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R2 )
     => ( ( member_a @ A @ ( field_a @ R2 ) )
       => ( ( member_a @ B @ ( field_a @ R2 ) )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ ( bNF_We3763454674811381836max2_a @ R2 @ A @ B ) ) @ R2 )
            & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ ( bNF_We3763454674811381836max2_a @ R2 @ A @ B ) ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_741_wo__rel_Omax2__greater,axiom,
    ! [R2: set_Pr8390707888732215751_a_d_c,A: produc6150846815813599699_a_d_c,B: produc6150846815813599699_a_d_c] :
      ( ( bNF_We920288868220767679_a_d_c @ R2 )
     => ( ( member2052822272342364412_a_d_c @ A @ ( field_6537103990374211872_a_d_c @ R2 ) )
       => ( ( member2052822272342364412_a_d_c @ B @ ( field_6537103990374211872_a_d_c @ R2 ) )
         => ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ A @ ( bNF_We4538586881274025761_a_d_c @ R2 @ A @ B ) ) @ R2 )
            & ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ B @ ( bNF_We4538586881274025761_a_d_c @ R2 @ A @ B ) ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_742_wo__rel_Omax2__greater,axiom,
    ! [R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
      ( ( bNF_We8432232079604507440tion_a @ R2 )
     => ( ( member_option_a @ A @ ( field_option_a @ R2 ) )
       => ( ( member_option_a @ B @ ( field_option_a @ R2 ) )
         => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ ( bNF_We4567742444881707410tion_a @ R2 @ A @ B ) ) @ R2 )
            & ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ ( bNF_We4567742444881707410tion_a @ R2 @ A @ B ) ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_743_underS__incl__iff,axiom,
    ! [R2: set_Product_prod_a_a,A: a,B: a] :
      ( ( order_8768733634509060147r_on_a @ ( field_a @ R2 ) @ R2 )
     => ( ( member_a @ A @ ( field_a @ R2 ) )
       => ( ( member_a @ B @ ( field_a @ R2 ) )
         => ( ( ord_less_eq_set_a @ ( order_underS_a @ R2 @ A ) @ ( order_underS_a @ R2 @ B ) )
            = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_744_underS__incl__iff,axiom,
    ! [R2: set_Pr8390707888732215751_a_d_c,A: produc6150846815813599699_a_d_c,B: produc6150846815813599699_a_d_c] :
      ( ( order_2462247862883012360_a_d_c @ ( field_6537103990374211872_a_d_c @ R2 ) @ R2 )
     => ( ( member2052822272342364412_a_d_c @ A @ ( field_6537103990374211872_a_d_c @ R2 ) )
       => ( ( member2052822272342364412_a_d_c @ B @ ( field_6537103990374211872_a_d_c @ R2 ) )
         => ( ( ord_le3761643085089054803_a_d_c @ ( order_4452426386760228761_a_d_c @ R2 @ A ) @ ( order_4452426386760228761_a_d_c @ R2 @ B ) )
            = ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ A @ B ) @ R2 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_745_underS__incl__iff,axiom,
    ! [R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
      ( ( order_7850372301378808569tion_a @ ( field_option_a @ R2 ) @ R2 )
     => ( ( member_option_a @ A @ ( field_option_a @ R2 ) )
       => ( ( member_option_a @ B @ ( field_option_a @ R2 ) )
         => ( ( ord_le1955136853071979460tion_a @ ( order_8525669848891258378tion_a @ R2 @ A ) @ ( order_8525669848891258378tion_a @ R2 @ B ) )
            = ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_746_max__ext_Omax__extI,axiom,
    ! [X6: set_Pr7868159745199425715_a_d_c,Y6: set_Pr7868159745199425715_a_d_c,R4: set_Pr8390707888732215751_a_d_c] :
      ( ( finite6538345537044216828_a_d_c @ X6 )
     => ( ( finite6538345537044216828_a_d_c @ Y6 )
       => ( ( Y6 != bot_bo1396921257525504799_a_d_c )
         => ( ! [X2: produc6150846815813599699_a_d_c] :
                ( ( member2052822272342364412_a_d_c @ X2 @ X6 )
               => ? [Xa2: produc6150846815813599699_a_d_c] :
                    ( ( member2052822272342364412_a_d_c @ Xa2 @ Y6 )
                    & ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ X2 @ Xa2 ) @ R4 ) ) )
           => ( member2235620415397146640_a_d_c @ ( produc1088311440089269847_a_d_c @ X6 @ Y6 ) @ ( max_ex8685825336233442702_a_d_c @ R4 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_747_max__ext_Omax__extI,axiom,
    ! [X6: set_a,Y6: set_a,R4: set_Product_prod_a_a] :
      ( ( finite_finite_a @ X6 )
     => ( ( finite_finite_a @ Y6 )
       => ( ( Y6 != bot_bot_set_a )
         => ( ! [X2: a] :
                ( ( member_a @ X2 @ X6 )
               => ? [Xa2: a] :
                    ( ( member_a @ Xa2 @ Y6 )
                    & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Xa2 ) @ R4 ) ) )
           => ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X6 @ Y6 ) @ ( max_ext_a @ R4 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_748_max__ext_Omax__extI,axiom,
    ! [X6: set_option_a,Y6: set_option_a,R4: set_Pr7585778909603769095tion_a] :
      ( ( finite1674126218327898605tion_a @ X6 )
     => ( ( finite1674126218327898605tion_a @ Y6 )
       => ( ( Y6 != bot_bot_set_option_a )
         => ( ! [X2: option_a] :
                ( ( member_option_a @ X2 @ X6 )
               => ? [Xa2: option_a] :
                    ( ( member_option_a @ Xa2 @ Y6 )
                    & ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Xa2 ) @ R4 ) ) )
           => ( member5358692782348450128tion_a @ ( produc8179951581375851543tion_a @ X6 @ Y6 ) @ ( max_ext_option_a @ R4 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_749_max__ext_Osimps,axiom,
    ! [A1: set_Pr7868159745199425715_a_d_c,A22: set_Pr7868159745199425715_a_d_c,R4: set_Pr8390707888732215751_a_d_c] :
      ( ( member2235620415397146640_a_d_c @ ( produc1088311440089269847_a_d_c @ A1 @ A22 ) @ ( max_ex8685825336233442702_a_d_c @ R4 ) )
      = ( ( finite6538345537044216828_a_d_c @ A1 )
        & ( finite6538345537044216828_a_d_c @ A22 )
        & ( A22 != bot_bo1396921257525504799_a_d_c )
        & ! [X: produc6150846815813599699_a_d_c] :
            ( ( member2052822272342364412_a_d_c @ X @ A1 )
           => ? [Y3: produc6150846815813599699_a_d_c] :
                ( ( member2052822272342364412_a_d_c @ Y3 @ A22 )
                & ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ X @ Y3 ) @ R4 ) ) ) ) ) ).

% max_ext.simps
thf(fact_750_max__ext_Osimps,axiom,
    ! [A1: set_a,A22: set_a,R4: set_Product_prod_a_a] :
      ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ A1 @ A22 ) @ ( max_ext_a @ R4 ) )
      = ( ( finite_finite_a @ A1 )
        & ( finite_finite_a @ A22 )
        & ( A22 != bot_bot_set_a )
        & ! [X: a] :
            ( ( member_a @ X @ A1 )
           => ? [Y3: a] :
                ( ( member_a @ Y3 @ A22 )
                & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R4 ) ) ) ) ) ).

% max_ext.simps
thf(fact_751_max__ext_Osimps,axiom,
    ! [A1: set_option_a,A22: set_option_a,R4: set_Pr7585778909603769095tion_a] :
      ( ( member5358692782348450128tion_a @ ( produc8179951581375851543tion_a @ A1 @ A22 ) @ ( max_ext_option_a @ R4 ) )
      = ( ( finite1674126218327898605tion_a @ A1 )
        & ( finite1674126218327898605tion_a @ A22 )
        & ( A22 != bot_bot_set_option_a )
        & ! [X: option_a] :
            ( ( member_option_a @ X @ A1 )
           => ? [Y3: option_a] :
                ( ( member_option_a @ Y3 @ A22 )
                & ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y3 ) @ R4 ) ) ) ) ) ).

% max_ext.simps
thf(fact_752_underS__E,axiom,
    ! [I: a,R4: set_Product_prod_a_a,J: a] :
      ( ( member_a @ I @ ( order_underS_a @ R4 @ J ) )
     => ( ( I != J )
        & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R4 ) ) ) ).

% underS_E
thf(fact_753_underS__E,axiom,
    ! [I: produc6150846815813599699_a_d_c,R4: set_Pr8390707888732215751_a_d_c,J: produc6150846815813599699_a_d_c] :
      ( ( member2052822272342364412_a_d_c @ I @ ( order_4452426386760228761_a_d_c @ R4 @ J ) )
     => ( ( I != J )
        & ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ I @ J ) @ R4 ) ) ) ).

% underS_E
thf(fact_754_underS__E,axiom,
    ! [I: option_a,R4: set_Pr7585778909603769095tion_a,J: option_a] :
      ( ( member_option_a @ I @ ( order_8525669848891258378tion_a @ R4 @ J ) )
     => ( ( I != J )
        & ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ I @ J ) @ R4 ) ) ) ).

% underS_E
thf(fact_755_underS__I,axiom,
    ! [I: a,J: a,R4: set_Product_prod_a_a] :
      ( ( I != J )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R4 )
       => ( member_a @ I @ ( order_underS_a @ R4 @ J ) ) ) ) ).

% underS_I
thf(fact_756_underS__I,axiom,
    ! [I: produc6150846815813599699_a_d_c,J: produc6150846815813599699_a_d_c,R4: set_Pr8390707888732215751_a_d_c] :
      ( ( I != J )
     => ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ I @ J ) @ R4 )
       => ( member2052822272342364412_a_d_c @ I @ ( order_4452426386760228761_a_d_c @ R4 @ J ) ) ) ) ).

% underS_I
thf(fact_757_underS__I,axiom,
    ! [I: option_a,J: option_a,R4: set_Pr7585778909603769095tion_a] :
      ( ( I != J )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ I @ J ) @ R4 )
       => ( member_option_a @ I @ ( order_8525669848891258378tion_a @ R4 @ J ) ) ) ) ).

% underS_I
thf(fact_758_max__ext_Ocases,axiom,
    ! [A1: set_Pr7868159745199425715_a_d_c,A22: set_Pr7868159745199425715_a_d_c,R4: set_Pr8390707888732215751_a_d_c] :
      ( ( member2235620415397146640_a_d_c @ ( produc1088311440089269847_a_d_c @ A1 @ A22 ) @ ( max_ex8685825336233442702_a_d_c @ R4 ) )
     => ~ ( ( finite6538345537044216828_a_d_c @ A1 )
         => ( ( finite6538345537044216828_a_d_c @ A22 )
           => ( ( A22 != bot_bo1396921257525504799_a_d_c )
             => ~ ! [X8: produc6150846815813599699_a_d_c] :
                    ( ( member2052822272342364412_a_d_c @ X8 @ A1 )
                   => ? [Xa: produc6150846815813599699_a_d_c] :
                        ( ( member2052822272342364412_a_d_c @ Xa @ A22 )
                        & ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ X8 @ Xa ) @ R4 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_759_max__ext_Ocases,axiom,
    ! [A1: set_a,A22: set_a,R4: set_Product_prod_a_a] :
      ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ A1 @ A22 ) @ ( max_ext_a @ R4 ) )
     => ~ ( ( finite_finite_a @ A1 )
         => ( ( finite_finite_a @ A22 )
           => ( ( A22 != bot_bot_set_a )
             => ~ ! [X8: a] :
                    ( ( member_a @ X8 @ A1 )
                   => ? [Xa: a] :
                        ( ( member_a @ Xa @ A22 )
                        & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X8 @ Xa ) @ R4 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_760_max__ext_Ocases,axiom,
    ! [A1: set_option_a,A22: set_option_a,R4: set_Pr7585778909603769095tion_a] :
      ( ( member5358692782348450128tion_a @ ( produc8179951581375851543tion_a @ A1 @ A22 ) @ ( max_ext_option_a @ R4 ) )
     => ~ ( ( finite1674126218327898605tion_a @ A1 )
         => ( ( finite1674126218327898605tion_a @ A22 )
           => ( ( A22 != bot_bot_set_option_a )
             => ~ ! [X8: option_a] :
                    ( ( member_option_a @ X8 @ A1 )
                   => ? [Xa: option_a] :
                        ( ( member_option_a @ Xa @ A22 )
                        & ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X8 @ Xa ) @ R4 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_761_underS__incr,axiom,
    ! [R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
      ( ( trans_on_option_a @ top_top_set_option_a @ R2 )
     => ( ( antisym_on_option_a @ top_top_set_option_a @ R2 )
       => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 )
         => ( ord_le1955136853071979460tion_a @ ( order_8525669848891258378tion_a @ R2 @ A ) @ ( order_8525669848891258378tion_a @ R2 @ B ) ) ) ) ) ).

% underS_incr
thf(fact_762_underS__incr,axiom,
    ! [R2: set_Product_prod_a_a,A: a,B: a] :
      ( ( trans_on_a @ top_top_set_a @ R2 )
     => ( ( antisym_on_a @ top_top_set_a @ R2 )
       => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
         => ( ord_less_eq_set_a @ ( order_underS_a @ R2 @ A ) @ ( order_underS_a @ R2 @ B ) ) ) ) ) ).

% underS_incr
thf(fact_763_wo__rel_OisMinim__def,axiom,
    ! [R2: set_Product_prod_a_a,A2: set_a,B: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R2 )
     => ( ( bNF_We6697304935525757620inim_a @ R2 @ A2 @ B )
        = ( ( member_a @ B @ A2 )
          & ! [X: a] :
              ( ( member_a @ X @ A2 )
             => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ X ) @ R2 ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_764_wo__rel_OisMinim__def,axiom,
    ! [R2: set_Pr8390707888732215751_a_d_c,A2: set_Pr7868159745199425715_a_d_c,B: produc6150846815813599699_a_d_c] :
      ( ( bNF_We920288868220767679_a_d_c @ R2 )
     => ( ( bNF_We4481243920461668745_a_d_c @ R2 @ A2 @ B )
        = ( ( member2052822272342364412_a_d_c @ B @ A2 )
          & ! [X: produc6150846815813599699_a_d_c] :
              ( ( member2052822272342364412_a_d_c @ X @ A2 )
             => ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ B @ X ) @ R2 ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_765_wo__rel_OisMinim__def,axiom,
    ! [R2: set_Pr7585778909603769095tion_a,A2: set_option_a,B: option_a] :
      ( ( bNF_We8432232079604507440tion_a @ R2 )
     => ( ( bNF_We2467337426749329402tion_a @ R2 @ A2 @ B )
        = ( ( member_option_a @ B @ A2 )
          & ! [X: option_a] :
              ( ( member_option_a @ X @ A2 )
             => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ X ) @ R2 ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_766_antisym__onI,axiom,
    ! [A2: set_a,R2: set_Product_prod_a_a] :
      ( ! [X2: a,Y4: a] :
          ( ( member_a @ X2 @ A2 )
         => ( ( member_a @ Y4 @ A2 )
           => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y4 ) @ R2 )
             => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X2 ) @ R2 )
               => ( X2 = Y4 ) ) ) ) )
     => ( antisym_on_a @ A2 @ R2 ) ) ).

% antisym_onI
thf(fact_767_antisym__onI,axiom,
    ! [A2: set_Pr7868159745199425715_a_d_c,R2: set_Pr8390707888732215751_a_d_c] :
      ( ! [X2: produc6150846815813599699_a_d_c,Y4: produc6150846815813599699_a_d_c] :
          ( ( member2052822272342364412_a_d_c @ X2 @ A2 )
         => ( ( member2052822272342364412_a_d_c @ Y4 @ A2 )
           => ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ X2 @ Y4 ) @ R2 )
             => ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ Y4 @ X2 ) @ R2 )
               => ( X2 = Y4 ) ) ) ) )
     => ( antisy3666119904615816886_a_d_c @ A2 @ R2 ) ) ).

% antisym_onI
thf(fact_768_antisym__onI,axiom,
    ! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a] :
      ( ! [X2: option_a,Y4: option_a] :
          ( ( member_option_a @ X2 @ A2 )
         => ( ( member_option_a @ Y4 @ A2 )
           => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y4 ) @ R2 )
             => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y4 @ X2 ) @ R2 )
               => ( X2 = Y4 ) ) ) ) )
     => ( antisym_on_option_a @ A2 @ R2 ) ) ).

% antisym_onI
thf(fact_769_antisym__onD,axiom,
    ! [A2: set_a,R2: set_Product_prod_a_a,X3: a,Y: a] :
      ( ( antisym_on_a @ A2 @ R2 )
     => ( ( member_a @ X3 @ A2 )
       => ( ( member_a @ Y @ A2 )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R2 )
           => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X3 ) @ R2 )
             => ( X3 = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_770_antisym__onD,axiom,
    ! [A2: set_Pr7868159745199425715_a_d_c,R2: set_Pr8390707888732215751_a_d_c,X3: produc6150846815813599699_a_d_c,Y: produc6150846815813599699_a_d_c] :
      ( ( antisy3666119904615816886_a_d_c @ A2 @ R2 )
     => ( ( member2052822272342364412_a_d_c @ X3 @ A2 )
       => ( ( member2052822272342364412_a_d_c @ Y @ A2 )
         => ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ X3 @ Y ) @ R2 )
           => ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ Y @ X3 ) @ R2 )
             => ( X3 = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_771_antisym__onD,axiom,
    ! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,X3: option_a,Y: option_a] :
      ( ( antisym_on_option_a @ A2 @ R2 )
     => ( ( member_option_a @ X3 @ A2 )
       => ( ( member_option_a @ Y @ A2 )
         => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y ) @ R2 )
           => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y @ X3 ) @ R2 )
             => ( X3 = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_772_image__strict__mono,axiom,
    ! [F2: option_a > a,B2: set_option_a,A2: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ B2 )
     => ( ( ord_le5631237216984945872tion_a @ A2 @ B2 )
       => ( ord_less_set_a @ ( image_option_a_a2 @ F2 @ A2 ) @ ( image_option_a_a2 @ F2 @ B2 ) ) ) ) ).

% image_strict_mono
thf(fact_773_image__strict__mono,axiom,
    ! [F2: a > option_a,B2: set_a,A2: set_a] :
      ( ( inj_on_a_option_a @ F2 @ B2 )
     => ( ( ord_less_set_a @ A2 @ B2 )
       => ( ord_le5631237216984945872tion_a @ ( image_a_option_a2 @ F2 @ A2 ) @ ( image_a_option_a2 @ F2 @ B2 ) ) ) ) ).

% image_strict_mono
thf(fact_774_antisymD,axiom,
    ! [R2: set_Pr7585778909603769095tion_a,X3: option_a,Y: option_a] :
      ( ( antisym_on_option_a @ top_top_set_option_a @ R2 )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y ) @ R2 )
       => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y @ X3 ) @ R2 )
         => ( X3 = Y ) ) ) ) ).

% antisymD
thf(fact_775_antisymD,axiom,
    ! [R2: set_Product_prod_a_a,X3: a,Y: a] :
      ( ( antisym_on_a @ top_top_set_a @ R2 )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R2 )
       => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X3 ) @ R2 )
         => ( X3 = Y ) ) ) ) ).

% antisymD
thf(fact_776_antisymI,axiom,
    ! [R2: set_Pr7585778909603769095tion_a] :
      ( ! [X2: option_a,Y4: option_a] :
          ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y4 ) @ R2 )
         => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y4 @ X2 ) @ R2 )
           => ( X2 = Y4 ) ) )
     => ( antisym_on_option_a @ top_top_set_option_a @ R2 ) ) ).

% antisymI
thf(fact_777_antisymI,axiom,
    ! [R2: set_Product_prod_a_a] :
      ( ! [X2: a,Y4: a] :
          ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y4 ) @ R2 )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X2 ) @ R2 )
           => ( X2 = Y4 ) ) )
     => ( antisym_on_a @ top_top_set_a @ R2 ) ) ).

% antisymI
thf(fact_778_in__finite__psubset,axiom,
    ! [A2: set_option_a,B2: set_option_a] :
      ( ( member5358692782348450128tion_a @ ( produc8179951581375851543tion_a @ A2 @ B2 ) @ finite4966134214920407047tion_a )
      = ( ( ord_le5631237216984945872tion_a @ A2 @ B2 )
        & ( finite1674126218327898605tion_a @ B2 ) ) ) ).

% in_finite_psubset
thf(fact_779_in__finite__psubset,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ A2 @ B2 ) @ finite_psubset_a )
      = ( ( ord_less_set_a @ A2 @ B2 )
        & ( finite_finite_a @ B2 ) ) ) ).

% in_finite_psubset
thf(fact_780_in__lex__prod,axiom,
    ! [A: product_prod_a_d_c,B: option2362401199305441953_a_d_c,A5: product_prod_a_d_c,B4: option2362401199305441953_a_d_c,R2: set_Pr2251131264138062051_a_d_c,S4: set_Pr2205332559432586627_a_d_c] :
      ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ ( produc5208860900648697099_a_d_c @ A @ B ) @ ( produc5208860900648697099_a_d_c @ A5 @ B4 ) ) @ ( lex_pr7013848916100950307_a_d_c @ R2 @ S4 ) )
      = ( ( member2237522729980725036_a_d_c @ ( produc3833742498670062139_a_d_c @ A @ A5 ) @ R2 )
        | ( ( A = A5 )
          & ( member7044238567156598732_a_d_c @ ( produc7954588475506768603_a_d_c @ B @ B4 ) @ S4 ) ) ) ) ).

% in_lex_prod
thf(fact_781_in__lex__prod,axiom,
    ! [A: a,B: d > c,A5: a,B4: d > c,R2: set_Product_prod_a_a,S4: set_Pr1239784501399087881_c_d_c] :
      ( ( member2237522729980725036_a_d_c @ ( produc3833742498670062139_a_d_c @ ( product_Pair_a_d_c @ A @ B ) @ ( product_Pair_a_d_c @ A5 @ B4 ) ) @ ( lex_prod_a_d_c @ R2 @ S4 ) )
      = ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A5 ) @ R2 )
        | ( ( A = A5 )
          & ( member3094720156254151762_c_d_c @ ( product_Pair_d_c_d_c @ B @ B4 ) @ S4 ) ) ) ) ).

% in_lex_prod
thf(fact_782_same__fstI,axiom,
    ! [P2: product_prod_a_d_c > $o,X3: product_prod_a_d_c,Y7: option2362401199305441953_a_d_c,Y: option2362401199305441953_a_d_c,R4: product_prod_a_d_c > set_Pr2205332559432586627_a_d_c] :
      ( ( P2 @ X3 )
     => ( ( member7044238567156598732_a_d_c @ ( produc7954588475506768603_a_d_c @ Y7 @ Y ) @ ( R4 @ X3 ) )
       => ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ ( produc5208860900648697099_a_d_c @ X3 @ Y7 ) @ ( produc5208860900648697099_a_d_c @ X3 @ Y ) ) @ ( same_f4766842598631432416_a_d_c @ P2 @ R4 ) ) ) ) ).

% same_fstI
thf(fact_783_same__fstI,axiom,
    ! [P2: a > $o,X3: a,Y7: d > c,Y: d > c,R4: a > set_Pr1239784501399087881_c_d_c] :
      ( ( P2 @ X3 )
     => ( ( member3094720156254151762_c_d_c @ ( product_Pair_d_c_d_c @ Y7 @ Y ) @ ( R4 @ X3 ) )
       => ( member2237522729980725036_a_d_c @ ( produc3833742498670062139_a_d_c @ ( product_Pair_a_d_c @ X3 @ Y7 ) @ ( product_Pair_a_d_c @ X3 @ Y ) ) @ ( same_fst_a_d_c @ P2 @ R4 ) ) ) ) ).

% same_fstI
thf(fact_784_wo__rel_Osuc__greater,axiom,
    ! [R2: set_Pr8390707888732215751_a_d_c,B2: set_Pr7868159745199425715_a_d_c,B: produc6150846815813599699_a_d_c] :
      ( ( bNF_We920288868220767679_a_d_c @ R2 )
     => ( ( ord_le3761643085089054803_a_d_c @ B2 @ ( field_6537103990374211872_a_d_c @ R2 ) )
       => ( ( ( order_1561669969411077278_a_d_c @ R2 @ B2 )
           != bot_bo1396921257525504799_a_d_c )
         => ( ( member2052822272342364412_a_d_c @ B @ B2 )
           => ( ( ( bNF_We1545902585215803188_a_d_c @ R2 @ B2 )
               != B )
              & ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ B @ ( bNF_We1545902585215803188_a_d_c @ R2 @ B2 ) ) @ R2 ) ) ) ) ) ) ).

% wo_rel.suc_greater
thf(fact_785_wo__rel_Osuc__greater,axiom,
    ! [R2: set_Product_prod_a_a,B2: set_a,B: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R2 )
     => ( ( ord_less_eq_set_a @ B2 @ ( field_a @ R2 ) )
       => ( ( ( order_AboveS_a @ R2 @ B2 )
           != bot_bot_set_a )
         => ( ( member_a @ B @ B2 )
           => ( ( ( bNF_We6154283375207884895_suc_a @ R2 @ B2 )
               != B )
              & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ ( bNF_We6154283375207884895_suc_a @ R2 @ B2 ) ) @ R2 ) ) ) ) ) ) ).

% wo_rel.suc_greater
thf(fact_786_wo__rel_Osuc__greater,axiom,
    ! [R2: set_Pr7585778909603769095tion_a,B2: set_option_a,B: option_a] :
      ( ( bNF_We8432232079604507440tion_a @ R2 )
     => ( ( ord_le1955136853071979460tion_a @ B2 @ ( field_option_a @ R2 ) )
       => ( ( ( order_6500638856667293583tion_a @ R2 @ B2 )
           != bot_bot_set_option_a )
         => ( ( member_option_a @ B @ B2 )
           => ( ( ( bNF_We5356091070762920229tion_a @ R2 @ B2 )
               != B )
              & ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ ( bNF_We5356091070762920229tion_a @ R2 @ B2 ) ) @ R2 ) ) ) ) ) ) ).

% wo_rel.suc_greater
thf(fact_787_wo__rel_Osuc__least__AboveS,axiom,
    ! [R2: set_Product_prod_a_a,A: a,B2: set_a] :
      ( ( bNF_We1162827675446709994_rel_a @ R2 )
     => ( ( member_a @ A @ ( order_AboveS_a @ R2 @ B2 ) )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( bNF_We6154283375207884895_suc_a @ R2 @ B2 ) @ A ) @ R2 ) ) ) ).

% wo_rel.suc_least_AboveS
thf(fact_788_wo__rel_Osuc__least__AboveS,axiom,
    ! [R2: set_Pr8390707888732215751_a_d_c,A: produc6150846815813599699_a_d_c,B2: set_Pr7868159745199425715_a_d_c] :
      ( ( bNF_We920288868220767679_a_d_c @ R2 )
     => ( ( member2052822272342364412_a_d_c @ A @ ( order_1561669969411077278_a_d_c @ R2 @ B2 ) )
       => ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ ( bNF_We1545902585215803188_a_d_c @ R2 @ B2 ) @ A ) @ R2 ) ) ) ).

% wo_rel.suc_least_AboveS
thf(fact_789_wo__rel_Osuc__least__AboveS,axiom,
    ! [R2: set_Pr7585778909603769095tion_a,A: option_a,B2: set_option_a] :
      ( ( bNF_We8432232079604507440tion_a @ R2 )
     => ( ( member_option_a @ A @ ( order_6500638856667293583tion_a @ R2 @ B2 ) )
       => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ ( bNF_We5356091070762920229tion_a @ R2 @ B2 ) @ A ) @ R2 ) ) ) ).

% wo_rel.suc_least_AboveS
thf(fact_790_wo__rel_Oequals__suc__AboveS,axiom,
    ! [R2: set_Product_prod_a_a,B2: set_a,A: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R2 )
     => ( ( ord_less_eq_set_a @ B2 @ ( field_a @ R2 ) )
       => ( ( member_a @ A @ ( order_AboveS_a @ R2 @ B2 ) )
         => ( ! [A6: a] :
                ( ( member_a @ A6 @ ( order_AboveS_a @ R2 @ B2 ) )
               => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A6 ) @ R2 ) )
           => ( A
              = ( bNF_We6154283375207884895_suc_a @ R2 @ B2 ) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
thf(fact_791_wo__rel_Oequals__suc__AboveS,axiom,
    ! [R2: set_Pr8390707888732215751_a_d_c,B2: set_Pr7868159745199425715_a_d_c,A: produc6150846815813599699_a_d_c] :
      ( ( bNF_We920288868220767679_a_d_c @ R2 )
     => ( ( ord_le3761643085089054803_a_d_c @ B2 @ ( field_6537103990374211872_a_d_c @ R2 ) )
       => ( ( member2052822272342364412_a_d_c @ A @ ( order_1561669969411077278_a_d_c @ R2 @ B2 ) )
         => ( ! [A6: produc6150846815813599699_a_d_c] :
                ( ( member2052822272342364412_a_d_c @ A6 @ ( order_1561669969411077278_a_d_c @ R2 @ B2 ) )
               => ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ A @ A6 ) @ R2 ) )
           => ( A
              = ( bNF_We1545902585215803188_a_d_c @ R2 @ B2 ) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
thf(fact_792_wo__rel_Oequals__suc__AboveS,axiom,
    ! [R2: set_Pr7585778909603769095tion_a,B2: set_option_a,A: option_a] :
      ( ( bNF_We8432232079604507440tion_a @ R2 )
     => ( ( ord_le1955136853071979460tion_a @ B2 @ ( field_option_a @ R2 ) )
       => ( ( member_option_a @ A @ ( order_6500638856667293583tion_a @ R2 @ B2 ) )
         => ( ! [A6: option_a] :
                ( ( member_option_a @ A6 @ ( order_6500638856667293583tion_a @ R2 @ B2 ) )
               => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ A6 ) @ R2 ) )
           => ( A
              = ( bNF_We5356091070762920229tion_a @ R2 @ B2 ) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
thf(fact_793_Not__Domain__rtrancl,axiom,
    ! [X3: a,R4: set_Product_prod_a_a,Y: a] :
      ( ~ ( member_a @ X3 @ ( domain_a_a @ R4 ) )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ ( transitive_rtrancl_a @ R4 ) )
        = ( X3 = Y ) ) ) ).

% Not_Domain_rtrancl
thf(fact_794_Not__Domain__rtrancl,axiom,
    ! [X3: produc6150846815813599699_a_d_c,R4: set_Pr8390707888732215751_a_d_c,Y: produc6150846815813599699_a_d_c] :
      ( ~ ( member2052822272342364412_a_d_c @ X3 @ ( domain3926269679852320972_a_d_c @ R4 ) )
     => ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ X3 @ Y ) @ ( transi6280719840288650198_a_d_c @ R4 ) )
        = ( X3 = Y ) ) ) ).

% Not_Domain_rtrancl
thf(fact_795_Not__Domain__rtrancl,axiom,
    ! [X3: option_a,R4: set_Pr7585778909603769095tion_a,Y: option_a] :
      ( ~ ( member_option_a @ X3 @ ( domain5649462347324568460tion_a @ R4 ) )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y ) @ ( transi330218190764880583tion_a @ R4 ) )
        = ( X3 = Y ) ) ) ).

