%------------------------------------------------------------------------------
% 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/0001_Combinability/prob_00127_005151__6573676_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1889 ( 473 unt; 608 typ;   0 def)
%            Number of atoms       : 4191 (1625 equ;   0 cnn)
%            Maximal formula atoms :   12 (   3 avg)
%            Number of connectives : 16071 ( 525   ~;  28   |; 281   &;13344   @)
%                                         (   0 <=>;1893  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   29 (   8 avg)
%            Number of types       :   84 (  83 usr)
%            Number of type conns  : 3025 (3025   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  528 ( 525 usr;  44 con; 0-7 aty)
%            Number of variables   : 4764 ( 116   ^;4520   !; 128   ?;4764   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 16:04:39.796
%------------------------------------------------------------------------------
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thf(ty_n_t__Option__Ooption_Itf__a_J,type,
    option_a: $tType ).

thf(ty_n_t__List__Olist_Itf__a_J,type,
    list_a: $tType ).

thf(ty_n_t__Set__Oset_Itf__d_J,type,
    set_d: $tType ).

thf(ty_n_t__Set__Oset_Itf__c_J,type,
    set_c: $tType ).

thf(ty_n_t__Set__Oset_Itf__a_J,type,
    set_a: $tType ).

thf(ty_n_t__Nat__Onat,type,
    nat: $tType ).

thf(ty_n_tf__d,type,
    d: $tType ).

thf(ty_n_tf__c,type,
    c: $tType ).

thf(ty_n_tf__b,type,
    b: $tType ).

thf(ty_n_tf__a,type,
    a: $tType ).

% Explicit typings (525)
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Option__Ooption_Itf__a_J_001t__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J,type,
    bNF_Gr2659421419784889954_a_c_d: set_option_a > ( option_a > product_prod_a_c_d ) > ( option_a > option6413918287372586467_a_c_d ) > set_Pr1275464188344874039_a_c_d ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Option__Ooption_Itf__a_J_001tf__a_001_062_Itf__c_Mtf__d_J,type,
    bNF_Gr1391775451562859130_a_c_d: set_option_a > ( option_a > a ) > ( option_a > c > d ) > set_Pr334309860168054345_a_c_d ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J_001t__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J,type,
    bNF_Gr589831376474905333_a_c_d: set_Pr1275464188344874039_a_c_d > ( produc5278197477302038359_a_c_d > product_prod_a_c_d ) > ( produc5278197477302038359_a_c_d > option6413918287372586467_a_c_d ) > set_Pr1275464188344874039_a_c_d ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J_001tf__a_001_062_Itf__c_Mtf__d_J,type,
    bNF_Gr4265814288748638247_a_c_d: set_Pr1275464188344874039_a_c_d > ( produc5278197477302038359_a_c_d > a ) > ( produc5278197477302038359_a_c_d > c > d ) > set_Pr334309860168054345_a_c_d ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001tf__a_001t__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J,type,
    bNF_Gr4981583572320052764_a_c_d: set_a > ( a > product_prod_a_c_d ) > ( a > option6413918287372586467_a_c_d ) > set_Pr1275464188344874039_a_c_d ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001tf__a_001tf__a_001_062_Itf__c_Mtf__d_J,type,
    bNF_Gr7437496851395112512_a_c_d: set_a > ( a > a ) > ( a > c > d ) > set_Pr334309860168054345_a_c_d ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Option__Ooption_Itf__a_J,type,
    bNF_We8432232079604507440tion_a: set_Pr7585778909603769095tion_a > $o ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J,type,
    bNF_We47639529709206339_a_c_d: set_Pr3774076249145843911_a_c_d > $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__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J,type,
    bNF_We3608594581950107405_a_c_d: set_Pr3774076249145843911_a_c_d > set_Pr1275464188344874039_a_c_d > produc5278197477302038359_a_c_d > $o ).

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

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2_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__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J,type,
    bNF_We3665937542762464421_a_c_d: set_Pr3774076249145843911_a_c_d > produc5278197477302038359_a_c_d > produc5278197477302038359_a_c_d > produc5278197477302038359_a_c_d ).

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__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_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__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J,type,
    bNF_We673253246704241848_a_c_d: set_Pr3774076249145843911_a_c_d > set_Pr1275464188344874039_a_c_d > produc5278197477302038359_a_c_d ).

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_Combinability_Ologic_Ocombinable_001tf__a_001tf__b_001tf__c_001tf__d,type,
    combinable_a_b_c_d: ( a > a > option_a ) > ( b > a > a ) > ( b > b > b ) > ( a > $o ) > ( ( c > d ) > set_a ) > assertion_a_b_d_c > $o ).

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__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J,type,
    equiv_777496545229528845_a_c_d: set_Pr1275464188344874039_a_c_d > set_Pr3774076249145843911_a_c_d > $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__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J,type,
    equiv_3313360957152018974_a_c_d: set_Pr1275464188344874039_a_c_d > set_Pr3774076249145843911_a_c_d > set_se3016068994149987607_a_c_d ).

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__c_Mtf__d_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__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J,type,
    finite5665696198532655488_a_c_d: set_Pr1275464188344874039_a_c_d > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J,type,
    finite6487500827408305578_a_c_d: set_Pr334309860168054345_a_c_d > $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__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J,type,
    bij_be7402712960780075529_a_c_d: ( option_a > produc5278197477302038359_a_c_d ) > set_option_a > set_Pr1275464188344874039_a_c_d > $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__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J_001t__Option__Ooption_Itf__a_J,type,
    bij_be8242579008920824995tion_a: ( produc5278197477302038359_a_c_d > option_a ) > set_Pr1275464188344874039_a_c_d > set_option_a > $o ).

thf(sy_c_Fun_Obij__betw_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J,type,
    bij_be3277052638744491318_a_c_d: ( produc5278197477302038359_a_c_d > produc5278197477302038359_a_c_d ) > set_Pr1275464188344874039_a_c_d > set_Pr1275464188344874039_a_c_d > $o ).

thf(sy_c_Fun_Obij__betw_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J_001tf__a,type,
    bij_be5244696088125194461_c_d_a: ( produc5278197477302038359_a_c_d > a ) > set_Pr1275464188344874039_a_c_d > 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__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_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__c_Mtf__d_J_J_J_001t__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_001t__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_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,
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thf(sy_c_Fun_Ocomp_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J_001tf__a,type,
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thf(sy_c_Fun_Ocomp_001t__Option__Ooption_Itf__a_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_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,
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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__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_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,
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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,
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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,
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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__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J,type,
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thf(sy_c_Fun_Ocomp_001tf__a_001t__Option__Ooption_Itf__a_J_001tf__a,type,
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thf(sy_c_Fun_Ocomp_001tf__a_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_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__d_001tf__d_001tf__c,type,
    comp_d_d_c: ( d > d ) > ( c > d ) > c > d ).

thf(sy_c_Fun_Ofun__upd_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J,type,
    fun_up5969202740897331946_a_c_d: ( option_a > option6413918287372586467_a_c_d ) > option_a > option6413918287372586467_a_c_d > option_a > option6413918287372586467_a_c_d ).

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__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J,type,
    fun_up4007221454163596925_a_c_d: ( produc5278197477302038359_a_c_d > option6413918287372586467_a_c_d ) > produc5278197477302038359_a_c_d > option6413918287372586467_a_c_d > produc5278197477302038359_a_c_d > option6413918287372586467_a_c_d ).

thf(sy_c_Fun_Ofun__upd_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J_001t__Option__Ooption_Itf__a_J,type,
    fun_up7463025390225626410tion_a: ( produc5278197477302038359_a_c_d > option_a ) > produc5278197477302038359_a_c_d > option_a > produc5278197477302038359_a_c_d > option_a ).

thf(sy_c_Fun_Ofun__upd_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J_001tf__a,type,
    fun_up2227350886784581604_c_d_a: ( produc5278197477302038359_a_c_d > a ) > produc5278197477302038359_a_c_d > a > produc5278197477302038359_a_c_d > a ).

thf(sy_c_Fun_Ofun__upd_001t__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_001t__Option__Ooption_It__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J,type,
    fun_up3914643275129436357_a_c_d: ( product_prod_a_c_d > option6212624336325788467_a_c_d ) > product_prod_a_c_d > option6212624336325788467_a_c_d > product_prod_a_c_d > option6212624336325788467_a_c_d ).

thf(sy_c_Fun_Ofun__upd_001tf__a_001t__Option__Ooption_I_062_Itf__c_Mtf__d_J_J,type,
    fun_upd_a_option_c_d: ( a > option_c_d ) > a > option_c_d > a > option_c_d ).

thf(sy_c_Fun_Ofun__upd_001tf__a_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J,type,
    fun_up6642239869823773348_a_c_d: ( a > option6413918287372586467_a_c_d ) > a > option6413918287372586467_a_c_d > a > option6413918287372586467_a_c_d ).

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__c_001tf__d,type,
    fun_upd_c_d: ( c > d ) > c > d > c > d ).

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__c_Mtf__d_J_J,type,
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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__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J,type,
    inj_on6212699291050112900_a_c_d: ( option_a > produc5278197477302038359_a_c_d ) > 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__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J_001t__Option__Ooption_Itf__a_J,type,
    inj_on7052565339190862366tion_a: ( produc5278197477302038359_a_c_d > option_a ) > set_Pr1275464188344874039_a_c_d > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J,type,
    inj_on7596037564690925361_a_c_d: ( produc5278197477302038359_a_c_d > produc5278197477302038359_a_c_d ) > set_Pr1275464188344874039_a_c_d > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J_001tf__a,type,
    inj_on5461451596835146456_c_d_a: ( produc5278197477302038359_a_c_d > a ) > set_Pr1275464188344874039_a_c_d > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J,type,
    inj_on7837763817969280361_a_c_d: ( product_prod_a_c_d > option6413918287372586467_a_c_d ) > set_Pr334309860168054345_a_c_d > $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__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J,type,
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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__c_001tf__d,type,
    inj_on_c_d: ( c > d ) > set_c > $o ).

thf(sy_c_Fun_Ooverride__on_001tf__c_001tf__d,type,
    override_on_c_d: ( c > d ) > ( c > d ) > set_c > c > d ).

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__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_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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thf(sy_c_Fun_Othe__inv__into_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_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__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_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__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J,type,
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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__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
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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__c_J,type,
    minus_minus_set_c: set_c > set_c > set_c ).

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__d,type,
    if_d: $o > d > d > d ).

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__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_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 ).

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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_Odistinct_001tf__a,type,
    distinct_a: list_a > $o ).

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

thf(sy_c_List_Olenlex_001t__Option__Ooption_Itf__a_J,type,
    lenlex_option_a: set_Pr7585778909603769095tion_a > set_Pr2064693230030669831tion_a ).

thf(sy_c_List_Olenlex_001tf__a,type,
    lenlex_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).

thf(sy_c_List_Olex_001tf__a,type,
    lex_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).

thf(sy_c_List_Olist_OCons_001_062_Itf__c_Mtf__d_J,type,
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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,
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thf(sy_c_List_Olist_Oset_001_062_Itf__c_Mtf__d_J,type,
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thf(sy_c_List_Olist_Oset_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J,type,
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thf(sy_c_List_Olist_Oset_001t__Option__Ooption_Itf__a_J,type,
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thf(sy_v_a1____,type,
    a1: a ).

thf(sy_v_a2____,type,
    a22: a ).

thf(sy_v_a____,type,
    a3: a ).

thf(sy_v_aa____,type,
    aa: a ).

thf(sy_v_aaa____,type,
    aaa: a ).

thf(sy_v_b____,type,
    b3: a ).

thf(sy_v_baa____,type,
    baa: 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_q____,type,
    q: b ).

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

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 ).

thf(sy_v_x____,type,
    x: a ).

% Relevant facts (1275)
thf(fact_0_equal__outside__def,axiom,
    ( equal_outside_c_d
    = ( ^ [S: c > d,S2: c > d,S3: set_c] :
        ! [X: c] :
          ( ~ ( member_c @ X @ S3 )
         => ( ( S @ X )
            = ( S2 @ X ) ) ) ) ) ).

% equal_outside_def
thf(fact_1_can__divide,axiom,
    ! [P: b,A: a,B: a] :
      ( ( ( mult @ P @ A )
        = ( mult @ P @ B ) )
     => ( A = B ) ) ).

% can_divide
thf(fact_2_unique__inv,axiom,
    ! [A: a,P: b,B: a] :
      ( ( A
        = ( mult @ P @ B ) )
      = ( B
        = ( mult @ ( sinv @ P ) @ A ) ) ) ).

% unique_inv
thf(fact_3_one__neutral,axiom,
    ! [A: a] :
      ( ( mult @ one @ A )
      = A ) ).

% one_neutral
thf(fact_4_double__mult,axiom,
    ! [P: b,Q: b,A: a] :
      ( ( mult @ P @ ( mult @ Q @ A ) )
      = ( mult @ ( smult @ P @ Q ) @ A ) ) ).

% double_mult
thf(fact_5__092_060open_062Some_Aa2_A_061_Ap_A_092_060odot_062_Aa_A_092_060oplus_062_Ap_A_092_060odot_062_Aaa_092_060close_062,axiom,
    ( ( some_a @ a22 )
    = ( plus @ ( mult @ p @ a3 ) @ ( mult @ p @ aa ) ) ) ).

% \<open>Some a2 = p \<odot> a \<oplus> p \<odot> aa\<close>
thf(fact_6__092_060open_062Some_Aaaa_A_061_Aa_A_092_060oplus_062_Aaa_092_060close_062,axiom,
    ( ( some_a @ aaa )
    = ( plus @ a3 @ aa ) ) ).

% \<open>Some aaa = a \<oplus> aa\<close>
thf(fact_7_calculation,axiom,
    ( ( some_a @ a1 )
    = ( plus @ a22 @ ( mult @ q @ b3 ) ) ) ).

% calculation
thf(fact_8__092_060open_062Some_Aa1_A_061_A_092_060sigma_062_A_092_060oplus_062_Ap_A_092_060odot_062_Aaa_092_060close_062,axiom,
    ( ( some_a @ a1 )
    = ( plus @ sigma2 @ ( mult @ p @ aa ) ) ) ).

% \<open>Some a1 = \<sigma> \<oplus> p \<odot> aa\<close>
thf(fact_9_smult__asso,axiom,
    ! [P: b,Q: b,R: b] :
      ( ( smult @ ( smult @ P @ Q ) @ R )
      = ( smult @ P @ ( smult @ Q @ R ) ) ) ).

% smult_asso
thf(fact_10_smult__comm,axiom,
    ! [P: b,Q: b] :
      ( ( smult @ P @ Q )
      = ( smult @ Q @ P ) ) ).

% smult_comm
thf(fact_11__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062a1_O_ASome_Aa1_A_061_A_092_060sigma_062_A_092_060oplus_062_Ap_A_092_060odot_062_Aaa_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [A1: a] :
        ( ( some_a @ A1 )
       != ( plus @ sigma2 @ ( mult @ p @ aa ) ) ) ).

% \<open>\<And>thesis. (\<And>a1. Some a1 = \<sigma> \<oplus> p \<odot> aa \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_12__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062a2_O_ASome_Aa2_A_061_Ap_A_092_060odot_062_Aa_A_092_060oplus_062_Ap_A_092_060odot_062_Aaa_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [A2: a] :
        ( ( some_a @ A2 )
       != ( plus @ ( mult @ p @ a3 ) @ ( mult @ p @ aa ) ) ) ).

% \<open>\<And>thesis. (\<And>a2. Some a2 = p \<odot> a \<oplus> p \<odot> aa \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_13_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_14_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_15_commutative,axiom,
    ! [A: a,B: a] :
      ( ( plus @ A @ B )
      = ( plus @ B @ A ) ) ).

% commutative
thf(fact_16_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_17_move__sum,axiom,
    ! [A: a,A12: a,A22: a,B: a,B1: a,B2: a,X2: a,X1: a,X22: a] :
      ( ( ( some_a @ A )
        = ( plus @ A12 @ A22 ) )
     => ( ( ( some_a @ B )
          = ( plus @ B1 @ B2 ) )
       => ( ( ( some_a @ X2 )
            = ( plus @ A @ B ) )
         => ( ( ( some_a @ X1 )
              = ( plus @ A12 @ B1 ) )
           => ( ( ( some_a @ X22 )
                = ( plus @ A22 @ B2 ) )
             => ( ( some_a @ X2 )
                = ( plus @ X1 @ X22 ) ) ) ) ) ) ) ).

% move_sum
thf(fact_18_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_19_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_20_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_21_larger__first__sum,axiom,
    ! [Y: a,A: a,B: a,X2: a] :
      ( ( ( some_a @ Y )
        = ( plus @ A @ B ) )
     => ( ( pre_larger_a @ plus @ X2 @ Y )
       => ? [A3: a] :
            ( ( ( some_a @ X2 )
              = ( plus @ A3 @ B ) )
            & ( pre_larger_a @ plus @ A3 @ A ) ) ) ) ).

% larger_first_sum
thf(fact_22_sum__both__larger,axiom,
    ! [X3: a,A4: a,B3: a,X2: a,A: a,B: a] :
      ( ( ( some_a @ X3 )
        = ( plus @ A4 @ B3 ) )
     => ( ( ( some_a @ X2 )
          = ( plus @ A @ B ) )
       => ( ( pre_larger_a @ plus @ A4 @ A )
         => ( ( pre_larger_a @ plus @ B3 @ B )
           => ( pre_larger_a @ plus @ X3 @ X2 ) ) ) ) ) ).