% Not_Domain_rtrancl
thf(fact_796_rtrancl__induct2,axiom,
    ! [Ax: product_prod_a_d_c,Ay: option2362401199305441953_a_d_c,Bx: product_prod_a_d_c,By: option2362401199305441953_a_d_c,R2: set_Pr8390707888732215751_a_d_c,P2: product_prod_a_d_c > option2362401199305441953_a_d_c > $o] :
      ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ ( produc5208860900648697099_a_d_c @ Ax @ Ay ) @ ( produc5208860900648697099_a_d_c @ Bx @ By ) ) @ ( transi6280719840288650198_a_d_c @ R2 ) )
     => ( ( P2 @ Ax @ Ay )
       => ( ! [A4: product_prod_a_d_c,B5: option2362401199305441953_a_d_c,Aa: product_prod_a_d_c,Ba: option2362401199305441953_a_d_c] :
              ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ ( produc5208860900648697099_a_d_c @ Ax @ Ay ) @ ( produc5208860900648697099_a_d_c @ A4 @ B5 ) ) @ ( transi6280719840288650198_a_d_c @ R2 ) )
             => ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ ( produc5208860900648697099_a_d_c @ A4 @ B5 ) @ ( produc5208860900648697099_a_d_c @ Aa @ Ba ) ) @ R2 )
               => ( ( P2 @ A4 @ B5 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_797_rtrancl__induct2,axiom,
    ! [Ax: a,Ay: d > c,Bx: a,By: d > c,R2: set_Pr2251131264138062051_a_d_c,P2: a > ( d > c ) > $o] :
      ( ( member2237522729980725036_a_d_c @ ( produc3833742498670062139_a_d_c @ ( product_Pair_a_d_c @ Ax @ Ay ) @ ( product_Pair_a_d_c @ Bx @ By ) ) @ ( transi984476200108537102_a_d_c @ R2 ) )
     => ( ( P2 @ Ax @ Ay )
       => ( ! [A4: a,B5: d > c,Aa: a,Ba: d > c] :
              ( ( member2237522729980725036_a_d_c @ ( produc3833742498670062139_a_d_c @ ( product_Pair_a_d_c @ Ax @ Ay ) @ ( product_Pair_a_d_c @ A4 @ B5 ) ) @ ( transi984476200108537102_a_d_c @ R2 ) )
             => ( ( member2237522729980725036_a_d_c @ ( produc3833742498670062139_a_d_c @ ( product_Pair_a_d_c @ A4 @ B5 ) @ ( product_Pair_a_d_c @ Aa @ Ba ) ) @ R2 )
               => ( ( P2 @ A4 @ B5 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_798_converse__rtranclE2,axiom,
    ! [Xa3: product_prod_a_d_c,Xb: option2362401199305441953_a_d_c,Za: product_prod_a_d_c,Zb: option2362401199305441953_a_d_c,R2: set_Pr8390707888732215751_a_d_c] :
      ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ ( produc5208860900648697099_a_d_c @ Xa3 @ Xb ) @ ( produc5208860900648697099_a_d_c @ Za @ Zb ) ) @ ( transi6280719840288650198_a_d_c @ R2 ) )
     => ( ( ( produc5208860900648697099_a_d_c @ Xa3 @ Xb )
         != ( produc5208860900648697099_a_d_c @ Za @ Zb ) )
       => ~ ! [A4: product_prod_a_d_c,B5: option2362401199305441953_a_d_c] :
              ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ ( produc5208860900648697099_a_d_c @ Xa3 @ Xb ) @ ( produc5208860900648697099_a_d_c @ A4 @ B5 ) ) @ R2 )
             => ~ ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ ( produc5208860900648697099_a_d_c @ A4 @ B5 ) @ ( produc5208860900648697099_a_d_c @ Za @ Zb ) ) @ ( transi6280719840288650198_a_d_c @ R2 ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_799_converse__rtranclE2,axiom,
    ! [Xa3: a,Xb: d > c,Za: a,Zb: d > c,R2: set_Pr2251131264138062051_a_d_c] :
      ( ( member2237522729980725036_a_d_c @ ( produc3833742498670062139_a_d_c @ ( product_Pair_a_d_c @ Xa3 @ Xb ) @ ( product_Pair_a_d_c @ Za @ Zb ) ) @ ( transi984476200108537102_a_d_c @ R2 ) )
     => ( ( ( product_Pair_a_d_c @ Xa3 @ Xb )
         != ( product_Pair_a_d_c @ Za @ Zb ) )
       => ~ ! [A4: a,B5: d > c] :
              ( ( member2237522729980725036_a_d_c @ ( produc3833742498670062139_a_d_c @ ( product_Pair_a_d_c @ Xa3 @ Xb ) @ ( product_Pair_a_d_c @ A4 @ B5 ) ) @ R2 )
             => ~ ( member2237522729980725036_a_d_c @ ( produc3833742498670062139_a_d_c @ ( product_Pair_a_d_c @ A4 @ B5 ) @ ( product_Pair_a_d_c @ Za @ Zb ) ) @ ( transi984476200108537102_a_d_c @ R2 ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_800_converse__rtrancl__induct2,axiom,
    ! [Ax: product_prod_a_d_c,Ay: option2362401199305441953_a_d_c,Bx: product_prod_a_d_c,By: option2362401199305441953_a_d_c,R2: set_Pr8390707888732215751_a_d_c,P2: product_prod_a_d_c > option2362401199305441953_a_d_c > $o] :
      ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ ( produc5208860900648697099_a_d_c @ Ax @ Ay ) @ ( produc5208860900648697099_a_d_c @ Bx @ By ) ) @ ( transi6280719840288650198_a_d_c @ R2 ) )
     => ( ( P2 @ Bx @ By )
       => ( ! [A4: product_prod_a_d_c,B5: option2362401199305441953_a_d_c,Aa: product_prod_a_d_c,Ba: option2362401199305441953_a_d_c] :
              ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ ( produc5208860900648697099_a_d_c @ A4 @ B5 ) @ ( produc5208860900648697099_a_d_c @ Aa @ Ba ) ) @ R2 )
             => ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ ( produc5208860900648697099_a_d_c @ Aa @ Ba ) @ ( produc5208860900648697099_a_d_c @ Bx @ By ) ) @ ( transi6280719840288650198_a_d_c @ R2 ) )
               => ( ( P2 @ Aa @ Ba )
                 => ( P2 @ A4 @ B5 ) ) ) )
         => ( P2 @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_801_converse__rtrancl__induct2,axiom,
    ! [Ax: a,Ay: d > c,Bx: a,By: d > c,R2: set_Pr2251131264138062051_a_d_c,P2: a > ( d > c ) > $o] :
      ( ( member2237522729980725036_a_d_c @ ( produc3833742498670062139_a_d_c @ ( product_Pair_a_d_c @ Ax @ Ay ) @ ( product_Pair_a_d_c @ Bx @ By ) ) @ ( transi984476200108537102_a_d_c @ R2 ) )
     => ( ( P2 @ Bx @ By )
       => ( ! [A4: a,B5: d > c,Aa: a,Ba: d > c] :
              ( ( member2237522729980725036_a_d_c @ ( produc3833742498670062139_a_d_c @ ( product_Pair_a_d_c @ A4 @ B5 ) @ ( product_Pair_a_d_c @ Aa @ Ba ) ) @ R2 )
             => ( ( member2237522729980725036_a_d_c @ ( produc3833742498670062139_a_d_c @ ( product_Pair_a_d_c @ Aa @ Ba ) @ ( product_Pair_a_d_c @ Bx @ By ) ) @ ( transi984476200108537102_a_d_c @ R2 ) )
               => ( ( P2 @ Aa @ Ba )
                 => ( P2 @ A4 @ B5 ) ) ) )
         => ( P2 @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_802_wo__rel_Osuc__ofilter__in,axiom,
    ! [R2: set_Pr8390707888732215751_a_d_c,A2: set_Pr7868159745199425715_a_d_c,B: produc6150846815813599699_a_d_c] :
      ( ( bNF_We920288868220767679_a_d_c @ R2 )
     => ( ( order_7127145726454921833_a_d_c @ R2 @ A2 )
       => ( ( ( order_1561669969411077278_a_d_c @ R2 @ A2 )
           != bot_bo1396921257525504799_a_d_c )
         => ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ B @ ( bNF_We1545902585215803188_a_d_c @ R2 @ A2 ) ) @ R2 )
           => ( ( B
               != ( bNF_We1545902585215803188_a_d_c @ R2 @ A2 ) )
             => ( member2052822272342364412_a_d_c @ B @ A2 ) ) ) ) ) ) ).

% wo_rel.suc_ofilter_in
thf(fact_803_wo__rel_Osuc__ofilter__in,axiom,
    ! [R2: set_Product_prod_a_a,A2: set_a,B: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R2 )
     => ( ( order_ofilter_a @ R2 @ A2 )
       => ( ( ( order_AboveS_a @ R2 @ A2 )
           != bot_bot_set_a )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ ( bNF_We6154283375207884895_suc_a @ R2 @ A2 ) ) @ R2 )
           => ( ( B
               != ( bNF_We6154283375207884895_suc_a @ R2 @ A2 ) )
             => ( member_a @ B @ A2 ) ) ) ) ) ) ).

% wo_rel.suc_ofilter_in
thf(fact_804_wo__rel_Osuc__ofilter__in,axiom,
    ! [R2: set_Pr7585778909603769095tion_a,A2: set_option_a,B: option_a] :
      ( ( bNF_We8432232079604507440tion_a @ R2 )
     => ( ( order_6420974439381506266tion_a @ R2 @ A2 )
       => ( ( ( order_6500638856667293583tion_a @ R2 @ A2 )
           != bot_bot_set_option_a )
         => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ ( bNF_We5356091070762920229tion_a @ R2 @ A2 ) ) @ R2 )
           => ( ( B
               != ( bNF_We5356091070762920229tion_a @ R2 @ A2 ) )
             => ( member_option_a @ B @ A2 ) ) ) ) ) ) ).

% wo_rel.suc_ofilter_in
thf(fact_805_vimage__subsetI,axiom,
    ! [F2: option_a > a,B2: set_a,A2: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ top_top_set_option_a )
     => ( ( ord_less_eq_set_a @ B2 @ ( image_option_a_a2 @ F2 @ A2 ) )
       => ( ord_le1955136853071979460tion_a @ ( vimage_option_a_a @ F2 @ B2 ) @ A2 ) ) ) ).

% vimage_subsetI
thf(fact_806_vimage__subsetI,axiom,
    ! [F2: a > option_a,B2: set_option_a,A2: set_a] :
      ( ( inj_on_a_option_a @ F2 @ top_top_set_a )
     => ( ( ord_le1955136853071979460tion_a @ B2 @ ( image_a_option_a2 @ F2 @ A2 ) )
       => ( ord_less_eq_set_a @ ( vimage_a_option_a @ F2 @ B2 ) @ A2 ) ) ) ).

% vimage_subsetI
thf(fact_807_image2__eqI,axiom,
    ! [B: product_prod_a_d_c,F2: a > product_prod_a_d_c,X3: a,C: option2362401199305441953_a_d_c,G: a > option2362401199305441953_a_d_c,A2: set_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C
          = ( G @ X3 ) )
       => ( ( member_a @ X3 @ A2 )
         => ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ B @ C ) @ ( bNF_Gr2096653962509776792_a_d_c @ A2 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_808_image2__eqI,axiom,
    ! [B: product_prod_a_d_c,F2: produc6150846815813599699_a_d_c > product_prod_a_d_c,X3: produc6150846815813599699_a_d_c,C: option2362401199305441953_a_d_c,G: produc6150846815813599699_a_d_c > option2362401199305441953_a_d_c,A2: set_Pr7868159745199425715_a_d_c] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C
          = ( G @ X3 ) )
       => ( ( member2052822272342364412_a_d_c @ X3 @ A2 )
         => ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ B @ C ) @ ( bNF_Gr5816929820373306605_a_d_c @ A2 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_809_image2__eqI,axiom,
    ! [B: product_prod_a_d_c,F2: option_a > product_prod_a_d_c,X3: option_a,C: option2362401199305441953_a_d_c,G: option_a > option2362401199305441953_a_d_c,A2: set_option_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C
          = ( G @ X3 ) )
       => ( ( member_option_a @ X3 @ A2 )
         => ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ B @ C ) @ ( bNF_Gr8997863846829389790_a_d_c @ A2 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_810_image2__eqI,axiom,
    ! [B: a,F2: a > a,X3: a,C: d > c,G: a > d > c,A2: set_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C
          = ( G @ X3 ) )
       => ( ( member_a @ X3 @ A2 )
         => ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ B @ C ) @ ( bNF_Gr4650041265119207678_a_d_c @ A2 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_811_image2__eqI,axiom,
    ! [B: a,F2: produc6150846815813599699_a_d_c > a,X3: produc6150846815813599699_a_d_c,C: d > c,G: produc6150846815813599699_a_d_c > d > c,A2: set_Pr7868159745199425715_a_d_c] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C
          = ( G @ X3 ) )
       => ( ( member2052822272342364412_a_d_c @ X3 @ A2 )
         => ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ B @ C ) @ ( bNF_Gr3902548584958070633_a_d_c @ A2 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_812_image2__eqI,axiom,
    ! [B: a,F2: option_a > a,X3: option_a,C: d > c,G: option_a > d > c,A2: set_option_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C
          = ( G @ X3 ) )
       => ( ( member_option_a @ X3 @ A2 )
         => ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ B @ C ) @ ( bNF_Gr7827691902141730104_a_d_c @ A2 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_813_option_Oinj__map,axiom,
    ! [F2: a > option_a] :
      ( ( inj_on_a_option_a @ F2 @ top_top_set_a )
     => ( inj_on2224753519991154999tion_a @ ( map_op2340691886215429841tion_a @ F2 ) @ top_top_set_option_a ) ) ).

% option.inj_map
thf(fact_814_option_Oinj__map,axiom,
    ! [F2: a > a] :
      ( ( inj_on_a_a @ F2 @ top_top_set_a )
     => ( inj_on8559383841115902449tion_a @ ( map_option_a_a @ F2 ) @ top_top_set_option_a ) ) ).

% option.inj_map
thf(fact_815_map__option__eq__Some,axiom,
    ! [F2: product_prod_a_d_c > a,Xo: option2362401199305441953_a_d_c,Y: a] :
      ( ( ( map_op484624876561133600_d_c_a @ F2 @ Xo )
        = ( some_a @ Y ) )
      = ( ? [Z: product_prod_a_d_c] :
            ( ( Xo
              = ( some_P377817780860425132_a_d_c @ Z ) )
            & ( ( F2 @ Z )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_816_map__option__eq__Some,axiom,
    ! [F2: a > product_prod_a_d_c,Xo: option_a,Y: product_prod_a_d_c] :
      ( ( ( map_op6893876686296609156_a_d_c @ F2 @ Xo )
        = ( some_P377817780860425132_a_d_c @ Y ) )
      = ( ? [Z: a] :
            ( ( Xo
              = ( some_a @ Z ) )
            & ( ( F2 @ Z )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_817_map__option__eq__Some,axiom,
    ! [F2: product_prod_a_d_c > product_prod_a_d_c,Xo: option2362401199305441953_a_d_c,Y: product_prod_a_d_c] :
      ( ( ( map_op2548020587289152815_a_d_c @ F2 @ Xo )
        = ( some_P377817780860425132_a_d_c @ Y ) )
      = ( ? [Z: product_prod_a_d_c] :
            ( ( Xo
              = ( some_P377817780860425132_a_d_c @ Z ) )
            & ( ( F2 @ Z )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_818_map__option__eq__Some,axiom,
    ! [F2: a > a,Xo: option_a,Y: a] :
      ( ( ( map_option_a_a @ F2 @ Xo )
        = ( some_a @ Y ) )
      = ( ? [Z: a] :
            ( ( Xo
              = ( some_a @ Z ) )
            & ( ( F2 @ Z )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_819_option_Omap__disc__iff,axiom,
    ! [F2: a > a,A: option_a] :
      ( ( ( map_option_a_a @ F2 @ A )
        = none_a )
      = ( A = none_a ) ) ).

% option.map_disc_iff
thf(fact_820_map__option__is__None,axiom,
    ! [F2: a > a,Opt: option_a] :
      ( ( ( map_option_a_a @ F2 @ Opt )
        = none_a )
      = ( Opt = none_a ) ) ).

% map_option_is_None
thf(fact_821_None__eq__map__option__iff,axiom,
    ! [F2: a > a,X3: option_a] :
      ( ( none_a
        = ( map_option_a_a @ F2 @ X3 ) )
      = ( X3 = none_a ) ) ).

% None_eq_map_option_iff
thf(fact_822_is__none__map__option,axiom,
    ! [F2: a > a,X3: option_a] :
      ( ( is_none_a @ ( map_option_a_a @ F2 @ X3 ) )
      = ( is_none_a @ X3 ) ) ).

% is_none_map_option
thf(fact_823_option_Osimps_I9_J,axiom,
    ! [F2: a > product_prod_a_d_c,X22: a] :
      ( ( map_op6893876686296609156_a_d_c @ F2 @ ( some_a @ X22 ) )
      = ( some_P377817780860425132_a_d_c @ ( F2 @ X22 ) ) ) ).

% option.simps(9)
thf(fact_824_option_Osimps_I9_J,axiom,
    ! [F2: product_prod_a_d_c > a,X22: product_prod_a_d_c] :
      ( ( map_op484624876561133600_d_c_a @ F2 @ ( some_P377817780860425132_a_d_c @ X22 ) )
      = ( some_a @ ( F2 @ X22 ) ) ) ).

% option.simps(9)
thf(fact_825_option_Osimps_I9_J,axiom,
    ! [F2: product_prod_a_d_c > product_prod_a_d_c,X22: product_prod_a_d_c] :
      ( ( map_op2548020587289152815_a_d_c @ F2 @ ( some_P377817780860425132_a_d_c @ X22 ) )
      = ( some_P377817780860425132_a_d_c @ ( F2 @ X22 ) ) ) ).

% option.simps(9)
thf(fact_826_option_Osimps_I9_J,axiom,
    ! [F2: a > a,X22: a] :
      ( ( map_option_a_a @ F2 @ ( some_a @ X22 ) )
      = ( some_a @ ( F2 @ X22 ) ) ) ).

% option.simps(9)
thf(fact_827_map__option__cong,axiom,
    ! [X3: option_a,Y: option_a,F2: a > a,G: a > a] :
      ( ( X3 = Y )
     => ( ! [A4: a] :
            ( ( Y
              = ( some_a @ A4 ) )
           => ( ( F2 @ A4 )
              = ( G @ A4 ) ) )
       => ( ( map_option_a_a @ F2 @ X3 )
          = ( map_option_a_a @ G @ Y ) ) ) ) ).

% map_option_cong
thf(fact_828_option_Osimps_I8_J,axiom,
    ! [F2: a > a] :
      ( ( map_option_a_a @ F2 @ none_a )
      = none_a ) ).

% option.simps(8)
thf(fact_829_map__option__idI,axiom,
    ! [X3: option4105738954736457113_a_d_c,F2: produc6150846815813599699_a_d_c > produc6150846815813599699_a_d_c] :
      ( ! [Y4: produc6150846815813599699_a_d_c] :
          ( ( member2052822272342364412_a_d_c @ Y4 @ ( set_op2111387851716960097_a_d_c @ X3 ) )
         => ( ( F2 @ Y4 )
            = Y4 ) )
     => ( ( map_op1777168665185447755_a_d_c @ F2 @ X3 )
        = X3 ) ) ).

% map_option_idI
thf(fact_830_map__option__idI,axiom,
    ! [X3: option_option_a,F2: option_a > option_a] :
      ( ! [Y4: option_a] :
          ( ( member_option_a @ Y4 @ ( set_option_option_a2 @ X3 ) )
         => ( ( F2 @ Y4 )
            = Y4 ) )
     => ( ( map_op788413144570152203tion_a @ F2 @ X3 )
        = X3 ) ) ).

% map_option_idI
thf(fact_831_map__option__idI,axiom,
    ! [X3: option_a,F2: a > a] :
      ( ! [Y4: a] :
          ( ( member_a @ Y4 @ ( set_option_a3 @ X3 ) )
         => ( ( F2 @ Y4 )
            = Y4 ) )
     => ( ( map_option_a_a @ F2 @ X3 )
        = X3 ) ) ).

% map_option_idI
thf(fact_832_option_Omap__ident__strong,axiom,
    ! [T: option4105738954736457113_a_d_c,F2: produc6150846815813599699_a_d_c > produc6150846815813599699_a_d_c] :
      ( ! [Z3: produc6150846815813599699_a_d_c] :
          ( ( member2052822272342364412_a_d_c @ Z3 @ ( set_op2111387851716960097_a_d_c @ T ) )
         => ( ( F2 @ Z3 )
            = Z3 ) )
     => ( ( map_op1777168665185447755_a_d_c @ F2 @ T )
        = T ) ) ).

% option.map_ident_strong
thf(fact_833_option_Omap__ident__strong,axiom,
    ! [T: option_option_a,F2: option_a > option_a] :
      ( ! [Z3: option_a] :
          ( ( member_option_a @ Z3 @ ( set_option_option_a2 @ T ) )
         => ( ( F2 @ Z3 )
            = Z3 ) )
     => ( ( map_op788413144570152203tion_a @ F2 @ T )
        = T ) ) ).

% option.map_ident_strong
thf(fact_834_option_Omap__ident__strong,axiom,
    ! [T: option_a,F2: a > a] :
      ( ! [Z3: a] :
          ( ( member_a @ Z3 @ ( set_option_a3 @ T ) )
         => ( ( F2 @ Z3 )
            = Z3 ) )
     => ( ( map_option_a_a @ F2 @ T )
        = T ) ) ).

% option.map_ident_strong
thf(fact_835_option_Oinj__map__strong,axiom,
    ! [X3: option_a,Xa3: option_a,F2: a > a,Fa: a > a] :
      ( ! [Z3: a,Za2: a] :
          ( ( member_a @ Z3 @ ( set_option_a3 @ X3 ) )
         => ( ( member_a @ Za2 @ ( set_option_a3 @ Xa3 ) )
           => ( ( ( F2 @ Z3 )
                = ( Fa @ Za2 ) )
             => ( Z3 = Za2 ) ) ) )
     => ( ( ( map_option_a_a @ F2 @ X3 )
          = ( map_option_a_a @ Fa @ Xa3 ) )
       => ( X3 = Xa3 ) ) ) ).

% option.inj_map_strong
thf(fact_836_option_Omap__cong0,axiom,
    ! [X3: option_a,F2: a > a,G: a > a] :
      ( ! [Z3: a] :
          ( ( member_a @ Z3 @ ( set_option_a3 @ X3 ) )
         => ( ( F2 @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( map_option_a_a @ F2 @ X3 )
        = ( map_option_a_a @ G @ X3 ) ) ) ).

% option.map_cong0
thf(fact_837_option_Omap__cong,axiom,
    ! [X3: option_a,Ya: option_a,F2: a > a,G: a > a] :
      ( ( X3 = Ya )
     => ( ! [Z3: a] :
            ( ( member_a @ Z3 @ ( set_option_a3 @ Ya ) )
           => ( ( F2 @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( map_option_a_a @ F2 @ X3 )
          = ( map_option_a_a @ G @ Ya ) ) ) ) ).

% option.map_cong
thf(fact_838_surj__image__vimage__eq,axiom,
    ! [F2: option_a > option_a,A2: set_option_a] :
      ( ( ( image_7439109396645324421tion_a @ F2 @ top_top_set_option_a )
        = top_top_set_option_a )
     => ( ( image_7439109396645324421tion_a @ F2 @ ( vimage1562710927270423099tion_a @ F2 @ A2 ) )
        = A2 ) ) ).

% surj_image_vimage_eq
thf(fact_839_surj__image__vimage__eq,axiom,
    ! [F2: option_a > a,A2: set_a] :
      ( ( ( image_option_a_a2 @ F2 @ top_top_set_option_a )
        = top_top_set_a )
     => ( ( image_option_a_a2 @ F2 @ ( vimage_option_a_a @ F2 @ A2 ) )
        = A2 ) ) ).

% surj_image_vimage_eq
thf(fact_840_surj__image__vimage__eq,axiom,
    ! [F2: a > option_a,A2: set_option_a] :
      ( ( ( image_a_option_a2 @ F2 @ top_top_set_a )
        = top_top_set_option_a )
     => ( ( image_a_option_a2 @ F2 @ ( vimage_a_option_a @ F2 @ A2 ) )
        = A2 ) ) ).

% surj_image_vimage_eq
thf(fact_841_surj__image__vimage__eq,axiom,
    ! [F2: a > a,A2: set_a] :
      ( ( ( image_a_a2 @ F2 @ top_top_set_a )
        = top_top_set_a )
     => ( ( image_a_a2 @ F2 @ ( vimage_a_a @ F2 @ A2 ) )
        = A2 ) ) ).

% surj_image_vimage_eq
thf(fact_842_option_Oset__map,axiom,
    ! [F2: a > option_a,V3: option_a] :
      ( ( set_option_option_a2 @ ( map_op2340691886215429841tion_a @ F2 @ V3 ) )
      = ( image_a_option_a2 @ F2 @ ( set_option_a3 @ V3 ) ) ) ).

% option.set_map
thf(fact_843_option_Oset__map,axiom,
    ! [F2: option_a > a,V3: option_option_a] :
      ( ( set_option_a3 @ ( map_op4563205767754224965on_a_a @ F2 @ V3 ) )
      = ( image_option_a_a2 @ F2 @ ( set_option_option_a2 @ V3 ) ) ) ).

% option.set_map
thf(fact_844_option_Oset__map,axiom,
    ! [F2: a > a,V3: option_a] :
      ( ( set_option_a3 @ ( map_option_a_a @ F2 @ V3 ) )
      = ( image_a_a2 @ F2 @ ( set_option_a3 @ V3 ) ) ) ).

% option.set_map
thf(fact_845_option_Omap__sel,axiom,
    ! [A: option_a,F2: a > a] :
      ( ( A != none_a )
     => ( ( the_a @ ( map_option_a_a @ F2 @ A ) )
        = ( F2 @ ( the_a @ A ) ) ) ) ).

% option.map_sel
thf(fact_846_the__map__option,axiom,
    ! [X3: option_a,F2: a > a] :
      ( ~ ( is_none_a @ X3 )
     => ( ( the_a @ ( map_option_a_a @ F2 @ X3 ) )
        = ( F2 @ ( the_a @ X3 ) ) ) ) ).

% the_map_option
thf(fact_847_surj__vimage__empty,axiom,
    ! [F2: option_a > option_a,A2: set_option_a] :
      ( ( ( image_7439109396645324421tion_a @ F2 @ top_top_set_option_a )
        = top_top_set_option_a )
     => ( ( ( vimage1562710927270423099tion_a @ F2 @ A2 )
          = bot_bot_set_option_a )
        = ( A2 = bot_bot_set_option_a ) ) ) ).

% surj_vimage_empty
thf(fact_848_surj__vimage__empty,axiom,
    ! [F2: option_a > a,A2: set_a] :
      ( ( ( image_option_a_a2 @ F2 @ top_top_set_option_a )
        = top_top_set_a )
     => ( ( ( vimage_option_a_a @ F2 @ A2 )
          = bot_bot_set_option_a )
        = ( A2 = bot_bot_set_a ) ) ) ).

% surj_vimage_empty
thf(fact_849_surj__vimage__empty,axiom,
    ! [F2: a > option_a,A2: set_option_a] :
      ( ( ( image_a_option_a2 @ F2 @ top_top_set_a )
        = top_top_set_option_a )
     => ( ( ( vimage_a_option_a @ F2 @ A2 )
          = bot_bot_set_a )
        = ( A2 = bot_bot_set_option_a ) ) ) ).

% surj_vimage_empty
thf(fact_850_surj__vimage__empty,axiom,
    ! [F2: a > a,A2: set_a] :
      ( ( ( image_a_a2 @ F2 @ top_top_set_a )
        = top_top_set_a )
     => ( ( ( vimage_a_a @ F2 @ A2 )
          = bot_bot_set_a )
        = ( A2 = bot_bot_set_a ) ) ) ).

% surj_vimage_empty
thf(fact_851_vimage__subsetD,axiom,
    ! [F2: option_a > option_a,B2: set_option_a,A2: set_option_a] :
      ( ( ( image_7439109396645324421tion_a @ F2 @ top_top_set_option_a )
        = top_top_set_option_a )
     => ( ( ord_le1955136853071979460tion_a @ ( vimage1562710927270423099tion_a @ F2 @ B2 ) @ A2 )
       => ( ord_le1955136853071979460tion_a @ B2 @ ( image_7439109396645324421tion_a @ F2 @ A2 ) ) ) ) ).

% vimage_subsetD
thf(fact_852_vimage__subsetD,axiom,
    ! [F2: option_a > a,B2: set_a,A2: set_option_a] :
      ( ( ( image_option_a_a2 @ F2 @ top_top_set_option_a )
        = top_top_set_a )
     => ( ( ord_le1955136853071979460tion_a @ ( vimage_option_a_a @ F2 @ B2 ) @ A2 )
       => ( ord_less_eq_set_a @ B2 @ ( image_option_a_a2 @ F2 @ A2 ) ) ) ) ).

% vimage_subsetD
thf(fact_853_vimage__subsetD,axiom,
    ! [F2: a > option_a,B2: set_option_a,A2: set_a] :
      ( ( ( image_a_option_a2 @ F2 @ top_top_set_a )
        = top_top_set_option_a )
     => ( ( ord_less_eq_set_a @ ( vimage_a_option_a @ F2 @ B2 ) @ A2 )
       => ( ord_le1955136853071979460tion_a @ B2 @ ( image_a_option_a2 @ F2 @ A2 ) ) ) ) ).

% vimage_subsetD
thf(fact_854_vimage__subsetD,axiom,
    ! [F2: a > a,B2: set_a,A2: set_a] :
      ( ( ( image_a_a2 @ F2 @ top_top_set_a )
        = top_top_set_a )
     => ( ( ord_less_eq_set_a @ ( vimage_a_a @ F2 @ B2 ) @ A2 )
       => ( ord_less_eq_set_a @ B2 @ ( image_a_a2 @ F2 @ A2 ) ) ) ) ).

% vimage_subsetD
thf(fact_855_inj__vimage__image__eq,axiom,
    ! [F2: option_a > a,A2: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ top_top_set_option_a )
     => ( ( vimage_option_a_a @ F2 @ ( image_option_a_a2 @ F2 @ A2 ) )
        = A2 ) ) ).

% inj_vimage_image_eq
thf(fact_856_inj__vimage__image__eq,axiom,
    ! [F2: a > option_a,A2: set_a] :
      ( ( inj_on_a_option_a @ F2 @ top_top_set_a )
     => ( ( vimage_a_option_a @ F2 @ ( image_a_option_a2 @ F2 @ A2 ) )
        = A2 ) ) ).

% inj_vimage_image_eq
thf(fact_857_vimage__subset__eq,axiom,
    ! [F2: option_a > option_a,B2: set_option_a,A2: set_option_a] :
      ( ( bij_be5431266891817924854tion_a @ F2 @ top_top_set_option_a @ top_top_set_option_a )
     => ( ( ord_le1955136853071979460tion_a @ ( vimage1562710927270423099tion_a @ F2 @ B2 ) @ A2 )
        = ( ord_le1955136853071979460tion_a @ B2 @ ( image_7439109396645324421tion_a @ F2 @ A2 ) ) ) ) ).

% vimage_subset_eq
thf(fact_858_vimage__subset__eq,axiom,
    ! [F2: option_a > a,B2: set_a,A2: set_option_a] :
      ( ( bij_betw_option_a_a @ F2 @ top_top_set_option_a @ top_top_set_a )
     => ( ( ord_le1955136853071979460tion_a @ ( vimage_option_a_a @ F2 @ B2 ) @ A2 )
        = ( ord_less_eq_set_a @ B2 @ ( image_option_a_a2 @ F2 @ A2 ) ) ) ) ).

% vimage_subset_eq
thf(fact_859_vimage__subset__eq,axiom,
    ! [F2: a > option_a,B2: set_option_a,A2: set_a] :
      ( ( bij_betw_a_option_a @ F2 @ top_top_set_a @ top_top_set_option_a )
     => ( ( ord_less_eq_set_a @ ( vimage_a_option_a @ F2 @ B2 ) @ A2 )
        = ( ord_le1955136853071979460tion_a @ B2 @ ( image_a_option_a2 @ F2 @ A2 ) ) ) ) ).

% vimage_subset_eq
thf(fact_860_vimage__subset__eq,axiom,
    ! [F2: a > a,B2: set_a,A2: set_a] :
      ( ( bij_betw_a_a @ F2 @ top_top_set_a @ top_top_set_a )
     => ( ( ord_less_eq_set_a @ ( vimage_a_a @ F2 @ B2 ) @ A2 )
        = ( ord_less_eq_set_a @ B2 @ ( image_a_a2 @ F2 @ A2 ) ) ) ) ).

% vimage_subset_eq
thf(fact_861_asymI,axiom,
    ! [R2: set_Pr7585778909603769095tion_a] :
      ( ! [X2: option_a,Y4: option_a] :
          ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y4 ) @ R2 )
         => ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y4 @ X2 ) @ R2 ) )
     => ( asym_on_option_a @ top_top_set_option_a @ R2 ) ) ).

% asymI
thf(fact_862_asymI,axiom,
    ! [R2: set_Product_prod_a_a] :
      ( ! [X2: a,Y4: a] :
          ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y4 ) @ R2 )
         => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X2 ) @ R2 ) )
     => ( asym_on_a @ top_top_set_a @ R2 ) ) ).

% asymI
thf(fact_863_comp__apply,axiom,
    ( comp_option_a_a_a
    = ( ^ [F: option_a > a,G2: a > option_a,X: a] : ( F @ ( G2 @ X ) ) ) ) ).

% comp_apply
thf(fact_864_asym__onI,axiom,
    ! [A2: set_a,R2: set_Product_prod_a_a] :
      ( ! [X2: a,Y4: a] :
          ( ( member_a @ X2 @ A2 )
         => ( ( member_a @ Y4 @ A2 )
           => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y4 ) @ R2 )
             => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X2 ) @ R2 ) ) ) )
     => ( asym_on_a @ A2 @ R2 ) ) ).

% asym_onI
thf(fact_865_asym__onI,axiom,
    ! [A2: set_Pr7868159745199425715_a_d_c,R2: set_Pr8390707888732215751_a_d_c] :
      ( ! [X2: produc6150846815813599699_a_d_c,Y4: produc6150846815813599699_a_d_c] :
          ( ( member2052822272342364412_a_d_c @ X2 @ A2 )
         => ( ( member2052822272342364412_a_d_c @ Y4 @ A2 )
           => ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ X2 @ Y4 ) @ R2 )
             => ~ ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ Y4 @ X2 ) @ R2 ) ) ) )
     => ( asym_o7473497389364382981_a_d_c @ A2 @ R2 ) ) ).

% asym_onI
thf(fact_866_asym__onI,axiom,
    ! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a] :
      ( ! [X2: option_a,Y4: option_a] :
          ( ( member_option_a @ X2 @ A2 )
         => ( ( member_option_a @ Y4 @ A2 )
           => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y4 ) @ R2 )
             => ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y4 @ X2 ) @ R2 ) ) ) )
     => ( asym_on_option_a @ A2 @ R2 ) ) ).

% asym_onI
thf(fact_867_case__map__option,axiom,
    ! [G: a,H: option_a > a,F2: a > option_a,X3: option_a] :
      ( ( case_o926465512965637841tion_a @ G @ H @ ( map_op2340691886215429841tion_a @ F2 @ X3 ) )
      = ( case_option_a_a @ G @ ( comp_option_a_a_a @ H @ F2 ) @ X3 ) ) ).

% case_map_option
thf(fact_868_case__map__option,axiom,
    ! [G: $o,H: a > $o,F2: a > a,X3: option_a] :
      ( ( case_option_o_a @ G @ H @ ( map_option_a_a @ F2 @ X3 ) )
      = ( case_option_o_a @ G @ ( comp_a_o_a @ H @ F2 ) @ X3 ) ) ).

% case_map_option
thf(fact_869_inj__on__imageI2,axiom,
    ! [F3: option_a > a,F2: a > option_a,A2: set_a] :
      ( ( inj_on_a_a @ ( comp_option_a_a_a @ F3 @ F2 ) @ A2 )
     => ( inj_on_a_option_a @ F2 @ A2 ) ) ).

% inj_on_imageI2
thf(fact_870_inj__on__imageI2,axiom,
    ! [F3: option_a > option_a,F2: a > option_a,A2: set_a] :
      ( ( inj_on_a_option_a @ ( comp_o6087033147929006299on_a_a @ F3 @ F2 ) @ A2 )
     => ( inj_on_a_option_a @ F2 @ A2 ) ) ).