% sum_both_larger
thf(fact_23_compatible__smaller,axiom,
    ! [A: a,B: a,X2: a] :
      ( ( pre_larger_a @ plus @ A @ B )
     => ( ( pre_compatible_a @ plus @ X2 @ A )
       => ( pre_compatible_a @ plus @ X2 @ B ) ) ) ).

% compatible_smaller
thf(fact_24_larger__implies__compatible,axiom,
    ! [X2: a,Y: a] :
      ( ( pre_larger_a @ plus @ X2 @ Y )
     => ( pre_compatible_a @ plus @ X2 @ Y ) ) ).

% larger_implies_compatible
thf(fact_25_sone__neutral,axiom,
    ! [P: b] :
      ( ( smult @ one @ P )
      = P ) ).

% sone_neutral
thf(fact_26_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_27_compatible__multiples,axiom,
    ! [P: b,A: a,Q: b,B: a] :
      ( ( pre_compatible_a @ plus @ ( mult @ P @ A ) @ ( mult @ Q @ B ) )
     => ( pre_compatible_a @ plus @ A @ B ) ) ).

% compatible_multiples
thf(fact_28_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_29__092_060open_062Some_Abaa_A_061_Ab_A_092_060oplus_062_Aaa_092_060close_062,axiom,
    ( ( some_a @ baa )
    = ( plus @ b3 @ aa ) ) ).

% \<open>Some baa = b \<oplus> aa\<close>
thf(fact_30__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062aaa_O_ASome_Aaaa_A_061_Aa_A_092_060oplus_062_Aaa_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Aaa: a] :
        ( ( some_a @ Aaa )
       != ( plus @ a3 @ aa ) ) ).

% \<open>\<And>thesis. (\<And>aaa. Some aaa = a \<oplus> aa \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_31__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062baa_O_ASome_Abaa_A_061_Ab_A_092_060oplus_062_Aaa_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Baa: a] :
        ( ( some_a @ Baa )
       != ( plus @ b3 @ aa ) ) ).

% \<open>\<And>thesis. (\<And>baa. Some baa = b \<oplus> aa \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_32__092_060open_062a_A_D_D_Aaa_092_060close_062,axiom,
    pre_compatible_a @ plus @ a3 @ aa ).

% \<open>a ## aa\<close>
thf(fact_33_sinv__inverse,axiom,
    ! [P: b] :
      ( ( smult @ P @ ( sinv @ P ) )
      = one ) ).

% sinv_inverse
thf(fact_34__092_060open_062b_A_D_D_Aaa_092_060close_062,axiom,
    pre_compatible_a @ plus @ b3 @ aa ).

% \<open>b ## aa\<close>
thf(fact_35__092_060open_062Some_A_Isadd_Ap_Aq_A_092_060odot_062_Aaa_J_A_061_Ap_A_092_060odot_062_Aaa_A_092_060oplus_062_Aq_A_092_060odot_062_Aaa_092_060close_062,axiom,
    ( ( some_a @ ( mult @ ( sadd @ p @ q ) @ aa ) )
    = ( plus @ ( mult @ p @ aa ) @ ( mult @ q @ aa ) ) ) ).

% \<open>Some (sadd p q \<odot> aa) = p \<odot> aa \<oplus> q \<odot> aa\<close>
thf(fact_36__092_060open_062Some_A_Isadd_Ap_Aq_A_092_060odot_062_A_092_060sigma_062_H_J_A_061_A_092_060sigma_062_A_092_060oplus_062_Asadd_Ap_Aq_A_092_060odot_062_Aaa_092_060close_062,axiom,
    ( ( some_a @ ( mult @ ( sadd @ p @ q ) @ sigma ) )
    = ( plus @ sigma2 @ ( mult @ ( sadd @ p @ q ) @ aa ) ) ) ).

% \<open>Some (sadd p q \<odot> \<sigma>') = \<sigma> \<oplus> sadd p q \<odot> aa\<close>
thf(fact_37_distrib__mult,axiom,
    ! [P: b,Q: b,X2: a] :
      ( ( some_a @ ( mult @ ( sadd @ P @ Q ) @ X2 ) )
      = ( plus @ ( mult @ P @ X2 ) @ ( mult @ Q @ X2 ) ) ) ).

% distrib_mult
thf(fact_38_option_Oinject,axiom,
    ! [X22: product_prod_a_c_d,Y2: product_prod_a_c_d] :
      ( ( ( some_P1084500821511757806_a_c_d @ X22 )
        = ( some_P1084500821511757806_a_c_d @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% option.inject
thf(fact_39_option_Oinject,axiom,
    ! [X22: a,Y2: a] :
      ( ( ( some_a @ X22 )
        = ( some_a @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% option.inject
thf(fact_40_compatible__def,axiom,
    ! [A: a,B: a] :
      ( ( pre_compatible_a @ plus @ A @ B )
      = ( ( plus @ A @ B )
       != none_a ) ) ).

% compatible_def
thf(fact_41_pre__logic_Olarger__def,axiom,
    ( pre_la7759296445815288060_a_c_d
    = ( ^ [Plus: product_prod_a_c_d > product_prod_a_c_d > option6413918287372586467_a_c_d,A5: product_prod_a_c_d,B4: product_prod_a_c_d] :
        ? [C2: product_prod_a_c_d] :
          ( ( some_P1084500821511757806_a_c_d @ A5 )
          = ( Plus @ B4 @ C2 ) ) ) ) ).

% pre_logic.larger_def
thf(fact_42_pre__logic_Olarger__def,axiom,
    ( pre_larger_a
    = ( ^ [Plus: a > a > option_a,A5: a,B4: a] :
        ? [C2: a] :
          ( ( some_a @ A5 )
          = ( Plus @ B4 @ C2 ) ) ) ) ).

% pre_logic.larger_def
thf(fact_43_smult__distrib,axiom,
    ! [P: b,Q: b,R: b] :
      ( ( smult @ P @ ( sadd @ Q @ R ) )
      = ( sadd @ ( smult @ P @ Q ) @ ( smult @ P @ R ) ) ) ).

% smult_distrib
thf(fact_44_pre__logic_Olarger_Ocong,axiom,
    pre_larger_a = pre_larger_a ).

% pre_logic.larger.cong
thf(fact_45_pre__logic_Ocompatible_Ocong,axiom,
    pre_compatible_a = pre_compatible_a ).

% pre_logic.compatible.cong
thf(fact_46_logic__axioms,axiom,
    logic_a_b @ plus @ mult @ smult @ sadd @ sinv @ one @ valid ).

% logic_axioms
thf(fact_47_mem__Collect__eq,axiom,
    ! [A: a,P2: a > $o] :
      ( ( member_a @ A @ ( collect_a @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_48_mem__Collect__eq,axiom,
    ! [A: produc5278197477302038359_a_c_d,P2: produc5278197477302038359_a_c_d > $o] :
      ( ( member1180172933830803072_a_c_d @ A @ ( collec245936426562692802_a_c_d @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_49_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_50_Collect__mem__eq,axiom,
    ! [A6: set_a] :
      ( ( collect_a
        @ ^ [X: a] : ( member_a @ X @ A6 ) )
      = A6 ) ).

% Collect_mem_eq
thf(fact_51_Collect__mem__eq,axiom,
    ! [A6: set_Pr1275464188344874039_a_c_d] :
      ( ( collec245936426562692802_a_c_d
        @ ^ [X: produc5278197477302038359_a_c_d] : ( member1180172933830803072_a_c_d @ X @ A6 ) )
      = A6 ) ).

% Collect_mem_eq
thf(fact_52_Collect__mem__eq,axiom,
    ! [A6: set_option_a] :
      ( ( collect_option_a
        @ ^ [X: option_a] : ( member_option_a @ X @ A6 ) )
      = A6 ) ).

% Collect_mem_eq
thf(fact_53_Collect__cong,axiom,
    ! [P2: option_a > $o,Q2: option_a > $o] :
      ( ! [X4: option_a] :
          ( ( P2 @ X4 )
          = ( Q2 @ X4 ) )
     => ( ( collect_option_a @ P2 )
        = ( collect_option_a @ Q2 ) ) ) ).

% Collect_cong
thf(fact_54_valid__mono,axiom,
    ! [A: a,B: a] :
      ( ( ( valid @ A )
        & ( pre_larger_a @ plus @ A @ B ) )
     => ( valid @ B ) ) ).

% valid_mono
thf(fact_55__092_060open_062a_M_As_M_A_092_060Delta_062_A_092_060Turnstile_062_AWand_AA_AB_A_092_060and_062_Ab_M_As_M_A_092_060Delta_062_A_092_060Turnstile_062_AWand_AA_AB_A_092_060and_062_ASome_A_092_060sigma_062_A_061_Ap_A_092_060odot_062_Aa_A_092_060oplus_062_Aq_A_092_060odot_062_Ab_A_092_060and_062_A_092_060sigma_062_A_061_Asadd_Ap_Aq_A_092_060odot_062_Ax_092_060close_062,axiom,
    ( ( sat_a_b_c_d @ plus @ mult @ valid @ a3 @ s @ delta @ ( wand_a_b_d_c @ a2 @ b2 ) )
    & ( sat_a_b_c_d @ plus @ mult @ valid @ b3 @ s @ delta @ ( wand_a_b_d_c @ a2 @ b2 ) )
    & ( ( some_a @ sigma2 )
      = ( plus @ ( mult @ p @ a3 ) @ ( mult @ q @ b3 ) ) )
    & ( sigma2
      = ( mult @ ( sadd @ p @ q ) @ x ) ) ) ).

% \<open>a, s, \<Delta> \<Turnstile> Wand A B \<and> b, s, \<Delta> \<Turnstile> Wand A B \<and> Some \<sigma> = p \<odot> a \<oplus> q \<odot> b \<and> \<sigma> = sadd p q \<odot> x\<close>
thf(fact_56_sadd__comm,axiom,
    ! [P: b,Q: b] :
      ( ( sadd @ P @ Q )
      = ( sadd @ Q @ P ) ) ).

% sadd_comm
thf(fact_57_assertion_Oinject_I4_J,axiom,
    ! [X41: assertion_a_b_d_c,X42: assertion_a_b_d_c,Y41: assertion_a_b_d_c,Y42: assertion_a_b_d_c] :
      ( ( ( wand_a_b_d_c @ X41 @ X42 )
        = ( wand_a_b_d_c @ Y41 @ Y42 ) )
      = ( ( X41 = Y41 )
        & ( X42 = Y42 ) ) ) ).

% assertion.inject(4)
thf(fact_58_not__Some__eq,axiom,
    ! [X2: option6413918287372586467_a_c_d] :
      ( ( ! [Y3: product_prod_a_c_d] :
            ( X2
           != ( some_P1084500821511757806_a_c_d @ Y3 ) ) )
      = ( X2 = none_P2239999354720063602_a_c_d ) ) ).

% not_Some_eq
thf(fact_59_not__Some__eq,axiom,
    ! [X2: option_a] :
      ( ( ! [Y3: a] :
            ( X2
           != ( some_a @ Y3 ) ) )
      = ( X2 = none_a ) ) ).

% not_Some_eq
thf(fact_60_not__None__eq,axiom,
    ! [X2: option6413918287372586467_a_c_d] :
      ( ( X2 != none_P2239999354720063602_a_c_d )
      = ( ? [Y3: product_prod_a_c_d] :
            ( X2
            = ( some_P1084500821511757806_a_c_d @ Y3 ) ) ) ) ).

% not_None_eq
thf(fact_61_not__None__eq,axiom,
    ! [X2: option_a] :
      ( ( X2 != none_a )
      = ( ? [Y3: a] :
            ( X2
            = ( some_a @ Y3 ) ) ) ) ).

% not_None_eq
thf(fact_62_sat__wand,axiom,
    ! [S4: c > d,Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c,Sigma: a,B5: assertion_a_b_d_c] :
      ( ! [A7: a,Sigma2: a] :
          ( ( ( sat_a_b_c_d @ plus @ mult @ valid @ A7 @ S4 @ Delta @ A6 )
            & ( ( some_a @ Sigma2 )
              = ( plus @ Sigma @ A7 ) ) )
         => ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma2 @ S4 @ Delta @ B5 ) )
     => ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( wand_a_b_d_c @ A6 @ B5 ) ) ) ).

% sat_wand
thf(fact_63_sat_Osimps_I3_J,axiom,
    ! [Sigma: a,S4: c > d,Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c,B5: assertion_a_b_d_c] :
      ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( wand_a_b_d_c @ A6 @ B5 ) )
      = ( ! [A5: a,Sigma3: a] :
            ( ( ( sat_a_b_c_d @ plus @ mult @ valid @ A5 @ S4 @ Delta @ A6 )
              & ( ( some_a @ Sigma3 )
                = ( plus @ Sigma @ A5 ) ) )
           => ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma3 @ S4 @ Delta @ B5 ) ) ) ) ).

% sat.simps(3)
thf(fact_64__092_060open_062aa_M_As_M_A_092_060Delta_062_A_092_060Turnstile_062_AA_A_092_060and_062_ASome_A_092_060sigma_062_H_A_061_Ax_A_092_060oplus_062_Aaa_092_060close_062,axiom,
    ( ( sat_a_b_c_d @ plus @ mult @ valid @ aa @ s @ delta @ a2 )
    & ( ( some_a @ sigma )
      = ( plus @ x @ aa ) ) ) ).

% \<open>aa, s, \<Delta> \<Turnstile> A \<and> Some \<sigma>' = x \<oplus> aa\<close>
thf(fact_65_assms,axiom,
    combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ delta @ b2 ).

% assms
thf(fact_66_pure__def,axiom,
    ! [A6: assertion_a_b_d_c] :
      ( ( pure_a_b_d_c @ plus @ mult @ valid @ A6 )
      = ( ! [Sigma4: a,Sigma3: a,S: c > d,Delta2: ( c > d ) > set_a,Delta3: ( c > d ) > set_a] :
            ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma4 @ S @ Delta2 @ A6 )
            = ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma3 @ S @ Delta3 @ A6 ) ) ) ) ).

% pure_def
thf(fact_67_equivalentI,axiom,
    ! [Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c,B5: assertion_a_b_d_c] :
      ( ! [Sigma5: a,S5: c > d] :
          ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S5 @ Delta @ A6 )
         => ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S5 @ Delta @ B5 ) )
     => ( ! [Sigma5: a,S5: c > d] :
            ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S5 @ Delta @ B5 )
           => ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S5 @ Delta @ A6 ) )
       => ( equivalent_a_b_d_c @ plus @ mult @ valid @ A6 @ Delta @ B5 ) ) ) ).

% equivalentI
thf(fact_68_intuitionistic__def,axiom,
    ! [S4: c > d,Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c] :
      ( ( intuit7508411120625971703_b_c_d @ plus @ mult @ valid @ S4 @ Delta @ A6 )
      = ( ! [A5: a,B4: a] :
            ( ( ( pre_larger_a @ plus @ A5 @ B4 )
              & ( sat_a_b_c_d @ plus @ mult @ valid @ B4 @ S4 @ Delta @ A6 ) )
           => ( sat_a_b_c_d @ plus @ mult @ valid @ A5 @ S4 @ Delta @ A6 ) ) ) ) ).

% intuitionistic_def
thf(fact_69_intuitionisticI,axiom,
    ! [S4: c > d,Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c] :
      ( ! [A7: a,B6: a] :
          ( ( ( pre_larger_a @ plus @ A7 @ B6 )
            & ( sat_a_b_c_d @ plus @ mult @ valid @ B6 @ S4 @ Delta @ A6 ) )
         => ( sat_a_b_c_d @ plus @ mult @ valid @ A7 @ S4 @ Delta @ A6 ) )
     => ( intuit7508411120625971703_b_c_d @ plus @ mult @ valid @ S4 @ Delta @ A6 ) ) ).

% intuitionisticI
thf(fact_70_not__in__fv__def,axiom,
    ! [A6: assertion_a_b_d_c,S6: set_c] :
      ( ( not_in_fv_a_b_d_c @ plus @ mult @ valid @ A6 @ S6 )
      = ( ! [Sigma4: a,S: c > d,Delta2: ( c > d ) > set_a,S2: c > d] :
            ( ( equal_outside_c_d @ S @ S2 @ S6 )
           => ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma4 @ S @ Delta2 @ A6 )
              = ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma4 @ S2 @ Delta2 @ A6 ) ) ) ) ) ).

% not_in_fv_def
thf(fact_71_logic_Osat_Osimps_I3_J,axiom,
    ! [Plus2: 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: c > d,Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c,B5: assertion_a_b_d_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( wand_a_b_d_c @ A6 @ B5 ) )
        = ( ! [A5: a,Sigma3: a] :
              ( ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A5 @ S4 @ Delta @ A6 )
                & ( ( some_a @ Sigma3 )
                  = ( Plus2 @ Sigma @ A5 ) ) )
             => ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma3 @ S4 @ Delta @ B5 ) ) ) ) ) ).