% inj_on_imageI2
thf(fact_871_bij__betw__imp__inj__on,axiom,
    ! [F2: a > option_a,A2: set_a,B2: set_option_a] :
      ( ( bij_betw_a_option_a @ F2 @ A2 @ B2 )
     => ( inj_on_a_option_a @ F2 @ A2 ) ) ).

% bij_betw_imp_inj_on
thf(fact_872_set_Ocomp,axiom,
    ! [F2: a > option_a,G: option_a > a] :
      ( ( comp_s1419921648917501825_set_a @ ( vimage_a_option_a @ F2 ) @ ( vimage_option_a_a @ G ) )
      = ( vimage_a_a @ ( comp_option_a_a_a @ G @ F2 ) ) ) ).

% set.comp
thf(fact_873_set_Ocompositionality,axiom,
    ! [F2: a > option_a,G: option_a > a,Set: set_a] :
      ( ( vimage_a_option_a @ F2 @ ( vimage_option_a_a @ G @ Set ) )
      = ( vimage_a_a @ ( comp_option_a_a_a @ G @ F2 ) @ Set ) ) ).

% set.compositionality
thf(fact_874_vimage__comp,axiom,
    ! [F2: a > option_a,G: option_a > a,X3: set_a] :
      ( ( vimage_a_option_a @ F2 @ ( vimage_option_a_a @ G @ X3 ) )
      = ( vimage_a_a @ ( comp_option_a_a_a @ G @ F2 ) @ X3 ) ) ).

% vimage_comp
thf(fact_875_map__option_Ocomp,axiom,
    ! [F2: option_a > a,G: a > option_a] :
      ( ( comp_o1254687777855551975tion_a @ ( map_op4563205767754224965on_a_a @ F2 ) @ ( map_op2340691886215429841tion_a @ G ) )
      = ( map_option_a_a @ ( comp_option_a_a_a @ F2 @ G ) ) ) ).

% map_option.comp
thf(fact_876_map__option_Ocomp,axiom,
    ! [F2: a > a,G: a > a] :
      ( ( comp_o3154387707078715297tion_a @ ( map_option_a_a @ F2 ) @ ( map_option_a_a @ G ) )
      = ( map_option_a_a @ ( comp_a_a_a @ F2 @ G ) ) ) ).

% map_option.comp
thf(fact_877_option_Omap__comp,axiom,
    ! [G: option_a > a,F2: a > option_a,V3: option_a] :
      ( ( map_op4563205767754224965on_a_a @ G @ ( map_op2340691886215429841tion_a @ F2 @ V3 ) )
      = ( map_option_a_a @ ( comp_option_a_a_a @ G @ F2 ) @ V3 ) ) ).

% option.map_comp
thf(fact_878_option_Omap__comp,axiom,
    ! [G: a > a,F2: a > a,V3: option_a] :
      ( ( map_option_a_a @ G @ ( map_option_a_a @ F2 @ V3 ) )
      = ( map_option_a_a @ ( comp_a_a_a @ G @ F2 ) @ V3 ) ) ).

% option.map_comp
thf(fact_879_map__option_Ocompositionality,axiom,
    ! [F2: option_a > a,G: a > option_a,Option: option_a] :
      ( ( map_op4563205767754224965on_a_a @ F2 @ ( map_op2340691886215429841tion_a @ G @ Option ) )
      = ( map_option_a_a @ ( comp_option_a_a_a @ F2 @ G ) @ Option ) ) ).

% map_option.compositionality
thf(fact_880_map__option_Ocompositionality,axiom,
    ! [F2: a > a,G: a > a,Option: option_a] :
      ( ( map_option_a_a @ F2 @ ( map_option_a_a @ G @ Option ) )
      = ( map_option_a_a @ ( comp_a_a_a @ F2 @ G ) @ Option ) ) ).

% map_option.compositionality
thf(fact_881_bij__betw__imp__surj__on,axiom,
    ! [F2: a > option_a,A2: set_a,B2: set_option_a] :
      ( ( bij_betw_a_option_a @ F2 @ A2 @ B2 )
     => ( ( image_a_option_a2 @ F2 @ A2 )
        = B2 ) ) ).

% bij_betw_imp_surj_on
thf(fact_882_bij__betw__imp__surj__on,axiom,
    ! [F2: option_a > a,A2: set_option_a,B2: set_a] :
      ( ( bij_betw_option_a_a @ F2 @ A2 @ B2 )
     => ( ( image_option_a_a2 @ F2 @ A2 )
        = B2 ) ) ).

% bij_betw_imp_surj_on
thf(fact_883_image__eq__imp__comp,axiom,
    ! [F2: option_a > option_a,A2: set_option_a,G: a > option_a,B2: set_a,H: option_a > a] :
      ( ( ( image_7439109396645324421tion_a @ F2 @ A2 )
        = ( image_a_option_a2 @ G @ B2 ) )
     => ( ( image_option_a_a2 @ ( comp_o3864519266390211175tion_a @ H @ F2 ) @ A2 )
        = ( image_a_a2 @ ( comp_option_a_a_a @ H @ G ) @ B2 ) ) ) ).

% image_eq_imp_comp
thf(fact_884_image__eq__imp__comp,axiom,
    ! [F2: a > a,A2: set_a,G: option_a > a,B2: set_option_a,H: a > option_a] :
      ( ( ( image_a_a2 @ F2 @ A2 )
        = ( image_option_a_a2 @ G @ B2 ) )
     => ( ( image_a_option_a2 @ ( comp_a_option_a_a @ H @ F2 ) @ A2 )
        = ( image_7439109396645324421tion_a @ ( comp_a6249931511552232923tion_a @ H @ G ) @ B2 ) ) ) ).

% image_eq_imp_comp
thf(fact_885_image__eq__imp__comp,axiom,
    ! [F2: a > option_a,A2: set_a,G: option_a > option_a,B2: set_option_a,H: option_a > a] :
      ( ( ( image_a_option_a2 @ F2 @ A2 )
        = ( image_7439109396645324421tion_a @ G @ B2 ) )
     => ( ( image_a_a2 @ ( comp_option_a_a_a @ H @ F2 ) @ A2 )
        = ( image_option_a_a2 @ ( comp_o3864519266390211175tion_a @ H @ G ) @ B2 ) ) ) ).

% image_eq_imp_comp
thf(fact_886_image__eq__imp__comp,axiom,
    ! [F2: a > option_a,A2: set_a,G: a > option_a,B2: set_a,H: option_a > a] :
      ( ( ( image_a_option_a2 @ F2 @ A2 )
        = ( image_a_option_a2 @ G @ B2 ) )
     => ( ( image_a_a2 @ ( comp_option_a_a_a @ H @ F2 ) @ A2 )
        = ( image_a_a2 @ ( comp_option_a_a_a @ H @ G ) @ B2 ) ) ) ).

% image_eq_imp_comp
thf(fact_887_image__eq__imp__comp,axiom,
    ! [F2: a > option_a,A2: set_a,G: a > option_a,B2: set_a,H: option_a > option_a] :
      ( ( ( image_a_option_a2 @ F2 @ A2 )
        = ( image_a_option_a2 @ G @ B2 ) )
     => ( ( image_a_option_a2 @ ( comp_o6087033147929006299on_a_a @ H @ F2 ) @ A2 )
        = ( image_a_option_a2 @ ( comp_o6087033147929006299on_a_a @ H @ G ) @ B2 ) ) ) ).

% image_eq_imp_comp
thf(fact_888_image__eq__imp__comp,axiom,
    ! [F2: option_a > a,A2: set_option_a,G: a > a,B2: set_a,H: a > option_a] :
      ( ( ( image_option_a_a2 @ F2 @ A2 )
        = ( image_a_a2 @ G @ B2 ) )
     => ( ( image_7439109396645324421tion_a @ ( comp_a6249931511552232923tion_a @ H @ F2 ) @ A2 )
        = ( image_a_option_a2 @ ( comp_a_option_a_a @ H @ G ) @ B2 ) ) ) ).

% image_eq_imp_comp
thf(fact_889_image__eq__imp__comp,axiom,
    ! [F2: option_a > a,A2: set_option_a,G: option_a > a,B2: set_option_a,H: a > a] :
      ( ( ( image_option_a_a2 @ F2 @ A2 )
        = ( image_option_a_a2 @ G @ B2 ) )
     => ( ( image_option_a_a2 @ ( comp_a_a_option_a @ H @ F2 ) @ A2 )
        = ( image_option_a_a2 @ ( comp_a_a_option_a @ H @ G ) @ B2 ) ) ) ).

% image_eq_imp_comp
thf(fact_890_image__comp,axiom,
    ! [F2: option_a > option_a,G: a > option_a,R2: set_a] :
      ( ( image_7439109396645324421tion_a @ F2 @ ( image_a_option_a2 @ G @ R2 ) )
      = ( image_a_option_a2 @ ( comp_o6087033147929006299on_a_a @ F2 @ G ) @ R2 ) ) ).

% image_comp
thf(fact_891_image__comp,axiom,
    ! [F2: a > a,G: option_a > a,R2: set_option_a] :
      ( ( image_a_a2 @ F2 @ ( image_option_a_a2 @ G @ R2 ) )
      = ( image_option_a_a2 @ ( comp_a_a_option_a @ F2 @ G ) @ R2 ) ) ).

% image_comp
thf(fact_892_image__comp,axiom,
    ! [F2: a > option_a,G: a > a,R2: set_a] :
      ( ( image_a_option_a2 @ F2 @ ( image_a_a2 @ G @ R2 ) )
      = ( image_a_option_a2 @ ( comp_a_option_a_a @ F2 @ G ) @ R2 ) ) ).

% image_comp
thf(fact_893_image__comp,axiom,
    ! [F2: a > option_a,G: option_a > a,R2: set_option_a] :
      ( ( image_a_option_a2 @ F2 @ ( image_option_a_a2 @ G @ R2 ) )
      = ( image_7439109396645324421tion_a @ ( comp_a6249931511552232923tion_a @ F2 @ G ) @ R2 ) ) ).

% image_comp
thf(fact_894_image__comp,axiom,
    ! [F2: option_a > a,G: option_a > option_a,R2: set_option_a] :
      ( ( image_option_a_a2 @ F2 @ ( image_7439109396645324421tion_a @ G @ R2 ) )
      = ( image_option_a_a2 @ ( comp_o3864519266390211175tion_a @ F2 @ G ) @ R2 ) ) ).

% image_comp
thf(fact_895_image__comp,axiom,
    ! [F2: option_a > a,G: a > option_a,R2: set_a] :
      ( ( image_option_a_a2 @ F2 @ ( image_a_option_a2 @ G @ R2 ) )
      = ( image_a_a2 @ ( comp_option_a_a_a @ F2 @ G ) @ R2 ) ) ).

% image_comp
thf(fact_896_bij__betw__iff__bijections,axiom,
    ( bij_betw_a_a
    = ( ^ [F: a > a,A7: set_a,B6: set_a] :
        ? [G2: a > a] :
          ( ! [X: a] :
              ( ( member_a @ X @ A7 )
             => ( ( member_a @ ( F @ X ) @ B6 )
                & ( ( G2 @ ( F @ X ) )
                  = X ) ) )
          & ! [X: a] :
              ( ( member_a @ X @ B6 )
             => ( ( member_a @ ( G2 @ X ) @ A7 )
                & ( ( F @ ( G2 @ X ) )
                  = X ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_897_bij__betw__iff__bijections,axiom,
    ( bij_be6485230601037006177_d_c_a
    = ( ^ [F: produc6150846815813599699_a_d_c > a,A7: set_Pr7868159745199425715_a_d_c,B6: set_a] :
        ? [G2: a > produc6150846815813599699_a_d_c] :
          ( ! [X: produc6150846815813599699_a_d_c] :
              ( ( member2052822272342364412_a_d_c @ X @ A7 )
             => ( ( member_a @ ( F @ X ) @ B6 )
                & ( ( G2 @ ( F @ X ) )
                  = X ) ) )
          & ! [X: a] :
              ( ( member_a @ X @ B6 )
             => ( ( member2052822272342364412_a_d_c @ ( G2 @ X ) @ A7 )
                & ( ( F @ ( G2 @ X ) )
                  = X ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_898_bij__betw__iff__bijections,axiom,
    ( bij_betw_option_a_a
    = ( ^ [F: option_a > a,A7: set_option_a,B6: set_a] :
        ? [G2: a > option_a] :
          ( ! [X: option_a] :
              ( ( member_option_a @ X @ A7 )
             => ( ( member_a @ ( F @ X ) @ B6 )
                & ( ( G2 @ ( F @ X ) )
                  = X ) ) )
          & ! [X: a] :
              ( ( member_a @ X @ B6 )
             => ( ( member_option_a @ ( G2 @ X ) @ A7 )
                & ( ( F @ ( G2 @ X ) )
                  = X ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_899_bij__betw__iff__bijections,axiom,
    ( bij_be3492573195200641995_a_d_c
    = ( ^ [F: a > produc6150846815813599699_a_d_c,A7: set_a,B6: set_Pr7868159745199425715_a_d_c] :
        ? [G2: produc6150846815813599699_a_d_c > a] :
          ( ! [X: a] :
              ( ( member_a @ X @ A7 )
             => ( ( member2052822272342364412_a_d_c @ ( F @ X ) @ B6 )
                & ( ( G2 @ ( F @ X ) )
                  = X ) ) )
          & ! [X: produc6150846815813599699_a_d_c] :
              ( ( member2052822272342364412_a_d_c @ X @ B6 )
             => ( ( member_a @ ( G2 @ X ) @ A7 )
                & ( ( F @ ( G2 @ X ) )
                  = X ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_900_bij__betw__iff__bijections,axiom,
    ( bij_be734363561790614582_a_d_c
    = ( ^ [F: produc6150846815813599699_a_d_c > produc6150846815813599699_a_d_c,A7: set_Pr7868159745199425715_a_d_c,B6: set_Pr7868159745199425715_a_d_c] :
        ? [G2: produc6150846815813599699_a_d_c > produc6150846815813599699_a_d_c] :
          ( ! [X: produc6150846815813599699_a_d_c] :
              ( ( member2052822272342364412_a_d_c @ X @ A7 )
             => ( ( member2052822272342364412_a_d_c @ ( F @ X ) @ B6 )
                & ( ( G2 @ ( F @ X ) )
                  = X ) ) )
          & ! [X: produc6150846815813599699_a_d_c] :
              ( ( member2052822272342364412_a_d_c @ X @ B6 )
             => ( ( member2052822272342364412_a_d_c @ ( G2 @ X ) @ A7 )
                & ( ( F @ ( G2 @ X ) )
                  = X ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_901_bij__betw__iff__bijections,axiom,
    ( bij_be8275362299291636869_a_d_c
    = ( ^ [F: option_a > produc6150846815813599699_a_d_c,A7: set_option_a,B6: set_Pr7868159745199425715_a_d_c] :
        ? [G2: produc6150846815813599699_a_d_c > option_a] :
          ( ! [X: option_a] :
              ( ( member_option_a @ X @ A7 )
             => ( ( member2052822272342364412_a_d_c @ ( F @ X ) @ B6 )
                & ( ( G2 @ ( F @ X ) )
                  = X ) ) )
          & ! [X: produc6150846815813599699_a_d_c] :
              ( ( member2052822272342364412_a_d_c @ X @ B6 )
             => ( ( member_option_a @ ( G2 @ X ) @ A7 )
                & ( ( F @ ( G2 @ X ) )
                  = X ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_902_bij__betw__iff__bijections,axiom,
    ( bij_betw_a_option_a
    = ( ^ [F: a > option_a,A7: set_a,B6: set_option_a] :
        ? [G2: option_a > a] :
          ( ! [X: a] :
              ( ( member_a @ X @ A7 )
             => ( ( member_option_a @ ( F @ X ) @ B6 )
                & ( ( G2 @ ( F @ X ) )
                  = X ) ) )
          & ! [X: option_a] :
              ( ( member_option_a @ X @ B6 )
             => ( ( member_a @ ( G2 @ X ) @ A7 )
                & ( ( F @ ( G2 @ X ) )
                  = X ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_903_bij__betw__iff__bijections,axiom,
    ( bij_be4831480119166671655tion_a
    = ( ^ [F: produc6150846815813599699_a_d_c > option_a,A7: set_Pr7868159745199425715_a_d_c,B6: set_option_a] :
        ? [G2: option_a > produc6150846815813599699_a_d_c] :
          ( ! [X: produc6150846815813599699_a_d_c] :
              ( ( member2052822272342364412_a_d_c @ X @ A7 )
             => ( ( member_option_a @ ( F @ X ) @ B6 )
                & ( ( G2 @ ( F @ X ) )
                  = X ) ) )
          & ! [X: option_a] :
              ( ( member_option_a @ X @ B6 )
             => ( ( member2052822272342364412_a_d_c @ ( G2 @ X ) @ A7 )
                & ( ( F @ ( G2 @ X ) )
                  = X ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_904_bij__betw__iff__bijections,axiom,
    ( bij_be5431266891817924854tion_a
    = ( ^ [F: option_a > option_a,A7: set_option_a,B6: set_option_a] :
        ? [G2: option_a > option_a] :
          ( ! [X: option_a] :
              ( ( member_option_a @ X @ A7 )
             => ( ( member_option_a @ ( F @ X ) @ B6 )
                & ( ( G2 @ ( F @ X ) )
                  = X ) ) )
          & ! [X: option_a] :
              ( ( member_option_a @ X @ B6 )
             => ( ( member_option_a @ ( G2 @ X ) @ A7 )
                & ( ( F @ ( G2 @ X ) )
                  = X ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_905_bij__betw__comp__iff,axiom,
    ! [F2: a > option_a,A2: set_a,A9: set_option_a,F3: option_a > a,A10: set_a] :
      ( ( bij_betw_a_option_a @ F2 @ A2 @ A9 )
     => ( ( bij_betw_option_a_a @ F3 @ A9 @ A10 )
        = ( bij_betw_a_a @ ( comp_option_a_a_a @ F3 @ F2 ) @ A2 @ A10 ) ) ) ).

% bij_betw_comp_iff
thf(fact_906_comp__eq__dest__lhs,axiom,
    ! [A: option_a > a,B: a > option_a,C: a > a,V3: a] :
      ( ( ( comp_option_a_a_a @ A @ B )
        = C )
     => ( ( A @ ( B @ V3 ) )
        = ( C @ V3 ) ) ) ).

% comp_eq_dest_lhs
thf(fact_907_bij__betw__trans,axiom,
    ! [F2: a > option_a,A2: set_a,B2: set_option_a,G: option_a > a,C3: set_a] :
      ( ( bij_betw_a_option_a @ F2 @ A2 @ B2 )
     => ( ( bij_betw_option_a_a @ G @ B2 @ C3 )
       => ( bij_betw_a_a @ ( comp_option_a_a_a @ G @ F2 ) @ A2 @ C3 ) ) ) ).

% bij_betw_trans
thf(fact_908_bij__betw__apply,axiom,
    ! [F2: a > a,A2: set_a,B2: set_a,A: a] :
      ( ( bij_betw_a_a @ F2 @ A2 @ B2 )
     => ( ( member_a @ A @ A2 )
       => ( member_a @ ( F2 @ A ) @ B2 ) ) ) ).

% bij_betw_apply
thf(fact_909_bij__betw__apply,axiom,
    ! [F2: a > produc6150846815813599699_a_d_c,A2: set_a,B2: set_Pr7868159745199425715_a_d_c,A: a] :
      ( ( bij_be3492573195200641995_a_d_c @ F2 @ A2 @ B2 )
     => ( ( member_a @ A @ A2 )
       => ( member2052822272342364412_a_d_c @ ( F2 @ A ) @ B2 ) ) ) ).

% bij_betw_apply
thf(fact_910_bij__betw__apply,axiom,
    ! [F2: a > option_a,A2: set_a,B2: set_option_a,A: a] :
      ( ( bij_betw_a_option_a @ F2 @ A2 @ B2 )
     => ( ( member_a @ A @ A2 )
       => ( member_option_a @ ( F2 @ A ) @ B2 ) ) ) ).

% bij_betw_apply
thf(fact_911_bij__betw__apply,axiom,
    ! [F2: produc6150846815813599699_a_d_c > a,A2: set_Pr7868159745199425715_a_d_c,B2: set_a,A: produc6150846815813599699_a_d_c] :
      ( ( bij_be6485230601037006177_d_c_a @ F2 @ A2 @ B2 )
     => ( ( member2052822272342364412_a_d_c @ A @ A2 )
       => ( member_a @ ( F2 @ A ) @ B2 ) ) ) ).

% bij_betw_apply
thf(fact_912_bij__betw__apply,axiom,
    ! [F2: produc6150846815813599699_a_d_c > produc6150846815813599699_a_d_c,A2: set_Pr7868159745199425715_a_d_c,B2: set_Pr7868159745199425715_a_d_c,A: produc6150846815813599699_a_d_c] :
      ( ( bij_be734363561790614582_a_d_c @ F2 @ A2 @ B2 )
     => ( ( member2052822272342364412_a_d_c @ A @ A2 )
       => ( member2052822272342364412_a_d_c @ ( F2 @ A ) @ B2 ) ) ) ).

% bij_betw_apply
thf(fact_913_bij__betw__apply,axiom,
    ! [F2: produc6150846815813599699_a_d_c > option_a,A2: set_Pr7868159745199425715_a_d_c,B2: set_option_a,A: produc6150846815813599699_a_d_c] :
      ( ( bij_be4831480119166671655tion_a @ F2 @ A2 @ B2 )
     => ( ( member2052822272342364412_a_d_c @ A @ A2 )
       => ( member_option_a @ ( F2 @ A ) @ B2 ) ) ) ).

% bij_betw_apply
thf(fact_914_bij__betw__apply,axiom,
    ! [F2: option_a > a,A2: set_option_a,B2: set_a,A: option_a] :
      ( ( bij_betw_option_a_a @ F2 @ A2 @ B2 )
     => ( ( member_option_a @ A @ A2 )
       => ( member_a @ ( F2 @ A ) @ B2 ) ) ) ).

% bij_betw_apply
thf(fact_915_bij__betw__apply,axiom,
    ! [F2: option_a > produc6150846815813599699_a_d_c,A2: set_option_a,B2: set_Pr7868159745199425715_a_d_c,A: option_a] :
      ( ( bij_be8275362299291636869_a_d_c @ F2 @ A2 @ B2 )
     => ( ( member_option_a @ A @ A2 )
       => ( member2052822272342364412_a_d_c @ ( F2 @ A ) @ B2 ) ) ) ).

% bij_betw_apply
thf(fact_916_bij__betw__apply,axiom,
    ! [F2: option_a > option_a,A2: set_option_a,B2: set_option_a,A: option_a] :
      ( ( bij_be5431266891817924854tion_a @ F2 @ A2 @ B2 )
     => ( ( member_option_a @ A @ A2 )
       => ( member_option_a @ ( F2 @ A ) @ B2 ) ) ) ).

% bij_betw_apply
thf(fact_917_comp__eq__elim,axiom,
    ! [A: option_a > a,B: a > option_a,C: option_a > a,D: a > option_a] :
      ( ( ( comp_option_a_a_a @ A @ B )
        = ( comp_option_a_a_a @ C @ D ) )
     => ! [V4: a] :
          ( ( A @ ( B @ V4 ) )
          = ( C @ ( D @ V4 ) ) ) ) ).

% comp_eq_elim
thf(fact_918_comp__eq__dest,axiom,
    ! [A: option_a > a,B: a > option_a,C: option_a > a,D: a > option_a,V3: a] :
      ( ( ( comp_option_a_a_a @ A @ B )
        = ( comp_option_a_a_a @ C @ D ) )
     => ( ( A @ ( B @ V3 ) )
        = ( C @ ( D @ V3 ) ) ) ) ).

% comp_eq_dest
thf(fact_919_comp__assoc,axiom,
    ! [F2: option_a > a,G: a > option_a,H: a > a] :
      ( ( comp_a_a_a @ ( comp_option_a_a_a @ F2 @ G ) @ H )
      = ( comp_option_a_a_a @ F2 @ ( comp_a_option_a_a @ G @ H ) ) ) ).

% comp_assoc
thf(fact_920_comp__assoc,axiom,
    ! [F2: a > a,G: option_a > a,H: a > option_a] :
      ( ( comp_option_a_a_a @ ( comp_a_a_option_a @ F2 @ G ) @ H )
      = ( comp_a_a_a @ F2 @ ( comp_option_a_a_a @ G @ H ) ) ) ).

% comp_assoc
thf(fact_921_comp__assoc,axiom,
    ! [F2: option_a > a,G: option_a > option_a,H: a > option_a] :
      ( ( comp_option_a_a_a @ ( comp_o3864519266390211175tion_a @ F2 @ G ) @ H )
      = ( comp_option_a_a_a @ F2 @ ( comp_o6087033147929006299on_a_a @ G @ H ) ) ) ).

% comp_assoc
thf(fact_922_comp__def,axiom,
    ( comp_option_a_a_a
    = ( ^ [F: option_a > a,G2: a > option_a,X: a] : ( F @ ( G2 @ X ) ) ) ) ).

% comp_def
thf(fact_923_fun__upd__comp,axiom,
    ! [F2: option_a > a,G: a > option_a,X3: a,Y: option_a] :
      ( ( comp_option_a_a_a @ F2 @ ( fun_upd_a_option_a @ G @ X3 @ Y ) )
      = ( fun_upd_a_a @ ( comp_option_a_a_a @ F2 @ G ) @ X3 @ ( F2 @ Y ) ) ) ).

% fun_upd_comp
thf(fact_924_fun__upd__comp,axiom,
    ! [F2: c > c,G: d > c,X3: d,Y: c] :
      ( ( comp_c_c_d @ F2 @ ( fun_upd_d_c @ G @ X3 @ Y ) )
      = ( fun_upd_d_c @ ( comp_c_c_d @ F2 @ G ) @ X3 @ ( F2 @ Y ) ) ) ).

% fun_upd_comp
thf(fact_925_bij__betw__empty2,axiom,
    ! [F2: a > a,A2: set_a] :
      ( ( bij_betw_a_a @ F2 @ A2 @ bot_bot_set_a )
     => ( A2 = bot_bot_set_a ) ) ).

% bij_betw_empty2
thf(fact_926_bij__betw__empty2,axiom,
    ! [F2: option_a > a,A2: set_option_a] :
      ( ( bij_betw_option_a_a @ F2 @ A2 @ bot_bot_set_a )
     => ( A2 = bot_bot_set_option_a ) ) ).

% bij_betw_empty2
thf(fact_927_bij__betw__empty2,axiom,
    ! [F2: a > option_a,A2: set_a] :
      ( ( bij_betw_a_option_a @ F2 @ A2 @ bot_bot_set_option_a )
     => ( A2 = bot_bot_set_a ) ) ).

% bij_betw_empty2
thf(fact_928_bij__betw__empty2,axiom,
    ! [F2: option_a > option_a,A2: set_option_a] :
      ( ( bij_be5431266891817924854tion_a @ F2 @ A2 @ bot_bot_set_option_a )
     => ( A2 = bot_bot_set_option_a ) ) ).

% bij_betw_empty2
thf(fact_929_bij__betw__empty1,axiom,
    ! [F2: a > a,A2: set_a] :
      ( ( bij_betw_a_a @ F2 @ bot_bot_set_a @ A2 )
     => ( A2 = bot_bot_set_a ) ) ).

% bij_betw_empty1
thf(fact_930_bij__betw__empty1,axiom,
    ! [F2: a > option_a,A2: set_option_a] :
      ( ( bij_betw_a_option_a @ F2 @ bot_bot_set_a @ A2 )
     => ( A2 = bot_bot_set_option_a ) ) ).

% bij_betw_empty1
thf(fact_931_bij__betw__empty1,axiom,
    ! [F2: option_a > a,A2: set_a] :
      ( ( bij_betw_option_a_a @ F2 @ bot_bot_set_option_a @ A2 )
     => ( A2 = bot_bot_set_a ) ) ).

% bij_betw_empty1
thf(fact_932_bij__betw__empty1,axiom,
    ! [F2: option_a > option_a,A2: set_option_a] :
      ( ( bij_be5431266891817924854tion_a @ F2 @ bot_bot_set_option_a @ A2 )
     => ( A2 = bot_bot_set_option_a ) ) ).

% bij_betw_empty1
thf(fact_933_involuntory__imp__bij,axiom,
    ! [F2: option_a > option_a] :
      ( ! [X2: option_a] :
          ( ( F2 @ ( F2 @ X2 ) )
          = X2 )
     => ( bij_be5431266891817924854tion_a @ F2 @ top_top_set_option_a @ top_top_set_option_a ) ) ).

% involuntory_imp_bij
thf(fact_934_involuntory__imp__bij,axiom,
    ! [F2: a > a] :
      ( ! [X2: a] :
          ( ( F2 @ ( F2 @ X2 ) )
          = X2 )
     => ( bij_betw_a_a @ F2 @ top_top_set_a @ top_top_set_a ) ) ).

% involuntory_imp_bij
thf(fact_935_bij__pointE,axiom,
    ! [F2: option_a > option_a,Y: option_a] :
      ( ( bij_be5431266891817924854tion_a @ F2 @ top_top_set_option_a @ top_top_set_option_a )
     => ~ ! [X2: option_a] :
            ( ( Y
              = ( F2 @ X2 ) )
           => ~ ! [X9: option_a] :
                  ( ( Y
                    = ( F2 @ X9 ) )
                 => ( X9 = X2 ) ) ) ) ).

% bij_pointE
thf(fact_936_bij__pointE,axiom,
    ! [F2: option_a > a,Y: a] :
      ( ( bij_betw_option_a_a @ F2 @ top_top_set_option_a @ top_top_set_a )
     => ~ ! [X2: option_a] :
            ( ( Y
              = ( F2 @ X2 ) )
           => ~ ! [X9: option_a] :
                  ( ( Y
                    = ( F2 @ X9 ) )
                 => ( X9 = X2 ) ) ) ) ).

% bij_pointE
thf(fact_937_bij__pointE,axiom,
    ! [F2: a > option_a,Y: option_a] :
      ( ( bij_betw_a_option_a @ F2 @ top_top_set_a @ top_top_set_option_a )
     => ~ ! [X2: a] :
            ( ( Y
              = ( F2 @ X2 ) )
           => ~ ! [X9: a] :
                  ( ( Y
                    = ( F2 @ X9 ) )
                 => ( X9 = X2 ) ) ) ) ).

% bij_pointE
thf(fact_938_bij__pointE,axiom,
    ! [F2: a > a,Y: a] :
      ( ( bij_betw_a_a @ F2 @ top_top_set_a @ top_top_set_a )
     => ~ ! [X2: a] :
            ( ( Y
              = ( F2 @ X2 ) )
           => ~ ! [X9: a] :
                  ( ( Y
                    = ( F2 @ X9 ) )
                 => ( X9 = X2 ) ) ) ) ).

% bij_pointE
thf(fact_939_bij__comp,axiom,
    ! [F2: option_a > option_a,G: option_a > option_a] :
      ( ( bij_be5431266891817924854tion_a @ F2 @ top_top_set_option_a @ top_top_set_option_a )
     => ( ( bij_be5431266891817924854tion_a @ G @ top_top_set_option_a @ top_top_set_option_a )
       => ( bij_be5431266891817924854tion_a @ ( comp_o3154387707078715297tion_a @ G @ F2 ) @ top_top_set_option_a @ top_top_set_option_a ) ) ) ).

% bij_comp
thf(fact_940_bij__comp,axiom,
    ! [F2: option_a > option_a,G: option_a > a] :
      ( ( bij_be5431266891817924854tion_a @ F2 @ top_top_set_option_a @ top_top_set_option_a )
     => ( ( bij_betw_option_a_a @ G @ top_top_set_option_a @ top_top_set_a )
       => ( bij_betw_option_a_a @ ( comp_o3864519266390211175tion_a @ G @ F2 ) @ top_top_set_option_a @ top_top_set_a ) ) ) ).

% bij_comp
thf(fact_941_bij__comp,axiom,
    ! [F2: option_a > a,G: a > option_a] :
      ( ( bij_betw_option_a_a @ F2 @ top_top_set_option_a @ top_top_set_a )
     => ( ( bij_betw_a_option_a @ G @ top_top_set_a @ top_top_set_option_a )
       => ( bij_be5431266891817924854tion_a @ ( comp_a6249931511552232923tion_a @ G @ F2 ) @ top_top_set_option_a @ top_top_set_option_a ) ) ) ).

% bij_comp
thf(fact_942_bij__comp,axiom,
    ! [F2: option_a > a,G: a > a] :
      ( ( bij_betw_option_a_a @ F2 @ top_top_set_option_a @ top_top_set_a )
     => ( ( bij_betw_a_a @ G @ top_top_set_a @ top_top_set_a )
       => ( bij_betw_option_a_a @ ( comp_a_a_option_a @ G @ F2 ) @ top_top_set_option_a @ top_top_set_a ) ) ) ).

% bij_comp
thf(fact_943_bij__comp,axiom,
    ! [F2: a > option_a,G: option_a > option_a] :
      ( ( bij_betw_a_option_a @ F2 @ top_top_set_a @ top_top_set_option_a )
     => ( ( bij_be5431266891817924854tion_a @ G @ top_top_set_option_a @ top_top_set_option_a )
       => ( bij_betw_a_option_a @ ( comp_o6087033147929006299on_a_a @ G @ F2 ) @ top_top_set_a @ top_top_set_option_a ) ) ) ).

% bij_comp
thf(fact_944_bij__comp,axiom,
    ! [F2: a > option_a,G: option_a > a] :
      ( ( bij_betw_a_option_a @ F2 @ top_top_set_a @ top_top_set_option_a )
     => ( ( bij_betw_option_a_a @ G @ top_top_set_option_a @ top_top_set_a )
       => ( bij_betw_a_a @ ( comp_option_a_a_a @ G @ F2 ) @ top_top_set_a @ top_top_set_a ) ) ) ).

% bij_comp
thf(fact_945_bij__comp,axiom,
    ! [F2: a > a,G: a > option_a] :
      ( ( bij_betw_a_a @ F2 @ top_top_set_a @ top_top_set_a )
     => ( ( bij_betw_a_option_a @ G @ top_top_set_a @ top_top_set_option_a )
       => ( bij_betw_a_option_a @ ( comp_a_option_a_a @ G @ F2 ) @ top_top_set_a @ top_top_set_option_a ) ) ) ).

% bij_comp
thf(fact_946_bij__comp,axiom,
    ! [F2: a > a,G: a > a] :
      ( ( bij_betw_a_a @ F2 @ top_top_set_a @ top_top_set_a )
     => ( ( bij_betw_a_a @ G @ top_top_set_a @ top_top_set_a )
       => ( bij_betw_a_a @ ( comp_a_a_a @ G @ F2 ) @ top_top_set_a @ top_top_set_a ) ) ) ).

% bij_comp
thf(fact_947_bij__iff,axiom,
    ! [F2: option_a > option_a] :
      ( ( bij_be5431266891817924854tion_a @ F2 @ top_top_set_option_a @ top_top_set_option_a )
      = ( ! [X: option_a] :
          ? [Y3: option_a] :
            ( ( ( F2 @ Y3 )
              = X )
            & ! [Z: option_a] :
                ( ( ( F2 @ Z )
                  = X )
               => ( Z = Y3 ) ) ) ) ) ).

% bij_iff
thf(fact_948_bij__iff,axiom,
    ! [F2: option_a > a] :
      ( ( bij_betw_option_a_a @ F2 @ top_top_set_option_a @ top_top_set_a )
      = ( ! [X: a] :
          ? [Y3: option_a] :
            ( ( ( F2 @ Y3 )
              = X )
            & ! [Z: option_a] :
                ( ( ( F2 @ Z )
                  = X )
               => ( Z = Y3 ) ) ) ) ) ).

% bij_iff
thf(fact_949_bij__iff,axiom,
    ! [F2: a > option_a] :
      ( ( bij_betw_a_option_a @ F2 @ top_top_set_a @ top_top_set_option_a )
      = ( ! [X: option_a] :
          ? [Y3: a] :
            ( ( ( F2 @ Y3 )
              = X )
            & ! [Z: a] :
                ( ( ( F2 @ Z )
                  = X )
               => ( Z = Y3 ) ) ) ) ) ).

% bij_iff
thf(fact_950_bij__iff,axiom,
    ! [F2: a > a] :
      ( ( bij_betw_a_a @ F2 @ top_top_set_a @ top_top_set_a )
      = ( ! [X: a] :
          ? [Y3: a] :
            ( ( ( F2 @ Y3 )
              = X )
            & ! [Z: a] :
                ( ( ( F2 @ Z )
                  = X )
               => ( Z = Y3 ) ) ) ) ) ).

% bij_iff
thf(fact_951_asym__onD,axiom,
    ! [A2: set_a,R2: set_Product_prod_a_a,X3: a,Y: a] :
      ( ( asym_on_a @ A2 @ R2 )
     => ( ( member_a @ X3 @ A2 )
       => ( ( member_a @ Y @ A2 )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R2 )
           => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X3 ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_952_asym__onD,axiom,
    ! [A2: set_Pr7868159745199425715_a_d_c,R2: set_Pr8390707888732215751_a_d_c,X3: produc6150846815813599699_a_d_c,Y: produc6150846815813599699_a_d_c] :
      ( ( asym_o7473497389364382981_a_d_c @ A2 @ R2 )
     => ( ( member2052822272342364412_a_d_c @ X3 @ A2 )
       => ( ( member2052822272342364412_a_d_c @ Y @ A2 )
         => ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ X3 @ Y ) @ R2 )
           => ~ ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ Y @ X3 ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_953_asym__onD,axiom,
    ! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,X3: option_a,Y: option_a] :
      ( ( asym_on_option_a @ A2 @ R2 )
     => ( ( member_option_a @ X3 @ A2 )
       => ( ( member_option_a @ Y @ A2 )
         => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y ) @ R2 )
           => ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y @ X3 ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_954_bij__betw__comp__iff2,axiom,
    ! [F3: option_a > a,A9: set_option_a,A10: set_a,F2: a > option_a,A2: set_a] :
      ( ( bij_betw_option_a_a @ F3 @ A9 @ A10 )
     => ( ( ord_le1955136853071979460tion_a @ ( image_a_option_a2 @ F2 @ A2 ) @ A9 )
       => ( ( bij_betw_a_option_a @ F2 @ A2 @ A9 )
          = ( bij_betw_a_a @ ( comp_option_a_a_a @ F3 @ F2 ) @ A2 @ A10 ) ) ) ) ).

% bij_betw_comp_iff2
thf(fact_955_bij__is__surj,axiom,
    ! [F2: option_a > option_a] :
      ( ( bij_be5431266891817924854tion_a @ F2 @ top_top_set_option_a @ top_top_set_option_a )
     => ( ( image_7439109396645324421tion_a @ F2 @ top_top_set_option_a )
        = top_top_set_option_a ) ) ).

% bij_is_surj
thf(fact_956_bij__is__surj,axiom,
    ! [F2: option_a > a] :
      ( ( bij_betw_option_a_a @ F2 @ top_top_set_option_a @ top_top_set_a )
     => ( ( image_option_a_a2 @ F2 @ top_top_set_option_a )
        = top_top_set_a ) ) ).

% bij_is_surj
thf(fact_957_bij__is__surj,axiom,
    ! [F2: a > option_a] :
      ( ( bij_betw_a_option_a @ F2 @ top_top_set_a @ top_top_set_option_a )
     => ( ( image_a_option_a2 @ F2 @ top_top_set_a )
        = top_top_set_option_a ) ) ).

% bij_is_surj
thf(fact_958_bij__is__surj,axiom,
    ! [F2: a > a] :
      ( ( bij_betw_a_a @ F2 @ top_top_set_a @ top_top_set_a )
     => ( ( image_a_a2 @ F2 @ top_top_set_a )
        = top_top_set_a ) ) ).

% bij_is_surj
thf(fact_959_bij__betw__imp__surj,axiom,
    ! [F2: option_a > option_a,A2: set_option_a] :
      ( ( bij_be5431266891817924854tion_a @ F2 @ A2 @ top_top_set_option_a )
     => ( ( image_7439109396645324421tion_a @ F2 @ top_top_set_option_a )
        = top_top_set_option_a ) ) ).

% bij_betw_imp_surj
thf(fact_960_bij__betw__imp__surj,axiom,
    ! [F2: a > option_a,A2: set_a] :
      ( ( bij_betw_a_option_a @ F2 @ A2 @ top_top_set_option_a )
     => ( ( image_a_option_a2 @ F2 @ top_top_set_a )
        = top_top_set_option_a ) ) ).

% bij_betw_imp_surj
thf(fact_961_bij__betw__imp__surj,axiom,
    ! [F2: option_a > a,A2: set_option_a] :
      ( ( bij_betw_option_a_a @ F2 @ A2 @ top_top_set_a )
     => ( ( image_option_a_a2 @ F2 @ top_top_set_option_a )
        = top_top_set_a ) ) ).

% bij_betw_imp_surj
thf(fact_962_bij__betw__imp__surj,axiom,
    ! [F2: a > a,A2: set_a] :
      ( ( bij_betw_a_a @ F2 @ A2 @ top_top_set_a )
     => ( ( image_a_a2 @ F2 @ top_top_set_a )
        = top_top_set_a ) ) ).

% bij_betw_imp_surj
thf(fact_963_bij__betw__byWitness,axiom,
    ! [A2: set_a,F3: option_a > a,F2: a > option_a,A9: set_option_a] :
      ( ! [X2: a] :
          ( ( member_a @ X2 @ A2 )
         => ( ( F3 @ ( F2 @ X2 ) )
            = X2 ) )
     => ( ! [X2: option_a] :
            ( ( member_option_a @ X2 @ A9 )
           => ( ( F2 @ ( F3 @ X2 ) )
              = X2 ) )
       => ( ( ord_le1955136853071979460tion_a @ ( image_a_option_a2 @ F2 @ A2 ) @ A9 )
         => ( ( ord_less_eq_set_a @ ( image_option_a_a2 @ F3 @ A9 ) @ A2 )
           => ( bij_betw_a_option_a @ F2 @ A2 @ A9 ) ) ) ) ) ).

% bij_betw_byWitness
thf(fact_964_bij__betw__byWitness,axiom,
    ! [A2: set_option_a,F3: a > option_a,F2: option_a > a,A9: set_a] :
      ( ! [X2: option_a] :
          ( ( member_option_a @ X2 @ A2 )
         => ( ( F3 @ ( F2 @ X2 ) )
            = X2 ) )
     => ( ! [X2: a] :
            ( ( member_a @ X2 @ A9 )
           => ( ( F2 @ ( F3 @ X2 ) )
              = X2 ) )
       => ( ( ord_less_eq_set_a @ ( image_option_a_a2 @ F2 @ A2 ) @ A9 )
         => ( ( ord_le1955136853071979460tion_a @ ( image_a_option_a2 @ F3 @ A9 ) @ A2 )
           => ( bij_betw_option_a_a @ F2 @ A2 @ A9 ) ) ) ) ) ).

% bij_betw_byWitness
thf(fact_965_bij__betw__subset,axiom,
    ! [F2: a > option_a,A2: set_a,A9: set_option_a,B2: set_a,B7: set_option_a] :
      ( ( bij_betw_a_option_a @ F2 @ A2 @ A9 )
     => ( ( ord_less_eq_set_a @ B2 @ A2 )
       => ( ( ( image_a_option_a2 @ F2 @ B2 )
            = B7 )
         => ( bij_betw_a_option_a @ F2 @ B2 @ B7 ) ) ) ) ).

% bij_betw_subset
thf(fact_966_bij__betw__subset,axiom,
    ! [F2: option_a > a,A2: set_option_a,A9: set_a,B2: set_option_a,B7: set_a] :
      ( ( bij_betw_option_a_a @ F2 @ A2 @ A9 )
     => ( ( ord_le1955136853071979460tion_a @ B2 @ A2 )
       => ( ( ( image_option_a_a2 @ F2 @ B2 )
            = B7 )
         => ( bij_betw_option_a_a @ F2 @ B2 @ B7 ) ) ) ) ).

% bij_betw_subset
thf(fact_967_bij__betw__def,axiom,
    ( bij_betw_option_a_a
    = ( ^ [F: option_a > a,A7: set_option_a,B6: set_a] :
          ( ( inj_on_option_a_a @ F @ A7 )
          & ( ( image_option_a_a2 @ F @ A7 )
            = B6 ) ) ) ) ).

% bij_betw_def
thf(fact_968_bij__betw__def,axiom,
    ( bij_betw_a_option_a
    = ( ^ [F: a > option_a,A7: set_a,B6: set_option_a] :
          ( ( inj_on_a_option_a @ F @ A7 )
          & ( ( image_a_option_a2 @ F @ A7 )
            = B6 ) ) ) ) ).

% bij_betw_def
thf(fact_969_bij__betw__imageI,axiom,
    ! [F2: option_a > a,A2: set_option_a,B2: set_a] :
      ( ( inj_on_option_a_a @ F2 @ A2 )
     => ( ( ( image_option_a_a2 @ F2 @ A2 )
          = B2 )
       => ( bij_betw_option_a_a @ F2 @ A2 @ B2 ) ) ) ).

% bij_betw_imageI
thf(fact_970_bij__betw__imageI,axiom,
    ! [F2: a > option_a,A2: set_a,B2: set_option_a] :
      ( ( inj_on_a_option_a @ F2 @ A2 )
     => ( ( ( image_a_option_a2 @ F2 @ A2 )
          = B2 )
       => ( bij_betw_a_option_a @ F2 @ A2 @ B2 ) ) ) ).

% bij_betw_imageI
thf(fact_971_inj__on__imp__bij__betw,axiom,
    ! [F2: option_a > a,A2: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ A2 )
     => ( bij_betw_option_a_a @ F2 @ A2 @ ( image_option_a_a2 @ F2 @ A2 ) ) ) ).

% inj_on_imp_bij_betw
thf(fact_972_inj__on__imp__bij__betw,axiom,
    ! [F2: a > option_a,A2: set_a] :
      ( ( inj_on_a_option_a @ F2 @ A2 )
     => ( bij_betw_a_option_a @ F2 @ A2 @ ( image_a_option_a2 @ F2 @ A2 ) ) ) ).

% inj_on_imp_bij_betw
thf(fact_973_bij__is__inj,axiom,
    ! [F2: option_a > option_a] :
      ( ( bij_be5431266891817924854tion_a @ F2 @ top_top_set_option_a @ top_top_set_option_a )
     => ( inj_on8559383841115902449tion_a @ F2 @ top_top_set_option_a ) ) ).

% bij_is_inj
thf(fact_974_bij__is__inj,axiom,
    ! [F2: option_a > a] :
      ( ( bij_betw_option_a_a @ F2 @ top_top_set_option_a @ top_top_set_a )
     => ( inj_on_option_a_a @ F2 @ top_top_set_option_a ) ) ).

% bij_is_inj
thf(fact_975_bij__is__inj,axiom,
    ! [F2: a > option_a] :
      ( ( bij_betw_a_option_a @ F2 @ top_top_set_a @ top_top_set_option_a )
     => ( inj_on_a_option_a @ F2 @ top_top_set_a ) ) ).

% bij_is_inj
thf(fact_976_bij__is__inj,axiom,
    ! [F2: a > a] :
      ( ( bij_betw_a_a @ F2 @ top_top_set_a @ top_top_set_a )
     => ( inj_on_a_a @ F2 @ top_top_set_a ) ) ).

% bij_is_inj
thf(fact_977_comp__surj,axiom,
    ! [F2: option_a > option_a,G: option_a > option_a] :
      ( ( ( image_7439109396645324421tion_a @ F2 @ top_top_set_option_a )
        = top_top_set_option_a )
     => ( ( ( image_7439109396645324421tion_a @ G @ top_top_set_option_a )
          = top_top_set_option_a )
       => ( ( image_7439109396645324421tion_a @ ( comp_o3154387707078715297tion_a @ G @ F2 ) @ top_top_set_option_a )
          = top_top_set_option_a ) ) ) ).

% comp_surj
thf(fact_978_comp__surj,axiom,
    ! [F2: option_a > option_a,G: option_a > a] :
      ( ( ( image_7439109396645324421tion_a @ F2 @ top_top_set_option_a )
        = top_top_set_option_a )
     => ( ( ( image_option_a_a2 @ G @ top_top_set_option_a )
          = top_top_set_a )
       => ( ( image_option_a_a2 @ ( comp_o3864519266390211175tion_a @ G @ F2 ) @ top_top_set_option_a )
          = top_top_set_a ) ) ) ).

% comp_surj
thf(fact_979_comp__surj,axiom,
    ! [F2: option_a > a,G: a > option_a] :
      ( ( ( image_option_a_a2 @ F2 @ top_top_set_option_a )
        = top_top_set_a )
     => ( ( ( image_a_option_a2 @ G @ top_top_set_a )
          = top_top_set_option_a )
       => ( ( image_7439109396645324421tion_a @ ( comp_a6249931511552232923tion_a @ G @ F2 ) @ top_top_set_option_a )
          = top_top_set_option_a ) ) ) ).

% comp_surj
thf(fact_980_comp__surj,axiom,
    ! [F2: option_a > a,G: a > a] :
      ( ( ( image_option_a_a2 @ F2 @ top_top_set_option_a )
        = top_top_set_a )
     => ( ( ( image_a_a2 @ G @ top_top_set_a )
          = top_top_set_a )
       => ( ( image_option_a_a2 @ ( comp_a_a_option_a @ G @ F2 ) @ top_top_set_option_a )
          = top_top_set_a ) ) ) ).

% comp_surj
thf(fact_981_comp__surj,axiom,
    ! [F2: a > option_a,G: option_a > option_a] :
      ( ( ( image_a_option_a2 @ F2 @ top_top_set_a )
        = top_top_set_option_a )
     => ( ( ( image_7439109396645324421tion_a @ G @ top_top_set_option_a )
          = top_top_set_option_a )
       => ( ( image_a_option_a2 @ ( comp_o6087033147929006299on_a_a @ G @ F2 ) @ top_top_set_a )
          = top_top_set_option_a ) ) ) ).

% comp_surj
thf(fact_982_comp__surj,axiom,
    ! [F2: a > option_a,G: option_a > a] :
      ( ( ( image_a_option_a2 @ F2 @ top_top_set_a )
        = top_top_set_option_a )
     => ( ( ( image_option_a_a2 @ G @ top_top_set_option_a )
          = top_top_set_a )
       => ( ( image_a_a2 @ ( comp_option_a_a_a @ G @ F2 ) @ top_top_set_a )
          = top_top_set_a ) ) ) ).

% comp_surj
thf(fact_983_comp__surj,axiom,
    ! [F2: a > a,G: a > option_a] :
      ( ( ( image_a_a2 @ F2 @ top_top_set_a )
        = top_top_set_a )
     => ( ( ( image_a_option_a2 @ G @ top_top_set_a )
          = top_top_set_option_a )
       => ( ( image_a_option_a2 @ ( comp_a_option_a_a @ G @ F2 ) @ top_top_set_a )
          = top_top_set_option_a ) ) ) ).

% comp_surj
thf(fact_984_comp__surj,axiom,
    ! [F2: a > a,G: a > a] :
      ( ( ( image_a_a2 @ F2 @ top_top_set_a )
        = top_top_set_a )
     => ( ( ( image_a_a2 @ G @ top_top_set_a )
          = top_top_set_a )
       => ( ( image_a_a2 @ ( comp_a_a_a @ G @ F2 ) @ top_top_set_a )
          = top_top_set_a ) ) ) ).

% comp_surj
thf(fact_985_comp__inj__on,axiom,
    ! [F2: option_a > a,A2: set_option_a,G: a > option_a] :
      ( ( inj_on_option_a_a @ F2 @ A2 )
     => ( ( inj_on_a_option_a @ G @ ( image_option_a_a2 @ F2 @ A2 ) )
       => ( inj_on8559383841115902449tion_a @ ( comp_a6249931511552232923tion_a @ G @ F2 ) @ A2 ) ) ) ).

% comp_inj_on
thf(fact_986_comp__inj__on,axiom,
    ! [F2: a > a,A2: set_a,G: a > option_a] :
      ( ( inj_on_a_a @ F2 @ A2 )
     => ( ( inj_on_a_option_a @ G @ ( image_a_a2 @ F2 @ A2 ) )
       => ( inj_on_a_option_a @ ( comp_a_option_a_a @ G @ F2 ) @ A2 ) ) ) ).

% comp_inj_on
thf(fact_987_comp__inj__on,axiom,
    ! [F2: a > option_a,A2: set_a,G: option_a > a] :
      ( ( inj_on_a_option_a @ F2 @ A2 )
     => ( ( inj_on_option_a_a @ G @ ( image_a_option_a2 @ F2 @ A2 ) )
       => ( inj_on_a_a @ ( comp_option_a_a_a @ G @ F2 ) @ A2 ) ) ) ).

% comp_inj_on
thf(fact_988_comp__inj__on,axiom,
    ! [F2: a > option_a,A2: set_a,G: option_a > option_a] :
      ( ( inj_on_a_option_a @ F2 @ A2 )
     => ( ( inj_on8559383841115902449tion_a @ G @ ( image_a_option_a2 @ F2 @ A2 ) )
       => ( inj_on_a_option_a @ ( comp_o6087033147929006299on_a_a @ G @ F2 ) @ A2 ) ) ) ).

% comp_inj_on
thf(fact_989_inj__on__imageI,axiom,
    ! [G: option_a > a,F2: a > option_a,A2: set_a] :
      ( ( inj_on_a_a @ ( comp_option_a_a_a @ G @ F2 ) @ A2 )
     => ( inj_on_option_a_a @ G @ ( image_a_option_a2 @ F2 @ A2 ) ) ) ).

% inj_on_imageI
thf(fact_990_inj__on__imageI,axiom,
    ! [G: a > option_a,F2: option_a > a,A2: set_option_a] :
      ( ( inj_on8559383841115902449tion_a @ ( comp_a6249931511552232923tion_a @ G @ F2 ) @ A2 )
     => ( inj_on_a_option_a @ G @ ( image_option_a_a2 @ F2 @ A2 ) ) ) ).

% inj_on_imageI
thf(fact_991_inj__on__imageI,axiom,
    ! [G: option_a > option_a,F2: a > option_a,A2: set_a] :
      ( ( inj_on_a_option_a @ ( comp_o6087033147929006299on_a_a @ G @ F2 ) @ A2 )
     => ( inj_on8559383841115902449tion_a @ G @ ( image_a_option_a2 @ F2 @ A2 ) ) ) ).

% inj_on_imageI
thf(fact_992_inj__on__imageI,axiom,
    ! [G: a > option_a,F2: a > a,A2: set_a] :
      ( ( inj_on_a_option_a @ ( comp_a_option_a_a @ G @ F2 ) @ A2 )
     => ( inj_on_a_option_a @ G @ ( image_a_a2 @ F2 @ A2 ) ) ) ).

% inj_on_imageI
thf(fact_993_comp__inj__on__iff,axiom,
    ! [F2: option_a > a,A2: set_option_a,F3: a > option_a] :
      ( ( inj_on_option_a_a @ F2 @ A2 )
     => ( ( inj_on_a_option_a @ F3 @ ( image_option_a_a2 @ F2 @ A2 ) )
        = ( inj_on8559383841115902449tion_a @ ( comp_a6249931511552232923tion_a @ F3 @ F2 ) @ A2 ) ) ) ).

% comp_inj_on_iff
thf(fact_994_comp__inj__on__iff,axiom,
    ! [F2: a > a,A2: set_a,F3: a > option_a] :
      ( ( inj_on_a_a @ F2 @ A2 )
     => ( ( inj_on_a_option_a @ F3 @ ( image_a_a2 @ F2 @ A2 ) )
        = ( inj_on_a_option_a @ ( comp_a_option_a_a @ F3 @ F2 ) @ A2 ) ) ) ).

% comp_inj_on_iff
thf(fact_995_comp__inj__on__iff,axiom,
    ! [F2: a > option_a,A2: set_a,F3: option_a > a] :
      ( ( inj_on_a_option_a @ F2 @ A2 )
     => ( ( inj_on_option_a_a @ F3 @ ( image_a_option_a2 @ F2 @ A2 ) )
        = ( inj_on_a_a @ ( comp_option_a_a_a @ F3 @ F2 ) @ A2 ) ) ) ).

% comp_inj_on_iff
thf(fact_996_comp__inj__on__iff,axiom,
    ! [F2: a > option_a,A2: set_a,F3: option_a > option_a] :
      ( ( inj_on_a_option_a @ F2 @ A2 )
     => ( ( inj_on8559383841115902449tion_a @ F3 @ ( image_a_option_a2 @ F2 @ A2 ) )
        = ( inj_on_a_option_a @ ( comp_o6087033147929006299on_a_a @ F3 @ F2 ) @ A2 ) ) ) ).

% comp_inj_on_iff
thf(fact_997_inj__compose,axiom,
    ! [F2: option_a > a,G: a > option_a] :
      ( ( inj_on_option_a_a @ F2 @ top_top_set_option_a )
     => ( ( inj_on_a_option_a @ G @ top_top_set_a )
       => ( inj_on_a_a @ ( comp_option_a_a_a @ F2 @ G ) @ top_top_set_a ) ) ) ).

% inj_compose
thf(fact_998_inj__compose,axiom,
    ! [F2: option_a > option_a,G: a > option_a] :
      ( ( inj_on8559383841115902449tion_a @ F2 @ top_top_set_option_a )
     => ( ( inj_on_a_option_a @ G @ top_top_set_a )
       => ( inj_on_a_option_a @ ( comp_o6087033147929006299on_a_a @ F2 @ G ) @ top_top_set_a ) ) ) ).

% inj_compose
thf(fact_999_inj__compose,axiom,
    ! [F2: a > option_a,G: option_a > a] :
      ( ( inj_on_a_option_a @ F2 @ top_top_set_a )
     => ( ( inj_on_option_a_a @ G @ top_top_set_option_a )
       => ( inj_on8559383841115902449tion_a @ ( comp_a6249931511552232923tion_a @ F2 @ G ) @ top_top_set_option_a ) ) ) ).

% inj_compose
thf(fact_1000_inj__compose,axiom,
    ! [F2: a > option_a,G: a > a] :
      ( ( inj_on_a_option_a @ F2 @ top_top_set_a )
     => ( ( inj_on_a_a @ G @ top_top_set_a )
       => ( inj_on_a_option_a @ ( comp_a_option_a_a @ F2 @ G ) @ top_top_set_a ) ) ) ).

% inj_compose
thf(fact_1001_bind__map__option,axiom,
    ! [F2: a > a,X3: option_a,G: a > option_a] :
      ( ( bind_a_a @ ( map_option_a_a @ F2 @ X3 ) @ G )
      = ( bind_a_a @ X3 @ ( comp_a_option_a_a @ G @ F2 ) ) ) ).

% bind_map_option
thf(fact_1002_map__option__bind,axiom,
    ! [F2: a > a,X3: option_a,G: a > option_a] :
      ( ( map_option_a_a @ F2 @ ( bind_a_a @ X3 @ G ) )
      = ( bind_a_a @ X3 @ ( comp_o6087033147929006299on_a_a @ ( map_option_a_a @ F2 ) @ G ) ) ) ).

% map_option_bind
thf(fact_1003_asymD,axiom,
    ! [R2: set_Pr7585778909603769095tion_a,X3: option_a,Y: option_a] :
      ( ( asym_on_option_a @ top_top_set_option_a @ R2 )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y ) @ R2 )
       => ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y @ X3 ) @ R2 ) ) ) ).

% asymD
thf(fact_1004_asymD,axiom,
    ! [R2: set_Product_prod_a_a,X3: a,Y: a] :
      ( ( asym_on_a @ top_top_set_a @ R2 )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R2 )
       => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X3 ) @ R2 ) ) ) ).

% asymD
thf(fact_1005_asym__iff,axiom,
    ! [R2: set_Pr7585778909603769095tion_a] :
      ( ( asym_on_option_a @ top_top_set_option_a @ R2 )
      = ( ! [X: option_a,Y3: option_a] :
            ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y3 ) @ R2 )
           => ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y3 @ X ) @ R2 ) ) ) ) ).

% asym_iff
thf(fact_1006_asym__iff,axiom,
    ! [R2: set_Product_prod_a_a] :
      ( ( asym_on_a @ top_top_set_a @ R2 )
      = ( ! [X: a,Y3: a] :
            ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R2 )
           => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X ) @ R2 ) ) ) ) ).

% asym_iff
thf(fact_1007_bij__def,axiom,
    ! [F2: option_a > option_a] :
      ( ( bij_be5431266891817924854tion_a @ F2 @ top_top_set_option_a @ top_top_set_option_a )
      = ( ( inj_on8559383841115902449tion_a @ F2 @ top_top_set_option_a )
        & ( ( image_7439109396645324421tion_a @ F2 @ top_top_set_option_a )
          = top_top_set_option_a ) ) ) ).

% bij_def
thf(fact_1008_bij__def,axiom,
    ! [F2: option_a > a] :
      ( ( bij_betw_option_a_a @ F2 @ top_top_set_option_a @ top_top_set_a )
      = ( ( inj_on_option_a_a @ F2 @ top_top_set_option_a )
        & ( ( image_option_a_a2 @ F2 @ top_top_set_option_a )
          = top_top_set_a ) ) ) ).

% bij_def
thf(fact_1009_bij__def,axiom,
    ! [F2: a > option_a] :
      ( ( bij_betw_a_option_a @ F2 @ top_top_set_a @ top_top_set_option_a )
      = ( ( inj_on_a_option_a @ F2 @ top_top_set_a )
        & ( ( image_a_option_a2 @ F2 @ top_top_set_a )
          = top_top_set_option_a ) ) ) ).

% bij_def
thf(fact_1010_bij__def,axiom,
    ! [F2: a > a] :
      ( ( bij_betw_a_a @ F2 @ top_top_set_a @ top_top_set_a )
      = ( ( inj_on_a_a @ F2 @ top_top_set_a )
        & ( ( image_a_a2 @ F2 @ top_top_set_a )
          = top_top_set_a ) ) ) ).

% bij_def
thf(fact_1011_bijI,axiom,
    ! [F2: option_a > option_a] :
      ( ( inj_on8559383841115902449tion_a @ F2 @ top_top_set_option_a )
     => ( ( ( image_7439109396645324421tion_a @ F2 @ top_top_set_option_a )
          = top_top_set_option_a )
       => ( bij_be5431266891817924854tion_a @ F2 @ top_top_set_option_a @ top_top_set_option_a ) ) ) ).

% bijI
thf(fact_1012_bijI,axiom,
    ! [F2: option_a > a] :
      ( ( inj_on_option_a_a @ F2 @ top_top_set_option_a )
     => ( ( ( image_option_a_a2 @ F2 @ top_top_set_option_a )
          = top_top_set_a )
       => ( bij_betw_option_a_a @ F2 @ top_top_set_option_a @ top_top_set_a ) ) ) ).

% bijI
thf(fact_1013_bijI,axiom,
    ! [F2: a > option_a] :
      ( ( inj_on_a_option_a @ F2 @ top_top_set_a )
     => ( ( ( image_a_option_a2 @ F2 @ top_top_set_a )
          = top_top_set_option_a )
       => ( bij_betw_a_option_a @ F2 @ top_top_set_a @ top_top_set_option_a ) ) ) ).

% bijI
thf(fact_1014_bijI,axiom,
    ! [F2: a > a] :
      ( ( inj_on_a_a @ F2 @ top_top_set_a )
     => ( ( ( image_a_a2 @ F2 @ top_top_set_a )
          = top_top_set_a )
       => ( bij_betw_a_a @ F2 @ top_top_set_a @ top_top_set_a ) ) ) ).

% bijI
thf(fact_1015_notIn__Un__bij__betw,axiom,
    ! [B: produc6150846815813599699_a_d_c,A2: set_Pr7868159745199425715_a_d_c,F2: produc6150846815813599699_a_d_c > produc6150846815813599699_a_d_c,A9: set_Pr7868159745199425715_a_d_c] :
      ( ~ ( member2052822272342364412_a_d_c @ B @ A2 )
     => ( ~ ( member2052822272342364412_a_d_c @ ( F2 @ B ) @ A9 )
       => ( ( bij_be734363561790614582_a_d_c @ F2 @ A2 @ A9 )
         => ( bij_be734363561790614582_a_d_c @ F2 @ ( sup_su826086822432945799_a_d_c @ A2 @ ( insert4944988292672327587_a_d_c @ B @ bot_bo1396921257525504799_a_d_c ) ) @ ( sup_su826086822432945799_a_d_c @ A9 @ ( insert4944988292672327587_a_d_c @ ( F2 @ B ) @ bot_bo1396921257525504799_a_d_c ) ) ) ) ) ) ).

% notIn_Un_bij_betw
thf(fact_1016_notIn__Un__bij__betw,axiom,
    ! [B: produc6150846815813599699_a_d_c,A2: set_Pr7868159745199425715_a_d_c,F2: produc6150846815813599699_a_d_c > a,A9: set_a] :
      ( ~ ( member2052822272342364412_a_d_c @ B @ A2 )
     => ( ~ ( member_a @ ( F2 @ B ) @ A9 )
       => ( ( bij_be6485230601037006177_d_c_a @ F2 @ A2 @ A9 )
         => ( bij_be6485230601037006177_d_c_a @ F2 @ ( sup_su826086822432945799_a_d_c @ A2 @ ( insert4944988292672327587_a_d_c @ B @ bot_bo1396921257525504799_a_d_c ) ) @ ( sup_sup_set_a @ A9 @ ( insert_a @ ( F2 @ B ) @ bot_bot_set_a ) ) ) ) ) ) ).

% notIn_Un_bij_betw
thf(fact_1017_notIn__Un__bij__betw,axiom,
    ! [B: produc6150846815813599699_a_d_c,A2: set_Pr7868159745199425715_a_d_c,F2: produc6150846815813599699_a_d_c > option_a,A9: set_option_a] :
      ( ~ ( member2052822272342364412_a_d_c @ B @ A2 )
     => ( ~ ( member_option_a @ ( F2 @ B ) @ A9 )
       => ( ( bij_be4831480119166671655tion_a @ F2 @ A2 @ A9 )
         => ( bij_be4831480119166671655tion_a @ F2 @ ( sup_su826086822432945799_a_d_c @ A2 @ ( insert4944988292672327587_a_d_c @ B @ bot_bo1396921257525504799_a_d_c ) ) @ ( sup_sup_set_option_a @ A9 @ ( insert_option_a @ ( F2 @ B ) @ bot_bot_set_option_a ) ) ) ) ) ) ).

% notIn_Un_bij_betw
thf(fact_1018_notIn__Un__bij__betw,axiom,
    ! [B: a,A2: set_a,F2: a > produc6150846815813599699_a_d_c,A9: set_Pr7868159745199425715_a_d_c] :
      ( ~ ( member_a @ B @ A2 )
     => ( ~ ( member2052822272342364412_a_d_c @ ( F2 @ B ) @ A9 )
       => ( ( bij_be3492573195200641995_a_d_c @ F2 @ A2 @ A9 )
         => ( bij_be3492573195200641995_a_d_c @ F2 @ ( sup_sup_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( sup_su826086822432945799_a_d_c @ A9 @ ( insert4944988292672327587_a_d_c @ ( F2 @ B ) @ bot_bo1396921257525504799_a_d_c ) ) ) ) ) ) ).

% notIn_Un_bij_betw
thf(fact_1019_notIn__Un__bij__betw,axiom,
    ! [B: a,A2: set_a,F2: a > a,A9: set_a] :
      ( ~ ( member_a @ B @ A2 )
     => ( ~ ( member_a @ ( F2 @ B ) @ A9 )
       => ( ( bij_betw_a_a @ F2 @ A2 @ A9 )
         => ( bij_betw_a_a @ F2 @ ( sup_sup_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( sup_sup_set_a @ A9 @ ( insert_a @ ( F2 @ B ) @ bot_bot_set_a ) ) ) ) ) ) ).