% logic.sat.simps(3)
thf(fact_72_logic_Osat__wand,axiom,
    ! [Plus2: 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: c > d,Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c,Sigma: a,B5: assertion_a_b_d_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ! [A7: a,Sigma2: a] :
            ( ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A7 @ S4 @ Delta @ A6 )
              & ( ( some_a @ Sigma2 )
                = ( Plus2 @ Sigma @ A7 ) ) )
           => ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma2 @ S4 @ Delta @ B5 ) )
       => ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( wand_a_b_d_c @ A6 @ B5 ) ) ) ) ).

% logic.sat_wand
thf(fact_73_logic_Osmult__distrib,axiom,
    ! [Plus2: 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,Q: b,R: b] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Smult @ P @ ( Sadd @ Q @ R ) )
        = ( Sadd @ ( Smult @ P @ Q ) @ ( Smult @ P @ R ) ) ) ) ).

% logic.smult_distrib
thf(fact_74_logic_Osone__neutral,axiom,
    ! [Plus2: 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 @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Smult @ One @ P )
        = P ) ) ).

% logic.sone_neutral
thf(fact_75_logic_Osinv__inverse,axiom,
    ! [Plus2: 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 @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Smult @ P @ ( Sinv @ P ) )
        = One ) ) ).

% logic.sinv_inverse
thf(fact_76_logic_Oone__neutral,axiom,
    ! [Plus2: 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 @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Mult @ One @ A )
        = A ) ) ).

% logic.one_neutral
thf(fact_77_logic_Odouble__mult,axiom,
    ! [Plus2: 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,Q: b,A: a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Mult @ P @ ( Mult @ Q @ A ) )
        = ( Mult @ ( Smult @ P @ Q ) @ A ) ) ) ).

% logic.double_mult
thf(fact_78_logic_Ocommutative,axiom,
    ! [Plus2: 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 @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Plus2 @ A @ B )
        = ( Plus2 @ B @ A ) ) ) ).

% logic.commutative
thf(fact_79_logic_Ounique__inv,axiom,
    ! [Plus2: 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 @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( A
          = ( Mult @ P @ B ) )
        = ( B
          = ( Mult @ ( Sinv @ P ) @ A ) ) ) ) ).

% logic.unique_inv
thf(fact_80_logic_Osmult__comm,axiom,
    ! [Plus2: 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,Q: b] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Smult @ P @ Q )
        = ( Smult @ Q @ P ) ) ) ).

% logic.smult_comm
thf(fact_81_logic_Osmult__asso,axiom,
    ! [Plus2: 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,Q: b,R: b] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Smult @ ( Smult @ P @ Q ) @ R )
        = ( Smult @ P @ ( Smult @ Q @ R ) ) ) ) ).

% logic.smult_asso
thf(fact_82_logic_Ocan__divide,axiom,
    ! [Plus2: 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 @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( Mult @ P @ A )
          = ( Mult @ P @ B ) )
       => ( A = B ) ) ) ).

% logic.can_divide
thf(fact_83_logic_Osadd__comm,axiom,
    ! [Plus2: 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,Q: b] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Sadd @ P @ Q )
        = ( Sadd @ Q @ P ) ) ) ).

% logic.sadd_comm
thf(fact_84_logic_Osat_Ocong,axiom,
    sat_a_b_c_d = sat_a_b_c_d ).

% logic.sat.cong
thf(fact_85_logic_Odistrib__mult,axiom,
    ! [Plus2: 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,Q: b,X2: a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( some_a @ ( Mult @ ( Sadd @ P @ Q ) @ X2 ) )
        = ( Plus2 @ ( Mult @ P @ X2 ) @ ( Mult @ Q @ X2 ) ) ) ) ).

% logic.distrib_mult
thf(fact_86_logic_Oplus__mult,axiom,
    ! [Plus2: 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 @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( some_a @ A )
          = ( Plus2 @ B @ C ) )
       => ( ( some_a @ ( Mult @ P @ A ) )
          = ( Plus2 @ ( Mult @ P @ B ) @ ( Mult @ P @ C ) ) ) ) ) ).

% logic.plus_mult
thf(fact_87_logic_Omove__sum,axiom,
    ! [Plus2: 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,A12: a,A22: a,B: a,B1: a,B2: a,X2: a,X1: a,X22: a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( some_a @ A )
          = ( Plus2 @ A12 @ A22 ) )
       => ( ( ( some_a @ B )
            = ( Plus2 @ B1 @ B2 ) )
         => ( ( ( some_a @ X2 )
              = ( Plus2 @ A @ B ) )
           => ( ( ( some_a @ X1 )
                = ( Plus2 @ A12 @ B1 ) )
             => ( ( ( some_a @ X22 )
                  = ( Plus2 @ A22 @ B2 ) )
               => ( ( some_a @ X2 )
                  = ( Plus2 @ X1 @ X22 ) ) ) ) ) ) ) ) ).

% logic.move_sum
thf(fact_88_logic_Oasso1,axiom,
    ! [Plus2: 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 @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( ( Plus2 @ A @ B )
            = ( some_a @ Ab ) )
          & ( ( Plus2 @ B @ C )
            = ( some_a @ Bc ) ) )
       => ( ( Plus2 @ Ab @ C )
          = ( Plus2 @ A @ Bc ) ) ) ) ).

% logic.asso1
thf(fact_89_combine__options__cases,axiom,
    ! [X2: option_a,P2: option_a > option6413918287372586467_a_c_d > $o,Y: option6413918287372586467_a_c_d] :
      ( ( ( X2 = none_a )
       => ( P2 @ X2 @ Y ) )
     => ( ( ( Y = none_P2239999354720063602_a_c_d )
         => ( P2 @ X2 @ Y ) )
       => ( ! [A7: a,B6: product_prod_a_c_d] :
              ( ( X2
                = ( some_a @ A7 ) )
             => ( ( Y
                  = ( some_P1084500821511757806_a_c_d @ B6 ) )
               => ( P2 @ X2 @ Y ) ) )
         => ( P2 @ X2 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_90_combine__options__cases,axiom,
    ! [X2: option6413918287372586467_a_c_d,P2: option6413918287372586467_a_c_d > option_a > $o,Y: option_a] :
      ( ( ( X2 = none_P2239999354720063602_a_c_d )
       => ( P2 @ X2 @ Y ) )
     => ( ( ( Y = none_a )
         => ( P2 @ X2 @ Y ) )
       => ( ! [A7: product_prod_a_c_d,B6: a] :
              ( ( X2
                = ( some_P1084500821511757806_a_c_d @ A7 ) )
             => ( ( Y
                  = ( some_a @ B6 ) )
               => ( P2 @ X2 @ Y ) ) )
         => ( P2 @ X2 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_91_combine__options__cases,axiom,
    ! [X2: option6413918287372586467_a_c_d,P2: option6413918287372586467_a_c_d > option6413918287372586467_a_c_d > $o,Y: option6413918287372586467_a_c_d] :
      ( ( ( X2 = none_P2239999354720063602_a_c_d )
       => ( P2 @ X2 @ Y ) )
     => ( ( ( Y = none_P2239999354720063602_a_c_d )
         => ( P2 @ X2 @ Y ) )
       => ( ! [A7: product_prod_a_c_d,B6: product_prod_a_c_d] :
              ( ( X2
                = ( some_P1084500821511757806_a_c_d @ A7 ) )
             => ( ( Y
                  = ( some_P1084500821511757806_a_c_d @ B6 ) )
               => ( P2 @ X2 @ Y ) ) )
         => ( P2 @ X2 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_92_combine__options__cases,axiom,
    ! [X2: option_a,P2: option_a > option_a > $o,Y: option_a] :
      ( ( ( X2 = none_a )
       => ( P2 @ X2 @ Y ) )
     => ( ( ( Y = none_a )
         => ( P2 @ X2 @ Y ) )
       => ( ! [A7: a,B6: a] :
              ( ( X2
                = ( some_a @ A7 ) )
             => ( ( Y
                  = ( some_a @ B6 ) )
               => ( P2 @ X2 @ Y ) ) )
         => ( P2 @ X2 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_93_split__option__all,axiom,
    ( ( ^ [P3: option6413918287372586467_a_c_d > $o] :
        ! [X5: option6413918287372586467_a_c_d] : ( P3 @ X5 ) )
    = ( ^ [P4: option6413918287372586467_a_c_d > $o] :
          ( ( P4 @ none_P2239999354720063602_a_c_d )
          & ! [X: product_prod_a_c_d] : ( P4 @ ( some_P1084500821511757806_a_c_d @ X ) ) ) ) ) ).

% split_option_all
thf(fact_94_split__option__all,axiom,
    ( ( ^ [P3: option_a > $o] :
        ! [X5: option_a] : ( P3 @ X5 ) )
    = ( ^ [P4: option_a > $o] :
          ( ( P4 @ none_a )
          & ! [X: a] : ( P4 @ ( some_a @ X ) ) ) ) ) ).

% split_option_all
thf(fact_95_split__option__ex,axiom,
    ( ( ^ [P3: option6413918287372586467_a_c_d > $o] :
        ? [X5: option6413918287372586467_a_c_d] : ( P3 @ X5 ) )
    = ( ^ [P4: option6413918287372586467_a_c_d > $o] :
          ( ( P4 @ none_P2239999354720063602_a_c_d )
          | ? [X: product_prod_a_c_d] : ( P4 @ ( some_P1084500821511757806_a_c_d @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_96_split__option__ex,axiom,
    ( ( ^ [P3: option_a > $o] :
        ? [X5: option_a] : ( P3 @ X5 ) )
    = ( ^ [P4: option_a > $o] :
          ( ( P4 @ none_a )
          | ? [X: a] : ( P4 @ ( some_a @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_97_option_Oexhaust,axiom,
    ! [Y: option6413918287372586467_a_c_d] :
      ( ( Y != none_P2239999354720063602_a_c_d )
     => ~ ! [X23: product_prod_a_c_d] :
            ( Y
           != ( some_P1084500821511757806_a_c_d @ X23 ) ) ) ).

% option.exhaust
thf(fact_98_option_Oexhaust,axiom,
    ! [Y: option_a] :
      ( ( Y != none_a )
     => ~ ! [X23: a] :
            ( Y
           != ( some_a @ X23 ) ) ) ).

% option.exhaust
thf(fact_99_option_OdiscI,axiom,
    ! [Option: option6413918287372586467_a_c_d,X22: product_prod_a_c_d] :
      ( ( Option
        = ( some_P1084500821511757806_a_c_d @ X22 ) )
     => ( Option != none_P2239999354720063602_a_c_d ) ) ).

% option.discI
thf(fact_100_option_OdiscI,axiom,
    ! [Option: option_a,X22: a] :
      ( ( Option
        = ( some_a @ X22 ) )
     => ( Option != none_a ) ) ).

% option.discI
thf(fact_101_option_Odistinct_I1_J,axiom,
    ! [X22: product_prod_a_c_d] :
      ( none_P2239999354720063602_a_c_d
     != ( some_P1084500821511757806_a_c_d @ X22 ) ) ).

% option.distinct(1)
thf(fact_102_option_Odistinct_I1_J,axiom,
    ! [X22: a] :
      ( none_a
     != ( some_a @ X22 ) ) ).

% option.distinct(1)
thf(fact_103_logic_Ocompatible__multiples,axiom,
    ! [Plus2: 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,Q: b,B: a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( pre_compatible_a @ Plus2 @ ( Mult @ P @ A ) @ ( Mult @ Q @ B ) )
       => ( pre_compatible_a @ Plus2 @ A @ B ) ) ) ).

% logic.compatible_multiples
thf(fact_104_logic_Ocompatible__imp,axiom,
    ! [Plus2: 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 @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( pre_compatible_a @ Plus2 @ A @ B )
       => ( pre_compatible_a @ Plus2 @ ( Mult @ P @ A ) @ ( Mult @ P @ B ) ) ) ) ).

% logic.compatible_imp
thf(fact_105_logic_Ocompatible__iff,axiom,
    ! [Plus2: 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 @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( pre_compatible_a @ Plus2 @ A @ B )
        = ( pre_compatible_a @ Plus2 @ ( Mult @ P @ A ) @ ( Mult @ P @ B ) ) ) ) ).

% logic.compatible_iff
thf(fact_106_logic_Olarger__same,axiom,
    ! [Plus2: 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 @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( pre_larger_a @ Plus2 @ A @ B )
        = ( pre_larger_a @ Plus2 @ ( Mult @ P @ A ) @ ( Mult @ P @ B ) ) ) ) ).

% logic.larger_same
thf(fact_107_logic_Ovalid__mono,axiom,
    ! [Plus2: 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 @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( Valid @ A )
          & ( pre_larger_a @ Plus2 @ A @ B ) )
       => ( Valid @ B ) ) ) ).

% logic.valid_mono
thf(fact_108_pre__logic_Ocompatible__def,axiom,
    ( pre_compatible_a
    = ( ^ [Plus: a > a > option_a,A5: a,B4: a] :
          ( ( Plus @ A5 @ B4 )
         != none_a ) ) ) ).

% pre_logic.compatible_def
thf(fact_109_logic_Oequal__outside__def,axiom,
    ! [Plus2: 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: c > d,S7: c > d,S6: set_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( equal_outside_c_d @ S4 @ S7 @ S6 )
        = ( ! [X: c] :
              ( ~ ( member_c @ X @ S6 )
             => ( ( S4 @ X )
                = ( S7 @ X ) ) ) ) ) ) ).

% logic.equal_outside_def
thf(fact_110_logic_Oasso3,axiom,
    ! [Plus2: 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 @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ~ ( pre_compatible_a @ Plus2 @ A @ B )
       => ( ( ( Plus2 @ B @ C )
            = ( some_a @ Bc ) )
         => ~ ( pre_compatible_a @ Plus2 @ A @ Bc ) ) ) ) ).

% logic.asso3
thf(fact_111_logic_Oasso2,axiom,
    ! [Plus2: 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 @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( ( Plus2 @ A @ B )
            = ( some_a @ Ab ) )
          & ~ ( pre_compatible_a @ Plus2 @ B @ C ) )
       => ~ ( pre_compatible_a @ Plus2 @ Ab @ C ) ) ) ).

% logic.asso2
thf(fact_112_logic_Olarger__first__sum,axiom,
    ! [Plus2: 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,X2: a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( some_a @ Y )
          = ( Plus2 @ A @ B ) )
       => ( ( pre_larger_a @ Plus2 @ X2 @ Y )
         => ? [A3: a] :
              ( ( ( some_a @ X2 )
                = ( Plus2 @ A3 @ B ) )
              & ( pre_larger_a @ Plus2 @ A3 @ A ) ) ) ) ) ).

% logic.larger_first_sum
thf(fact_113_logic_Osum__both__larger,axiom,
    ! [Plus2: 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,A4: a,B3: a,X2: a,A: a,B: a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( some_a @ X3 )
          = ( Plus2 @ A4 @ B3 ) )
       => ( ( ( some_a @ X2 )
            = ( Plus2 @ A @ B ) )
         => ( ( pre_larger_a @ Plus2 @ A4 @ A )
           => ( ( pre_larger_a @ Plus2 @ B3 @ B )
             => ( pre_larger_a @ Plus2 @ X3 @ X2 ) ) ) ) ) ) ).

% logic.sum_both_larger
thf(fact_114_logic_Olarger__implies__compatible,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X2: a,Y: a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( pre_larger_a @ Plus2 @ X2 @ Y )
       => ( pre_compatible_a @ Plus2 @ X2 @ Y ) ) ) ).

% logic.larger_implies_compatible
thf(fact_115_logic_Ocompatible__smaller,axiom,
    ! [Plus2: 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,X2: a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( pre_larger_a @ Plus2 @ A @ B )
       => ( ( pre_compatible_a @ Plus2 @ X2 @ A )
         => ( pre_compatible_a @ Plus2 @ X2 @ B ) ) ) ) ).

% logic.compatible_smaller
thf(fact_116_logic__def,axiom,
    ( logic_a_b
    = ( ^ [Plus: a > a > option_a,Mult2: b > a > a,Smult2: b > b > b,Sadd2: b > b > b,Sinv2: b > b,One2: b,Valid2: a > $o] :
          ( ! [A5: a,B4: a] :
              ( ( Plus @ A5 @ B4 )
              = ( Plus @ B4 @ A5 ) )
          & ! [A5: a,B4: a,Ab2: a,C2: a,Bc2: a] :
              ( ( ( ( Plus @ A5 @ B4 )
                  = ( some_a @ Ab2 ) )
                & ( ( Plus @ B4 @ C2 )
                  = ( some_a @ Bc2 ) ) )
             => ( ( Plus @ Ab2 @ C2 )
                = ( Plus @ A5 @ Bc2 ) ) )
          & ! [A5: a,B4: a,Ab2: a,C2: a] :
              ( ( ( ( Plus @ A5 @ B4 )
                  = ( some_a @ Ab2 ) )
                & ~ ( pre_compatible_a @ Plus @ B4 @ C2 ) )
             => ~ ( pre_compatible_a @ Plus @ Ab2 @ C2 ) )
          & ! [P5: b] :
              ( ( Smult2 @ P5 @ ( Sinv2 @ P5 ) )
              = One2 )
          & ! [P5: b] :
              ( ( Smult2 @ One2 @ P5 )
              = P5 )
          & ! [P5: b,Q3: b] :
              ( ( Sadd2 @ P5 @ Q3 )
              = ( Sadd2 @ Q3 @ P5 ) )
          & ! [P5: b,Q3: b] :
              ( ( Smult2 @ P5 @ Q3 )
              = ( Smult2 @ Q3 @ P5 ) )
          & ! [P5: b,Q3: b,R2: b] :
              ( ( Smult2 @ P5 @ ( Sadd2 @ Q3 @ R2 ) )
              = ( Sadd2 @ ( Smult2 @ P5 @ Q3 ) @ ( Smult2 @ P5 @ R2 ) ) )
          & ! [P5: b,Q3: b,R2: b] :
              ( ( Smult2 @ ( Smult2 @ P5 @ Q3 ) @ R2 )
              = ( Smult2 @ P5 @ ( Smult2 @ Q3 @ R2 ) ) )
          & ! [P5: b,Q3: b,A5: a] :
              ( ( Mult2 @ P5 @ ( Mult2 @ Q3 @ A5 ) )
              = ( Mult2 @ ( Smult2 @ P5 @ Q3 ) @ A5 ) )
          & ! [A5: a,B4: a,C2: a,P5: b] :
              ( ( ( some_a @ A5 )
                = ( Plus @ B4 @ C2 ) )
             => ( ( some_a @ ( Mult2 @ P5 @ A5 ) )
                = ( Plus @ ( Mult2 @ P5 @ B4 ) @ ( Mult2 @ P5 @ C2 ) ) ) )
          & ! [P5: b,Q3: b,X: a] :
              ( ( some_a @ ( Mult2 @ ( Sadd2 @ P5 @ Q3 ) @ X ) )
              = ( Plus @ ( Mult2 @ P5 @ X ) @ ( Mult2 @ Q3 @ X ) ) )
          & ! [A5: a] :
              ( ( Mult2 @ One2 @ A5 )
              = A5 )
          & ! [A5: a,B4: a] :
              ( ( ( Valid2 @ A5 )
                & ( pre_larger_a @ Plus @ A5 @ B4 ) )
             => ( Valid2 @ B4 ) ) ) ) ) ).

% logic_def
thf(fact_117_combinable__instantiate__one,axiom,
    ! [Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c,A: a,S4: c > d,B: a,X2: a,P: b,Q: b] :
      ( ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ A6 )
     => ( ( sat_a_b_c_d @ plus @ mult @ valid @ A @ S4 @ Delta @ A6 )
       => ( ( sat_a_b_c_d @ plus @ mult @ valid @ B @ S4 @ Delta @ A6 )
         => ( ( ( some_a @ X2 )
              = ( plus @ ( mult @ P @ A ) @ ( mult @ Q @ B ) ) )
           => ( ( ( sadd @ P @ Q )
                = one )
             => ( sat_a_b_c_d @ plus @ mult @ valid @ X2 @ S4 @ Delta @ A6 ) ) ) ) ) ) ).

% combinable_instantiate_one
thf(fact_118_combinableI__old,axiom,
    ! [Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c] :
      ( ! [A7: a,B6: a,P6: b,Q4: b,X4: a,Sigma5: a,S5: c > d] :
          ( ( ( sat_a_b_c_d @ plus @ mult @ valid @ A7 @ S5 @ Delta @ A6 )
            & ( sat_a_b_c_d @ plus @ mult @ valid @ B6 @ S5 @ Delta @ A6 )
            & ( ( some_a @ Sigma5 )
              = ( plus @ ( mult @ P6 @ A7 ) @ ( mult @ Q4 @ B6 ) ) )
            & ( Sigma5
              = ( mult @ ( sadd @ P6 @ Q4 ) @ X4 ) ) )
         => ( sat_a_b_c_d @ plus @ mult @ valid @ X4 @ S5 @ Delta @ A6 ) )
     => ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ A6 ) ) ).

% combinableI_old
thf(fact_119_entails__def,axiom,
    ! [A6: assertion_a_b_d_c,Delta: ( c > d ) > set_a,B5: assertion_a_b_d_c] :
      ( ( entails_a_b_d_c @ plus @ mult @ valid @ A6 @ Delta @ B5 )
      = ( ! [Sigma4: a,S: c > d] :
            ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma4 @ S @ Delta @ A6 )
           => ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma4 @ S @ Delta @ B5 ) ) ) ) ).

% entails_def
thf(fact_120_entailsI,axiom,
    ! [Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c,B5: assertion_a_b_d_c] :
      ( ! [Sigma5: a,S5: c > d] :
          ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S5 @ Delta @ A6 )
         => ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S5 @ Delta @ B5 ) )
     => ( entails_a_b_d_c @ plus @ mult @ valid @ A6 @ Delta @ B5 ) ) ).

% entailsI
thf(fact_121_sat_Osimps_I10_J,axiom,
    ! [Sigma: a,S4: c > d,Delta: ( c > d ) > set_a] :
      ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ pred_a_b_d_c )
      = ( member_a @ Sigma @ ( Delta @ S4 ) ) ) ).

% sat.simps(10)
thf(fact_122_sat_Osimps_I2_J,axiom,
    ! [Sigma: a,S4: c > d,Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c,B5: assertion_a_b_d_c] :
      ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( star_a_b_d_c @ A6 @ B5 ) )
      = ( ? [A5: a,B4: a] :
            ( ( ( some_a @ Sigma )
              = ( plus @ A5 @ B4 ) )
            & ( sat_a_b_c_d @ plus @ mult @ valid @ A5 @ S4 @ Delta @ A6 )
            & ( sat_a_b_c_d @ plus @ mult @ valid @ B4 @ S4 @ Delta @ B5 ) ) ) ) ).

% sat.simps(2)
thf(fact_123_sat_Osimps_I12_J,axiom,
    ! [Sigma: a,S4: c > d,Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c] :
      ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( wildcard_a_b_d_c @ A6 ) )
      = ( ? [A5: a,P5: b] :
            ( ( Sigma
              = ( mult @ P5 @ A5 ) )
            & ( sat_a_b_c_d @ plus @ mult @ valid @ A5 @ S4 @ Delta @ A6 ) ) ) ) ).

% sat.simps(12)
thf(fact_124_sat_Osimps_I6_J,axiom,
    ! [Sigma: a,S4: c > d,Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c,B5: assertion_a_b_d_c] :
      ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( or_a_b_d_c @ A6 @ B5 ) )
      = ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ A6 )
        | ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ B5 ) ) ) ).

% sat.simps(6)
thf(fact_125_sat_Osimps_I7_J,axiom,
    ! [Sigma: a,S4: c > d,Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c,B5: assertion_a_b_d_c] :
      ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( and_a_b_d_c @ A6 @ B5 ) )
      = ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ A6 )
        & ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ B5 ) ) ) ).

% sat.simps(7)
thf(fact_126_sat__imp,axiom,
    ! [Sigma: a,S4: c > d,Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c,B5: assertion_a_b_d_c] :
      ( ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ A6 )
       => ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ B5 ) )
     => ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( imp_a_b_d_c @ A6 @ B5 ) ) ) ).

% sat_imp
thf(fact_127_assertion_Oinject_I3_J,axiom,
    ! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c,Y31: assertion_a_b_d_c,Y32: assertion_a_b_d_c] :
      ( ( ( star_a_b_d_c @ X31 @ X32 )
        = ( star_a_b_d_c @ Y31 @ Y32 ) )
      = ( ( X31 = Y31 )
        & ( X32 = Y32 ) ) ) ).

% assertion.inject(3)
thf(fact_128_assertion_Oinject_I7_J,axiom,
    ! [X71: assertion_a_b_d_c,X72: assertion_a_b_d_c,Y71: assertion_a_b_d_c,Y72: assertion_a_b_d_c] :
      ( ( ( imp_a_b_d_c @ X71 @ X72 )
        = ( imp_a_b_d_c @ Y71 @ Y72 ) )
      = ( ( X71 = Y71 )
        & ( X72 = Y72 ) ) ) ).

% assertion.inject(7)
thf(fact_129_assertion_Oinject_I5_J,axiom,
    ! [X51: assertion_a_b_d_c,X52: assertion_a_b_d_c,Y51: assertion_a_b_d_c,Y52: assertion_a_b_d_c] :
      ( ( ( or_a_b_d_c @ X51 @ X52 )
        = ( or_a_b_d_c @ Y51 @ Y52 ) )
      = ( ( X51 = Y51 )
        & ( X52 = Y52 ) ) ) ).

% assertion.inject(5)
thf(fact_130_assertion_Oinject_I6_J,axiom,
    ! [X61: assertion_a_b_d_c,X62: assertion_a_b_d_c,Y61: assertion_a_b_d_c,Y62: assertion_a_b_d_c] :
      ( ( ( and_a_b_d_c @ X61 @ X62 )
        = ( and_a_b_d_c @ Y61 @ Y62 ) )
      = ( ( X61 = Y61 )
        & ( X62 = Y62 ) ) ) ).

% assertion.inject(6)
thf(fact_131_assertion_Oinject_I11_J,axiom,
    ! [X12: assertion_a_b_d_c,Y12: assertion_a_b_d_c] :
      ( ( ( wildcard_a_b_d_c @ X12 )
        = ( wildcard_a_b_d_c @ Y12 ) )
      = ( X12 = Y12 ) ) ).

% assertion.inject(11)
thf(fact_132_sat_Osimps_I5_J,axiom,
    ! [Sigma: a,S4: c > d,Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c,B5: assertion_a_b_d_c] :
      ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( imp_a_b_d_c @ A6 @ B5 ) )
      = ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ A6 )
       => ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ B5 ) ) ) ).

% sat.simps(5)
thf(fact_133_equivalent__def,axiom,
    ! [A6: assertion_a_b_d_c,Delta: ( c > d ) > set_a,B5: assertion_a_b_d_c] :
      ( ( equivalent_a_b_d_c @ plus @ mult @ valid @ A6 @ Delta @ B5 )
      = ( ( entails_a_b_d_c @ plus @ mult @ valid @ A6 @ Delta @ B5 )
        & ( entails_a_b_d_c @ plus @ mult @ valid @ B5 @ Delta @ A6 ) ) ) ).

% equivalent_def
thf(fact_134_assertion_Odistinct_I45_J,axiom,
    ! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c,X51: assertion_a_b_d_c,X52: assertion_a_b_d_c] :
      ( ( star_a_b_d_c @ X31 @ X32 )
     != ( or_a_b_d_c @ X51 @ X52 ) ) ).

% assertion.distinct(45)
thf(fact_135_assertion_Odistinct_I47_J,axiom,
    ! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c,X61: assertion_a_b_d_c,X62: assertion_a_b_d_c] :
      ( ( star_a_b_d_c @ X31 @ X32 )
     != ( and_a_b_d_c @ X61 @ X62 ) ) ).

% assertion.distinct(47)
thf(fact_136_assertion_Odistinct_I49_J,axiom,
    ! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c,X71: assertion_a_b_d_c,X72: assertion_a_b_d_c] :
      ( ( star_a_b_d_c @ X31 @ X32 )
     != ( imp_a_b_d_c @ X71 @ X72 ) ) ).

% assertion.distinct(49)
thf(fact_137_assertion_Odistinct_I55_J,axiom,
    ! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c] :
      ( ( star_a_b_d_c @ X31 @ X32 )
     != pred_a_b_d_c ) ).

% assertion.distinct(55)
thf(fact_138_assertion_Odistinct_I59_J,axiom,
    ! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c,X12: assertion_a_b_d_c] :
      ( ( star_a_b_d_c @ X31 @ X32 )
     != ( wildcard_a_b_d_c @ X12 ) ) ).

% assertion.distinct(59)
thf(fact_139_assertion_Odistinct_I77_J,axiom,
    ! [X51: assertion_a_b_d_c,X52: assertion_a_b_d_c,X61: assertion_a_b_d_c,X62: assertion_a_b_d_c] :
      ( ( or_a_b_d_c @ X51 @ X52 )
     != ( and_a_b_d_c @ X61 @ X62 ) ) ).

% assertion.distinct(77)
thf(fact_140_assertion_Odistinct_I79_J,axiom,
    ! [X51: assertion_a_b_d_c,X52: assertion_a_b_d_c,X71: assertion_a_b_d_c,X72: assertion_a_b_d_c] :
      ( ( or_a_b_d_c @ X51 @ X52 )
     != ( imp_a_b_d_c @ X71 @ X72 ) ) ).

% assertion.distinct(79)
thf(fact_141_assertion_Odistinct_I85_J,axiom,
    ! [X51: assertion_a_b_d_c,X52: assertion_a_b_d_c] :
      ( ( or_a_b_d_c @ X51 @ X52 )
     != pred_a_b_d_c ) ).

% assertion.distinct(85)
thf(fact_142_assertion_Odistinct_I89_J,axiom,
    ! [X51: assertion_a_b_d_c,X52: assertion_a_b_d_c,X12: assertion_a_b_d_c] :
      ( ( or_a_b_d_c @ X51 @ X52 )
     != ( wildcard_a_b_d_c @ X12 ) ) ).

% assertion.distinct(89)
thf(fact_143_assertion_Odistinct_I91_J,axiom,
    ! [X61: assertion_a_b_d_c,X62: assertion_a_b_d_c,X71: assertion_a_b_d_c,X72: assertion_a_b_d_c] :
      ( ( and_a_b_d_c @ X61 @ X62 )
     != ( imp_a_b_d_c @ X71 @ X72 ) ) ).

% assertion.distinct(91)
thf(fact_144_assertion_Odistinct_I97_J,axiom,
    ! [X61: assertion_a_b_d_c,X62: assertion_a_b_d_c] :
      ( ( and_a_b_d_c @ X61 @ X62 )
     != pred_a_b_d_c ) ).

% assertion.distinct(97)
thf(fact_145_assertion_Odistinct_I101_J,axiom,
    ! [X61: assertion_a_b_d_c,X62: assertion_a_b_d_c,X12: assertion_a_b_d_c] :
      ( ( and_a_b_d_c @ X61 @ X62 )
     != ( wildcard_a_b_d_c @ X12 ) ) ).

% assertion.distinct(101)
thf(fact_146_assertion_Odistinct_I107_J,axiom,
    ! [X71: assertion_a_b_d_c,X72: assertion_a_b_d_c] :
      ( ( imp_a_b_d_c @ X71 @ X72 )
     != pred_a_b_d_c ) ).

% assertion.distinct(107)
thf(fact_147_assertion_Odistinct_I111_J,axiom,
    ! [X71: assertion_a_b_d_c,X72: assertion_a_b_d_c,X12: assertion_a_b_d_c] :
      ( ( imp_a_b_d_c @ X71 @ X72 )
     != ( wildcard_a_b_d_c @ X12 ) ) ).

% assertion.distinct(111)
thf(fact_148_assertion_Odistinct_I129_J,axiom,
    ! [X12: assertion_a_b_d_c] :
      ( pred_a_b_d_c
     != ( wildcard_a_b_d_c @ X12 ) ) ).

% assertion.distinct(129)
thf(fact_149_logic_Ocombinable_Ocong,axiom,
    combinable_a_b_c_d = combinable_a_b_c_d ).

% logic.combinable.cong
thf(fact_150_logic_Oentails_Ocong,axiom,
    entails_a_b_d_c = entails_a_b_d_c ).

% logic.entails.cong
thf(fact_151_logic_Onot__in__fv_Ocong,axiom,
    not_in_fv_a_b_d_c = not_in_fv_a_b_d_c ).

% logic.not_in_fv.cong
thf(fact_152_logic_Oequivalent_Ocong,axiom,
    equivalent_a_b_d_c = equivalent_a_b_d_c ).

% logic.equivalent.cong
thf(fact_153_logic_Ointuitionistic_Ocong,axiom,
    intuit7508411120625971703_b_c_d = intuit7508411120625971703_b_c_d ).

% logic.intuitionistic.cong
thf(fact_154_logic_Opure_Ocong,axiom,
    pure_a_b_d_c = pure_a_b_d_c ).

% logic.pure.cong
thf(fact_155_logic_Oequivalent__def,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A6: assertion_a_b_d_c,Delta: ( c > d ) > set_a,B5: assertion_a_b_d_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( equivalent_a_b_d_c @ Plus2 @ Mult @ Valid @ A6 @ Delta @ B5 )
        = ( ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ A6 @ Delta @ B5 )
          & ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ B5 @ Delta @ A6 ) ) ) ) ).

% logic.equivalent_def
thf(fact_156_assertion_Odistinct_I43_J,axiom,
    ! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c,X41: assertion_a_b_d_c,X42: assertion_a_b_d_c] :
      ( ( star_a_b_d_c @ X31 @ X32 )
     != ( wand_a_b_d_c @ X41 @ X42 ) ) ).

% assertion.distinct(43)
thf(fact_157_assertion_Odistinct_I65_J,axiom,
    ! [X41: assertion_a_b_d_c,X42: assertion_a_b_d_c,X71: assertion_a_b_d_c,X72: assertion_a_b_d_c] :
      ( ( wand_a_b_d_c @ X41 @ X42 )
     != ( imp_a_b_d_c @ X71 @ X72 ) ) ).

% assertion.distinct(65)
thf(fact_158_assertion_Odistinct_I63_J,axiom,
    ! [X41: assertion_a_b_d_c,X42: assertion_a_b_d_c,X61: assertion_a_b_d_c,X62: assertion_a_b_d_c] :
      ( ( wand_a_b_d_c @ X41 @ X42 )
     != ( and_a_b_d_c @ X61 @ X62 ) ) ).