% notIn_Un_bij_betw
thf(fact_1020_notIn__Un__bij__betw,axiom,
    ! [B: a,A2: set_a,F2: a > option_a,A9: set_option_a] :
      ( ~ ( member_a @ B @ A2 )
     => ( ~ ( member_option_a @ ( F2 @ B ) @ A9 )
       => ( ( bij_betw_a_option_a @ F2 @ A2 @ A9 )
         => ( bij_betw_a_option_a @ F2 @ ( sup_sup_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( sup_sup_set_option_a @ A9 @ ( insert_option_a @ ( F2 @ B ) @ bot_bot_set_option_a ) ) ) ) ) ) ).

% notIn_Un_bij_betw
thf(fact_1021_notIn__Un__bij__betw,axiom,
    ! [B: option_a,A2: set_option_a,F2: option_a > produc6150846815813599699_a_d_c,A9: set_Pr7868159745199425715_a_d_c] :
      ( ~ ( member_option_a @ B @ A2 )
     => ( ~ ( member2052822272342364412_a_d_c @ ( F2 @ B ) @ A9 )
       => ( ( bij_be8275362299291636869_a_d_c @ F2 @ A2 @ A9 )
         => ( bij_be8275362299291636869_a_d_c @ F2 @ ( sup_sup_set_option_a @ A2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( sup_su826086822432945799_a_d_c @ A9 @ ( insert4944988292672327587_a_d_c @ ( F2 @ B ) @ bot_bo1396921257525504799_a_d_c ) ) ) ) ) ) ).

% notIn_Un_bij_betw
thf(fact_1022_notIn__Un__bij__betw,axiom,
    ! [B: option_a,A2: set_option_a,F2: option_a > a,A9: set_a] :
      ( ~ ( member_option_a @ B @ A2 )
     => ( ~ ( member_a @ ( F2 @ B ) @ A9 )
       => ( ( bij_betw_option_a_a @ F2 @ A2 @ A9 )
         => ( bij_betw_option_a_a @ F2 @ ( sup_sup_set_option_a @ A2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( sup_sup_set_a @ A9 @ ( insert_a @ ( F2 @ B ) @ bot_bot_set_a ) ) ) ) ) ) ).

% notIn_Un_bij_betw
thf(fact_1023_notIn__Un__bij__betw,axiom,
    ! [B: option_a,A2: set_option_a,F2: option_a > option_a,A9: set_option_a] :
      ( ~ ( member_option_a @ B @ A2 )
     => ( ~ ( member_option_a @ ( F2 @ B ) @ A9 )
       => ( ( bij_be5431266891817924854tion_a @ F2 @ A2 @ A9 )
         => ( bij_be5431266891817924854tion_a @ F2 @ ( sup_sup_set_option_a @ A2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( sup_sup_set_option_a @ A9 @ ( insert_option_a @ ( F2 @ B ) @ bot_bot_set_option_a ) ) ) ) ) ) ).

% notIn_Un_bij_betw
thf(fact_1024_notIn__Un__bij__betw3,axiom,
    ! [B: produc6150846815813599699_a_d_c,A2: set_Pr7868159745199425715_a_d_c,F2: produc6150846815813599699_a_d_c > produc6150846815813599699_a_d_c,A9: set_Pr7868159745199425715_a_d_c] :
      ( ~ ( member2052822272342364412_a_d_c @ B @ A2 )
     => ( ~ ( member2052822272342364412_a_d_c @ ( F2 @ B ) @ A9 )
       => ( ( bij_be734363561790614582_a_d_c @ F2 @ A2 @ A9 )
          = ( bij_be734363561790614582_a_d_c @ F2 @ ( sup_su826086822432945799_a_d_c @ A2 @ ( insert4944988292672327587_a_d_c @ B @ bot_bo1396921257525504799_a_d_c ) ) @ ( sup_su826086822432945799_a_d_c @ A9 @ ( insert4944988292672327587_a_d_c @ ( F2 @ B ) @ bot_bo1396921257525504799_a_d_c ) ) ) ) ) ) ).

% notIn_Un_bij_betw3
thf(fact_1025_notIn__Un__bij__betw3,axiom,
    ! [B: produc6150846815813599699_a_d_c,A2: set_Pr7868159745199425715_a_d_c,F2: produc6150846815813599699_a_d_c > a,A9: set_a] :
      ( ~ ( member2052822272342364412_a_d_c @ B @ A2 )
     => ( ~ ( member_a @ ( F2 @ B ) @ A9 )
       => ( ( bij_be6485230601037006177_d_c_a @ F2 @ A2 @ A9 )
          = ( bij_be6485230601037006177_d_c_a @ F2 @ ( sup_su826086822432945799_a_d_c @ A2 @ ( insert4944988292672327587_a_d_c @ B @ bot_bo1396921257525504799_a_d_c ) ) @ ( sup_sup_set_a @ A9 @ ( insert_a @ ( F2 @ B ) @ bot_bot_set_a ) ) ) ) ) ) ).

% notIn_Un_bij_betw3
thf(fact_1026_notIn__Un__bij__betw3,axiom,
    ! [B: produc6150846815813599699_a_d_c,A2: set_Pr7868159745199425715_a_d_c,F2: produc6150846815813599699_a_d_c > option_a,A9: set_option_a] :
      ( ~ ( member2052822272342364412_a_d_c @ B @ A2 )
     => ( ~ ( member_option_a @ ( F2 @ B ) @ A9 )
       => ( ( bij_be4831480119166671655tion_a @ F2 @ A2 @ A9 )
          = ( bij_be4831480119166671655tion_a @ F2 @ ( sup_su826086822432945799_a_d_c @ A2 @ ( insert4944988292672327587_a_d_c @ B @ bot_bo1396921257525504799_a_d_c ) ) @ ( sup_sup_set_option_a @ A9 @ ( insert_option_a @ ( F2 @ B ) @ bot_bot_set_option_a ) ) ) ) ) ) ).

% notIn_Un_bij_betw3
thf(fact_1027_notIn__Un__bij__betw3,axiom,
    ! [B: a,A2: set_a,F2: a > produc6150846815813599699_a_d_c,A9: set_Pr7868159745199425715_a_d_c] :
      ( ~ ( member_a @ B @ A2 )
     => ( ~ ( member2052822272342364412_a_d_c @ ( F2 @ B ) @ A9 )
       => ( ( bij_be3492573195200641995_a_d_c @ F2 @ A2 @ A9 )
          = ( bij_be3492573195200641995_a_d_c @ F2 @ ( sup_sup_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( sup_su826086822432945799_a_d_c @ A9 @ ( insert4944988292672327587_a_d_c @ ( F2 @ B ) @ bot_bo1396921257525504799_a_d_c ) ) ) ) ) ) ).

% notIn_Un_bij_betw3
thf(fact_1028_notIn__Un__bij__betw3,axiom,
    ! [B: a,A2: set_a,F2: a > a,A9: set_a] :
      ( ~ ( member_a @ B @ A2 )
     => ( ~ ( member_a @ ( F2 @ B ) @ A9 )
       => ( ( bij_betw_a_a @ F2 @ A2 @ A9 )
          = ( bij_betw_a_a @ F2 @ ( sup_sup_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( sup_sup_set_a @ A9 @ ( insert_a @ ( F2 @ B ) @ bot_bot_set_a ) ) ) ) ) ) ).

% notIn_Un_bij_betw3
thf(fact_1029_notIn__Un__bij__betw3,axiom,
    ! [B: a,A2: set_a,F2: a > option_a,A9: set_option_a] :
      ( ~ ( member_a @ B @ A2 )
     => ( ~ ( member_option_a @ ( F2 @ B ) @ A9 )
       => ( ( bij_betw_a_option_a @ F2 @ A2 @ A9 )
          = ( bij_betw_a_option_a @ F2 @ ( sup_sup_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( sup_sup_set_option_a @ A9 @ ( insert_option_a @ ( F2 @ B ) @ bot_bot_set_option_a ) ) ) ) ) ) ).

% notIn_Un_bij_betw3
thf(fact_1030_notIn__Un__bij__betw3,axiom,
    ! [B: option_a,A2: set_option_a,F2: option_a > produc6150846815813599699_a_d_c,A9: set_Pr7868159745199425715_a_d_c] :
      ( ~ ( member_option_a @ B @ A2 )
     => ( ~ ( member2052822272342364412_a_d_c @ ( F2 @ B ) @ A9 )
       => ( ( bij_be8275362299291636869_a_d_c @ F2 @ A2 @ A9 )
          = ( bij_be8275362299291636869_a_d_c @ F2 @ ( sup_sup_set_option_a @ A2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( sup_su826086822432945799_a_d_c @ A9 @ ( insert4944988292672327587_a_d_c @ ( F2 @ B ) @ bot_bo1396921257525504799_a_d_c ) ) ) ) ) ) ).

% notIn_Un_bij_betw3
thf(fact_1031_notIn__Un__bij__betw3,axiom,
    ! [B: option_a,A2: set_option_a,F2: option_a > a,A9: set_a] :
      ( ~ ( member_option_a @ B @ A2 )
     => ( ~ ( member_a @ ( F2 @ B ) @ A9 )
       => ( ( bij_betw_option_a_a @ F2 @ A2 @ A9 )
          = ( bij_betw_option_a_a @ F2 @ ( sup_sup_set_option_a @ A2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( sup_sup_set_a @ A9 @ ( insert_a @ ( F2 @ B ) @ bot_bot_set_a ) ) ) ) ) ) ).

% notIn_Un_bij_betw3
thf(fact_1032_notIn__Un__bij__betw3,axiom,
    ! [B: option_a,A2: set_option_a,F2: option_a > option_a,A9: set_option_a] :
      ( ~ ( member_option_a @ B @ A2 )
     => ( ~ ( member_option_a @ ( F2 @ B ) @ A9 )
       => ( ( bij_be5431266891817924854tion_a @ F2 @ A2 @ A9 )
          = ( bij_be5431266891817924854tion_a @ F2 @ ( sup_sup_set_option_a @ A2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( sup_sup_set_option_a @ A9 @ ( insert_option_a @ ( F2 @ B ) @ bot_bot_set_option_a ) ) ) ) ) ) ).

% notIn_Un_bij_betw3
thf(fact_1033_bij__betw__partition,axiom,
    ! [F2: a > a,A2: set_a,C3: set_a,B2: set_a,D2: set_a] :
      ( ( bij_betw_a_a @ F2 @ ( sup_sup_set_a @ A2 @ C3 ) @ ( sup_sup_set_a @ B2 @ D2 ) )
     => ( ( bij_betw_a_a @ F2 @ C3 @ D2 )
       => ( ( ( inf_inf_set_a @ A2 @ C3 )
            = bot_bot_set_a )
         => ( ( ( inf_inf_set_a @ B2 @ D2 )
              = bot_bot_set_a )
           => ( bij_betw_a_a @ F2 @ A2 @ B2 ) ) ) ) ) ).

% bij_betw_partition
thf(fact_1034_bij__betw__partition,axiom,
    ! [F2: a > option_a,A2: set_a,C3: set_a,B2: set_option_a,D2: set_option_a] :
      ( ( bij_betw_a_option_a @ F2 @ ( sup_sup_set_a @ A2 @ C3 ) @ ( sup_sup_set_option_a @ B2 @ D2 ) )
     => ( ( bij_betw_a_option_a @ F2 @ C3 @ D2 )
       => ( ( ( inf_inf_set_a @ A2 @ C3 )
            = bot_bot_set_a )
         => ( ( ( inf_inf_set_option_a @ B2 @ D2 )
              = bot_bot_set_option_a )
           => ( bij_betw_a_option_a @ F2 @ A2 @ B2 ) ) ) ) ) ).

% bij_betw_partition
thf(fact_1035_bij__betw__partition,axiom,
    ! [F2: option_a > a,A2: set_option_a,C3: set_option_a,B2: set_a,D2: set_a] :
      ( ( bij_betw_option_a_a @ F2 @ ( sup_sup_set_option_a @ A2 @ C3 ) @ ( sup_sup_set_a @ B2 @ D2 ) )
     => ( ( bij_betw_option_a_a @ F2 @ C3 @ D2 )
       => ( ( ( inf_inf_set_option_a @ A2 @ C3 )
            = bot_bot_set_option_a )
         => ( ( ( inf_inf_set_a @ B2 @ D2 )
              = bot_bot_set_a )
           => ( bij_betw_option_a_a @ F2 @ A2 @ B2 ) ) ) ) ) ).

% bij_betw_partition
thf(fact_1036_bij__betw__partition,axiom,
    ! [F2: option_a > option_a,A2: set_option_a,C3: set_option_a,B2: set_option_a,D2: set_option_a] :
      ( ( bij_be5431266891817924854tion_a @ F2 @ ( sup_sup_set_option_a @ A2 @ C3 ) @ ( sup_sup_set_option_a @ B2 @ D2 ) )
     => ( ( bij_be5431266891817924854tion_a @ F2 @ C3 @ D2 )
       => ( ( ( inf_inf_set_option_a @ A2 @ C3 )
            = bot_bot_set_option_a )
         => ( ( ( inf_inf_set_option_a @ B2 @ D2 )
              = bot_bot_set_option_a )
           => ( bij_be5431266891817924854tion_a @ F2 @ A2 @ B2 ) ) ) ) ) ).

% bij_betw_partition
thf(fact_1037_bij__image__Compl__eq,axiom,
    ! [F2: option_a > option_a,A2: set_option_a] :
      ( ( bij_be5431266891817924854tion_a @ F2 @ top_top_set_option_a @ top_top_set_option_a )
     => ( ( image_7439109396645324421tion_a @ F2 @ ( uminus6205308855922866075tion_a @ A2 ) )
        = ( uminus6205308855922866075tion_a @ ( image_7439109396645324421tion_a @ F2 @ A2 ) ) ) ) ).

% bij_image_Compl_eq
thf(fact_1038_bij__image__Compl__eq,axiom,
    ! [F2: option_a > a,A2: set_option_a] :
      ( ( bij_betw_option_a_a @ F2 @ top_top_set_option_a @ top_top_set_a )
     => ( ( image_option_a_a2 @ F2 @ ( uminus6205308855922866075tion_a @ A2 ) )
        = ( uminus_uminus_set_a @ ( image_option_a_a2 @ F2 @ A2 ) ) ) ) ).

% bij_image_Compl_eq
thf(fact_1039_bij__image__Compl__eq,axiom,
    ! [F2: a > option_a,A2: set_a] :
      ( ( bij_betw_a_option_a @ F2 @ top_top_set_a @ top_top_set_option_a )
     => ( ( image_a_option_a2 @ F2 @ ( uminus_uminus_set_a @ A2 ) )
        = ( uminus6205308855922866075tion_a @ ( image_a_option_a2 @ F2 @ A2 ) ) ) ) ).

% bij_image_Compl_eq
thf(fact_1040_bij__image__Compl__eq,axiom,
    ! [F2: a > a,A2: set_a] :
      ( ( bij_betw_a_a @ F2 @ top_top_set_a @ top_top_set_a )
     => ( ( image_a_a2 @ F2 @ ( uminus_uminus_set_a @ A2 ) )
        = ( uminus_uminus_set_a @ ( image_a_a2 @ F2 @ A2 ) ) ) ) ).

% bij_image_Compl_eq
thf(fact_1041_the__inv__into__comp,axiom,
    ! [F2: a > a,G: option_a > a,A2: set_option_a,X3: a] :
      ( ( inj_on_a_a @ F2 @ ( image_option_a_a2 @ G @ A2 ) )
     => ( ( inj_on_option_a_a @ G @ A2 )
       => ( ( member_a @ X3 @ ( image_a_a2 @ F2 @ ( image_option_a_a2 @ G @ A2 ) ) )
         => ( ( the_in1757154643552616557on_a_a @ A2 @ ( comp_a_a_option_a @ F2 @ G ) @ X3 )
            = ( comp_a_option_a_a @ ( the_in1757154643552616557on_a_a @ A2 @ G ) @ ( the_inv_into_a_a @ ( image_option_a_a2 @ G @ A2 ) @ F2 ) @ X3 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1042_the__inv__into__comp,axiom,
    ! [F2: option_a > a,G: a > option_a,A2: set_a,X3: a] :
      ( ( inj_on_option_a_a @ F2 @ ( image_a_option_a2 @ G @ A2 ) )
     => ( ( inj_on_a_option_a @ G @ A2 )
       => ( ( member_a @ X3 @ ( image_option_a_a2 @ F2 @ ( image_a_option_a2 @ G @ A2 ) ) )
         => ( ( the_inv_into_a_a @ A2 @ ( comp_option_a_a_a @ F2 @ G ) @ X3 )
            = ( comp_option_a_a_a @ ( the_in8758012798868597241tion_a @ A2 @ G ) @ ( the_in1757154643552616557on_a_a @ ( image_a_option_a2 @ G @ A2 ) @ F2 ) @ X3 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1043_the__inv__into__comp,axiom,
    ! [F2: a > produc6150846815813599699_a_d_c,G: option_a > a,A2: set_option_a,X3: produc6150846815813599699_a_d_c] :
      ( ( inj_on3709328703910593990_a_d_c @ F2 @ ( image_option_a_a2 @ G @ A2 ) )
     => ( ( inj_on_option_a_a @ G @ A2 )
       => ( ( member2052822272342364412_a_d_c @ X3 @ ( image_6884406963069720154_a_d_c @ F2 @ ( image_option_a_a2 @ G @ A2 ) ) )
         => ( ( the_in5027906823813023554_a_d_c @ A2 @ ( comp_a8250252551939253132tion_a @ F2 @ G ) @ X3 )
            = ( comp_a2470762695209442538_a_d_c @ ( the_in1757154643552616557on_a_a @ A2 @ G ) @ ( the_in8611454470807686664_a_d_c @ ( image_option_a_a2 @ G @ A2 ) @ F2 ) @ X3 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1044_the__inv__into__comp,axiom,
    ! [F2: option_a > produc6150846815813599699_a_d_c,G: a > option_a,A2: set_a,X3: produc6150846815813599699_a_d_c] :
      ( ( inj_on7085348629561674240_a_d_c @ F2 @ ( image_a_option_a2 @ G @ A2 ) )
     => ( ( inj_on_a_option_a @ G @ A2 )
       => ( ( member2052822272342364412_a_d_c @ X3 @ ( image_191531556693044372_a_d_c @ F2 @ ( image_a_option_a2 @ G @ A2 ) ) )
         => ( ( the_in8611454470807686664_a_d_c @ A2 @ ( comp_o7359491733365814540_d_c_a @ F2 @ G ) @ X3 )
            = ( comp_o4366834327529450358_a_d_c @ ( the_in8758012798868597241tion_a @ A2 @ G ) @ ( the_in5027906823813023554_a_d_c @ ( image_a_option_a2 @ G @ A2 ) @ F2 ) @ X3 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1045_the__inv__into__comp,axiom,
    ! [F2: option_a > option_a,G: a > option_a,A2: set_a,X3: option_a] :
      ( ( inj_on8559383841115902449tion_a @ F2 @ ( image_a_option_a2 @ G @ A2 ) )
     => ( ( inj_on_a_option_a @ G @ A2 )
       => ( ( member_option_a @ X3 @ ( image_7439109396645324421tion_a @ F2 @ ( image_a_option_a2 @ G @ A2 ) ) )
         => ( ( the_in8758012798868597241tion_a @ A2 @ ( comp_o6087033147929006299on_a_a @ F2 @ G ) @ X3 )
            = ( comp_o3864519266390211175tion_a @ ( the_in8758012798868597241tion_a @ A2 @ G ) @ ( the_in2538339130118444083tion_a @ ( image_a_option_a2 @ G @ A2 ) @ F2 ) @ X3 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1046_the__inv__into__comp,axiom,
    ! [F2: a > option_a,G: option_a > a,A2: set_option_a,X3: option_a] :
      ( ( inj_on_a_option_a @ F2 @ ( image_option_a_a2 @ G @ A2 ) )
     => ( ( inj_on_option_a_a @ G @ A2 )
       => ( ( member_option_a @ X3 @ ( image_a_option_a2 @ F2 @ ( image_option_a_a2 @ G @ A2 ) ) )
         => ( ( the_in2538339130118444083tion_a @ A2 @ ( comp_a6249931511552232923tion_a @ F2 @ G ) @ X3 )
            = ( comp_a6249931511552232923tion_a @ ( the_in1757154643552616557on_a_a @ A2 @ G ) @ ( the_in8758012798868597241tion_a @ ( image_option_a_a2 @ G @ A2 ) @ F2 ) @ X3 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1047_map__conv__bind__option,axiom,
    ( map_option_a_a
    = ( ^ [F: a > a,X: option_a] : ( bind_a_a @ X @ ( comp_a_option_a_a @ some_a @ F ) ) ) ) ).

% map_conv_bind_option
thf(fact_1048_comp__the__Some,axiom,
    ( ( comp_o8084098562076169384_a_d_c @ the_Pr6609529173119848253_a_d_c @ some_P377817780860425132_a_d_c )
    = id_Pro4159844622578998816_a_d_c ) ).

% comp_the_Some
thf(fact_1049_comp__the__Some,axiom,
    ( ( comp_option_a_a_a @ the_a @ some_a )
    = id_a ) ).

% comp_the_Some
thf(fact_1050_Image__singleton__iff,axiom,
    ! [B: option2362401199305441953_a_d_c,R2: set_Pr7868159745199425715_a_d_c,A: product_prod_a_d_c] :
      ( ( member1874962706598213560_a_d_c @ B @ ( image_6557094986464633159_a_d_c @ R2 @ ( insert2041379952318371585_a_d_c @ A @ bot_bo5281969120672942491_a_d_c ) ) )
      = ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ A @ B ) @ R2 ) ) ).

% Image_singleton_iff
thf(fact_1051_Image__singleton__iff,axiom,
    ! [B: a,R2: set_Product_prod_a_a,A: a] :
      ( ( member_a @ B @ ( image_a_a @ R2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
      = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 ) ) ).

% Image_singleton_iff
thf(fact_1052_Image__singleton__iff,axiom,
    ! [B: produc6150846815813599699_a_d_c,R2: set_Pr3248570043410859740_a_d_c,A: a] :
      ( ( member2052822272342364412_a_d_c @ B @ ( image_6703077481432038952_a_d_c @ R2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
      = ( member4023273573022285733_a_d_c @ ( produc1782254833265735148_a_d_c @ A @ B ) @ R2 ) ) ).

% Image_singleton_iff
thf(fact_1053_Image__singleton__iff,axiom,
    ! [B: option_a,R2: set_Pr3411724424142761165tion_a,A: a] :
      ( ( member_option_a @ B @ ( image_a_option_a @ R2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
      = ( member6937434987665551382tion_a @ ( produc1224194096085666781tion_a @ A @ B ) @ R2 ) ) ).

% Image_singleton_iff
thf(fact_1054_Image__singleton__iff,axiom,
    ! [B: d > c,R2: set_Pr5506164808955685639_a_d_c,A: a] :
      ( ( member_d_c @ B @ ( image_a_d_c @ R2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
      = ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ A @ B ) @ R2 ) ) ).

% Image_singleton_iff
thf(fact_1055_Image__singleton__iff,axiom,
    ! [B: a,R2: set_Pr6308966090954093121on_a_a,A: option_a] :
      ( ( member_a @ B @ ( image_option_a_a @ R2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) )
      = ( member6056235002698166154on_a_a @ ( produc3446707977624461905on_a_a @ A @ B ) @ R2 ) ) ).

% Image_singleton_iff
thf(fact_1056_Image__singleton__iff,axiom,
    ! [B: produc6150846815813599699_a_d_c,R2: set_Pr8153776533466944534_a_d_c,A: option_a] :
      ( ( member2052822272342364412_a_d_c @ B @ ( image_1093898981003806562_a_d_c @ R2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) )
      = ( member6161829431480814559_a_d_c @ ( produc4753234341022239014_a_d_c @ A @ B ) @ R2 ) ) ).

% Image_singleton_iff
thf(fact_1057_Image__singleton__iff,axiom,
    ! [B: option_a,R2: set_Pr7585778909603769095tion_a,A: option_a] :
      ( ( member_option_a @ B @ ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) )
      = ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 ) ) ).

% Image_singleton_iff
thf(fact_1058_id__apply,axiom,
    ( id_a
    = ( ^ [X: a] : X ) ) ).

% id_apply
thf(fact_1059_image__id,axiom,
    ( ( image_a_a2 @ id_a )
    = id_set_a ) ).

% image_id
thf(fact_1060_ImageI,axiom,
    ! [A: a,B: a,R2: set_Product_prod_a_a,A2: set_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
     => ( ( member_a @ A @ A2 )
       => ( member_a @ B @ ( image_a_a @ R2 @ A2 ) ) ) ) ).

% ImageI
thf(fact_1061_ImageI,axiom,
    ! [A: a,B: option_a,R2: set_Pr3411724424142761165tion_a,A2: set_a] :
      ( ( member6937434987665551382tion_a @ ( produc1224194096085666781tion_a @ A @ B ) @ R2 )
     => ( ( member_a @ A @ A2 )
       => ( member_option_a @ B @ ( image_a_option_a @ R2 @ A2 ) ) ) ) ).

% ImageI
thf(fact_1062_ImageI,axiom,
    ! [A: option_a,B: a,R2: set_Pr6308966090954093121on_a_a,A2: set_option_a] :
      ( ( member6056235002698166154on_a_a @ ( produc3446707977624461905on_a_a @ A @ B ) @ R2 )
     => ( ( member_option_a @ A @ A2 )
       => ( member_a @ B @ ( image_option_a_a @ R2 @ A2 ) ) ) ) ).

% ImageI
thf(fact_1063_ImageI,axiom,
    ! [A: option_a,B: option_a,R2: set_Pr7585778909603769095tion_a,A2: set_option_a] :
      ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 )
     => ( ( member_option_a @ A @ A2 )
       => ( member_option_a @ B @ ( image_4442594622209975379tion_a @ R2 @ A2 ) ) ) ) ).

% ImageI
thf(fact_1064_ImageI,axiom,
    ! [A: a,B: d > c,R2: set_Pr5506164808955685639_a_d_c,A2: set_a] :
      ( ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ A @ B ) @ R2 )
     => ( ( member_a @ A @ A2 )
       => ( member_d_c @ B @ ( image_a_d_c @ R2 @ A2 ) ) ) ) ).

% ImageI
thf(fact_1065_ImageI,axiom,
    ! [A: product_prod_a_d_c,B: option2362401199305441953_a_d_c,R2: set_Pr7868159745199425715_a_d_c,A2: set_Pr5506164808955685639_a_d_c] :
      ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ A @ B ) @ R2 )
     => ( ( member8140300833826434152_a_d_c @ A @ A2 )
       => ( member1874962706598213560_a_d_c @ B @ ( image_6557094986464633159_a_d_c @ R2 @ A2 ) ) ) ) ).

% ImageI
thf(fact_1066_ImageI,axiom,
    ! [A: a,B: produc6150846815813599699_a_d_c,R2: set_Pr3248570043410859740_a_d_c,A2: set_a] :
      ( ( member4023273573022285733_a_d_c @ ( produc1782254833265735148_a_d_c @ A @ B ) @ R2 )
     => ( ( member_a @ A @ A2 )
       => ( member2052822272342364412_a_d_c @ B @ ( image_6703077481432038952_a_d_c @ R2 @ A2 ) ) ) ) ).

% ImageI
thf(fact_1067_ImageI,axiom,
    ! [A: produc6150846815813599699_a_d_c,B: a,R2: set_Pr1516632107838038002_d_c_a,A2: set_Pr7868159745199425715_a_d_c] :
      ( ( member2251515730839929787_d_c_a @ ( produc4774912239102099330_d_c_a @ A @ B ) @ R2 )
     => ( ( member2052822272342364412_a_d_c @ A @ A2 )
       => ( member_a @ B @ ( image_472362850413627326_d_c_a @ R2 @ A2 ) ) ) ) ).

% ImageI
thf(fact_1068_ImageI,axiom,
    ! [A: produc6150846815813599699_a_d_c,B: option_a,R2: set_Pr5439073823713721144tion_a,A2: set_Pr7868159745199425715_a_d_c] :
      ( ( member1057245869843309569tion_a @ ( produc1309352160897273800tion_a @ A @ B ) @ R2 )
     => ( ( member2052822272342364412_a_d_c @ A @ A2 )
       => ( member_option_a @ B @ ( image_6873388837733617156tion_a @ R2 @ A2 ) ) ) ) ).

% ImageI
thf(fact_1069_ImageI,axiom,
    ! [A: option_a,B: produc6150846815813599699_a_d_c,R2: set_Pr8153776533466944534_a_d_c,A2: set_option_a] :
      ( ( member6161829431480814559_a_d_c @ ( produc4753234341022239014_a_d_c @ A @ B ) @ R2 )
     => ( ( member_option_a @ A @ A2 )
       => ( member2052822272342364412_a_d_c @ B @ ( image_1093898981003806562_a_d_c @ R2 @ A2 ) ) ) ) ).

% ImageI
thf(fact_1070_bij__betw__id,axiom,
    ! [A2: set_a] : ( bij_betw_a_a @ id_a @ A2 @ A2 ) ).

% bij_betw_id
thf(fact_1071_vimage__id,axiom,
    ( ( vimage_a_a @ id_a )
    = id_set_a ) ).

% vimage_id
thf(fact_1072_comp__eq__id__dest,axiom,
    ! [A: option_a > a,B: a > option_a,C: a > a,V3: a] :
      ( ( ( comp_option_a_a_a @ A @ B )
        = ( comp_a_a_a @ id_a @ C ) )
     => ( ( A @ ( B @ V3 ) )
        = ( C @ V3 ) ) ) ).

% comp_eq_id_dest
thf(fact_1073_bij__betw__id__iff,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( bij_betw_a_a @ id_a @ A2 @ B2 )
      = ( A2 = B2 ) ) ).

% bij_betw_id_iff
thf(fact_1074_rev__ImageI,axiom,
    ! [A: a,A2: set_a,B: a,R2: set_Product_prod_a_a] :
      ( ( member_a @ A @ A2 )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
       => ( member_a @ B @ ( image_a_a @ R2 @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_1075_rev__ImageI,axiom,
    ! [A: a,A2: set_a,B: option_a,R2: set_Pr3411724424142761165tion_a] :
      ( ( member_a @ A @ A2 )
     => ( ( member6937434987665551382tion_a @ ( produc1224194096085666781tion_a @ A @ B ) @ R2 )
       => ( member_option_a @ B @ ( image_a_option_a @ R2 @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_1076_rev__ImageI,axiom,
    ! [A: option_a,A2: set_option_a,B: a,R2: set_Pr6308966090954093121on_a_a] :
      ( ( member_option_a @ A @ A2 )
     => ( ( member6056235002698166154on_a_a @ ( produc3446707977624461905on_a_a @ A @ B ) @ R2 )
       => ( member_a @ B @ ( image_option_a_a @ R2 @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_1077_rev__ImageI,axiom,
    ! [A: option_a,A2: set_option_a,B: option_a,R2: set_Pr7585778909603769095tion_a] :
      ( ( member_option_a @ A @ A2 )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 )
       => ( member_option_a @ B @ ( image_4442594622209975379tion_a @ R2 @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_1078_rev__ImageI,axiom,
    ! [A: a,A2: set_a,B: d > c,R2: set_Pr5506164808955685639_a_d_c] :
      ( ( member_a @ A @ A2 )
     => ( ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ A @ B ) @ R2 )
       => ( member_d_c @ B @ ( image_a_d_c @ R2 @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_1079_rev__ImageI,axiom,
    ! [A: product_prod_a_d_c,A2: set_Pr5506164808955685639_a_d_c,B: option2362401199305441953_a_d_c,R2: set_Pr7868159745199425715_a_d_c] :
      ( ( member8140300833826434152_a_d_c @ A @ A2 )
     => ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ A @ B ) @ R2 )
       => ( member1874962706598213560_a_d_c @ B @ ( image_6557094986464633159_a_d_c @ R2 @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_1080_rev__ImageI,axiom,
    ! [A: a,A2: set_a,B: produc6150846815813599699_a_d_c,R2: set_Pr3248570043410859740_a_d_c] :
      ( ( member_a @ A @ A2 )
     => ( ( member4023273573022285733_a_d_c @ ( produc1782254833265735148_a_d_c @ A @ B ) @ R2 )
       => ( member2052822272342364412_a_d_c @ B @ ( image_6703077481432038952_a_d_c @ R2 @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_1081_rev__ImageI,axiom,
    ! [A: produc6150846815813599699_a_d_c,A2: set_Pr7868159745199425715_a_d_c,B: a,R2: set_Pr1516632107838038002_d_c_a] :
      ( ( member2052822272342364412_a_d_c @ A @ A2 )
     => ( ( member2251515730839929787_d_c_a @ ( produc4774912239102099330_d_c_a @ A @ B ) @ R2 )
       => ( member_a @ B @ ( image_472362850413627326_d_c_a @ R2 @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_1082_rev__ImageI,axiom,
    ! [A: produc6150846815813599699_a_d_c,A2: set_Pr7868159745199425715_a_d_c,B: option_a,R2: set_Pr5439073823713721144tion_a] :
      ( ( member2052822272342364412_a_d_c @ A @ A2 )
     => ( ( member1057245869843309569tion_a @ ( produc1309352160897273800tion_a @ A @ B ) @ R2 )
       => ( member_option_a @ B @ ( image_6873388837733617156tion_a @ R2 @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_1083_rev__ImageI,axiom,
    ! [A: option_a,A2: set_option_a,B: produc6150846815813599699_a_d_c,R2: set_Pr8153776533466944534_a_d_c] :
      ( ( member_option_a @ A @ A2 )
     => ( ( member6161829431480814559_a_d_c @ ( produc4753234341022239014_a_d_c @ A @ B ) @ R2 )
       => ( member2052822272342364412_a_d_c @ B @ ( image_1093898981003806562_a_d_c @ R2 @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_1084_Image__iff,axiom,
    ! [B: option2362401199305441953_a_d_c,R2: set_Pr7868159745199425715_a_d_c,A2: set_Pr5506164808955685639_a_d_c] :
      ( ( member1874962706598213560_a_d_c @ B @ ( image_6557094986464633159_a_d_c @ R2 @ A2 ) )
      = ( ? [X: product_prod_a_d_c] :
            ( ( member8140300833826434152_a_d_c @ X @ A2 )
            & ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ X @ B ) @ R2 ) ) ) ) ).

% Image_iff
thf(fact_1085_Image__iff,axiom,
    ! [B: d > c,R2: set_Pr5506164808955685639_a_d_c,A2: set_a] :
      ( ( member_d_c @ B @ ( image_a_d_c @ R2 @ A2 ) )
      = ( ? [X: a] :
            ( ( member_a @ X @ A2 )
            & ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ X @ B ) @ R2 ) ) ) ) ).

% Image_iff
thf(fact_1086_ImageE,axiom,
    ! [B: a,R2: set_Product_prod_a_a,A2: set_a] :
      ( ( member_a @ B @ ( image_a_a @ R2 @ A2 ) )
     => ~ ! [X2: a] :
            ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ B ) @ R2 )
           => ~ ( member_a @ X2 @ A2 ) ) ) ).

% ImageE
thf(fact_1087_ImageE,axiom,
    ! [B: a,R2: set_Pr6308966090954093121on_a_a,A2: set_option_a] :
      ( ( member_a @ B @ ( image_option_a_a @ R2 @ A2 ) )
     => ~ ! [X2: option_a] :
            ( ( member6056235002698166154on_a_a @ ( produc3446707977624461905on_a_a @ X2 @ B ) @ R2 )
           => ~ ( member_option_a @ X2 @ A2 ) ) ) ).

% ImageE
thf(fact_1088_ImageE,axiom,
    ! [B: option_a,R2: set_Pr3411724424142761165tion_a,A2: set_a] :
      ( ( member_option_a @ B @ ( image_a_option_a @ R2 @ A2 ) )
     => ~ ! [X2: a] :
            ( ( member6937434987665551382tion_a @ ( produc1224194096085666781tion_a @ X2 @ B ) @ R2 )
           => ~ ( member_a @ X2 @ A2 ) ) ) ).

% ImageE
thf(fact_1089_ImageE,axiom,
    ! [B: option_a,R2: set_Pr7585778909603769095tion_a,A2: set_option_a] :
      ( ( member_option_a @ B @ ( image_4442594622209975379tion_a @ R2 @ A2 ) )
     => ~ ! [X2: option_a] :
            ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ B ) @ R2 )
           => ~ ( member_option_a @ X2 @ A2 ) ) ) ).

% ImageE
thf(fact_1090_ImageE,axiom,
    ! [B: d > c,R2: set_Pr5506164808955685639_a_d_c,A2: set_a] :
      ( ( member_d_c @ B @ ( image_a_d_c @ R2 @ A2 ) )
     => ~ ! [X2: a] :
            ( ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ X2 @ B ) @ R2 )
           => ~ ( member_a @ X2 @ A2 ) ) ) ).

% ImageE
thf(fact_1091_ImageE,axiom,
    ! [B: option2362401199305441953_a_d_c,R2: set_Pr7868159745199425715_a_d_c,A2: set_Pr5506164808955685639_a_d_c] :
      ( ( member1874962706598213560_a_d_c @ B @ ( image_6557094986464633159_a_d_c @ R2 @ A2 ) )
     => ~ ! [X2: product_prod_a_d_c] :
            ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ X2 @ B ) @ R2 )
           => ~ ( member8140300833826434152_a_d_c @ X2 @ A2 ) ) ) ).

% ImageE
thf(fact_1092_ImageE,axiom,
    ! [B: a,R2: set_Pr1516632107838038002_d_c_a,A2: set_Pr7868159745199425715_a_d_c] :
      ( ( member_a @ B @ ( image_472362850413627326_d_c_a @ R2 @ A2 ) )
     => ~ ! [X2: produc6150846815813599699_a_d_c] :
            ( ( member2251515730839929787_d_c_a @ ( produc4774912239102099330_d_c_a @ X2 @ B ) @ R2 )
           => ~ ( member2052822272342364412_a_d_c @ X2 @ A2 ) ) ) ).

% ImageE
thf(fact_1093_ImageE,axiom,
    ! [B: produc6150846815813599699_a_d_c,R2: set_Pr3248570043410859740_a_d_c,A2: set_a] :
      ( ( member2052822272342364412_a_d_c @ B @ ( image_6703077481432038952_a_d_c @ R2 @ A2 ) )
     => ~ ! [X2: a] :
            ( ( member4023273573022285733_a_d_c @ ( produc1782254833265735148_a_d_c @ X2 @ B ) @ R2 )
           => ~ ( member_a @ X2 @ A2 ) ) ) ).

% ImageE
thf(fact_1094_ImageE,axiom,
    ! [B: produc6150846815813599699_a_d_c,R2: set_Pr8153776533466944534_a_d_c,A2: set_option_a] :
      ( ( member2052822272342364412_a_d_c @ B @ ( image_1093898981003806562_a_d_c @ R2 @ A2 ) )
     => ~ ! [X2: option_a] :
            ( ( member6161829431480814559_a_d_c @ ( produc4753234341022239014_a_d_c @ X2 @ B ) @ R2 )
           => ~ ( member_option_a @ X2 @ A2 ) ) ) ).

% ImageE
thf(fact_1095_ImageE,axiom,
    ! [B: option_a,R2: set_Pr5439073823713721144tion_a,A2: set_Pr7868159745199425715_a_d_c] :
      ( ( member_option_a @ B @ ( image_6873388837733617156tion_a @ R2 @ A2 ) )
     => ~ ! [X2: produc6150846815813599699_a_d_c] :
            ( ( member1057245869843309569tion_a @ ( produc1309352160897273800tion_a @ X2 @ B ) @ R2 )
           => ~ ( member2052822272342364412_a_d_c @ X2 @ A2 ) ) ) ).

% ImageE
thf(fact_1096_id__def,axiom,
    ( id_a
    = ( ^ [X: a] : X ) ) ).

% id_def
thf(fact_1097_eq__id__iff,axiom,
    ! [F2: a > a] :
      ( ( ! [X: a] :
            ( ( F2 @ X )
            = X ) )
      = ( F2 = id_a ) ) ).

% eq_id_iff
thf(fact_1098_option_Omap__id0,axiom,
    ( ( map_option_a_a @ id_a )
    = id_option_a ) ).

% option.map_id0
thf(fact_1099_option_Omap__id,axiom,
    ! [T: option_a] :
      ( ( map_option_a_a @ id_a @ T )
      = T ) ).

% option.map_id
thf(fact_1100_inj__on__id,axiom,
    ! [A2: set_a] : ( inj_on_a_a @ id_a @ A2 ) ).

% inj_on_id
thf(fact_1101_surj__id,axiom,
    ( ( image_7439109396645324421tion_a @ id_option_a @ top_top_set_option_a )
    = top_top_set_option_a ) ).

% surj_id
thf(fact_1102_surj__id,axiom,
    ( ( image_a_a2 @ id_a @ top_top_set_a )
    = top_top_set_a ) ).

% surj_id
thf(fact_1103_bij__id,axiom,
    bij_be5431266891817924854tion_a @ id_option_a @ top_top_set_option_a @ top_top_set_option_a ).

% bij_id
thf(fact_1104_bij__id,axiom,
    bij_betw_a_a @ id_a @ top_top_set_a @ top_top_set_a ).

% bij_id
thf(fact_1105_subset__Image1__Image1__iff,axiom,
    ! [R2: set_Pr8390707888732215751_a_d_c,A: produc6150846815813599699_a_d_c,B: produc6150846815813599699_a_d_c] :
      ( ( order_8382585884380629482_a_d_c @ ( field_6537103990374211872_a_d_c @ R2 ) @ R2 )
     => ( ( member2052822272342364412_a_d_c @ A @ ( field_6537103990374211872_a_d_c @ R2 ) )
       => ( ( member2052822272342364412_a_d_c @ B @ ( field_6537103990374211872_a_d_c @ R2 ) )
         => ( ( ord_le3761643085089054803_a_d_c @ ( image_3317614778096461459_a_d_c @ R2 @ ( insert4944988292672327587_a_d_c @ A @ bot_bo1396921257525504799_a_d_c ) ) @ ( image_3317614778096461459_a_d_c @ R2 @ ( insert4944988292672327587_a_d_c @ B @ bot_bo1396921257525504799_a_d_c ) ) )
            = ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ B @ A ) @ R2 ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_1106_subset__Image1__Image1__iff,axiom,
    ! [R2: set_Product_prod_a_a,A: a,B: a] :
      ( ( order_preorder_on_a @ ( field_a @ R2 ) @ R2 )
     => ( ( member_a @ A @ ( field_a @ R2 ) )
       => ( ( member_a @ B @ ( field_a @ R2 ) )
         => ( ( ord_less_eq_set_a @ ( image_a_a @ R2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( image_a_a @ R2 @ ( insert_a @ B @ bot_bot_set_a ) ) )
            = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ A ) @ R2 ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_1107_subset__Image1__Image1__iff,axiom,
    ! [R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
      ( ( order_4134995541221112539tion_a @ ( field_option_a @ R2 ) @ R2 )
     => ( ( member_option_a @ A @ ( field_option_a @ R2 ) )
       => ( ( member_option_a @ B @ ( field_option_a @ R2 ) )
         => ( ( ord_le1955136853071979460tion_a @ ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) @ ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) )
            = ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ A ) @ R2 ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_1108_equiv__class__nondisjoint,axiom,
    ! [A2: set_Pr7868159745199425715_a_d_c,R2: set_Pr8390707888732215751_a_d_c,X3: produc6150846815813599699_a_d_c,A: produc6150846815813599699_a_d_c,B: produc6150846815813599699_a_d_c] :
      ( ( equiv_1650145883741090185_a_d_c @ A2 @ R2 )
     => ( ( member2052822272342364412_a_d_c @ X3 @ ( inf_in5769963971169539745_a_d_c @ ( image_3317614778096461459_a_d_c @ R2 @ ( insert4944988292672327587_a_d_c @ A @ bot_bo1396921257525504799_a_d_c ) ) @ ( image_3317614778096461459_a_d_c @ R2 @ ( insert4944988292672327587_a_d_c @ B @ bot_bo1396921257525504799_a_d_c ) ) ) )
       => ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ A @ B ) @ R2 ) ) ) ).

% equiv_class_nondisjoint
thf(fact_1109_equiv__class__nondisjoint,axiom,
    ! [A2: set_a,R2: set_Product_prod_a_a,X3: a,A: a,B: a] :
      ( ( equiv_equiv_a @ A2 @ R2 )
     => ( ( member_a @ X3 @ ( inf_inf_set_a @ ( image_a_a @ R2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( image_a_a @ R2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 ) ) ) ).

% equiv_class_nondisjoint
thf(fact_1110_equiv__class__nondisjoint,axiom,
    ! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,X3: option_a,A: option_a,B: option_a] :
      ( ( equiv_equiv_option_a @ A2 @ R2 )
     => ( ( member_option_a @ X3 @ ( inf_inf_set_option_a @ ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) @ ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) ) )
       => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 ) ) ) ).

% equiv_class_nondisjoint
thf(fact_1111_equiv__class__eq__iff,axiom,
    ! [A2: set_Pr7868159745199425715_a_d_c,R2: set_Pr8390707888732215751_a_d_c,X3: produc6150846815813599699_a_d_c,Y: produc6150846815813599699_a_d_c] :
      ( ( equiv_1650145883741090185_a_d_c @ A2 @ R2 )
     => ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ X3 @ Y ) @ R2 )
        = ( ( ( image_3317614778096461459_a_d_c @ R2 @ ( insert4944988292672327587_a_d_c @ X3 @ bot_bo1396921257525504799_a_d_c ) )
            = ( image_3317614778096461459_a_d_c @ R2 @ ( insert4944988292672327587_a_d_c @ Y @ bot_bo1396921257525504799_a_d_c ) ) )
          & ( member2052822272342364412_a_d_c @ X3 @ A2 )
          & ( member2052822272342364412_a_d_c @ Y @ A2 ) ) ) ) ).

% equiv_class_eq_iff
thf(fact_1112_equiv__class__eq__iff,axiom,
    ! [A2: set_a,R2: set_Product_prod_a_a,X3: a,Y: a] :
      ( ( equiv_equiv_a @ A2 @ R2 )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R2 )
        = ( ( ( image_a_a @ R2 @ ( insert_a @ X3 @ bot_bot_set_a ) )
            = ( image_a_a @ R2 @ ( insert_a @ Y @ bot_bot_set_a ) ) )
          & ( member_a @ X3 @ A2 )
          & ( member_a @ Y @ A2 ) ) ) ) ).

% equiv_class_eq_iff
thf(fact_1113_equiv__class__eq__iff,axiom,
    ! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,X3: option_a,Y: option_a] :
      ( ( equiv_equiv_option_a @ A2 @ R2 )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y ) @ R2 )
        = ( ( ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ X3 @ bot_bot_set_option_a ) )
            = ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ Y @ bot_bot_set_option_a ) ) )
          & ( member_option_a @ X3 @ A2 )
          & ( member_option_a @ Y @ A2 ) ) ) ) ).

% equiv_class_eq_iff
thf(fact_1114_eq__equiv__class__iff,axiom,
    ! [A2: set_Pr7868159745199425715_a_d_c,R2: set_Pr8390707888732215751_a_d_c,X3: produc6150846815813599699_a_d_c,Y: produc6150846815813599699_a_d_c] :
      ( ( equiv_1650145883741090185_a_d_c @ A2 @ R2 )
     => ( ( member2052822272342364412_a_d_c @ X3 @ A2 )
       => ( ( member2052822272342364412_a_d_c @ Y @ A2 )
         => ( ( ( image_3317614778096461459_a_d_c @ R2 @ ( insert4944988292672327587_a_d_c @ X3 @ bot_bo1396921257525504799_a_d_c ) )
              = ( image_3317614778096461459_a_d_c @ R2 @ ( insert4944988292672327587_a_d_c @ Y @ bot_bo1396921257525504799_a_d_c ) ) )
            = ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ X3 @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff
thf(fact_1115_eq__equiv__class__iff,axiom,
    ! [A2: set_a,R2: set_Product_prod_a_a,X3: a,Y: a] :
      ( ( equiv_equiv_a @ A2 @ R2 )
     => ( ( member_a @ X3 @ A2 )
       => ( ( member_a @ Y @ A2 )
         => ( ( ( image_a_a @ R2 @ ( insert_a @ X3 @ bot_bot_set_a ) )
              = ( image_a_a @ R2 @ ( insert_a @ Y @ bot_bot_set_a ) ) )
            = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff
thf(fact_1116_eq__equiv__class__iff,axiom,
    ! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,X3: option_a,Y: option_a] :
      ( ( equiv_equiv_option_a @ A2 @ R2 )
     => ( ( member_option_a @ X3 @ A2 )
       => ( ( member_option_a @ Y @ A2 )
         => ( ( ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ X3 @ bot_bot_set_option_a ) )
              = ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ Y @ bot_bot_set_option_a ) ) )
            = ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff
thf(fact_1117_equiv__class__eq,axiom,
    ! [A2: set_a,R2: set_Product_prod_a_a,A: a,B: a] :
      ( ( equiv_equiv_a @ A2 @ R2 )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
       => ( ( image_a_a @ R2 @ ( insert_a @ A @ bot_bot_set_a ) )
          = ( image_a_a @ R2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ) ).

% equiv_class_eq
thf(fact_1118_equiv__class__eq,axiom,
    ! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
      ( ( equiv_equiv_option_a @ A2 @ R2 )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 )
       => ( ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) )
          = ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) ) ) ) ).

% equiv_class_eq
thf(fact_1119_eq__equiv__class,axiom,
    ! [R2: set_Pr8390707888732215751_a_d_c,A: produc6150846815813599699_a_d_c,B: produc6150846815813599699_a_d_c,A2: set_Pr7868159745199425715_a_d_c] :
      ( ( ( image_3317614778096461459_a_d_c @ R2 @ ( insert4944988292672327587_a_d_c @ A @ bot_bo1396921257525504799_a_d_c ) )
        = ( image_3317614778096461459_a_d_c @ R2 @ ( insert4944988292672327587_a_d_c @ B @ bot_bo1396921257525504799_a_d_c ) ) )
     => ( ( equiv_1650145883741090185_a_d_c @ A2 @ R2 )
       => ( ( member2052822272342364412_a_d_c @ B @ A2 )
         => ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ A @ B ) @ R2 ) ) ) ) ).

% eq_equiv_class
thf(fact_1120_eq__equiv__class,axiom,
    ! [R2: set_Product_prod_a_a,A: a,B: a,A2: set_a] :
      ( ( ( image_a_a @ R2 @ ( insert_a @ A @ bot_bot_set_a ) )
        = ( image_a_a @ R2 @ ( insert_a @ B @ bot_bot_set_a ) ) )
     => ( ( equiv_equiv_a @ A2 @ R2 )
       => ( ( member_a @ B @ A2 )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 ) ) ) ) ).

% eq_equiv_class
thf(fact_1121_eq__equiv__class,axiom,
    ! [R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a,A2: set_option_a] :
      ( ( ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) )
        = ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) )
     => ( ( equiv_equiv_option_a @ A2 @ R2 )
       => ( ( member_option_a @ B @ A2 )
         => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 ) ) ) ) ).

% eq_equiv_class
thf(fact_1122_subset__equiv__class,axiom,
    ! [A2: set_Pr7868159745199425715_a_d_c,R2: set_Pr8390707888732215751_a_d_c,B: produc6150846815813599699_a_d_c,A: produc6150846815813599699_a_d_c] :
      ( ( equiv_1650145883741090185_a_d_c @ A2 @ R2 )
     => ( ( ord_le3761643085089054803_a_d_c @ ( image_3317614778096461459_a_d_c @ R2 @ ( insert4944988292672327587_a_d_c @ B @ bot_bo1396921257525504799_a_d_c ) ) @ ( image_3317614778096461459_a_d_c @ R2 @ ( insert4944988292672327587_a_d_c @ A @ bot_bo1396921257525504799_a_d_c ) ) )
       => ( ( member2052822272342364412_a_d_c @ B @ A2 )
         => ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ A @ B ) @ R2 ) ) ) ) ).

% subset_equiv_class
thf(fact_1123_subset__equiv__class,axiom,
    ! [A2: set_a,R2: set_Product_prod_a_a,B: a,A: a] :
      ( ( equiv_equiv_a @ A2 @ R2 )
     => ( ( ord_less_eq_set_a @ ( image_a_a @ R2 @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( image_a_a @ R2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
       => ( ( member_a @ B @ A2 )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 ) ) ) ) ).

% subset_equiv_class
thf(fact_1124_subset__equiv__class,axiom,
    ! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,B: option_a,A: option_a] :
      ( ( equiv_equiv_option_a @ A2 @ R2 )
     => ( ( ord_le1955136853071979460tion_a @ ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) )
       => ( ( member_option_a @ B @ A2 )
         => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 ) ) ) ) ).

% subset_equiv_class
thf(fact_1125_equiv__class__subset,axiom,
    ! [A2: set_a,R2: set_Product_prod_a_a,A: a,B: a] :
      ( ( equiv_equiv_a @ A2 @ R2 )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
       => ( ord_less_eq_set_a @ ( image_a_a @ R2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( image_a_a @ R2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ) ).

% equiv_class_subset
thf(fact_1126_equiv__class__subset,axiom,
    ! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
      ( ( equiv_equiv_option_a @ A2 @ R2 )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 )
       => ( ord_le1955136853071979460tion_a @ ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) @ ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) ) ) ) ).

% equiv_class_subset
thf(fact_1127_proj__iff,axiom,
    ! [A2: set_a,R2: set_Product_prod_a_a,X3: a,Y: a] :
      ( ( equiv_equiv_a @ A2 @ R2 )
     => ( ( ord_less_eq_set_a @ ( insert_a @ X3 @ ( insert_a @ Y @ bot_bot_set_a ) ) @ A2 )
       => ( ( ( equiv_proj_a_a @ R2 @ X3 )
            = ( equiv_proj_a_a @ R2 @ Y ) )
          = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R2 ) ) ) ) ).

% proj_iff
thf(fact_1128_proj__iff,axiom,
    ! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,X3: option_a,Y: option_a] :
      ( ( equiv_equiv_option_a @ A2 @ R2 )
     => ( ( ord_le1955136853071979460tion_a @ ( insert_option_a @ X3 @ ( insert_option_a @ Y @ bot_bot_set_option_a ) ) @ A2 )
       => ( ( ( equiv_6865337221296424970tion_a @ R2 @ X3 )
            = ( equiv_6865337221296424970tion_a @ R2 @ Y ) )
          = ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y ) @ R2 ) ) ) ) ).

% proj_iff
thf(fact_1129_disjnt__equiv__class,axiom,
    ! [A2: set_a,R2: set_Product_prod_a_a,A: a,B: a] :
      ( ( equiv_equiv_a @ A2 @ R2 )
     => ( ( disjnt_a @ ( image_a_a @ R2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( image_a_a @ R2 @ ( insert_a @ B @ bot_bot_set_a ) ) )
        = ( ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 ) ) ) ) ).

% disjnt_equiv_class
thf(fact_1130_disjnt__equiv__class,axiom,
    ! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
      ( ( equiv_equiv_option_a @ A2 @ R2 )
     => ( ( disjnt_option_a @ ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) @ ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) )
        = ( ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 ) ) ) ) ).

% disjnt_equiv_class
thf(fact_1131_in__quotient__imp__in__rel,axiom,
    ! [A2: set_a,R2: set_Product_prod_a_a,X6: set_a,X3: a,Y: a] :
      ( ( equiv_equiv_a @ A2 @ R2 )
     => ( ( member_set_a @ X6 @ ( equiv_quotient_a @ A2 @ R2 ) )
       => ( ( ord_less_eq_set_a @ ( insert_a @ X3 @ ( insert_a @ Y @ bot_bot_set_a ) ) @ X6 )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R2 ) ) ) ) ).

% in_quotient_imp_in_rel
thf(fact_1132_in__quotient__imp__in__rel,axiom,
    ! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,X6: set_option_a,X3: option_a,Y: option_a] :
      ( ( equiv_equiv_option_a @ A2 @ R2 )
     => ( ( member_set_option_a @ X6 @ ( equiv_2859340374733651339tion_a @ A2 @ R2 ) )
       => ( ( ord_le1955136853071979460tion_a @ ( insert_option_a @ X3 @ ( insert_option_a @ Y @ bot_bot_set_option_a ) ) @ X6 )
         => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y ) @ R2 ) ) ) ) ).

% in_quotient_imp_in_rel
thf(fact_1133_quotient__eqI,axiom,
    ! [A2: set_a,R2: set_Product_prod_a_a,X6: set_a,Y6: set_a,X3: a,Y: a] :
      ( ( equiv_equiv_a @ A2 @ R2 )
     => ( ( member_set_a @ X6 @ ( equiv_quotient_a @ A2 @ R2 ) )
       => ( ( member_set_a @ Y6 @ ( equiv_quotient_a @ A2 @ R2 ) )
         => ( ( member_a @ X3 @ X6 )
           => ( ( member_a @ Y @ Y6 )
             => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R2 )
               => ( X6 = Y6 ) ) ) ) ) ) ) ).

% quotient_eqI
thf(fact_1134_quotient__eqI,axiom,
    ! [A2: set_Pr7868159745199425715_a_d_c,R2: set_Pr8390707888732215751_a_d_c,X6: set_Pr7868159745199425715_a_d_c,Y6: set_Pr7868159745199425715_a_d_c,X3: produc6150846815813599699_a_d_c,Y: produc6150846815813599699_a_d_c] :
      ( ( equiv_1650145883741090185_a_d_c @ A2 @ R2 )
     => ( ( member7718464485209093084_a_d_c @ X6 @ ( equiv_4186010295663580314_a_d_c @ A2 @ R2 ) )
       => ( ( member7718464485209093084_a_d_c @ Y6 @ ( equiv_4186010295663580314_a_d_c @ A2 @ R2 ) )
         => ( ( member2052822272342364412_a_d_c @ X3 @ X6 )
           => ( ( member2052822272342364412_a_d_c @ Y @ Y6 )
             => ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ X3 @ Y ) @ R2 )
               => ( X6 = Y6 ) ) ) ) ) ) ) ).

% quotient_eqI
thf(fact_1135_quotient__eqI,axiom,
    ! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,X6: set_option_a,Y6: set_option_a,X3: option_a,Y: option_a] :
      ( ( equiv_equiv_option_a @ A2 @ R2 )
     => ( ( member_set_option_a @ X6 @ ( equiv_2859340374733651339tion_a @ A2 @ R2 ) )
       => ( ( member_set_option_a @ Y6 @ ( equiv_2859340374733651339tion_a @ A2 @ R2 ) )
         => ( ( member_option_a @ X3 @ X6 )
           => ( ( member_option_a @ Y @ Y6 )
             => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y ) @ R2 )
               => ( X6 = Y6 ) ) ) ) ) ) ) ).

% quotient_eqI
thf(fact_1136_quotient__eq__iff,axiom,
    ! [A2: set_a,R2: set_Product_prod_a_a,X6: set_a,Y6: set_a,X3: a,Y: a] :
      ( ( equiv_equiv_a @ A2 @ R2 )
     => ( ( member_set_a @ X6 @ ( equiv_quotient_a @ A2 @ R2 ) )
       => ( ( member_set_a @ Y6 @ ( equiv_quotient_a @ A2 @ R2 ) )
         => ( ( member_a @ X3 @ X6 )
           => ( ( member_a @ Y @ Y6 )
             => ( ( X6 = Y6 )
                = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R2 ) ) ) ) ) ) ) ).

% quotient_eq_iff
thf(fact_1137_quotient__eq__iff,axiom,
    ! [A2: set_Pr7868159745199425715_a_d_c,R2: set_Pr8390707888732215751_a_d_c,X6: set_Pr7868159745199425715_a_d_c,Y6: set_Pr7868159745199425715_a_d_c,X3: produc6150846815813599699_a_d_c,Y: produc6150846815813599699_a_d_c] :
      ( ( equiv_1650145883741090185_a_d_c @ A2 @ R2 )
     => ( ( member7718464485209093084_a_d_c @ X6 @ ( equiv_4186010295663580314_a_d_c @ A2 @ R2 ) )
       => ( ( member7718464485209093084_a_d_c @ Y6 @ ( equiv_4186010295663580314_a_d_c @ A2 @ R2 ) )
         => ( ( member2052822272342364412_a_d_c @ X3 @ X6 )
           => ( ( member2052822272342364412_a_d_c @ Y @ Y6 )
             => ( ( X6 = Y6 )
                = ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ X3 @ Y ) @ R2 ) ) ) ) ) ) ) ).

% quotient_eq_iff
thf(fact_1138_quotient__eq__iff,axiom,
    ! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,X6: set_option_a,Y6: set_option_a,X3: option_a,Y: option_a] :
      ( ( equiv_equiv_option_a @ A2 @ R2 )
     => ( ( member_set_option_a @ X6 @ ( equiv_2859340374733651339tion_a @ A2 @ R2 ) )
       => ( ( member_set_option_a @ Y6 @ ( equiv_2859340374733651339tion_a @ A2 @ R2 ) )
         => ( ( member_option_a @ X3 @ X6 )
           => ( ( member_option_a @ Y @ Y6 )
             => ( ( X6 = Y6 )
                = ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y ) @ R2 ) ) ) ) ) ) ) ).

% quotient_eq_iff
thf(fact_1139_in__quotient__imp__closed,axiom,
    ! [A2: set_a,R2: set_Product_prod_a_a,X6: set_a,X3: a,Y: a] :
      ( ( equiv_equiv_a @ A2 @ R2 )
     => ( ( member_set_a @ X6 @ ( equiv_quotient_a @ A2 @ R2 ) )
       => ( ( member_a @ X3 @ X6 )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R2 )
           => ( member_a @ Y @ X6 ) ) ) ) ) ).

% in_quotient_imp_closed
thf(fact_1140_in__quotient__imp__closed,axiom,
    ! [A2: set_Pr7868159745199425715_a_d_c,R2: set_Pr8390707888732215751_a_d_c,X6: set_Pr7868159745199425715_a_d_c,X3: produc6150846815813599699_a_d_c,Y: produc6150846815813599699_a_d_c] :
      ( ( equiv_1650145883741090185_a_d_c @ A2 @ R2 )
     => ( ( member7718464485209093084_a_d_c @ X6 @ ( equiv_4186010295663580314_a_d_c @ A2 @ R2 ) )
       => ( ( member2052822272342364412_a_d_c @ X3 @ X6 )
         => ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ X3 @ Y ) @ R2 )
           => ( member2052822272342364412_a_d_c @ Y @ X6 ) ) ) ) ) ).

% in_quotient_imp_closed
thf(fact_1141_in__quotient__imp__closed,axiom,
    ! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,X6: set_option_a,X3: option_a,Y: option_a] :
      ( ( equiv_equiv_option_a @ A2 @ R2 )
     => ( ( member_set_option_a @ X6 @ ( equiv_2859340374733651339tion_a @ A2 @ R2 ) )
       => ( ( member_option_a @ X3 @ X6 )
         => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y ) @ R2 )
           => ( member_option_a @ Y @ X6 ) ) ) ) ) ).

% in_quotient_imp_closed
thf(fact_1142_eq__equiv__class__iff2,axiom,
    ! [A2: set_Pr7868159745199425715_a_d_c,R2: set_Pr8390707888732215751_a_d_c,X3: produc6150846815813599699_a_d_c,Y: produc6150846815813599699_a_d_c] :
      ( ( equiv_1650145883741090185_a_d_c @ A2 @ R2 )
     => ( ( member2052822272342364412_a_d_c @ X3 @ A2 )
       => ( ( member2052822272342364412_a_d_c @ Y @ A2 )
         => ( ( ( equiv_4186010295663580314_a_d_c @ ( insert4944988292672327587_a_d_c @ X3 @ bot_bo1396921257525504799_a_d_c ) @ R2 )
              = ( equiv_4186010295663580314_a_d_c @ ( insert4944988292672327587_a_d_c @ Y @ bot_bo1396921257525504799_a_d_c ) @ R2 ) )
            = ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ X3 @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff2
thf(fact_1143_eq__equiv__class__iff2,axiom,
    ! [A2: set_a,R2: set_Product_prod_a_a,X3: a,Y: a] :
      ( ( equiv_equiv_a @ A2 @ R2 )
     => ( ( member_a @ X3 @ A2 )
       => ( ( member_a @ Y @ A2 )
         => ( ( ( equiv_quotient_a @ ( insert_a @ X3 @ bot_bot_set_a ) @ R2 )
              = ( equiv_quotient_a @ ( insert_a @ Y @ bot_bot_set_a ) @ R2 ) )
            = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff2
thf(fact_1144_eq__equiv__class__iff2,axiom,
    ! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,X3: option_a,Y: option_a] :
      ( ( equiv_equiv_option_a @ A2 @ R2 )
     => ( ( member_option_a @ X3 @ A2 )
       => ( ( member_option_a @ Y @ A2 )
         => ( ( ( equiv_2859340374733651339tion_a @ ( insert_option_a @ X3 @ bot_bot_set_option_a ) @ R2 )
              = ( equiv_2859340374733651339tion_a @ ( insert_option_a @ Y @ bot_bot_set_option_a ) @ R2 ) )
            = ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff2
thf(fact_1145_option_Osize__gen__o__map,axiom,
    ! [F2: a > nat,G: a > a] :
      ( ( comp_o8583038678572498833tion_a @ ( size_option_a @ F2 ) @ ( map_option_a_a @ G ) )
      = ( size_option_a @ ( comp_a_nat_a @ F2 @ G ) ) ) ).

% option.size_gen_o_map
thf(fact_1146_map__of__Cons__code_I2_J,axiom,
    ! [L: product_prod_a_d_c,K: product_prod_a_d_c,V3: option2362401199305441953_a_d_c,Ps: list_P8522657617046293465_a_d_c] :
      ( ( ( L = K )
       => ( ( map_of2687971689722742103_a_d_c @ ( cons_P5713675596995722633_a_d_c @ ( produc5208860900648697099_a_d_c @ L @ V3 ) @ Ps ) @ K )
          = ( some_o4835035200596650236_a_d_c @ V3 ) ) )
      & ( ( L != K )
       => ( ( map_of2687971689722742103_a_d_c @ ( cons_P5713675596995722633_a_d_c @ ( produc5208860900648697099_a_d_c @ L @ V3 ) @ Ps ) @ K )
          = ( map_of2687971689722742103_a_d_c @ Ps @ K ) ) ) ) ).

% map_of_Cons_code(2)
thf(fact_1147_map__of__Cons__code_I2_J,axiom,
    ! [L: a,K: a,V3: d > c,Ps: list_P7605689430634331233_a_d_c] :
      ( ( ( L = K )
       => ( ( map_of_a_d_c @ ( cons_P982999123993234843_a_d_c @ ( product_Pair_a_d_c @ L @ V3 ) @ Ps ) @ K )
          = ( some_d_c @ V3 ) ) )
      & ( ( L != K )
       => ( ( map_of_a_d_c @ ( cons_P982999123993234843_a_d_c @ ( product_Pair_a_d_c @ L @ V3 ) @ Ps ) @ K )
          = ( map_of_a_d_c @ Ps @ K ) ) ) ) ).

% map_of_Cons_code(2)
thf(fact_1148_single__valuedD,axiom,
    ! [R2: set_Pr7868159745199425715_a_d_c,X3: product_prod_a_d_c,Y: option2362401199305441953_a_d_c,Z2: option2362401199305441953_a_d_c] :
      ( ( single7842414664564753655_a_d_c @ R2 )
     => ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ X3 @ Y ) @ R2 )
       => ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ X3 @ Z2 ) @ R2 )
         => ( Y = Z2 ) ) ) ) ).