% assertion.distinct(63)
thf(fact_159_assertion_Odistinct_I61_J,axiom,
    ! [X41: assertion_a_b_d_c,X42: assertion_a_b_d_c,X51: assertion_a_b_d_c,X52: assertion_a_b_d_c] :
      ( ( wand_a_b_d_c @ X41 @ X42 )
     != ( or_a_b_d_c @ X51 @ X52 ) ) ).

% assertion.distinct(61)
thf(fact_160_assertion_Odistinct_I75_J,axiom,
    ! [X41: assertion_a_b_d_c,X42: assertion_a_b_d_c,X12: assertion_a_b_d_c] :
      ( ( wand_a_b_d_c @ X41 @ X42 )
     != ( wildcard_a_b_d_c @ X12 ) ) ).

% assertion.distinct(75)
thf(fact_161_assertion_Odistinct_I71_J,axiom,
    ! [X41: assertion_a_b_d_c,X42: assertion_a_b_d_c] :
      ( ( wand_a_b_d_c @ X41 @ X42 )
     != pred_a_b_d_c ) ).

% assertion.distinct(71)
thf(fact_162_logic_Osat_Osimps_I5_J,axiom,
    ! [Plus2: 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: c > d,Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c,B5: assertion_a_b_d_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( imp_a_b_d_c @ A6 @ B5 ) )
        = ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S4 @ Delta @ A6 )
         => ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S4 @ Delta @ B5 ) ) ) ) ).

% logic.sat.simps(5)
thf(fact_163_logic_Osat__imp,axiom,
    ! [Plus2: 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: c > d,Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c,B5: assertion_a_b_d_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S4 @ Delta @ A6 )
         => ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S4 @ Delta @ B5 ) )
       => ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( imp_a_b_d_c @ A6 @ B5 ) ) ) ) ).

% logic.sat_imp
thf(fact_164_logic_Osat_Osimps_I7_J,axiom,
    ! [Plus2: 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: c > d,Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c,B5: assertion_a_b_d_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( and_a_b_d_c @ A6 @ B5 ) )
        = ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S4 @ Delta @ A6 )
          & ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S4 @ Delta @ B5 ) ) ) ) ).

% logic.sat.simps(7)
thf(fact_165_logic_Osat_Osimps_I6_J,axiom,
    ! [Plus2: 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: c > d,Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c,B5: assertion_a_b_d_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( or_a_b_d_c @ A6 @ B5 ) )
        = ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S4 @ Delta @ A6 )
          | ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S4 @ Delta @ B5 ) ) ) ) ).

% logic.sat.simps(6)
thf(fact_166_logic_Osat_Osimps_I12_J,axiom,
    ! [Plus2: 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: c > d,Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( wildcard_a_b_d_c @ A6 ) )
        = ( ? [A5: a,P5: b] :
              ( ( Sigma
                = ( Mult @ P5 @ A5 ) )
              & ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A5 @ S4 @ Delta @ A6 ) ) ) ) ) ).

% logic.sat.simps(12)
thf(fact_167_logic_Osat_Osimps_I10_J,axiom,
    ! [Plus2: 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: c > d,Delta: ( c > d ) > set_a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S4 @ Delta @ pred_a_b_d_c )
        = ( member_a @ Sigma @ ( Delta @ S4 ) ) ) ) ).

% logic.sat.simps(10)
thf(fact_168_logic_Osat_Osimps_I2_J,axiom,
    ! [Plus2: 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: c > d,Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c,B5: assertion_a_b_d_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( star_a_b_d_c @ A6 @ B5 ) )
        = ( ? [A5: a,B4: a] :
              ( ( ( some_a @ Sigma )
                = ( Plus2 @ A5 @ B4 ) )
              & ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A5 @ S4 @ Delta @ A6 )
              & ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ B4 @ S4 @ Delta @ B5 ) ) ) ) ) ).

% logic.sat.simps(2)
thf(fact_169_logic_OentailsI,axiom,
    ! [Plus2: 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: ( c > d ) > set_a,A6: assertion_a_b_d_c,B5: assertion_a_b_d_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ! [Sigma5: a,S5: c > d] :
            ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S5 @ Delta @ A6 )
           => ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S5 @ Delta @ B5 ) )
       => ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ A6 @ Delta @ B5 ) ) ) ).

% logic.entailsI
thf(fact_170_logic_Oentails__def,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A6: assertion_a_b_d_c,Delta: ( c > d ) > set_a,B5: assertion_a_b_d_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ A6 @ Delta @ B5 )
        = ( ! [Sigma4: a,S: c > d] :
              ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma4 @ S @ Delta @ A6 )
             => ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma4 @ S @ Delta @ B5 ) ) ) ) ) ).

% logic.entails_def
thf(fact_171_logic_OequivalentI,axiom,
    ! [Plus2: 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: ( c > d ) > set_a,A6: assertion_a_b_d_c,B5: assertion_a_b_d_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ! [Sigma5: a,S5: c > d] :
            ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S5 @ Delta @ A6 )
           => ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S5 @ Delta @ B5 ) )
       => ( ! [Sigma5: a,S5: c > d] :
              ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S5 @ Delta @ B5 )
             => ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S5 @ Delta @ A6 ) )
         => ( equivalent_a_b_d_c @ Plus2 @ Mult @ Valid @ A6 @ Delta @ B5 ) ) ) ) ).

% logic.equivalentI
thf(fact_172_logic_Ocombinable__instantiate__one,axiom,
    ! [Plus2: 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: ( c > d ) > set_a,A6: assertion_a_b_d_c,A: a,S4: c > d,B: a,X2: a,P: b,Q: b] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ A6 )
       => ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A @ S4 @ Delta @ A6 )
         => ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ B @ S4 @ Delta @ A6 )
           => ( ( ( some_a @ X2 )
                = ( Plus2 @ ( Mult @ P @ A ) @ ( Mult @ Q @ B ) ) )
             => ( ( ( Sadd @ P @ Q )
                  = One )
               => ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ X2 @ S4 @ Delta @ A6 ) ) ) ) ) ) ) ).

% logic.combinable_instantiate_one
thf(fact_173_logic_OcombinableI__old,axiom,
    ! [Plus2: 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: ( c > d ) > set_a,A6: assertion_a_b_d_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ! [A7: a,B6: a,P6: b,Q4: b,X4: a,Sigma5: a,S5: c > d] :
            ( ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A7 @ S5 @ Delta @ A6 )
              & ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ B6 @ S5 @ Delta @ A6 )
              & ( ( some_a @ Sigma5 )
                = ( Plus2 @ ( Mult @ P6 @ A7 ) @ ( Mult @ Q4 @ B6 ) ) )
              & ( Sigma5
                = ( Mult @ ( Sadd @ P6 @ Q4 ) @ X4 ) ) )
           => ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ X4 @ S5 @ Delta @ A6 ) )
       => ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ A6 ) ) ) ).

% logic.combinableI_old
thf(fact_174_logic_Opure__def,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A6: assertion_a_b_d_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( pure_a_b_d_c @ Plus2 @ Mult @ Valid @ A6 )
        = ( ! [Sigma4: a,Sigma3: a,S: c > d,Delta2: ( c > d ) > set_a,Delta3: ( c > d ) > set_a] :
              ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma4 @ S @ Delta2 @ A6 )
              = ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma3 @ S @ Delta3 @ A6 ) ) ) ) ) ).

% logic.pure_def
thf(fact_175_logic_Ointuitionistic__def,axiom,
    ! [Plus2: 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: c > d,Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( intuit7508411120625971703_b_c_d @ Plus2 @ Mult @ Valid @ S4 @ Delta @ A6 )
        = ( ! [A5: a,B4: a] :
              ( ( ( pre_larger_a @ Plus2 @ A5 @ B4 )
                & ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ B4 @ S4 @ Delta @ A6 ) )
             => ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A5 @ S4 @ Delta @ A6 ) ) ) ) ) ).

% logic.intuitionistic_def
thf(fact_176_logic_OintuitionisticI,axiom,
    ! [Plus2: 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: c > d,Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ! [A7: a,B6: a] :
            ( ( ( pre_larger_a @ Plus2 @ A7 @ B6 )
              & ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ B6 @ S4 @ Delta @ A6 ) )
           => ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A7 @ S4 @ Delta @ A6 ) )
       => ( intuit7508411120625971703_b_c_d @ Plus2 @ Mult @ Valid @ S4 @ Delta @ A6 ) ) ) ).

% logic.intuitionisticI
thf(fact_177_logic_Onot__in__fv__def,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A6: assertion_a_b_d_c,S6: set_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( not_in_fv_a_b_d_c @ Plus2 @ Mult @ Valid @ A6 @ S6 )
        = ( ! [Sigma4: a,S: c > d,Delta2: ( c > d ) > set_a,S2: c > d] :
              ( ( equal_outside_c_d @ S @ S2 @ S6 )
             => ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma4 @ S @ Delta2 @ A6 )
                = ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma4 @ S2 @ Delta2 @ A6 ) ) ) ) ) ) ).

% logic.not_in_fv_def
thf(fact_178_combinable__def,axiom,
    ! [Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c] :
      ( ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ A6 )
      = ( ! [P5: b,Q3: b] : ( entails_a_b_d_c @ plus @ mult @ valid @ ( star_a_b_d_c @ ( mult_b_a_d_c @ P5 @ A6 ) @ ( mult_b_a_d_c @ Q3 @ A6 ) ) @ Delta @ ( mult_b_a_d_c @ ( sadd @ P5 @ Q3 ) @ A6 ) ) ) ) ).

% combinable_def
thf(fact_179_combinable__instantiate,axiom,
    ! [Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c,A: a,S4: c > d,B: a,X2: a,P: b,Q: b] :
      ( ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ A6 )
     => ( ( sat_a_b_c_d @ plus @ mult @ valid @ A @ S4 @ Delta @ A6 )
       => ( ( sat_a_b_c_d @ plus @ mult @ valid @ B @ S4 @ Delta @ A6 )
         => ( ( ( some_a @ X2 )
              = ( plus @ ( mult @ P @ A ) @ ( mult @ Q @ B ) ) )
           => ( sat_a_b_c_d @ plus @ mult @ valid @ X2 @ S4 @ Delta @ ( mult_b_a_d_c @ ( sadd @ P @ Q ) @ A6 ) ) ) ) ) ) ).

% combinable_instantiate
thf(fact_180_sat_Osimps_I11_J,axiom,
    ! [Sigma: a,S4: c > d,Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c] :
      ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( bounded_a_b_d_c @ A6 ) )
      = ( ( valid @ Sigma )
       => ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ A6 ) ) ) ).

% sat.simps(11)
thf(fact_181_sat_Osimps_I1_J,axiom,
    ! [Sigma: a,S4: c > d,Delta: ( c > d ) > set_a,P: b,A6: assertion_a_b_d_c] :
      ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( mult_b_a_d_c @ P @ A6 ) )
      = ( ? [A5: a] :
            ( ( Sigma
              = ( mult @ P @ A5 ) )
            & ( sat_a_b_c_d @ plus @ mult @ valid @ A5 @ S4 @ Delta @ A6 ) ) ) ) ).

% sat.simps(1)
thf(fact_182_sat__mult,axiom,
    ! [Sigma: a,P: b,S4: c > d,Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c] :
      ( ! [A7: a] :
          ( ( Sigma
            = ( mult @ P @ A7 ) )
         => ( sat_a_b_c_d @ plus @ mult @ valid @ A7 @ S4 @ Delta @ A6 ) )
     => ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( mult_b_a_d_c @ P @ A6 ) ) ) ).

% sat_mult
thf(fact_183_sat_Osimps_I4_J,axiom,
    ! [Sigma: a,S4: c > d,Delta: ( c > d ) > set_a,B: ( c > d ) > a > $o] :
      ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( sem_c_d_a_b @ B ) )
      = ( B @ S4 @ Sigma ) ) ).

% sat.simps(4)
thf(fact_184_logic_Ocombinable__def,axiom,
    ! [Plus2: 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: ( c > d ) > set_a,A6: assertion_a_b_d_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ A6 )
        = ( ! [P5: b,Q3: b] : ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ ( star_a_b_d_c @ ( mult_b_a_d_c @ P5 @ A6 ) @ ( mult_b_a_d_c @ Q3 @ A6 ) ) @ Delta @ ( mult_b_a_d_c @ ( Sadd @ P5 @ Q3 ) @ A6 ) ) ) ) ) ).

% logic.combinable_def
thf(fact_185_sat_Osimps_I8_J,axiom,
    ! [Sigma: a,S4: c > d,Delta: ( c > d ) > set_a,X2: c,A6: assertion_a_b_d_c] :
      ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( exists_c_a_b_d @ X2 @ A6 ) )
      = ( ? [V: d] : ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ ( fun_upd_c_d @ S4 @ X2 @ V ) @ Delta @ A6 ) ) ) ).

% sat.simps(8)
thf(fact_186_sat_Osimps_I9_J,axiom,
    ! [Sigma: a,S4: c > d,Delta: ( c > d ) > set_a,X2: c,A6: assertion_a_b_d_c] :
      ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( forall_c_a_b_d @ X2 @ A6 ) )
      = ( ! [V: d] : ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ ( fun_upd_c_d @ S4 @ X2 @ V ) @ Delta @ A6 ) ) ) ).

% sat.simps(9)
thf(fact_187_assertion_Oinject_I2_J,axiom,
    ! [X21: b,X222: assertion_a_b_d_c,Y21: b,Y22: assertion_a_b_d_c] :
      ( ( ( mult_b_a_d_c @ X21 @ X222 )
        = ( mult_b_a_d_c @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% assertion.inject(2)
thf(fact_188_assertion_Oinject_I10_J,axiom,
    ! [X11: assertion_a_b_d_c,Y11: assertion_a_b_d_c] :
      ( ( ( bounded_a_b_d_c @ X11 )
        = ( bounded_a_b_d_c @ Y11 ) )
      = ( X11 = Y11 ) ) ).

% assertion.inject(10)
thf(fact_189_assertion_Oinject_I9_J,axiom,
    ! [X91: c,X92: assertion_a_b_d_c,Y91: c,Y92: assertion_a_b_d_c] :
      ( ( ( forall_c_a_b_d @ X91 @ X92 )
        = ( forall_c_a_b_d @ Y91 @ Y92 ) )
      = ( ( X91 = Y91 )
        & ( X92 = Y92 ) ) ) ).

% assertion.inject(9)
thf(fact_190_assertion_Oinject_I8_J,axiom,
    ! [X81: c,X82: assertion_a_b_d_c,Y81: c,Y82: assertion_a_b_d_c] :
      ( ( ( exists_c_a_b_d @ X81 @ X82 )
        = ( exists_c_a_b_d @ Y81 @ Y82 ) )
      = ( ( X81 = Y81 )
        & ( X82 = Y82 ) ) ) ).

% assertion.inject(8)
thf(fact_191_assertion_Oinject_I1_J,axiom,
    ! [X1: ( c > d ) > a > $o,Y1: ( c > d ) > a > $o] :
      ( ( ( sem_c_d_a_b @ X1 )
        = ( sem_c_d_a_b @ Y1 ) )
      = ( X1 = Y1 ) ) ).

% assertion.inject(1)
thf(fact_192_sat__forall,axiom,
    ! [Sigma: a,S4: c > d,X2: c,Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c] :
      ( ! [V2: d] : ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ ( fun_upd_c_d @ S4 @ X2 @ V2 ) @ Delta @ A6 )
     => ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S4 @ Delta @ ( forall_c_a_b_d @ X2 @ A6 ) ) ) ).

% sat_forall
thf(fact_193_assertion_Odistinct_I123_J,axiom,
    ! [X91: c,X92: assertion_a_b_d_c,X11: assertion_a_b_d_c] :
      ( ( forall_c_a_b_d @ X91 @ X92 )
     != ( bounded_a_b_d_c @ X11 ) ) ).

% assertion.distinct(123)
thf(fact_194_assertion_Odistinct_I117_J,axiom,
    ! [X81: c,X82: assertion_a_b_d_c,X11: assertion_a_b_d_c] :
      ( ( exists_c_a_b_d @ X81 @ X82 )
     != ( bounded_a_b_d_c @ X11 ) ) ).

% assertion.distinct(117)
thf(fact_195_assertion_Odistinct_I113_J,axiom,
    ! [X81: c,X82: assertion_a_b_d_c,X91: c,X92: assertion_a_b_d_c] :
      ( ( exists_c_a_b_d @ X81 @ X82 )
     != ( forall_c_a_b_d @ X91 @ X92 ) ) ).

% assertion.distinct(113)
thf(fact_196_assertion_Odistinct_I39_J,axiom,
    ! [X21: b,X222: assertion_a_b_d_c,X11: assertion_a_b_d_c] :
      ( ( mult_b_a_d_c @ X21 @ X222 )
     != ( bounded_a_b_d_c @ X11 ) ) ).

% assertion.distinct(39)
thf(fact_197_assertion_Odistinct_I35_J,axiom,
    ! [X21: b,X222: assertion_a_b_d_c,X91: c,X92: assertion_a_b_d_c] :
      ( ( mult_b_a_d_c @ X21 @ X222 )
     != ( forall_c_a_b_d @ X91 @ X92 ) ) ).

% assertion.distinct(35)
thf(fact_198_assertion_Odistinct_I33_J,axiom,
    ! [X21: b,X222: assertion_a_b_d_c,X81: c,X82: assertion_a_b_d_c] :
      ( ( mult_b_a_d_c @ X21 @ X222 )
     != ( exists_c_a_b_d @ X81 @ X82 ) ) ).