% single_valuedD
thf(fact_1149_single__valuedD,axiom,
    ! [R2: set_Pr5506164808955685639_a_d_c,X3: a,Y: d > c,Z2: d > c] :
      ( ( single_valued_a_d_c @ R2 )
     => ( ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ X3 @ Y ) @ R2 )
       => ( ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ X3 @ Z2 ) @ R2 )
         => ( Y = Z2 ) ) ) ) ).

% single_valuedD
thf(fact_1150_single__valuedI,axiom,
    ! [R2: set_Pr7868159745199425715_a_d_c] :
      ( ! [X2: product_prod_a_d_c,Y4: option2362401199305441953_a_d_c,Z3: option2362401199305441953_a_d_c] :
          ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ X2 @ Y4 ) @ R2 )
         => ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ X2 @ Z3 ) @ R2 )
           => ( Y4 = Z3 ) ) )
     => ( single7842414664564753655_a_d_c @ R2 ) ) ).

% single_valuedI
thf(fact_1151_single__valuedI,axiom,
    ! [R2: set_Pr5506164808955685639_a_d_c] :
      ( ! [X2: a,Y4: d > c,Z3: d > c] :
          ( ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ X2 @ Y4 ) @ R2 )
         => ( ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ X2 @ Z3 ) @ R2 )
           => ( Y4 = Z3 ) ) )
     => ( single_valued_a_d_c @ R2 ) ) ).

% single_valuedI
thf(fact_1152_single__valued__def,axiom,
    ( single7842414664564753655_a_d_c
    = ( ^ [R3: set_Pr7868159745199425715_a_d_c] :
        ! [X: product_prod_a_d_c,Y3: option2362401199305441953_a_d_c] :
          ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ X @ Y3 ) @ R3 )
         => ! [Z: option2362401199305441953_a_d_c] :
              ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ X @ Z ) @ R3 )
             => ( Y3 = Z ) ) ) ) ) ).

% single_valued_def
thf(fact_1153_single__valued__def,axiom,
    ( single_valued_a_d_c
    = ( ^ [R3: set_Pr5506164808955685639_a_d_c] :
        ! [X: a,Y3: d > c] :
          ( ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ X @ Y3 ) @ R3 )
         => ! [Z: d > c] :
              ( ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ X @ Z ) @ R3 )
             => ( Y3 = Z ) ) ) ) ) ).

% single_valued_def
thf(fact_1154_map__upds__Cons,axiom,
    ! [M: a > option_a,A: a,As: list_a,B: a,Bs: list_a] :
      ( ( map_upds_a_a @ M @ ( cons_a @ A @ As ) @ ( cons_a @ B @ Bs ) )
      = ( map_upds_a_a @ ( fun_upd_a_option_a @ M @ A @ ( some_a @ B ) ) @ As @ Bs ) ) ).

% map_upds_Cons
thf(fact_1155_map__upds__Cons,axiom,
    ! [M: a > option2362401199305441953_a_d_c,A: a,As: list_a,B: product_prod_a_d_c,Bs: list_P7605689430634331233_a_d_c] :
      ( ( map_up5911110495159793881_a_d_c @ M @ ( cons_a @ A @ As ) @ ( cons_P982999123993234843_a_d_c @ B @ Bs ) )
      = ( map_up5911110495159793881_a_d_c @ ( fun_up2590722781756628834_a_d_c @ M @ A @ ( some_P377817780860425132_a_d_c @ B ) ) @ As @ Bs ) ) ).

% map_upds_Cons
thf(fact_1156_weak__map__of__SomeI,axiom,
    ! [K: product_prod_a_d_c,X3: option2362401199305441953_a_d_c,L: list_P8522657617046293465_a_d_c] :
      ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ K @ X3 ) @ ( set_Pr111223993967229800_a_d_c @ L ) )
     => ? [X2: option2362401199305441953_a_d_c] :
          ( ( map_of2687971689722742103_a_d_c @ L @ K )
          = ( some_o4835035200596650236_a_d_c @ X2 ) ) ) ).

% weak_map_of_SomeI
thf(fact_1157_weak__map__of__SomeI,axiom,
    ! [K: a,X3: d > c,L: list_P7605689430634331233_a_d_c] :
      ( ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ K @ X3 ) @ ( set_Pr1306003423349932796_a_d_c @ L ) )
     => ? [X2: d > c] :
          ( ( map_of_a_d_c @ L @ K )
          = ( some_d_c @ X2 ) ) ) ).

% weak_map_of_SomeI
thf(fact_1158_map__upds__twist,axiom,
    ! [A: produc6150846815813599699_a_d_c,As: list_P8522657617046293465_a_d_c,M: produc6150846815813599699_a_d_c > option_a,B: a,Bs: list_a] :
      ( ~ ( member2052822272342364412_a_d_c @ A @ ( set_Pr111223993967229800_a_d_c @ As ) )
     => ( ( map_up1327547345322571105_d_c_a @ ( fun_up4051926500471473070tion_a @ M @ A @ ( some_a @ B ) ) @ As @ Bs )
        = ( fun_up4051926500471473070tion_a @ ( map_up1327547345322571105_d_c_a @ M @ As @ Bs ) @ A @ ( some_a @ B ) ) ) ) ).

% map_upds_twist
thf(fact_1159_map__upds__twist,axiom,
    ! [A: option_a,As: list_option_a,M: option_a > option_a,B: a,Bs: list_a] :
      ( ~ ( member_option_a @ A @ ( set_option_a2 @ As ) )
     => ( ( map_upds_option_a_a @ ( fun_up1079276522633388797tion_a @ M @ A @ ( some_a @ B ) ) @ As @ Bs )
        = ( fun_up1079276522633388797tion_a @ ( map_upds_option_a_a @ M @ As @ Bs ) @ A @ ( some_a @ B ) ) ) ) ).

% map_upds_twist
thf(fact_1160_map__upds__twist,axiom,
    ! [A: a,As: list_a,M: a > option_a,B: a,Bs: list_a] :
      ( ~ ( member_a @ A @ ( set_a2 @ As ) )
     => ( ( map_upds_a_a @ ( fun_upd_a_option_a @ M @ A @ ( some_a @ B ) ) @ As @ Bs )
        = ( fun_upd_a_option_a @ ( map_upds_a_a @ M @ As @ Bs ) @ A @ ( some_a @ B ) ) ) ) ).

% map_upds_twist
thf(fact_1161_map__upds__twist,axiom,
    ! [A: produc6150846815813599699_a_d_c,As: list_P8522657617046293465_a_d_c,M: produc6150846815813599699_a_d_c > option2362401199305441953_a_d_c,B: product_prod_a_d_c,Bs: list_P7605689430634331233_a_d_c] :
      ( ~ ( member2052822272342364412_a_d_c @ A @ ( set_Pr111223993967229800_a_d_c @ As ) )
     => ( ( map_up3260227830181942446_a_d_c @ ( fun_up4469873178457468855_a_d_c @ M @ A @ ( some_P377817780860425132_a_d_c @ B ) ) @ As @ Bs )
        = ( fun_up4469873178457468855_a_d_c @ ( map_up3260227830181942446_a_d_c @ M @ As @ Bs ) @ A @ ( some_P377817780860425132_a_d_c @ B ) ) ) ) ).

% map_upds_twist
thf(fact_1162_map__upds__twist,axiom,
    ! [A: option_a,As: list_option_a,M: option_a > option2362401199305441953_a_d_c,B: product_prod_a_d_c,Bs: list_P7605689430634331233_a_d_c] :
      ( ~ ( member_option_a @ A @ ( set_option_a2 @ As ) )
     => ( ( map_up1599184479267697567_a_d_c @ ( fun_up1917685652830187432_a_d_c @ M @ A @ ( some_P377817780860425132_a_d_c @ B ) ) @ As @ Bs )
        = ( fun_up1917685652830187432_a_d_c @ ( map_up1599184479267697567_a_d_c @ M @ As @ Bs ) @ A @ ( some_P377817780860425132_a_d_c @ B ) ) ) ) ).

% map_upds_twist
thf(fact_1163_map__upds__twist,axiom,
    ! [A: a,As: list_a,M: a > option2362401199305441953_a_d_c,B: product_prod_a_d_c,Bs: list_P7605689430634331233_a_d_c] :
      ( ~ ( member_a @ A @ ( set_a2 @ As ) )
     => ( ( map_up5911110495159793881_a_d_c @ ( fun_up2590722781756628834_a_d_c @ M @ A @ ( some_P377817780860425132_a_d_c @ B ) ) @ As @ Bs )
        = ( fun_up2590722781756628834_a_d_c @ ( map_up5911110495159793881_a_d_c @ M @ As @ Bs ) @ A @ ( some_P377817780860425132_a_d_c @ B ) ) ) ) ).

% map_upds_twist
thf(fact_1164_map__of__SomeD,axiom,
    ! [Xs: list_P8522657617046293465_a_d_c,K: product_prod_a_d_c,Y: option2362401199305441953_a_d_c] :
      ( ( ( map_of2687971689722742103_a_d_c @ Xs @ K )
        = ( some_o4835035200596650236_a_d_c @ Y ) )
     => ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ K @ Y ) @ ( set_Pr111223993967229800_a_d_c @ Xs ) ) ) ).

% map_of_SomeD
thf(fact_1165_map__of__SomeD,axiom,
    ! [Xs: list_P7605689430634331233_a_d_c,K: a,Y: d > c] :
      ( ( ( map_of_a_d_c @ Xs @ K )
        = ( some_d_c @ Y ) )
     => ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ K @ Y ) @ ( set_Pr1306003423349932796_a_d_c @ Xs ) ) ) ).

% map_of_SomeD
thf(fact_1166_restrict__map__upds,axiom,
    ! [Xs: list_a,Ys: list_a,D2: set_a,M: a > option_a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ D2 )
       => ( ( restrict_map_a_a @ ( map_upds_a_a @ M @ Xs @ Ys ) @ D2 )
          = ( map_upds_a_a @ ( restrict_map_a_a @ M @ ( minus_minus_set_a @ D2 @ ( set_a2 @ Xs ) ) ) @ Xs @ Ys ) ) ) ) ).

% restrict_map_upds
thf(fact_1167_graph__domD,axiom,
    ! [X3: produc6150846815813599699_a_d_c,M: product_prod_a_d_c > option5918686196580481265_a_d_c] :
      ( ( member2052822272342364412_a_d_c @ X3 @ ( graph_4360052036846219122_a_d_c @ M ) )
     => ( member8140300833826434152_a_d_c @ ( produc4568198246480768759_a_d_c @ X3 ) @ ( dom_Pr5389592109506216262_a_d_c @ M ) ) ) ).

% graph_domD
thf(fact_1168_fst__eqD,axiom,
    ! [X3: product_prod_a_d_c,Y: option2362401199305441953_a_d_c,A: product_prod_a_d_c] :
      ( ( ( produc4568198246480768759_a_d_c @ ( produc5208860900648697099_a_d_c @ X3 @ Y ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_1169_fst__eqD,axiom,
    ! [X3: a,Y: d > c,A: a] :
      ( ( ( product_fst_a_d_c @ ( product_Pair_a_d_c @ X3 @ Y ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_1170_fst__conv,axiom,
    ! [X1: product_prod_a_d_c,X22: option2362401199305441953_a_d_c] :
      ( ( produc4568198246480768759_a_d_c @ ( produc5208860900648697099_a_d_c @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_1171_fst__conv,axiom,
    ! [X1: a,X22: d > c] :
      ( ( product_fst_a_d_c @ ( product_Pair_a_d_c @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_1172_map__of__eq__None__iff,axiom,
    ! [Xys: list_P1396940483166286381od_a_a,X3: a] :
      ( ( ( map_of_a_a @ Xys @ X3 )
        = none_a )
      = ( ~ ( member_a @ X3 @ ( image_3437945252899457948_a_a_a @ product_fst_a_a @ ( set_Product_prod_a_a2 @ Xys ) ) ) ) ) ).

% map_of_eq_None_iff
thf(fact_1173_map__of__eq__None__iff,axiom,
    ! [Xys: list_P300064107090267416_d_c_a,X3: produc6150846815813599699_a_d_c] :
      ( ( ( map_of6548332200603098062_d_c_a @ Xys @ X3 )
        = none_a )
      = ( ~ ( member2052822272342364412_a_d_c @ X3 @ ( image_1998283260340132550_a_d_c @ produc4175853562096117102_d_c_a @ ( set_Pr1686552729088582183_d_c_a @ Xys ) ) ) ) ) ).

% map_of_eq_None_iff
thf(fact_1174_map__of__eq__None__iff,axiom,
    ! [Xys: list_P6260409590414597735on_a_a,X3: option_a] :
      ( ( ( map_of_option_a_a @ Xys @ X3 )
        = none_a )
      = ( ~ ( member_option_a @ X3 @ ( image_3098826861768462248tion_a @ produc8941638570267940413on_a_a @ ( set_Pr1233600038994746358on_a_a @ Xys ) ) ) ) ) ).

% map_of_eq_None_iff
thf(fact_1175_eq__fst__iff,axiom,
    ! [A: product_prod_a_d_c,P: produc6150846815813599699_a_d_c] :
      ( ( A
        = ( produc4568198246480768759_a_d_c @ P ) )
      = ( ? [B3: option2362401199305441953_a_d_c] :
            ( P
            = ( produc5208860900648697099_a_d_c @ A @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_1176_eq__fst__iff,axiom,
    ! [A: a,P: product_prod_a_d_c] :
      ( ( A
        = ( product_fst_a_d_c @ P ) )
      = ( ? [B3: d > c] :
            ( P
            = ( product_Pair_a_d_c @ A @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_1177_fstI,axiom,
    ! [X3: produc6150846815813599699_a_d_c,Y: product_prod_a_d_c,Z2: option2362401199305441953_a_d_c] :
      ( ( X3
        = ( produc5208860900648697099_a_d_c @ Y @ Z2 ) )
     => ( ( produc4568198246480768759_a_d_c @ X3 )
        = Y ) ) ).

% fstI
thf(fact_1178_fstI,axiom,
    ! [X3: product_prod_a_d_c,Y: a,Z2: d > c] :
      ( ( X3
        = ( product_Pair_a_d_c @ Y @ Z2 ) )
     => ( ( product_fst_a_d_c @ X3 )
        = Y ) ) ).

% fstI
thf(fact_1179_dom__map__upds,axiom,
    ! [M: a > option_a,Xs: list_a,Ys: list_a] :
      ( ( dom_a_a @ ( map_upds_a_a @ M @ Xs @ Ys ) )
      = ( sup_sup_set_a @ ( set_a2 @ ( take_a @ ( size_size_list_a @ Ys ) @ Xs ) ) @ ( dom_a_a @ M ) ) ) ).

% dom_map_upds
thf(fact_1180_prod_Ocollapse,axiom,
    ! [Prod: produc6150846815813599699_a_d_c] :
      ( ( produc5208860900648697099_a_d_c @ ( produc4568198246480768759_a_d_c @ Prod ) @ ( produc6534629525395632185_a_d_c @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_1181_prod_Ocollapse,axiom,
    ! [Prod: product_prod_a_d_c] :
      ( ( product_Pair_a_d_c @ ( product_fst_a_d_c @ Prod ) @ ( product_snd_a_d_c @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_1182_sndI,axiom,
    ! [X3: produc6150846815813599699_a_d_c,Y: product_prod_a_d_c,Z2: option2362401199305441953_a_d_c] :
      ( ( X3
        = ( produc5208860900648697099_a_d_c @ Y @ Z2 ) )
     => ( ( produc6534629525395632185_a_d_c @ X3 )
        = Z2 ) ) ).

% sndI
thf(fact_1183_sndI,axiom,
    ! [X3: product_prod_a_d_c,Y: a,Z2: d > c] :
      ( ( X3
        = ( product_Pair_a_d_c @ Y @ Z2 ) )
     => ( ( product_snd_a_d_c @ X3 )
        = Z2 ) ) ).

% sndI
thf(fact_1184_eq__snd__iff,axiom,
    ! [B: option2362401199305441953_a_d_c,P: produc6150846815813599699_a_d_c] :
      ( ( B
        = ( produc6534629525395632185_a_d_c @ P ) )
      = ( ? [A3: product_prod_a_d_c] :
            ( P
            = ( produc5208860900648697099_a_d_c @ A3 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_1185_eq__snd__iff,axiom,
    ! [B: d > c,P: product_prod_a_d_c] :
      ( ( B
        = ( product_snd_a_d_c @ P ) )
      = ( ? [A3: a] :
            ( P
            = ( product_Pair_a_d_c @ A3 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_1186_snd__eqD,axiom,
    ! [X3: product_prod_a_d_c,Y: option2362401199305441953_a_d_c,A: option2362401199305441953_a_d_c] :
      ( ( ( produc6534629525395632185_a_d_c @ ( produc5208860900648697099_a_d_c @ X3 @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_1187_snd__eqD,axiom,
    ! [X3: a,Y: d > c,A: d > c] :
      ( ( ( product_snd_a_d_c @ ( product_Pair_a_d_c @ X3 @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_1188_snd__conv,axiom,
    ! [X1: product_prod_a_d_c,X22: option2362401199305441953_a_d_c] :
      ( ( produc6534629525395632185_a_d_c @ ( produc5208860900648697099_a_d_c @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_1189_snd__conv,axiom,
    ! [X1: a,X22: d > c] :
      ( ( product_snd_a_d_c @ ( product_Pair_a_d_c @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_1190_prod_Oexhaust__sel,axiom,
    ! [Prod: produc6150846815813599699_a_d_c] :
      ( Prod
      = ( produc5208860900648697099_a_d_c @ ( produc4568198246480768759_a_d_c @ Prod ) @ ( produc6534629525395632185_a_d_c @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_1191_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_a_d_c] :
      ( Prod
      = ( product_Pair_a_d_c @ ( product_fst_a_d_c @ Prod ) @ ( product_snd_a_d_c @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_1192_surjective__pairing,axiom,
    ! [T: produc6150846815813599699_a_d_c] :
      ( T
      = ( produc5208860900648697099_a_d_c @ ( produc4568198246480768759_a_d_c @ T ) @ ( produc6534629525395632185_a_d_c @ T ) ) ) ).

% surjective_pairing
thf(fact_1193_surjective__pairing,axiom,
    ! [T: product_prod_a_d_c] :
      ( T
      = ( product_Pair_a_d_c @ ( product_fst_a_d_c @ T ) @ ( product_snd_a_d_c @ T ) ) ) ).

% surjective_pairing
thf(fact_1194_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: product_prod_a_d_c > option2362401199305441953_a_d_c > $o,X3: product_prod_a_d_c,Y: option2362401199305441953_a_d_c,A: produc6150846815813599699_a_d_c] :
      ( ( P2 @ X3 @ Y )
     => ( ( A
          = ( produc5208860900648697099_a_d_c @ X3 @ Y ) )
       => ( P2 @ ( produc4568198246480768759_a_d_c @ A ) @ ( produc6534629525395632185_a_d_c @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1195_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: a > ( d > c ) > $o,X3: a,Y: d > c,A: product_prod_a_d_c] :
      ( ( P2 @ X3 @ Y )
     => ( ( A
          = ( product_Pair_a_d_c @ X3 @ Y ) )
       => ( P2 @ ( product_fst_a_d_c @ A ) @ ( product_snd_a_d_c @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1196_graph__ranD,axiom,
    ! [X3: produc6150846815813599699_a_d_c,M: product_prod_a_d_c > option5918686196580481265_a_d_c] :
      ( ( member2052822272342364412_a_d_c @ X3 @ ( graph_4360052036846219122_a_d_c @ M ) )
     => ( member1874962706598213560_a_d_c @ ( produc6534629525395632185_a_d_c @ X3 ) @ ( ran_Pr6010444625066082659_a_d_c @ M ) ) ) ).

% graph_ranD
thf(fact_1197_map__upd__upds__conv__if,axiom,
    ! [X3: produc6150846815813599699_a_d_c,Ys: list_P7605689430634331233_a_d_c,Xs: list_P8522657617046293465_a_d_c,F2: produc6150846815813599699_a_d_c > option2362401199305441953_a_d_c,Y: product_prod_a_d_c] :
      ( ( ( member2052822272342364412_a_d_c @ X3 @ ( set_Pr111223993967229800_a_d_c @ ( take_P186955347922782657_a_d_c @ ( size_s1607940409241614069_a_d_c @ Ys ) @ Xs ) ) )
       => ( ( map_up3260227830181942446_a_d_c @ ( fun_up4469873178457468855_a_d_c @ F2 @ X3 @ ( some_P377817780860425132_a_d_c @ Y ) ) @ Xs @ Ys )
          = ( map_up3260227830181942446_a_d_c @ F2 @ Xs @ Ys ) ) )
      & ( ~ ( member2052822272342364412_a_d_c @ X3 @ ( set_Pr111223993967229800_a_d_c @ ( take_P186955347922782657_a_d_c @ ( size_s1607940409241614069_a_d_c @ Ys ) @ Xs ) ) )
       => ( ( map_up3260227830181942446_a_d_c @ ( fun_up4469873178457468855_a_d_c @ F2 @ X3 @ ( some_P377817780860425132_a_d_c @ Y ) ) @ Xs @ Ys )
          = ( fun_up4469873178457468855_a_d_c @ ( map_up3260227830181942446_a_d_c @ F2 @ Xs @ Ys ) @ X3 @ ( some_P377817780860425132_a_d_c @ Y ) ) ) ) ) ).

% map_upd_upds_conv_if
thf(fact_1198_map__upd__upds__conv__if,axiom,
    ! [X3: option_a,Ys: list_P7605689430634331233_a_d_c,Xs: list_option_a,F2: option_a > option2362401199305441953_a_d_c,Y: product_prod_a_d_c] :
      ( ( ( member_option_a @ X3 @ ( set_option_a2 @ ( take_option_a @ ( size_s1607940409241614069_a_d_c @ Ys ) @ Xs ) ) )
       => ( ( map_up1599184479267697567_a_d_c @ ( fun_up1917685652830187432_a_d_c @ F2 @ X3 @ ( some_P377817780860425132_a_d_c @ Y ) ) @ Xs @ Ys )
          = ( map_up1599184479267697567_a_d_c @ F2 @ Xs @ Ys ) ) )
      & ( ~ ( member_option_a @ X3 @ ( set_option_a2 @ ( take_option_a @ ( size_s1607940409241614069_a_d_c @ Ys ) @ Xs ) ) )
       => ( ( map_up1599184479267697567_a_d_c @ ( fun_up1917685652830187432_a_d_c @ F2 @ X3 @ ( some_P377817780860425132_a_d_c @ Y ) ) @ Xs @ Ys )
          = ( fun_up1917685652830187432_a_d_c @ ( map_up1599184479267697567_a_d_c @ F2 @ Xs @ Ys ) @ X3 @ ( some_P377817780860425132_a_d_c @ Y ) ) ) ) ) ).

% map_upd_upds_conv_if
thf(fact_1199_map__upd__upds__conv__if,axiom,
    ! [X3: a,Ys: list_P7605689430634331233_a_d_c,Xs: list_a,F2: a > option2362401199305441953_a_d_c,Y: product_prod_a_d_c] :
      ( ( ( member_a @ X3 @ ( set_a2 @ ( take_a @ ( size_s1607940409241614069_a_d_c @ Ys ) @ Xs ) ) )
       => ( ( map_up5911110495159793881_a_d_c @ ( fun_up2590722781756628834_a_d_c @ F2 @ X3 @ ( some_P377817780860425132_a_d_c @ Y ) ) @ Xs @ Ys )
          = ( map_up5911110495159793881_a_d_c @ F2 @ Xs @ Ys ) ) )
      & ( ~ ( member_a @ X3 @ ( set_a2 @ ( take_a @ ( size_s1607940409241614069_a_d_c @ Ys ) @ Xs ) ) )
       => ( ( map_up5911110495159793881_a_d_c @ ( fun_up2590722781756628834_a_d_c @ F2 @ X3 @ ( some_P377817780860425132_a_d_c @ Y ) ) @ Xs @ Ys )
          = ( fun_up2590722781756628834_a_d_c @ ( map_up5911110495159793881_a_d_c @ F2 @ Xs @ Ys ) @ X3 @ ( some_P377817780860425132_a_d_c @ Y ) ) ) ) ) ).

% map_upd_upds_conv_if
thf(fact_1200_map__upd__upds__conv__if,axiom,
    ! [X3: produc6150846815813599699_a_d_c,Ys: list_a,Xs: list_P8522657617046293465_a_d_c,F2: produc6150846815813599699_a_d_c > option_a,Y: a] :
      ( ( ( member2052822272342364412_a_d_c @ X3 @ ( set_Pr111223993967229800_a_d_c @ ( take_P186955347922782657_a_d_c @ ( size_size_list_a @ Ys ) @ Xs ) ) )
       => ( ( map_up1327547345322571105_d_c_a @ ( fun_up4051926500471473070tion_a @ F2 @ X3 @ ( some_a @ Y ) ) @ Xs @ Ys )
          = ( map_up1327547345322571105_d_c_a @ F2 @ Xs @ Ys ) ) )
      & ( ~ ( member2052822272342364412_a_d_c @ X3 @ ( set_Pr111223993967229800_a_d_c @ ( take_P186955347922782657_a_d_c @ ( size_size_list_a @ Ys ) @ Xs ) ) )
       => ( ( map_up1327547345322571105_d_c_a @ ( fun_up4051926500471473070tion_a @ F2 @ X3 @ ( some_a @ Y ) ) @ Xs @ Ys )
          = ( fun_up4051926500471473070tion_a @ ( map_up1327547345322571105_d_c_a @ F2 @ Xs @ Ys ) @ X3 @ ( some_a @ Y ) ) ) ) ) ).

% map_upd_upds_conv_if
thf(fact_1201_map__upd__upds__conv__if,axiom,
    ! [X3: option_a,Ys: list_a,Xs: list_option_a,F2: option_a > option_a,Y: a] :
      ( ( ( member_option_a @ X3 @ ( set_option_a2 @ ( take_option_a @ ( size_size_list_a @ Ys ) @ Xs ) ) )
       => ( ( map_upds_option_a_a @ ( fun_up1079276522633388797tion_a @ F2 @ X3 @ ( some_a @ Y ) ) @ Xs @ Ys )
          = ( map_upds_option_a_a @ F2 @ Xs @ Ys ) ) )
      & ( ~ ( member_option_a @ X3 @ ( set_option_a2 @ ( take_option_a @ ( size_size_list_a @ Ys ) @ Xs ) ) )
       => ( ( map_upds_option_a_a @ ( fun_up1079276522633388797tion_a @ F2 @ X3 @ ( some_a @ Y ) ) @ Xs @ Ys )
          = ( fun_up1079276522633388797tion_a @ ( map_upds_option_a_a @ F2 @ Xs @ Ys ) @ X3 @ ( some_a @ Y ) ) ) ) ) ).

% map_upd_upds_conv_if
thf(fact_1202_map__upd__upds__conv__if,axiom,
    ! [X3: a,Ys: list_a,Xs: list_a,F2: a > option_a,Y: a] :
      ( ( ( member_a @ X3 @ ( set_a2 @ ( take_a @ ( size_size_list_a @ Ys ) @ Xs ) ) )
       => ( ( map_upds_a_a @ ( fun_upd_a_option_a @ F2 @ X3 @ ( some_a @ Y ) ) @ Xs @ Ys )
          = ( map_upds_a_a @ F2 @ Xs @ Ys ) ) )
      & ( ~ ( member_a @ X3 @ ( set_a2 @ ( take_a @ ( size_size_list_a @ Ys ) @ Xs ) ) )
       => ( ( map_upds_a_a @ ( fun_upd_a_option_a @ F2 @ X3 @ ( some_a @ Y ) ) @ Xs @ Ys )
          = ( fun_upd_a_option_a @ ( map_upds_a_a @ F2 @ Xs @ Ys ) @ X3 @ ( some_a @ Y ) ) ) ) ) ).

% map_upd_upds_conv_if
thf(fact_1203_exI__realizer,axiom,
    ! [P2: option2362401199305441953_a_d_c > product_prod_a_d_c > $o,Y: option2362401199305441953_a_d_c,X3: product_prod_a_d_c] :
      ( ( P2 @ Y @ X3 )
     => ( P2 @ ( produc6534629525395632185_a_d_c @ ( produc5208860900648697099_a_d_c @ X3 @ Y ) ) @ ( produc4568198246480768759_a_d_c @ ( produc5208860900648697099_a_d_c @ X3 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_1204_exI__realizer,axiom,
    ! [P2: ( d > c ) > a > $o,Y: d > c,X3: a] :
      ( ( P2 @ Y @ X3 )
     => ( P2 @ ( product_snd_a_d_c @ ( product_Pair_a_d_c @ X3 @ Y ) ) @ ( product_fst_a_d_c @ ( product_Pair_a_d_c @ X3 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_1205_conjI__realizer,axiom,
    ! [P2: product_prod_a_d_c > $o,P: product_prod_a_d_c,Q: option2362401199305441953_a_d_c > $o,Q2: option2362401199305441953_a_d_c] :
      ( ( P2 @ P )
     => ( ( Q @ Q2 )
       => ( ( P2 @ ( produc4568198246480768759_a_d_c @ ( produc5208860900648697099_a_d_c @ P @ Q2 ) ) )
          & ( Q @ ( produc6534629525395632185_a_d_c @ ( produc5208860900648697099_a_d_c @ P @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_1206_conjI__realizer,axiom,
    ! [P2: a > $o,P: a,Q: ( d > c ) > $o,Q2: d > c] :
      ( ( P2 @ P )
     => ( ( Q @ Q2 )
       => ( ( P2 @ ( product_fst_a_d_c @ ( product_Pair_a_d_c @ P @ Q2 ) ) )
          & ( Q @ ( product_snd_a_d_c @ ( product_Pair_a_d_c @ P @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_1207_prod_Oswap__def,axiom,
    ( produc3400401144285396395_a_d_c
    = ( ^ [P3: produc3649636187029808851_a_d_c] : ( produc5208860900648697099_a_d_c @ ( produc5033949577835205433_a_d_c @ P3 ) @ ( produc3067518298920342007_a_d_c @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_1208_prod_Oswap__def,axiom,
    ( product_swap_d_c_a
    = ( ^ [P3: product_prod_d_c_a] : ( product_Pair_a_d_c @ ( product_snd_d_c_a @ P3 ) @ ( product_fst_d_c_a @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_1209_map__of__zip__upd,axiom,
    ! [Ys: list_P7605689430634331233_a_d_c,Xs: list_P8522657617046293465_a_d_c,Zs: list_P7605689430634331233_a_d_c,X3: produc6150846815813599699_a_d_c,Y: product_prod_a_d_c,Z2: product_prod_a_d_c] :
      ( ( ( size_s1607940409241614069_a_d_c @ Ys )
        = ( size_s4922472463745700421_a_d_c @ Xs ) )
     => ( ( ( size_s1607940409241614069_a_d_c @ Zs )
          = ( size_s4922472463745700421_a_d_c @ Xs ) )
       => ( ~ ( member2052822272342364412_a_d_c @ X3 @ ( set_Pr111223993967229800_a_d_c @ Xs ) )
         => ( ( ( fun_up4469873178457468855_a_d_c @ ( map_of1408915295682243585_a_d_c @ ( zip_Pr2959984412008898605_a_d_c @ Xs @ Ys ) ) @ X3 @ ( some_P377817780860425132_a_d_c @ Y ) )
              = ( fun_up4469873178457468855_a_d_c @ ( map_of1408915295682243585_a_d_c @ ( zip_Pr2959984412008898605_a_d_c @ Xs @ Zs ) ) @ X3 @ ( some_P377817780860425132_a_d_c @ Z2 ) ) )
           => ( ( map_of1408915295682243585_a_d_c @ ( zip_Pr2959984412008898605_a_d_c @ Xs @ Ys ) )
              = ( map_of1408915295682243585_a_d_c @ ( zip_Pr2959984412008898605_a_d_c @ Xs @ Zs ) ) ) ) ) ) ) ).

% map_of_zip_upd
thf(fact_1210_map__of__zip__upd,axiom,
    ! [Ys: list_P7605689430634331233_a_d_c,Xs: list_option_a,Zs: list_P7605689430634331233_a_d_c,X3: option_a,Y: product_prod_a_d_c,Z2: product_prod_a_d_c] :
      ( ( ( size_s1607940409241614069_a_d_c @ Ys )
        = ( size_s3078493964004954806tion_a @ Xs ) )
     => ( ( ( size_s1607940409241614069_a_d_c @ Zs )
          = ( size_s3078493964004954806tion_a @ Xs ) )
       => ( ~ ( member_option_a @ X3 @ ( set_option_a2 @ Xs ) )
         => ( ( ( fun_up1917685652830187432_a_d_c @ ( map_of3662478155680615026_a_d_c @ ( zip_op5654269838831021214_a_d_c @ Xs @ Ys ) ) @ X3 @ ( some_P377817780860425132_a_d_c @ Y ) )
              = ( fun_up1917685652830187432_a_d_c @ ( map_of3662478155680615026_a_d_c @ ( zip_op5654269838831021214_a_d_c @ Xs @ Zs ) ) @ X3 @ ( some_P377817780860425132_a_d_c @ Z2 ) ) )
           => ( ( map_of3662478155680615026_a_d_c @ ( zip_op5654269838831021214_a_d_c @ Xs @ Ys ) )
              = ( map_of3662478155680615026_a_d_c @ ( zip_op5654269838831021214_a_d_c @ Xs @ Zs ) ) ) ) ) ) ) ).