% assertion.distinct(33)
thf(fact_199_assertion_Odistinct_I19_J,axiom,
    ! [X1: ( c > d ) > a > $o,X11: assertion_a_b_d_c] :
      ( ( sem_c_d_a_b @ X1 )
     != ( bounded_a_b_d_c @ X11 ) ) ).

% assertion.distinct(19)
thf(fact_200_assertion_Odistinct_I15_J,axiom,
    ! [X1: ( c > d ) > a > $o,X91: c,X92: assertion_a_b_d_c] :
      ( ( sem_c_d_a_b @ X1 )
     != ( forall_c_a_b_d @ X91 @ X92 ) ) ).

% assertion.distinct(15)
thf(fact_201_assertion_Odistinct_I13_J,axiom,
    ! [X1: ( c > d ) > a > $o,X81: c,X82: assertion_a_b_d_c] :
      ( ( sem_c_d_a_b @ X1 )
     != ( exists_c_a_b_d @ X81 @ X82 ) ) ).

% assertion.distinct(13)
thf(fact_202_assertion_Odistinct_I1_J,axiom,
    ! [X1: ( c > d ) > a > $o,X21: b,X222: assertion_a_b_d_c] :
      ( ( sem_c_d_a_b @ X1 )
     != ( mult_b_a_d_c @ X21 @ X222 ) ) ).

% assertion.distinct(1)
thf(fact_203_logic_Osat__forall,axiom,
    ! [Plus2: 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: c > d,X2: c,Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ! [V2: d] : ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ ( fun_upd_c_d @ S4 @ X2 @ V2 ) @ Delta @ A6 )
       => ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( forall_c_a_b_d @ X2 @ A6 ) ) ) ) ).

% logic.sat_forall
thf(fact_204_logic_Osat_Osimps_I9_J,axiom,
    ! [Plus2: 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: c > d,Delta: ( c > d ) > set_a,X2: c,A6: assertion_a_b_d_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( forall_c_a_b_d @ X2 @ A6 ) )
        = ( ! [V: d] : ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ ( fun_upd_c_d @ S4 @ X2 @ V ) @ Delta @ A6 ) ) ) ) ).

% logic.sat.simps(9)
thf(fact_205_logic_Osat_Osimps_I8_J,axiom,
    ! [Plus2: 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: c > d,Delta: ( c > d ) > set_a,X2: c,A6: assertion_a_b_d_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( exists_c_a_b_d @ X2 @ A6 ) )
        = ( ? [V: d] : ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ ( fun_upd_c_d @ S4 @ X2 @ V ) @ Delta @ A6 ) ) ) ) ).

% logic.sat.simps(8)
thf(fact_206_assertion_Odistinct_I3_J,axiom,
    ! [X1: ( c > d ) > a > $o,X31: assertion_a_b_d_c,X32: assertion_a_b_d_c] :
      ( ( sem_c_d_a_b @ X1 )
     != ( star_a_b_d_c @ X31 @ X32 ) ) ).

% assertion.distinct(3)
thf(fact_207_assertion_Odistinct_I5_J,axiom,
    ! [X1: ( c > d ) > a > $o,X41: assertion_a_b_d_c,X42: assertion_a_b_d_c] :
      ( ( sem_c_d_a_b @ X1 )
     != ( wand_a_b_d_c @ X41 @ X42 ) ) ).

% assertion.distinct(5)
thf(fact_208_assertion_Odistinct_I11_J,axiom,
    ! [X1: ( c > d ) > a > $o,X71: assertion_a_b_d_c,X72: assertion_a_b_d_c] :
      ( ( sem_c_d_a_b @ X1 )
     != ( imp_a_b_d_c @ X71 @ X72 ) ) ).

% assertion.distinct(11)
thf(fact_209_assertion_Odistinct_I9_J,axiom,
    ! [X1: ( c > d ) > a > $o,X61: assertion_a_b_d_c,X62: assertion_a_b_d_c] :
      ( ( sem_c_d_a_b @ X1 )
     != ( and_a_b_d_c @ X61 @ X62 ) ) ).

% assertion.distinct(9)
thf(fact_210_assertion_Odistinct_I7_J,axiom,
    ! [X1: ( c > d ) > a > $o,X51: assertion_a_b_d_c,X52: assertion_a_b_d_c] :
      ( ( sem_c_d_a_b @ X1 )
     != ( or_a_b_d_c @ X51 @ X52 ) ) ).

% assertion.distinct(7)
thf(fact_211_assertion_Odistinct_I21_J,axiom,
    ! [X1: ( c > d ) > a > $o,X12: assertion_a_b_d_c] :
      ( ( sem_c_d_a_b @ X1 )
     != ( wildcard_a_b_d_c @ X12 ) ) ).

% assertion.distinct(21)
thf(fact_212_assertion_Odistinct_I17_J,axiom,
    ! [X1: ( c > d ) > a > $o] :
      ( ( sem_c_d_a_b @ X1 )
     != pred_a_b_d_c ) ).

% assertion.distinct(17)
thf(fact_213_assertion_Odistinct_I23_J,axiom,
    ! [X21: b,X222: assertion_a_b_d_c,X31: assertion_a_b_d_c,X32: assertion_a_b_d_c] :
      ( ( mult_b_a_d_c @ X21 @ X222 )
     != ( star_a_b_d_c @ X31 @ X32 ) ) ).

% assertion.distinct(23)
thf(fact_214_assertion_Odistinct_I25_J,axiom,
    ! [X21: b,X222: assertion_a_b_d_c,X41: assertion_a_b_d_c,X42: assertion_a_b_d_c] :
      ( ( mult_b_a_d_c @ X21 @ X222 )
     != ( wand_a_b_d_c @ X41 @ X42 ) ) ).

% assertion.distinct(25)
thf(fact_215_assertion_Odistinct_I31_J,axiom,
    ! [X21: b,X222: assertion_a_b_d_c,X71: assertion_a_b_d_c,X72: assertion_a_b_d_c] :
      ( ( mult_b_a_d_c @ X21 @ X222 )
     != ( imp_a_b_d_c @ X71 @ X72 ) ) ).

% assertion.distinct(31)
thf(fact_216_assertion_Odistinct_I29_J,axiom,
    ! [X21: b,X222: assertion_a_b_d_c,X61: assertion_a_b_d_c,X62: assertion_a_b_d_c] :
      ( ( mult_b_a_d_c @ X21 @ X222 )
     != ( and_a_b_d_c @ X61 @ X62 ) ) ).

% assertion.distinct(29)
thf(fact_217_assertion_Odistinct_I27_J,axiom,
    ! [X21: b,X222: assertion_a_b_d_c,X51: assertion_a_b_d_c,X52: assertion_a_b_d_c] :
      ( ( mult_b_a_d_c @ X21 @ X222 )
     != ( or_a_b_d_c @ X51 @ X52 ) ) ).

% assertion.distinct(27)
thf(fact_218_assertion_Odistinct_I41_J,axiom,
    ! [X21: b,X222: assertion_a_b_d_c,X12: assertion_a_b_d_c] :
      ( ( mult_b_a_d_c @ X21 @ X222 )
     != ( wildcard_a_b_d_c @ X12 ) ) ).

% assertion.distinct(41)
thf(fact_219_assertion_Odistinct_I37_J,axiom,
    ! [X21: b,X222: assertion_a_b_d_c] :
      ( ( mult_b_a_d_c @ X21 @ X222 )
     != pred_a_b_d_c ) ).

% assertion.distinct(37)
thf(fact_220_assertion_Odistinct_I57_J,axiom,
    ! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c,X11: assertion_a_b_d_c] :
      ( ( star_a_b_d_c @ X31 @ X32 )
     != ( bounded_a_b_d_c @ X11 ) ) ).

% assertion.distinct(57)
thf(fact_221_assertion_Odistinct_I73_J,axiom,
    ! [X41: assertion_a_b_d_c,X42: assertion_a_b_d_c,X11: assertion_a_b_d_c] :
      ( ( wand_a_b_d_c @ X41 @ X42 )
     != ( bounded_a_b_d_c @ X11 ) ) ).

% assertion.distinct(73)
thf(fact_222_assertion_Odistinct_I109_J,axiom,
    ! [X71: assertion_a_b_d_c,X72: assertion_a_b_d_c,X11: assertion_a_b_d_c] :
      ( ( imp_a_b_d_c @ X71 @ X72 )
     != ( bounded_a_b_d_c @ X11 ) ) ).

% assertion.distinct(109)
thf(fact_223_assertion_Odistinct_I53_J,axiom,
    ! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c,X91: c,X92: assertion_a_b_d_c] :
      ( ( star_a_b_d_c @ X31 @ X32 )
     != ( forall_c_a_b_d @ X91 @ X92 ) ) ).

% assertion.distinct(53)
thf(fact_224_assertion_Odistinct_I99_J,axiom,
    ! [X61: assertion_a_b_d_c,X62: assertion_a_b_d_c,X11: assertion_a_b_d_c] :
      ( ( and_a_b_d_c @ X61 @ X62 )
     != ( bounded_a_b_d_c @ X11 ) ) ).

% assertion.distinct(99)
thf(fact_225_assertion_Odistinct_I87_J,axiom,
    ! [X51: assertion_a_b_d_c,X52: assertion_a_b_d_c,X11: assertion_a_b_d_c] :
      ( ( or_a_b_d_c @ X51 @ X52 )
     != ( bounded_a_b_d_c @ X11 ) ) ).

% assertion.distinct(87)
thf(fact_226_assertion_Odistinct_I69_J,axiom,
    ! [X41: assertion_a_b_d_c,X42: assertion_a_b_d_c,X91: c,X92: assertion_a_b_d_c] :
      ( ( wand_a_b_d_c @ X41 @ X42 )
     != ( forall_c_a_b_d @ X91 @ X92 ) ) ).

% assertion.distinct(69)
thf(fact_227_assertion_Odistinct_I131_J,axiom,
    ! [X11: assertion_a_b_d_c,X12: assertion_a_b_d_c] :
      ( ( bounded_a_b_d_c @ X11 )
     != ( wildcard_a_b_d_c @ X12 ) ) ).

% assertion.distinct(131)
thf(fact_228_assertion_Odistinct_I105_J,axiom,
    ! [X71: assertion_a_b_d_c,X72: assertion_a_b_d_c,X91: c,X92: assertion_a_b_d_c] :
      ( ( imp_a_b_d_c @ X71 @ X72 )
     != ( forall_c_a_b_d @ X91 @ X92 ) ) ).

% assertion.distinct(105)
thf(fact_229_assertion_Odistinct_I51_J,axiom,
    ! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c,X81: c,X82: assertion_a_b_d_c] :
      ( ( star_a_b_d_c @ X31 @ X32 )
     != ( exists_c_a_b_d @ X81 @ X82 ) ) ).

% assertion.distinct(51)
thf(fact_230_assertion_Odistinct_I95_J,axiom,
    ! [X61: assertion_a_b_d_c,X62: assertion_a_b_d_c,X91: c,X92: assertion_a_b_d_c] :
      ( ( and_a_b_d_c @ X61 @ X62 )
     != ( forall_c_a_b_d @ X91 @ X92 ) ) ).

% assertion.distinct(95)
thf(fact_231_assertion_Odistinct_I83_J,axiom,
    ! [X51: assertion_a_b_d_c,X52: assertion_a_b_d_c,X91: c,X92: assertion_a_b_d_c] :
      ( ( or_a_b_d_c @ X51 @ X52 )
     != ( forall_c_a_b_d @ X91 @ X92 ) ) ).

% assertion.distinct(83)
thf(fact_232_assertion_Odistinct_I67_J,axiom,
    ! [X41: assertion_a_b_d_c,X42: assertion_a_b_d_c,X81: c,X82: assertion_a_b_d_c] :
      ( ( wand_a_b_d_c @ X41 @ X42 )
     != ( exists_c_a_b_d @ X81 @ X82 ) ) ).

% assertion.distinct(67)
thf(fact_233_assertion_Odistinct_I125_J,axiom,
    ! [X91: c,X92: assertion_a_b_d_c,X12: assertion_a_b_d_c] :
      ( ( forall_c_a_b_d @ X91 @ X92 )
     != ( wildcard_a_b_d_c @ X12 ) ) ).

% assertion.distinct(125)
thf(fact_234_assertion_Odistinct_I103_J,axiom,
    ! [X71: assertion_a_b_d_c,X72: assertion_a_b_d_c,X81: c,X82: assertion_a_b_d_c] :
      ( ( imp_a_b_d_c @ X71 @ X72 )
     != ( exists_c_a_b_d @ X81 @ X82 ) ) ).

% assertion.distinct(103)
thf(fact_235_assertion_Odistinct_I93_J,axiom,
    ! [X61: assertion_a_b_d_c,X62: assertion_a_b_d_c,X81: c,X82: assertion_a_b_d_c] :
      ( ( and_a_b_d_c @ X61 @ X62 )
     != ( exists_c_a_b_d @ X81 @ X82 ) ) ).

% assertion.distinct(93)
thf(fact_236_assertion_Odistinct_I81_J,axiom,
    ! [X51: assertion_a_b_d_c,X52: assertion_a_b_d_c,X81: c,X82: assertion_a_b_d_c] :
      ( ( or_a_b_d_c @ X51 @ X52 )
     != ( exists_c_a_b_d @ X81 @ X82 ) ) ).

% assertion.distinct(81)
thf(fact_237_assertion_Odistinct_I127_J,axiom,
    ! [X11: assertion_a_b_d_c] :
      ( pred_a_b_d_c
     != ( bounded_a_b_d_c @ X11 ) ) ).

% assertion.distinct(127)
thf(fact_238_assertion_Odistinct_I119_J,axiom,
    ! [X81: c,X82: assertion_a_b_d_c,X12: assertion_a_b_d_c] :
      ( ( exists_c_a_b_d @ X81 @ X82 )
     != ( wildcard_a_b_d_c @ X12 ) ) ).

% assertion.distinct(119)
thf(fact_239_assertion_Odistinct_I121_J,axiom,
    ! [X91: c,X92: assertion_a_b_d_c] :
      ( ( forall_c_a_b_d @ X91 @ X92 )
     != pred_a_b_d_c ) ).

% assertion.distinct(121)
thf(fact_240_assertion_Odistinct_I115_J,axiom,
    ! [X81: c,X82: assertion_a_b_d_c] :
      ( ( exists_c_a_b_d @ X81 @ X82 )
     != pred_a_b_d_c ) ).

% assertion.distinct(115)
thf(fact_241_assertion_Oexhaust,axiom,
    ! [Y: assertion_a_b_d_c] :
      ( ! [X13: ( c > d ) > a > $o] :
          ( Y
         != ( sem_c_d_a_b @ X13 ) )
     => ( ! [X212: b,X223: assertion_a_b_d_c] :
            ( Y
           != ( mult_b_a_d_c @ X212 @ X223 ) )
       => ( ! [X312: assertion_a_b_d_c,X322: assertion_a_b_d_c] :
              ( Y
             != ( star_a_b_d_c @ X312 @ X322 ) )
         => ( ! [X412: assertion_a_b_d_c,X422: assertion_a_b_d_c] :
                ( Y
               != ( wand_a_b_d_c @ X412 @ X422 ) )
           => ( ! [X512: assertion_a_b_d_c,X522: assertion_a_b_d_c] :
                  ( Y
                 != ( or_a_b_d_c @ X512 @ X522 ) )
             => ( ! [X612: assertion_a_b_d_c,X622: assertion_a_b_d_c] :
                    ( Y
                   != ( and_a_b_d_c @ X612 @ X622 ) )
               => ( ! [X712: assertion_a_b_d_c,X722: assertion_a_b_d_c] :
                      ( Y
                     != ( imp_a_b_d_c @ X712 @ X722 ) )
                 => ( ! [X812: c,X822: assertion_a_b_d_c] :
                        ( Y
                       != ( exists_c_a_b_d @ X812 @ X822 ) )
                   => ( ! [X912: c,X922: assertion_a_b_d_c] :
                          ( Y
                         != ( forall_c_a_b_d @ X912 @ X922 ) )
                     => ( ( Y != pred_a_b_d_c )
                       => ( ! [X112: assertion_a_b_d_c] :
                              ( Y
                             != ( bounded_a_b_d_c @ X112 ) )
                         => ~ ! [X122: assertion_a_b_d_c] :
                                ( Y
                               != ( wildcard_a_b_d_c @ X122 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% assertion.exhaust
thf(fact_242_logic_Osat_Osimps_I4_J,axiom,
    ! [Plus2: 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: c > d,Delta: ( c > d ) > set_a,B: ( c > d ) > a > $o] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( sem_c_d_a_b @ B ) )
        = ( B @ S4 @ Sigma ) ) ) ).

% logic.sat.simps(4)
thf(fact_243_logic_Osat__mult,axiom,
    ! [Plus2: 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: c > d,Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ! [A7: a] :
            ( ( Sigma
              = ( Mult @ P @ A7 ) )
           => ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A7 @ S4 @ Delta @ A6 ) )
       => ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( mult_b_a_d_c @ P @ A6 ) ) ) ) ).

% logic.sat_mult
thf(fact_244_logic_Osat_Osimps_I1_J,axiom,
    ! [Plus2: 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: c > d,Delta: ( c > d ) > set_a,P: b,A6: assertion_a_b_d_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( mult_b_a_d_c @ P @ A6 ) )
        = ( ? [A5: a] :
              ( ( Sigma
                = ( Mult @ P @ A5 ) )
              & ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A5 @ S4 @ Delta @ A6 ) ) ) ) ) ).

% logic.sat.simps(1)
thf(fact_245_logic_Osat_Osimps_I11_J,axiom,
    ! [Plus2: 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: c > d,Delta: ( c > d ) > set_a,A6: assertion_a_b_d_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S4 @ Delta @ ( bounded_a_b_d_c @ A6 ) )
        = ( ( Valid @ Sigma )
         => ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S4 @ Delta @ A6 ) ) ) ) ).

% logic.sat.simps(11)
thf(fact_246_logic_Ocombinable__instantiate,axiom,
    ! [Plus2: 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: ( c > d ) > set_a,A6: assertion_a_b_d_c,A: a,S4: c > d,B: a,X2: a,P: b,Q: b] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ A6 )
       => ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A @ S4 @ Delta @ A6 )
         => ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ B @ S4 @ Delta @ A6 )
           => ( ( ( some_a @ X2 )
                = ( Plus2 @ ( Mult @ P @ A ) @ ( Mult @ Q @ B ) ) )
             => ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ X2 @ S4 @ Delta @ ( mult_b_a_d_c @ ( Sadd @ P @ Q ) @ A6 ) ) ) ) ) ) ) ).

% logic.combinable_instantiate
thf(fact_247_hoare__triple__input,axiom,
    ! [P2: assertion_a_b_d_c,C: set_Pr1275464188344874039_a_c_d,Q2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
      ( ( valid_6037315502795721655_b_d_c @ plus @ mult @ valid @ P2 @ C @ Q2 @ Delta )
      = ( valid_6037315502795721655_b_d_c @ plus @ mult @ valid @ ( bounded_a_b_d_c @ P2 ) @ C @ Q2 @ Delta ) ) ).

% hoare_triple_input
thf(fact_248_fun__upd__apply,axiom,
    ( fun_upd_c_d
    = ( ^ [F: c > d,X: c,Y3: d,Z: c] : ( if_d @ ( Z = X ) @ Y3 @ ( F @ Z ) ) ) ) ).

% fun_upd_apply
thf(fact_249_fun__upd__triv,axiom,
    ! [F2: c > d,X2: c] :
      ( ( fun_upd_c_d @ F2 @ X2 @ ( F2 @ X2 ) )
      = F2 ) ).

% fun_upd_triv
thf(fact_250_fun__upd__upd,axiom,
    ! [F2: c > d,X2: c,Y: d,Z2: d] :
      ( ( fun_upd_c_d @ ( fun_upd_c_d @ F2 @ X2 @ Y ) @ X2 @ Z2 )
      = ( fun_upd_c_d @ F2 @ X2 @ Z2 ) ) ).

% fun_upd_upd
thf(fact_251_hoare__triple__output,axiom,
    ! [C: set_Pr1275464188344874039_a_c_d,P2: assertion_a_b_d_c,Q2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
      ( ( valid_command_a_c_d @ valid @ C )
     => ( ( valid_6037315502795721655_b_d_c @ plus @ mult @ valid @ P2 @ C @ Q2 @ Delta )
        = ( valid_6037315502795721655_b_d_c @ plus @ mult @ valid @ P2 @ C @ ( bounded_a_b_d_c @ Q2 ) @ Delta ) ) ) ).

% hoare_triple_output
thf(fact_252_logic_Ohoare__triple__input,axiom,
    ! [Plus2: 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_d_c,C: set_Pr1275464188344874039_a_c_d,Q2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( valid_6037315502795721655_b_d_c @ Plus2 @ Mult @ Valid @ P2 @ C @ Q2 @ Delta )
        = ( valid_6037315502795721655_b_d_c @ Plus2 @ Mult @ Valid @ ( bounded_a_b_d_c @ P2 ) @ C @ Q2 @ Delta ) ) ) ).

% logic.hoare_triple_input
thf(fact_253_not__in__fv__mod,axiom,
    ! [A6: assertion_a_b_d_c,C: set_Pr1275464188344874039_a_c_d,Sigma: a,S4: c > d,Sigma6: a,S7: c > d,X2: a,Delta: ( c > d ) > set_a] :
      ( ( not_in_fv_a_b_d_c @ plus @ mult @ valid @ A6 @ ( modified_a_c_d @ C ) )
     => ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma @ S4 ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma6 @ S7 ) ) ) @ C )
       => ( ( sat_a_b_c_d @ plus @ mult @ valid @ X2 @ S4 @ Delta @ A6 )
          = ( sat_a_b_c_d @ plus @ mult @ valid @ X2 @ S7 @ Delta @ A6 ) ) ) ) ).

% not_in_fv_mod
thf(fact_254_valid__command__def,axiom,
    ! [C: set_Pr1275464188344874039_a_c_d] :
      ( ( valid_command_a_c_d @ valid @ C )
      = ( ! [A5: a,B4: a,Sa: c > d,Sb: c > d] :
            ( ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ A5 @ Sa ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ B4 @ Sb ) ) ) @ C )
              & ( valid @ A5 ) )
           => ( valid @ B4 ) ) ) ) ).

% valid_command_def
thf(fact_255_logic_Ovalid__hoare__triple_Ocong,axiom,
    valid_6037315502795721655_b_d_c = valid_6037315502795721655_b_d_c ).

% logic.valid_hoare_triple.cong
thf(fact_256_logic_Ovalid__command__def,axiom,
    ! [Plus2: 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_Pr1275464188344874039_a_c_d] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( valid_command_a_c_d @ Valid @ C )
        = ( ! [A5: a,B4: a,Sa: c > d,Sb: c > d] :
              ( ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ A5 @ Sa ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ B4 @ Sb ) ) ) @ C )
                & ( Valid @ A5 ) )
             => ( Valid @ B4 ) ) ) ) ) ).

% logic.valid_command_def
thf(fact_257_logic_Onot__in__fv__mod,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A6: assertion_a_b_d_c,C: set_Pr1275464188344874039_a_c_d,Sigma: a,S4: c > d,Sigma6: a,S7: c > d,X2: a,Delta: ( c > d ) > set_a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( not_in_fv_a_b_d_c @ Plus2 @ Mult @ Valid @ A6 @ ( modified_a_c_d @ C ) )
       => ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma @ S4 ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma6 @ S7 ) ) ) @ C )
         => ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ X2 @ S4 @ Delta @ A6 )
            = ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ X2 @ S7 @ Delta @ A6 ) ) ) ) ) ).

% logic.not_in_fv_mod
thf(fact_258_logic_Ohoare__triple__output,axiom,
    ! [Plus2: 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_Pr1275464188344874039_a_c_d,P2: assertion_a_b_d_c,Q2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( valid_command_a_c_d @ Valid @ C )
       => ( ( valid_6037315502795721655_b_d_c @ Plus2 @ Mult @ Valid @ P2 @ C @ Q2 @ Delta )
          = ( valid_6037315502795721655_b_d_c @ Plus2 @ Mult @ Valid @ P2 @ C @ ( bounded_a_b_d_c @ Q2 ) @ Delta ) ) ) ) ).

% logic.hoare_triple_output
thf(fact_259_fun__upd__def,axiom,
    ( fun_upd_c_d
    = ( ^ [F: c > d,A5: c,B4: d,X: c] : ( if_d @ ( X = A5 ) @ B4 @ ( F @ X ) ) ) ) ).

% fun_upd_def
thf(fact_260_fun__upd__eqD,axiom,
    ! [F2: c > d,X2: c,Y: d,G: c > d,Z2: d] :
      ( ( ( fun_upd_c_d @ F2 @ X2 @ Y )
        = ( fun_upd_c_d @ G @ X2 @ Z2 ) )
     => ( Y = Z2 ) ) ).

% fun_upd_eqD
thf(fact_261_fun__upd__idem,axiom,
    ! [F2: c > d,X2: c,Y: d] :
      ( ( ( F2 @ X2 )
        = Y )
     => ( ( fun_upd_c_d @ F2 @ X2 @ Y )
        = F2 ) ) ).

% fun_upd_idem
thf(fact_262_fun__upd__same,axiom,
    ! [F2: c > d,X2: c,Y: d] :
      ( ( fun_upd_c_d @ F2 @ X2 @ Y @ X2 )
      = Y ) ).

% fun_upd_same
thf(fact_263_fun__upd__other,axiom,
    ! [Z2: c,X2: c,F2: c > d,Y: d] :
      ( ( Z2 != X2 )
     => ( ( fun_upd_c_d @ F2 @ X2 @ Y @ Z2 )
        = ( F2 @ Z2 ) ) ) ).

% fun_upd_other
thf(fact_264_fun__upd__twist,axiom,
    ! [A: c,C: c,M: c > d,B: d,D: d] :
      ( ( A != C )
     => ( ( fun_upd_c_d @ ( fun_upd_c_d @ M @ A @ B ) @ C @ D )
        = ( fun_upd_c_d @ ( fun_upd_c_d @ M @ C @ D ) @ A @ B ) ) ) ).

% fun_upd_twist
thf(fact_265_fun__upd__idem__iff,axiom,
    ! [F2: c > d,X2: c,Y: d] :
      ( ( ( fun_upd_c_d @ F2 @ X2 @ Y )
        = F2 )
      = ( ( F2 @ X2 )
        = Y ) ) ).

% fun_upd_idem_iff
thf(fact_266_valid__hoare__triple__def,axiom,
    ! [P2: assertion_a_b_d_c,C: set_Pr1275464188344874039_a_c_d,Q2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
      ( ( valid_6037315502795721655_b_d_c @ plus @ mult @ valid @ P2 @ C @ Q2 @ Delta )
      = ( ! [Sigma4: a,S: c > d] :
            ( ( ( valid @ Sigma4 )
              & ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma4 @ S @ Delta @ P2 ) )
           => ( ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma4 @ S ) )
              & ! [Sigma3: a,S2: c > d] :
                  ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma4 @ S ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma3 @ S2 ) ) ) @ C )
                 => ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma3 @ S2 @ Delta @ Q2 ) ) ) ) ) ) ).

% valid_hoare_triple_def
thf(fact_267_valid__hoare__tripleI,axiom,
    ! [Delta: ( c > d ) > set_a,P2: assertion_a_b_d_c,C: set_Pr1275464188344874039_a_c_d,Q2: assertion_a_b_d_c] :
      ( ! [Sigma5: a,S5: c > d] :
          ( ( ( valid @ Sigma5 )
            & ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S5 @ Delta @ P2 ) )
         => ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma5 @ S5 ) ) )
     => ( ! [Sigma5: a,S5: c > d,Sigma2: a,S8: c > d] :
            ( ( ( valid @ Sigma5 )
              & ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S5 @ Delta @ P2 ) )
           => ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma5 @ S5 ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma2 @ S8 ) ) ) @ C )
             => ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma2 @ S8 @ Delta @ Q2 ) ) )
       => ( valid_6037315502795721655_b_d_c @ plus @ mult @ valid @ P2 @ C @ Q2 @ Delta ) ) ) ).

% valid_hoare_tripleI
thf(fact_268_safe__def,axiom,
    ( safe_a_c_d
    = ( ^ [C2: set_Pr1275464188344874039_a_c_d,Sigma4: product_prod_a_c_d] :
          ~ ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ Sigma4 @ none_P2239999354720063602_a_c_d ) @ C2 ) ) ) ).

% safe_def
thf(fact_269_frame__rule,axiom,
    ! [C: set_Pr1275464188344874039_a_c_d,P2: assertion_a_b_d_c,Q2: assertion_a_b_d_c,Delta: ( c > d ) > set_a,R3: assertion_a_b_d_c] :
      ( ( valid_command_a_c_d @ valid @ C )
     => ( ( safety844553430189520448_a_c_d @ plus @ valid @ C )
       => ( ( frame_property_a_c_d @ plus @ valid @ C )
         => ( ( valid_6037315502795721655_b_d_c @ plus @ mult @ valid @ P2 @ C @ Q2 @ Delta )
           => ( ( not_in_fv_a_b_d_c @ plus @ mult @ valid @ R3 @ ( modified_a_c_d @ C ) )
             => ( valid_6037315502795721655_b_d_c @ plus @ mult @ valid @ ( star_a_b_d_c @ P2 @ R3 ) @ C @ ( star_a_b_d_c @ Q2 @ R3 ) @ Delta ) ) ) ) ) ) ).

% frame_rule
thf(fact_270_frame__property__def,axiom,
    ! [C: set_Pr1275464188344874039_a_c_d] :
      ( ( frame_property_a_c_d @ plus @ valid @ C )
      = ( ! [Sigma4: a,Sigma_0: a,R2: a,Sigma3: a,S: c > d,S2: c > d] :
            ( ( ( valid @ Sigma4 )
              & ( valid @ Sigma3 )
              & ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma_0 @ S ) )
              & ( ( some_a @ Sigma4 )
                = ( plus @ Sigma_0 @ R2 ) )
              & ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma4 @ S ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma3 @ S2 ) ) ) @ C ) )
           => ? [Sigma_02: a] :
                ( ( ( some_a @ Sigma3 )
                  = ( plus @ Sigma_02 @ R2 ) )
                & ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma_0 @ S ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma_02 @ S2 ) ) ) @ C ) ) ) ) ) ).

% frame_property_def
thf(fact_271_logic_Ovalid__hoare__tripleI,axiom,
    ! [Plus2: 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: ( c > d ) > set_a,P2: assertion_a_b_d_c,C: set_Pr1275464188344874039_a_c_d,Q2: assertion_a_b_d_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ! [Sigma5: a,S5: c > d] :
            ( ( ( Valid @ Sigma5 )
              & ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S5 @ Delta @ P2 ) )
           => ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma5 @ S5 ) ) )
       => ( ! [Sigma5: a,S5: c > d,Sigma2: a,S8: c > d] :
              ( ( ( Valid @ Sigma5 )
                & ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S5 @ Delta @ P2 ) )
             => ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma5 @ S5 ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma2 @ S8 ) ) ) @ C )
               => ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma2 @ S8 @ Delta @ Q2 ) ) )
         => ( valid_6037315502795721655_b_d_c @ Plus2 @ Mult @ Valid @ P2 @ C @ Q2 @ Delta ) ) ) ) ).

% logic.valid_hoare_tripleI
thf(fact_272_logic_Ovalid__hoare__triple__def,axiom,
    ! [Plus2: 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_d_c,C: set_Pr1275464188344874039_a_c_d,Q2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( valid_6037315502795721655_b_d_c @ Plus2 @ Mult @ Valid @ P2 @ C @ Q2 @ Delta )
        = ( ! [Sigma4: a,S: c > d] :
              ( ( ( Valid @ Sigma4 )
                & ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma4 @ S @ Delta @ P2 ) )
             => ( ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma4 @ S ) )
                & ! [Sigma3: a,S2: c > d] :
                    ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma4 @ S ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma3 @ S2 ) ) ) @ C )
                   => ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma3 @ S2 @ Delta @ Q2 ) ) ) ) ) ) ) ).

% logic.valid_hoare_triple_def
thf(fact_273_safety__monotonicity__def,axiom,
    ! [C: set_Pr1275464188344874039_a_c_d] :
      ( ( safety844553430189520448_a_c_d @ plus @ valid @ C )
      = ( ! [Sigma4: a,Sigma3: a,S: c > d] :
            ( ( ( valid @ Sigma3 )
              & ( pre_larger_a @ plus @ Sigma3 @ Sigma4 )
              & ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma4 @ S ) ) )
           => ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma3 @ S ) ) ) ) ) ).

% safety_monotonicity_def
thf(fact_274_logic_Osafety__monotonicity__def,axiom,
    ! [Plus2: 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_Pr1275464188344874039_a_c_d] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( safety844553430189520448_a_c_d @ Plus2 @ Valid @ C )
        = ( ! [Sigma4: a,Sigma3: a,S: c > d] :
              ( ( ( Valid @ Sigma3 )
                & ( pre_larger_a @ Plus2 @ Sigma3 @ Sigma4 )
                & ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma4 @ S ) ) )
             => ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma3 @ S ) ) ) ) ) ) ).

% logic.safety_monotonicity_def
thf(fact_275_logic_Osafe__def,axiom,
    ! [Plus2: 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_Pr1275464188344874039_a_c_d,Sigma: product_prod_a_c_d] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( safe_a_c_d @ C @ Sigma )
        = ( ~ ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ Sigma @ none_P2239999354720063602_a_c_d ) @ C ) ) ) ) ).

% logic.safe_def
thf(fact_276_logic_Oframe__property__def,axiom,
    ! [Plus2: 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_Pr1275464188344874039_a_c_d] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( frame_property_a_c_d @ Plus2 @ Valid @ C )
        = ( ! [Sigma4: a,Sigma_0: a,R2: a,Sigma3: a,S: c > d,S2: c > d] :
              ( ( ( Valid @ Sigma4 )
                & ( Valid @ Sigma3 )
                & ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma_0 @ S ) )
                & ( ( some_a @ Sigma4 )
                  = ( Plus2 @ Sigma_0 @ R2 ) )
                & ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma4 @ S ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma3 @ S2 ) ) ) @ C ) )
             => ? [Sigma_02: a] :
                  ( ( ( some_a @ Sigma3 )
                    = ( Plus2 @ Sigma_02 @ R2 ) )
                  & ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma_0 @ S ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma_02 @ S2 ) ) ) @ C ) ) ) ) ) ) ).