% map_of_zip_upd
thf(fact_1211_map__of__zip__upd,axiom,
    ! [Ys: list_P7605689430634331233_a_d_c,Xs: list_a,Zs: list_P7605689430634331233_a_d_c,X3: a,Y: product_prod_a_d_c,Z2: product_prod_a_d_c] :
      ( ( ( size_s1607940409241614069_a_d_c @ Ys )
        = ( size_size_list_a @ Xs ) )
     => ( ( ( size_s1607940409241614069_a_d_c @ Zs )
          = ( size_size_list_a @ Xs ) )
       => ( ~ ( member_a @ X3 @ ( set_a2 @ Xs ) )
         => ( ( ( fun_up2590722781756628834_a_d_c @ ( map_of6345765247428855852_a_d_c @ ( zip_a_594264266538851416_a_d_c @ Xs @ Ys ) ) @ X3 @ ( some_P377817780860425132_a_d_c @ Y ) )
              = ( fun_up2590722781756628834_a_d_c @ ( map_of6345765247428855852_a_d_c @ ( zip_a_594264266538851416_a_d_c @ Xs @ Zs ) ) @ X3 @ ( some_P377817780860425132_a_d_c @ Z2 ) ) )
           => ( ( map_of6345765247428855852_a_d_c @ ( zip_a_594264266538851416_a_d_c @ Xs @ Ys ) )
              = ( map_of6345765247428855852_a_d_c @ ( zip_a_594264266538851416_a_d_c @ Xs @ Zs ) ) ) ) ) ) ) ).

% map_of_zip_upd
thf(fact_1212_map__of__zip__upd,axiom,
    ! [Ys: list_a,Xs: list_P8522657617046293465_a_d_c,Zs: list_a,X3: produc6150846815813599699_a_d_c,Y: a,Z2: a] :
      ( ( ( size_size_list_a @ Ys )
        = ( size_s4922472463745700421_a_d_c @ Xs ) )
     => ( ( ( size_size_list_a @ Zs )
          = ( size_s4922472463745700421_a_d_c @ Xs ) )
       => ( ~ ( member2052822272342364412_a_d_c @ X3 @ ( set_Pr111223993967229800_a_d_c @ Xs ) )
         => ( ( ( fun_up4051926500471473070tion_a @ ( map_of6548332200603098062_d_c_a @ ( zip_Pr7553708875772114594_d_c_a @ Xs @ Ys ) ) @ X3 @ ( some_a @ Y ) )
              = ( fun_up4051926500471473070tion_a @ ( map_of6548332200603098062_d_c_a @ ( zip_Pr7553708875772114594_d_c_a @ Xs @ Zs ) ) @ X3 @ ( some_a @ Z2 ) ) )
           => ( ( map_of6548332200603098062_d_c_a @ ( zip_Pr7553708875772114594_d_c_a @ Xs @ Ys ) )
              = ( map_of6548332200603098062_d_c_a @ ( zip_Pr7553708875772114594_d_c_a @ Xs @ Zs ) ) ) ) ) ) ) ).

% map_of_zip_upd
thf(fact_1213_map__of__zip__upd,axiom,
    ! [Ys: list_a,Xs: list_option_a,Zs: list_a,X3: option_a,Y: a,Z2: a] :
      ( ( ( size_size_list_a @ Ys )
        = ( size_s3078493964004954806tion_a @ Xs ) )
     => ( ( ( size_size_list_a @ Zs )
          = ( size_s3078493964004954806tion_a @ Xs ) )
       => ( ~ ( member_option_a @ X3 @ ( set_option_a2 @ Xs ) )
         => ( ( ( fun_up1079276522633388797tion_a @ ( map_of_option_a_a @ ( zip_option_a_a @ Xs @ Ys ) ) @ X3 @ ( some_a @ Y ) )
              = ( fun_up1079276522633388797tion_a @ ( map_of_option_a_a @ ( zip_option_a_a @ Xs @ Zs ) ) @ X3 @ ( some_a @ Z2 ) ) )
           => ( ( map_of_option_a_a @ ( zip_option_a_a @ Xs @ Ys ) )
              = ( map_of_option_a_a @ ( zip_option_a_a @ Xs @ Zs ) ) ) ) ) ) ) ).

% map_of_zip_upd
thf(fact_1214_map__of__zip__upd,axiom,
    ! [Ys: list_a,Xs: list_a,Zs: list_a,X3: a,Y: a,Z2: a] :
      ( ( ( size_size_list_a @ Ys )
        = ( size_size_list_a @ Xs ) )
     => ( ( ( size_size_list_a @ Zs )
          = ( size_size_list_a @ Xs ) )
       => ( ~ ( member_a @ X3 @ ( set_a2 @ Xs ) )
         => ( ( ( fun_upd_a_option_a @ ( map_of_a_a @ ( zip_a_a @ Xs @ Ys ) ) @ X3 @ ( some_a @ Y ) )
              = ( fun_upd_a_option_a @ ( map_of_a_a @ ( zip_a_a @ Xs @ Zs ) ) @ X3 @ ( some_a @ Z2 ) ) )
           => ( ( map_of_a_a @ ( zip_a_a @ Xs @ Ys ) )
              = ( map_of_a_a @ ( zip_a_a @ Xs @ Zs ) ) ) ) ) ) ) ).

% map_of_zip_upd
thf(fact_1215_pair__in__swap__image,axiom,
    ! [Y: option2362401199305441953_a_d_c,X3: product_prod_a_d_c,A2: set_Pr7868159745199425715_a_d_c] :
      ( ( member8774983680413349372_a_d_c @ ( produc3708180953088270347_a_d_c @ Y @ X3 ) @ ( image_7270912687184270277_a_d_c @ produc4901081091845823147_a_d_c @ A2 ) )
      = ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ X3 @ Y ) @ A2 ) ) ).

% pair_in_swap_image
thf(fact_1216_pair__in__swap__image,axiom,
    ! [Y: d > c,X3: a,A2: set_Pr5506164808955685639_a_d_c] :
      ( ( member3659673801472585190_d_c_a @ ( product_Pair_d_c_a @ Y @ X3 ) @ ( image_3421666265304065831_d_c_a @ product_swap_a_d_c @ A2 ) )
      = ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ X3 @ Y ) @ A2 ) ) ).

% pair_in_swap_image
thf(fact_1217_pair__in__swap__image,axiom,
    ! [Y: product_prod_a_d_c,X3: option2362401199305441953_a_d_c,A2: set_Pr7666514177632660915_a_d_c] :
      ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ Y @ X3 ) @ ( image_6522827584173558213_a_d_c @ produc3400401144285396395_a_d_c @ A2 ) )
      = ( member8774983680413349372_a_d_c @ ( produc3708180953088270347_a_d_c @ X3 @ Y ) @ A2 ) ) ).

% pair_in_swap_image
thf(fact_1218_pair__in__swap__image,axiom,
    ! [Y: a,X3: d > c,A2: set_Pr6048285399905979781_d_c_a] :
      ( ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ Y @ X3 ) @ ( image_8905139229092693287_a_d_c @ product_swap_d_c_a @ A2 ) )
      = ( member3659673801472585190_d_c_a @ ( product_Pair_d_c_a @ X3 @ Y ) @ A2 ) ) ).

% pair_in_swap_image
thf(fact_1219_swap__simp,axiom,
    ! [X3: option2362401199305441953_a_d_c,Y: product_prod_a_d_c] :
      ( ( produc3400401144285396395_a_d_c @ ( produc3708180953088270347_a_d_c @ X3 @ Y ) )
      = ( produc5208860900648697099_a_d_c @ Y @ X3 ) ) ).

% swap_simp
thf(fact_1220_swap__simp,axiom,
    ! [X3: d > c,Y: a] :
      ( ( product_swap_d_c_a @ ( product_Pair_d_c_a @ X3 @ Y ) )
      = ( product_Pair_a_d_c @ Y @ X3 ) ) ).

% swap_simp
thf(fact_1221_swap__simp,axiom,
    ! [X3: product_prod_a_d_c,Y: option2362401199305441953_a_d_c] :
      ( ( produc4901081091845823147_a_d_c @ ( produc5208860900648697099_a_d_c @ X3 @ Y ) )
      = ( produc3708180953088270347_a_d_c @ Y @ X3 ) ) ).

% swap_simp
thf(fact_1222_swap__simp,axiom,
    ! [X3: a,Y: d > c] :
      ( ( product_swap_a_d_c @ ( product_Pair_a_d_c @ X3 @ Y ) )
      = ( product_Pair_d_c_a @ Y @ X3 ) ) ).

% swap_simp
thf(fact_1223_map__of__zip__is__None,axiom,
    ! [Xs: list_P8522657617046293465_a_d_c,Ys: list_a,X3: produc6150846815813599699_a_d_c] :
      ( ( ( size_s4922472463745700421_a_d_c @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( ( map_of6548332200603098062_d_c_a @ ( zip_Pr7553708875772114594_d_c_a @ Xs @ Ys ) @ X3 )
          = none_a )
        = ( ~ ( member2052822272342364412_a_d_c @ X3 @ ( set_Pr111223993967229800_a_d_c @ Xs ) ) ) ) ) ).

% map_of_zip_is_None
thf(fact_1224_map__of__zip__is__None,axiom,
    ! [Xs: list_option_a,Ys: list_a,X3: option_a] :
      ( ( ( size_s3078493964004954806tion_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( ( map_of_option_a_a @ ( zip_option_a_a @ Xs @ Ys ) @ X3 )
          = none_a )
        = ( ~ ( member_option_a @ X3 @ ( set_option_a2 @ Xs ) ) ) ) ) ).

% map_of_zip_is_None
thf(fact_1225_map__of__zip__is__None,axiom,
    ! [Xs: list_a,Ys: list_a,X3: a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( ( map_of_a_a @ ( zip_a_a @ Xs @ Ys ) @ X3 )
          = none_a )
        = ( ~ ( member_a @ X3 @ ( set_a2 @ Xs ) ) ) ) ) ).

% map_of_zip_is_None
thf(fact_1226_dom__map__of__zip,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( dom_a_a @ ( map_of_a_a @ ( zip_a_a @ Xs @ Ys ) ) )
        = ( set_a2 @ Xs ) ) ) ).

% dom_map_of_zip
thf(fact_1227_map__of__zip__is__Some,axiom,
    ! [Xs: list_P8522657617046293465_a_d_c,Ys: list_P7605689430634331233_a_d_c,X3: produc6150846815813599699_a_d_c] :
      ( ( ( size_s4922472463745700421_a_d_c @ Xs )
        = ( size_s1607940409241614069_a_d_c @ Ys ) )
     => ( ( member2052822272342364412_a_d_c @ X3 @ ( set_Pr111223993967229800_a_d_c @ Xs ) )
        = ( ? [Y3: product_prod_a_d_c] :
              ( ( map_of1408915295682243585_a_d_c @ ( zip_Pr2959984412008898605_a_d_c @ Xs @ Ys ) @ X3 )
              = ( some_P377817780860425132_a_d_c @ Y3 ) ) ) ) ) ).

% map_of_zip_is_Some
thf(fact_1228_map__of__zip__is__Some,axiom,
    ! [Xs: list_option_a,Ys: list_P7605689430634331233_a_d_c,X3: option_a] :
      ( ( ( size_s3078493964004954806tion_a @ Xs )
        = ( size_s1607940409241614069_a_d_c @ Ys ) )
     => ( ( member_option_a @ X3 @ ( set_option_a2 @ Xs ) )
        = ( ? [Y3: product_prod_a_d_c] :
              ( ( map_of3662478155680615026_a_d_c @ ( zip_op5654269838831021214_a_d_c @ Xs @ Ys ) @ X3 )
              = ( some_P377817780860425132_a_d_c @ Y3 ) ) ) ) ) ).

% map_of_zip_is_Some
thf(fact_1229_map__of__zip__is__Some,axiom,
    ! [Xs: list_P8522657617046293465_a_d_c,Ys: list_a,X3: produc6150846815813599699_a_d_c] :
      ( ( ( size_s4922472463745700421_a_d_c @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member2052822272342364412_a_d_c @ X3 @ ( set_Pr111223993967229800_a_d_c @ Xs ) )
        = ( ? [Y3: a] :
              ( ( map_of6548332200603098062_d_c_a @ ( zip_Pr7553708875772114594_d_c_a @ Xs @ Ys ) @ X3 )
              = ( some_a @ Y3 ) ) ) ) ) ).

% map_of_zip_is_Some
thf(fact_1230_map__of__zip__is__Some,axiom,
    ! [Xs: list_option_a,Ys: list_a,X3: option_a] :
      ( ( ( size_s3078493964004954806tion_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member_option_a @ X3 @ ( set_option_a2 @ Xs ) )
        = ( ? [Y3: a] :
              ( ( map_of_option_a_a @ ( zip_option_a_a @ Xs @ Ys ) @ X3 )
              = ( some_a @ Y3 ) ) ) ) ) ).

% map_of_zip_is_Some
thf(fact_1231_map__of__zip__is__Some,axiom,
    ! [Xs: list_a,Ys: list_P7605689430634331233_a_d_c,X3: a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_s1607940409241614069_a_d_c @ Ys ) )
     => ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
        = ( ? [Y3: product_prod_a_d_c] :
              ( ( map_of6345765247428855852_a_d_c @ ( zip_a_594264266538851416_a_d_c @ Xs @ Ys ) @ X3 )
              = ( some_P377817780860425132_a_d_c @ Y3 ) ) ) ) ) ).

% map_of_zip_is_Some
thf(fact_1232_map__of__zip__is__Some,axiom,
    ! [Xs: list_a,Ys: list_a,X3: a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
        = ( ? [Y3: a] :
              ( ( map_of_a_a @ ( zip_a_a @ Xs @ Ys ) @ X3 )
              = ( some_a @ Y3 ) ) ) ) ) ).

% map_of_zip_is_Some
thf(fact_1233_zip__Cons__Cons,axiom,
    ! [X3: a,Xs: list_a,Y: a,Ys: list_a] :
      ( ( zip_a_a @ ( cons_a @ X3 @ Xs ) @ ( cons_a @ Y @ Ys ) )
      = ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ ( zip_a_a @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_1234_zip__Cons__Cons,axiom,
    ! [X3: product_prod_a_d_c,Xs: list_P7605689430634331233_a_d_c,Y: option2362401199305441953_a_d_c,Ys: list_o3221539067586843313_a_d_c] :
      ( ( zip_Pr6767014853595796523_a_d_c @ ( cons_P982999123993234843_a_d_c @ X3 @ Xs ) @ ( cons_o7888461836264444011_a_d_c @ Y @ Ys ) )
      = ( cons_P5713675596995722633_a_d_c @ ( produc5208860900648697099_a_d_c @ X3 @ Y ) @ ( zip_Pr6767014853595796523_a_d_c @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_1235_zip__Cons__Cons,axiom,
    ! [X3: a,Xs: list_a,Y: d > c,Ys: list_d_c] :
      ( ( zip_a_d_c @ ( cons_a @ X3 @ Xs ) @ ( cons_d_c @ Y @ Ys ) )
      = ( cons_P982999123993234843_a_d_c @ ( product_Pair_a_d_c @ X3 @ Y ) @ ( zip_a_d_c @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_1236_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_P7605689430634331233_a_d_c,Ys: list_o3221539067586843313_a_d_c,X3: product_prod_a_d_c] :
      ( ( ( size_s1607940409241614069_a_d_c @ Xs )
        = ( size_s3965190101641584453_a_d_c @ Ys ) )
     => ( ( member8140300833826434152_a_d_c @ X3 @ ( set_Pr1306003423349932796_a_d_c @ Xs ) )
       => ~ ! [Y4: option2362401199305441953_a_d_c] :
              ~ ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ X3 @ Y4 ) @ ( set_Pr111223993967229800_a_d_c @ ( zip_Pr6767014853595796523_a_d_c @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1237_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_P8522657617046293465_a_d_c,Ys: list_a,X3: produc6150846815813599699_a_d_c] :
      ( ( ( size_s4922472463745700421_a_d_c @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member2052822272342364412_a_d_c @ X3 @ ( set_Pr111223993967229800_a_d_c @ Xs ) )
       => ~ ! [Y4: a] :
              ~ ( member2251515730839929787_d_c_a @ ( produc4774912239102099330_d_c_a @ X3 @ Y4 ) @ ( set_Pr1686552729088582183_d_c_a @ ( zip_Pr7553708875772114594_d_c_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1238_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_option_a,Ys: list_a,X3: option_a] :
      ( ( ( size_s3078493964004954806tion_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member_option_a @ X3 @ ( set_option_a2 @ Xs ) )
       => ~ ! [Y4: a] :
              ~ ( member6056235002698166154on_a_a @ ( produc3446707977624461905on_a_a @ X3 @ Y4 ) @ ( set_Pr1233600038994746358on_a_a @ ( zip_option_a_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1239_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_a,Ys: list_d_c,X3: a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_d_c @ Ys ) )
     => ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
       => ~ ! [Y4: d > c] :
              ~ ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ X3 @ Y4 ) @ ( set_Pr1306003423349932796_a_d_c @ ( zip_a_d_c @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1240_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_a,Ys: list_a,X3: a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
       => ~ ! [Y4: a] :
              ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1241_zip__same,axiom,
    ! [A: produc6150846815813599699_a_d_c,B: produc6150846815813599699_a_d_c,Xs: list_P8522657617046293465_a_d_c] :
      ( ( member3575411683413719056_a_d_c @ ( produc1995845767055916631_a_d_c @ A @ B ) @ ( set_Pr342166883923555964_a_d_c @ ( zip_Pr7710295530879645047_a_d_c @ Xs @ Xs ) ) )
      = ( ( member2052822272342364412_a_d_c @ A @ ( set_Pr111223993967229800_a_d_c @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_1242_zip__same,axiom,
    ! [A: option_a,B: option_a,Xs: list_option_a] :
      ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ ( set_Pr948796958549772220tion_a @ ( zip_op6411647709037274935tion_a @ Xs @ Xs ) ) )
      = ( ( member_option_a @ A @ ( set_option_a2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_1243_zip__same,axiom,
    ! [A: a,B: a,Xs: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Xs ) ) )
      = ( ( member_a @ A @ ( set_a2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_1244_in__set__zipE,axiom,
    ! [X3: a,Y: a,Xs: list_a,Ys: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) )
     => ~ ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
         => ~ ( member_a @ Y @ ( set_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1245_in__set__zipE,axiom,
    ! [X3: option_a,Y: a,Xs: list_option_a,Ys: list_a] :
      ( ( member6056235002698166154on_a_a @ ( produc3446707977624461905on_a_a @ X3 @ Y ) @ ( set_Pr1233600038994746358on_a_a @ ( zip_option_a_a @ Xs @ Ys ) ) )
     => ~ ( ( member_option_a @ X3 @ ( set_option_a2 @ Xs ) )
         => ~ ( member_a @ Y @ ( set_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1246_in__set__zipE,axiom,
    ! [X3: a,Y: option_a,Xs: list_a,Ys: list_option_a] :
      ( ( member6937434987665551382tion_a @ ( produc1224194096085666781tion_a @ X3 @ Y ) @ ( set_Pr2114800023962131586tion_a @ ( zip_a_option_a @ Xs @ Ys ) ) )
     => ~ ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
         => ~ ( member_option_a @ Y @ ( set_option_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1247_in__set__zipE,axiom,
    ! [X3: option_a,Y: option_a,Xs: list_option_a,Ys: list_option_a] :
      ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y ) @ ( set_Pr948796958549772220tion_a @ ( zip_op6411647709037274935tion_a @ Xs @ Ys ) ) )
     => ~ ( ( member_option_a @ X3 @ ( set_option_a2 @ Xs ) )
         => ~ ( member_option_a @ Y @ ( set_option_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1248_in__set__zipE,axiom,
    ! [X3: a,Y: d > c,Xs: list_a,Ys: list_d_c] :
      ( ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ X3 @ Y ) @ ( set_Pr1306003423349932796_a_d_c @ ( zip_a_d_c @ Xs @ Ys ) ) )
     => ~ ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
         => ~ ( member_d_c @ Y @ ( set_d_c2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1249_in__set__zipE,axiom,
    ! [X3: product_prod_a_d_c,Y: option2362401199305441953_a_d_c,Xs: list_P7605689430634331233_a_d_c,Ys: list_o3221539067586843313_a_d_c] :
      ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ X3 @ Y ) @ ( set_Pr111223993967229800_a_d_c @ ( zip_Pr6767014853595796523_a_d_c @ Xs @ Ys ) ) )
     => ~ ( ( member8140300833826434152_a_d_c @ X3 @ ( set_Pr1306003423349932796_a_d_c @ Xs ) )
         => ~ ( member1874962706598213560_a_d_c @ Y @ ( set_op7679606080901865548_a_d_c @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1250_in__set__zipE,axiom,
    ! [X3: produc6150846815813599699_a_d_c,Y: a,Xs: list_P8522657617046293465_a_d_c,Ys: list_a] :
      ( ( member2251515730839929787_d_c_a @ ( produc4774912239102099330_d_c_a @ X3 @ Y ) @ ( set_Pr1686552729088582183_d_c_a @ ( zip_Pr7553708875772114594_d_c_a @ Xs @ Ys ) ) )
     => ~ ( ( member2052822272342364412_a_d_c @ X3 @ ( set_Pr111223993967229800_a_d_c @ Xs ) )
         => ~ ( member_a @ Y @ ( set_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1251_in__set__zipE,axiom,
    ! [X3: a,Y: produc6150846815813599699_a_d_c,Xs: list_a,Ys: list_P8522657617046293465_a_d_c] :
      ( ( member4023273573022285733_a_d_c @ ( produc1782254833265735148_a_d_c @ X3 @ Y ) @ ( set_Pr3458310571270938129_a_d_c @ ( zip_a_4561051469935750412_a_d_c @ Xs @ Ys ) ) )
     => ~ ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
         => ~ ( member2052822272342364412_a_d_c @ Y @ ( set_Pr111223993967229800_a_d_c @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1252_in__set__zipE,axiom,
    ! [X3: produc6150846815813599699_a_d_c,Y: option_a,Xs: list_P8522657617046293465_a_d_c,Ys: list_option_a] :
      ( ( member1057245869843309569tion_a @ ( produc1309352160897273800tion_a @ X3 @ Y ) @ ( set_Pr5011486240240414829tion_a @ ( zip_Pr7235565932068673256tion_a @ Xs @ Ys ) ) )
     => ~ ( ( member2052822272342364412_a_d_c @ X3 @ ( set_Pr111223993967229800_a_d_c @ Xs ) )
         => ~ ( member_option_a @ Y @ ( set_option_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1253_in__set__zipE,axiom,
    ! [X3: option_a,Y: produc6150846815813599699_a_d_c,Xs: list_option_a,Ys: list_P8522657617046293465_a_d_c] :
      ( ( member6161829431480814559_a_d_c @ ( produc4753234341022239014_a_d_c @ X3 @ Y ) @ ( set_Pr892697765023144011_a_d_c @ ( zip_op1456076075338862662_a_d_c @ Xs @ Ys ) ) )
     => ~ ( ( member_option_a @ X3 @ ( set_option_a2 @ Xs ) )
         => ~ ( member2052822272342364412_a_d_c @ Y @ ( set_Pr111223993967229800_a_d_c @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1254_set__zip__leftD,axiom,
    ! [X3: product_prod_a_d_c,Y: option2362401199305441953_a_d_c,Xs: list_P7605689430634331233_a_d_c,Ys: list_o3221539067586843313_a_d_c] :
      ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ X3 @ Y ) @ ( set_Pr111223993967229800_a_d_c @ ( zip_Pr6767014853595796523_a_d_c @ Xs @ Ys ) ) )
     => ( member8140300833826434152_a_d_c @ X3 @ ( set_Pr1306003423349932796_a_d_c @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1255_set__zip__leftD,axiom,
    ! [X3: a,Y: d > c,Xs: list_a,Ys: list_d_c] :
      ( ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ X3 @ Y ) @ ( set_Pr1306003423349932796_a_d_c @ ( zip_a_d_c @ Xs @ Ys ) ) )
     => ( member_a @ X3 @ ( set_a2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1256_set__zip__rightD,axiom,
    ! [X3: product_prod_a_d_c,Y: option2362401199305441953_a_d_c,Xs: list_P7605689430634331233_a_d_c,Ys: list_o3221539067586843313_a_d_c] :
      ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ X3 @ Y ) @ ( set_Pr111223993967229800_a_d_c @ ( zip_Pr6767014853595796523_a_d_c @ Xs @ Ys ) ) )
     => ( member1874962706598213560_a_d_c @ Y @ ( set_op7679606080901865548_a_d_c @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1257_set__zip__rightD,axiom,
    ! [X3: a,Y: d > c,Xs: list_a,Ys: list_d_c] :
      ( ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ X3 @ Y ) @ ( set_Pr1306003423349932796_a_d_c @ ( zip_a_d_c @ Xs @ Ys ) ) )
     => ( member_d_c @ Y @ ( set_d_c2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1258_zip__eq__ConsE,axiom,
    ! [Xs: list_a,Ys: list_a,Xy: product_prod_a_a,Xys: list_P1396940483166286381od_a_a] :
      ( ( ( zip_a_a @ Xs @ Ys )
        = ( cons_P7316939126706565853od_a_a @ Xy @ Xys ) )
     => ~ ! [X2: a,Xs2: list_a] :
            ( ( Xs
              = ( cons_a @ X2 @ Xs2 ) )
           => ! [Y4: a,Ys2: list_a] :
                ( ( Ys
                  = ( cons_a @ Y4 @ Ys2 ) )
               => ( ( Xy
                    = ( product_Pair_a_a @ X2 @ Y4 ) )
                 => ( Xys
                   != ( zip_a_a @ Xs2 @ Ys2 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_1259_zip__eq__ConsE,axiom,
    ! [Xs: list_P7605689430634331233_a_d_c,Ys: list_o3221539067586843313_a_d_c,Xy: produc6150846815813599699_a_d_c,Xys: list_P8522657617046293465_a_d_c] :
      ( ( ( zip_Pr6767014853595796523_a_d_c @ Xs @ Ys )
        = ( cons_P5713675596995722633_a_d_c @ Xy @ Xys ) )
     => ~ ! [X2: product_prod_a_d_c,Xs2: list_P7605689430634331233_a_d_c] :
            ( ( Xs
              = ( cons_P982999123993234843_a_d_c @ X2 @ Xs2 ) )
           => ! [Y4: option2362401199305441953_a_d_c,Ys2: list_o3221539067586843313_a_d_c] :
                ( ( Ys
                  = ( cons_o7888461836264444011_a_d_c @ Y4 @ Ys2 ) )
               => ( ( Xy
                    = ( produc5208860900648697099_a_d_c @ X2 @ Y4 ) )
                 => ( Xys
                   != ( zip_Pr6767014853595796523_a_d_c @ Xs2 @ Ys2 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_1260_zip__eq__ConsE,axiom,
    ! [Xs: list_a,Ys: list_d_c,Xy: product_prod_a_d_c,Xys: list_P7605689430634331233_a_d_c] :
      ( ( ( zip_a_d_c @ Xs @ Ys )
        = ( cons_P982999123993234843_a_d_c @ Xy @ Xys ) )
     => ~ ! [X2: a,Xs2: list_a] :
            ( ( Xs
              = ( cons_a @ X2 @ Xs2 ) )
           => ! [Y4: d > c,Ys2: list_d_c] :
                ( ( Ys
                  = ( cons_d_c @ Y4 @ Ys2 ) )
               => ( ( Xy
                    = ( product_Pair_a_d_c @ X2 @ Y4 ) )
                 => ( Xys
                   != ( zip_a_d_c @ Xs2 @ Ys2 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_1261_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_P7605689430634331233_a_d_c,Ys: list_o3221539067586843313_a_d_c,Y: option2362401199305441953_a_d_c] :
      ( ( ( size_s1607940409241614069_a_d_c @ Xs )
        = ( size_s3965190101641584453_a_d_c @ Ys ) )
     => ( ( member1874962706598213560_a_d_c @ Y @ ( set_op7679606080901865548_a_d_c @ Ys ) )
       => ~ ! [X2: product_prod_a_d_c] :
              ~ ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ X2 @ Y ) @ ( set_Pr111223993967229800_a_d_c @ ( zip_Pr6767014853595796523_a_d_c @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1262_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_a,Ys: list_P8522657617046293465_a_d_c,Y: produc6150846815813599699_a_d_c] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_s4922472463745700421_a_d_c @ Ys ) )
     => ( ( member2052822272342364412_a_d_c @ Y @ ( set_Pr111223993967229800_a_d_c @ Ys ) )
       => ~ ! [X2: a] :
              ~ ( member4023273573022285733_a_d_c @ ( produc1782254833265735148_a_d_c @ X2 @ Y ) @ ( set_Pr3458310571270938129_a_d_c @ ( zip_a_4561051469935750412_a_d_c @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1263_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_a,Ys: list_option_a,Y: option_a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_s3078493964004954806tion_a @ Ys ) )
     => ( ( member_option_a @ Y @ ( set_option_a2 @ Ys ) )
       => ~ ! [X2: a] :
              ~ ( member6937434987665551382tion_a @ ( produc1224194096085666781tion_a @ X2 @ Y ) @ ( set_Pr2114800023962131586tion_a @ ( zip_a_option_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1264_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_a,Ys: list_d_c,Y: d > c] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_d_c @ Ys ) )
     => ( ( member_d_c @ Y @ ( set_d_c2 @ Ys ) )
       => ~ ! [X2: a] :
              ~ ( member8140300833826434152_a_d_c @ ( product_Pair_a_d_c @ X2 @ Y ) @ ( set_Pr1306003423349932796_a_d_c @ ( zip_a_d_c @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1265_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_a,Ys: list_a,Y: a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member_a @ Y @ ( set_a2 @ Ys ) )
       => ~ ! [X2: a] :
              ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1266_find__Some__iff,axiom,
    ! [P2: a > $o,Xs: list_a,X3: a] :
      ( ( ( find_a @ P2 @ Xs )
        = ( some_a @ X3 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
            & ( P2 @ ( nth_a @ Xs @ I2 ) )
            & ( X3
              = ( nth_a @ Xs @ I2 ) )
            & ! [J2: nat] :
                ( ( ord_less_nat @ J2 @ I2 )
               => ~ ( P2 @ ( nth_a @ Xs @ J2 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_1267_find__Some__iff2,axiom,
    ! [X3: a,P2: a > $o,Xs: list_a] :
      ( ( ( some_a @ X3 )
        = ( find_a @ P2 @ Xs ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
            & ( P2 @ ( nth_a @ Xs @ I2 ) )
            & ( X3
              = ( nth_a @ Xs @ I2 ) )
            & ! [J2: nat] :
                ( ( ord_less_nat @ J2 @ I2 )
               => ~ ( P2 @ ( nth_a @ Xs @ J2 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_1268_find_Osimps_I2_J,axiom,
    ! [P2: a > $o,X3: a,Xs: list_a] :
      ( ( ( P2 @ X3 )
       => ( ( find_a @ P2 @ ( cons_a @ X3 @ Xs ) )
          = ( some_a @ X3 ) ) )
      & ( ~ ( P2 @ X3 )
       => ( ( find_a @ P2 @ ( cons_a @ X3 @ Xs ) )
          = ( find_a @ P2 @ Xs ) ) ) ) ).

% find.simps(2)
thf(fact_1269_find__None__iff2,axiom,
    ! [P2: a > $o,Xs: list_a] :
      ( ( none_a
        = ( find_a @ P2 @ Xs ) )
      = ( ~ ? [X: a] :
              ( ( member_a @ X @ ( set_a2 @ Xs ) )
              & ( P2 @ X ) ) ) ) ).

% find_None_iff2
thf(fact_1270_find__None__iff,axiom,
    ! [P2: a > $o,Xs: list_a] :
      ( ( ( find_a @ P2 @ Xs )
        = none_a )
      = ( ~ ? [X: a] :
              ( ( member_a @ X @ ( set_a2 @ Xs ) )
              & ( P2 @ X ) ) ) ) ).

% find_None_iff
thf(fact_1271_find_Osimps_I1_J,axiom,
    ! [Uu: a > $o] :
      ( ( find_a @ Uu @ nil_a )
      = none_a ) ).

% find.simps(1)
thf(fact_1272_case__optionE,axiom,
    ! [P2: $o,Q: a > $o,X3: option_a] :
      ( ( case_option_o_a @ P2 @ Q @ X3 )
     => ( ( ( X3 = none_a )
         => ~ P2 )
       => ~ ! [Y4: a] :
              ( ( X3
                = ( some_a @ Y4 ) )
             => ~ ( Q @ Y4 ) ) ) ) ).

% case_optionE
thf(fact_1273_Some__image__these__eq,axiom,
    ! [A2: set_option_a] :
      ( ( image_a_option_a2 @ some_a @ ( these_a @ A2 ) )
      = ( collect_option_a
        @ ^ [X: option_a] :
            ( ( member_option_a @ X @ A2 )
            & ( X != none_a ) ) ) ) ).

% Some_image_these_eq
thf(fact_1274_Option_Othese__def,axiom,
    ( these_a
    = ( ^ [A7: set_option_a] :
          ( image_option_a_a2 @ the_a
          @ ( collect_option_a
            @ ^ [X: option_a] :
                ( ( member_option_a @ X @ A7 )
                & ( X != none_a ) ) ) ) ) ) ).

% Option.these_def
thf(fact_1275_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option_a] :
      ( ( Option != none_a )
      = ( case_option_o_a @ $false
        @ ^ [Uu2: a] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_1276_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option_a] :
      ( ( Option = none_a )
      = ( case_option_o_a @ $true
        @ ^ [Uu2: a] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)

% Helper facts (5)
thf(help_If_2_1_If_001tf__c_T,axiom,
    ! [X3: c,Y: c] :
      ( ( if_c @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001tf__c_T,axiom,
    ! [X3: c,Y: c] :
      ( ( if_c @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_If_3_1_If_001t__Option__Ooption_Itf__a_J_T,axiom,
    ! [P2: $o] :
      ( ( P2 = $true )
      | ( P2 = $false ) ) ).

thf(help_If_2_1_If_001t__Option__Ooption_Itf__a_J_T,axiom,
    ! [X3: option_a,Y: option_a] :
      ( ( if_option_a @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_Itf__a_J_T,axiom,
    ! [X3: option_a,Y: option_a] :
      ( ( if_option_a @ $true @ X3 @ Y )
      = X3 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    sat_a_b_d_c @ plus @ mult @ valid @ sigma @ s @ delta @ ( mult_b_a_c_d @ p @ ( and_a_b_c_d @ a2 @ b2 ) ) ).

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