% logic.frame_property_def
thf(fact_277_logic_Oframe__rule,axiom,
    ! [Plus2: 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_Pr1275464188344874039_a_c_d,P2: assertion_a_b_d_c,Q2: assertion_a_b_d_c,Delta: ( c > d ) > set_a,R3: assertion_a_b_d_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( valid_command_a_c_d @ Valid @ C )
       => ( ( safety844553430189520448_a_c_d @ Plus2 @ Valid @ C )
         => ( ( frame_property_a_c_d @ Plus2 @ Valid @ C )
           => ( ( valid_6037315502795721655_b_d_c @ Plus2 @ Mult @ Valid @ P2 @ C @ Q2 @ Delta )
             => ( ( not_in_fv_a_b_d_c @ Plus2 @ Mult @ Valid @ R3 @ ( modified_a_c_d @ C ) )
               => ( valid_6037315502795721655_b_d_c @ Plus2 @ Mult @ Valid @ ( star_a_b_d_c @ P2 @ R3 ) @ C @ ( star_a_b_d_c @ Q2 @ R3 ) @ Delta ) ) ) ) ) ) ) ).

% logic.frame_rule
thf(fact_278_old_Oprod_Oinject,axiom,
    ! [A: product_prod_a_c_d,B: option6413918287372586467_a_c_d,A4: product_prod_a_c_d,B3: option6413918287372586467_a_c_d] :
      ( ( ( produc8093790510458973071_a_c_d @ A @ B )
        = ( produc8093790510458973071_a_c_d @ A4 @ B3 ) )
      = ( ( A = A4 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_279_old_Oprod_Oinject,axiom,
    ! [A: a,B: c > d,A4: a,B3: c > d] :
      ( ( ( product_Pair_a_c_d @ A @ B )
        = ( product_Pair_a_c_d @ A4 @ B3 ) )
      = ( ( A = A4 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_280_prod_Oinject,axiom,
    ! [X1: product_prod_a_c_d,X22: option6413918287372586467_a_c_d,Y1: product_prod_a_c_d,Y2: option6413918287372586467_a_c_d] :
      ( ( ( produc8093790510458973071_a_c_d @ X1 @ X22 )
        = ( produc8093790510458973071_a_c_d @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_281_prod_Oinject,axiom,
    ! [X1: a,X22: c > d,Y1: a,Y2: c > d] :
      ( ( ( product_Pair_a_c_d @ X1 @ X22 )
        = ( product_Pair_a_c_d @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_282_ssubst__Pair__rhs,axiom,
    ! [R: product_prod_a_c_d,S4: option6413918287372586467_a_c_d,R3: set_Pr1275464188344874039_a_c_d,S7: option6413918287372586467_a_c_d] :
      ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ R @ S4 ) @ R3 )
     => ( ( S7 = S4 )
       => ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ R @ S7 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_283_ssubst__Pair__rhs,axiom,
    ! [R: a,S4: c > d,R3: set_Pr334309860168054345_a_c_d,S7: c > d] :
      ( ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ R @ S4 ) @ R3 )
     => ( ( S7 = S4 )
       => ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ R @ S7 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_284_old_Oprod_Oexhaust,axiom,
    ! [Y: produc5278197477302038359_a_c_d] :
      ~ ! [A7: product_prod_a_c_d,B6: option6413918287372586467_a_c_d] :
          ( Y
         != ( produc8093790510458973071_a_c_d @ A7 @ B6 ) ) ).

% old.prod.exhaust
thf(fact_285_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_a_c_d] :
      ~ ! [A7: a,B6: c > d] :
          ( Y
         != ( product_Pair_a_c_d @ A7 @ B6 ) ) ).

% old.prod.exhaust
thf(fact_286_surj__pair,axiom,
    ! [P: produc5278197477302038359_a_c_d] :
    ? [X4: product_prod_a_c_d,Y4: option6413918287372586467_a_c_d] :
      ( P
      = ( produc8093790510458973071_a_c_d @ X4 @ Y4 ) ) ).

% surj_pair
thf(fact_287_surj__pair,axiom,
    ! [P: product_prod_a_c_d] :
    ? [X4: a,Y4: c > d] :
      ( P
      = ( product_Pair_a_c_d @ X4 @ Y4 ) ) ).

% surj_pair
thf(fact_288_prod__cases,axiom,
    ! [P2: produc5278197477302038359_a_c_d > $o,P: produc5278197477302038359_a_c_d] :
      ( ! [A7: product_prod_a_c_d,B6: option6413918287372586467_a_c_d] : ( P2 @ ( produc8093790510458973071_a_c_d @ A7 @ B6 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_289_prod__cases,axiom,
    ! [P2: product_prod_a_c_d > $o,P: product_prod_a_c_d] :
      ( ! [A7: a,B6: c > d] : ( P2 @ ( product_Pair_a_c_d @ A7 @ B6 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_290_Pair__inject,axiom,
    ! [A: product_prod_a_c_d,B: option6413918287372586467_a_c_d,A4: product_prod_a_c_d,B3: option6413918287372586467_a_c_d] :
      ( ( ( produc8093790510458973071_a_c_d @ A @ B )
        = ( produc8093790510458973071_a_c_d @ A4 @ B3 ) )
     => ~ ( ( A = A4 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_291_Pair__inject,axiom,
    ! [A: a,B: c > d,A4: a,B3: c > d] :
      ( ( ( product_Pair_a_c_d @ A @ B )
        = ( product_Pair_a_c_d @ A4 @ B3 ) )
     => ~ ( ( A = A4 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_292_option_Ocollapse,axiom,
    ! [Option: option6413918287372586467_a_c_d] :
      ( ( Option != none_P2239999354720063602_a_c_d )
     => ( ( some_P1084500821511757806_a_c_d @ ( the_Pr7316212213771180927_a_c_d @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_293_option_Ocollapse,axiom,
    ! [Option: option_a] :
      ( ( Option != none_a )
     => ( ( some_a @ ( the_a @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_294_curryI,axiom,
    ! [F2: produc5278197477302038359_a_c_d > $o,A: product_prod_a_c_d,B: option6413918287372586467_a_c_d] :
      ( ( F2 @ ( produc8093790510458973071_a_c_d @ A @ B ) )
     => ( produc7638649735877690389_c_d_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_295_curryI,axiom,
    ! [F2: product_prod_a_c_d > $o,A: a,B: c > d] :
      ( ( F2 @ ( product_Pair_a_c_d @ A @ B ) )
     => ( produc4973722101612664189_c_d_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_296_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_297_fun__upd__None__if__notin__dom,axiom,
    ! [K: produc5278197477302038359_a_c_d,M: produc5278197477302038359_a_c_d > option_a] :
      ( ~ ( member1180172933830803072_a_c_d @ K @ ( dom_Pr210953022859676729_c_d_a @ M ) )
     => ( ( fun_up7463025390225626410tion_a @ M @ K @ none_a )
        = M ) ) ).

% fun_upd_None_if_notin_dom
thf(fact_298_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_299_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_300_map__add__upd__left,axiom,
    ! [M: produc5278197477302038359_a_c_d,E2: produc5278197477302038359_a_c_d > option_a,E1: produc5278197477302038359_a_c_d > option_a,U1: a] :
      ( ~ ( member1180172933830803072_a_c_d @ M @ ( dom_Pr210953022859676729_c_d_a @ E2 ) )
     => ( ( map_ad5220634294393796252_c_d_a @ ( fun_up7463025390225626410tion_a @ E1 @ M @ ( some_a @ U1 ) ) @ E2 )
        = ( fun_up7463025390225626410tion_a @ ( map_ad5220634294393796252_c_d_a @ E1 @ E2 ) @ M @ ( some_a @ U1 ) ) ) ) ).

% map_add_upd_left
thf(fact_301_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_302_map__add__upd__left,axiom,
    ! [M: a,E2: a > option6413918287372586467_a_c_d,E1: a > option6413918287372586467_a_c_d,U1: product_prod_a_c_d] :
      ( ~ ( member_a @ M @ ( dom_a_3529858286862933695_a_c_d @ E2 ) )
     => ( ( map_ad2004115526166715036_a_c_d @ ( fun_up6642239869823773348_a_c_d @ E1 @ M @ ( some_P1084500821511757806_a_c_d @ U1 ) ) @ E2 )
        = ( fun_up6642239869823773348_a_c_d @ ( map_ad2004115526166715036_a_c_d @ E1 @ E2 ) @ M @ ( some_P1084500821511757806_a_c_d @ U1 ) ) ) ) ).

% map_add_upd_left
thf(fact_303_map__add__upd__left,axiom,
    ! [M: produc5278197477302038359_a_c_d,E2: produc5278197477302038359_a_c_d > option6413918287372586467_a_c_d,E1: produc5278197477302038359_a_c_d > option6413918287372586467_a_c_d,U1: product_prod_a_c_d] :
      ( ~ ( member1180172933830803072_a_c_d @ M @ ( dom_Pr2275460399045561368_a_c_d @ E2 ) )
     => ( ( map_ad4724009413847597045_a_c_d @ ( fun_up4007221454163596925_a_c_d @ E1 @ M @ ( some_P1084500821511757806_a_c_d @ U1 ) ) @ E2 )
        = ( fun_up4007221454163596925_a_c_d @ ( map_ad4724009413847597045_a_c_d @ E1 @ E2 ) @ M @ ( some_P1084500821511757806_a_c_d @ U1 ) ) ) ) ).

% map_add_upd_left
thf(fact_304_map__add__upd__left,axiom,
    ! [M: option_a,E2: option_a > option6413918287372586467_a_c_d,E1: option_a > option6413918287372586467_a_c_d,U1: product_prod_a_c_d] :
      ( ~ ( member_option_a @ M @ ( dom_op6156779764657312645_a_c_d @ E2 ) )
     => ( ( map_ad3766086351437021154_a_c_d @ ( fun_up5969202740897331946_a_c_d @ E1 @ M @ ( some_P1084500821511757806_a_c_d @ U1 ) ) @ E2 )
        = ( fun_up5969202740897331946_a_c_d @ ( map_ad3766086351437021154_a_c_d @ E1 @ E2 ) @ M @ ( some_P1084500821511757806_a_c_d @ U1 ) ) ) ) ).

% map_add_upd_left
thf(fact_305_domD,axiom,
    ! [A: a,M: a > option_a] :
      ( ( member_a @ A @ ( dom_a_a @ M ) )
     => ? [B6: a] :
          ( ( M @ A )
          = ( some_a @ B6 ) ) ) ).

% domD
thf(fact_306_domD,axiom,
    ! [A: produc5278197477302038359_a_c_d,M: produc5278197477302038359_a_c_d > option_a] :
      ( ( member1180172933830803072_a_c_d @ A @ ( dom_Pr210953022859676729_c_d_a @ M ) )
     => ? [B6: a] :
          ( ( M @ A )
          = ( some_a @ B6 ) ) ) ).

% domD
thf(fact_307_domD,axiom,
    ! [A: option_a,M: option_a > option_a] :
      ( ( member_option_a @ A @ ( dom_option_a_a @ M ) )
     => ? [B6: a] :
          ( ( M @ A )
          = ( some_a @ B6 ) ) ) ).

% domD
thf(fact_308_domD,axiom,
    ! [A: a,M: a > option6413918287372586467_a_c_d] :
      ( ( member_a @ A @ ( dom_a_3529858286862933695_a_c_d @ M ) )
     => ? [B6: product_prod_a_c_d] :
          ( ( M @ A )
          = ( some_P1084500821511757806_a_c_d @ B6 ) ) ) ).

% domD
thf(fact_309_domD,axiom,
    ! [A: produc5278197477302038359_a_c_d,M: produc5278197477302038359_a_c_d > option6413918287372586467_a_c_d] :
      ( ( member1180172933830803072_a_c_d @ A @ ( dom_Pr2275460399045561368_a_c_d @ M ) )
     => ? [B6: product_prod_a_c_d] :
          ( ( M @ A )
          = ( some_P1084500821511757806_a_c_d @ B6 ) ) ) ).

% domD
thf(fact_310_domD,axiom,
    ! [A: option_a,M: option_a > option6413918287372586467_a_c_d] :
      ( ( member_option_a @ A @ ( dom_op6156779764657312645_a_c_d @ M ) )
     => ? [B6: product_prod_a_c_d] :
          ( ( M @ A )
          = ( some_P1084500821511757806_a_c_d @ B6 ) ) ) ).

% domD
thf(fact_311_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_312_domI,axiom,
    ! [M: produc5278197477302038359_a_c_d > option_a,A: produc5278197477302038359_a_c_d,B: a] :
      ( ( ( M @ A )
        = ( some_a @ B ) )
     => ( member1180172933830803072_a_c_d @ A @ ( dom_Pr210953022859676729_c_d_a @ M ) ) ) ).

% domI
thf(fact_313_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_314_domI,axiom,
    ! [M: a > option6413918287372586467_a_c_d,A: a,B: product_prod_a_c_d] :
      ( ( ( M @ A )
        = ( some_P1084500821511757806_a_c_d @ B ) )
     => ( member_a @ A @ ( dom_a_3529858286862933695_a_c_d @ M ) ) ) ).

% domI
thf(fact_315_domI,axiom,
    ! [M: produc5278197477302038359_a_c_d > option6413918287372586467_a_c_d,A: produc5278197477302038359_a_c_d,B: product_prod_a_c_d] :
      ( ( ( M @ A )
        = ( some_P1084500821511757806_a_c_d @ B ) )
     => ( member1180172933830803072_a_c_d @ A @ ( dom_Pr2275460399045561368_a_c_d @ M ) ) ) ).

% domI
thf(fact_316_domI,axiom,
    ! [M: option_a > option6413918287372586467_a_c_d,A: option_a,B: product_prod_a_c_d] :
      ( ( ( M @ A )
        = ( some_P1084500821511757806_a_c_d @ B ) )
     => ( member_option_a @ A @ ( dom_op6156779764657312645_a_c_d @ M ) ) ) ).

% domI
thf(fact_317_domIff,axiom,
    ! [A: a,M: a > option_a] :
      ( ( member_a @ A @ ( dom_a_a @ M ) )
      = ( ( M @ A )
       != none_a ) ) ).

% domIff
thf(fact_318_domIff,axiom,
    ! [A: produc5278197477302038359_a_c_d,M: produc5278197477302038359_a_c_d > option_a] :
      ( ( member1180172933830803072_a_c_d @ A @ ( dom_Pr210953022859676729_c_d_a @ M ) )
      = ( ( M @ A )
       != none_a ) ) ).

% domIff
thf(fact_319_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_320_option_Osel,axiom,
    ! [X22: product_prod_a_c_d] :
      ( ( the_Pr7316212213771180927_a_c_d @ ( some_P1084500821511757806_a_c_d @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_321_option_Osel,axiom,
    ! [X22: a] :
      ( ( the_a @ ( some_a @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_322_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_323_curryE,axiom,
    ! [F2: produc5278197477302038359_a_c_d > $o,A: product_prod_a_c_d,B: option6413918287372586467_a_c_d] :
      ( ( produc7638649735877690389_c_d_o @ F2 @ A @ B )
     => ( F2 @ ( produc8093790510458973071_a_c_d @ A @ B ) ) ) ).

% curryE
thf(fact_324_curryE,axiom,
    ! [F2: product_prod_a_c_d > $o,A: a,B: c > d] :
      ( ( produc4973722101612664189_c_d_o @ F2 @ A @ B )
     => ( F2 @ ( product_Pair_a_c_d @ A @ B ) ) ) ).

% curryE
thf(fact_325_curryD,axiom,
    ! [F2: produc5278197477302038359_a_c_d > $o,A: product_prod_a_c_d,B: option6413918287372586467_a_c_d] :
      ( ( produc7638649735877690389_c_d_o @ F2 @ A @ B )
     => ( F2 @ ( produc8093790510458973071_a_c_d @ A @ B ) ) ) ).

% curryD
thf(fact_326_curryD,axiom,
    ! [F2: product_prod_a_c_d > $o,A: a,B: c > d] :
      ( ( produc4973722101612664189_c_d_o @ F2 @ A @ B )
     => ( F2 @ ( product_Pair_a_c_d @ A @ B ) ) ) ).

% curryD
thf(fact_327_option_Oexhaust__sel,axiom,
    ! [Option: option6413918287372586467_a_c_d] :
      ( ( Option != none_P2239999354720063602_a_c_d )
     => ( Option
        = ( some_P1084500821511757806_a_c_d @ ( the_Pr7316212213771180927_a_c_d @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_328_option_Oexhaust__sel,axiom,
    ! [Option: option_a] :
      ( ( Option != none_a )
     => ( Option
        = ( some_a @ ( the_a @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_329_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_330_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_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_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_333_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_334_option_Oset__sel,axiom,
    ! [A: option6736415434736681245_a_c_d] :
      ( ( A != none_P6708852886386159800_a_c_d )
     => ( member1180172933830803072_a_c_d @ ( the_Pr8658966078822770667_a_c_d @ A ) @ ( set_op1238738513205398757_a_c_d @ A ) ) ) ).

% option.set_sel
thf(fact_335_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_336_option_Oset__sel,axiom,
    ! [A: option_a] :
      ( ( A != none_a )
     => ( member_a @ ( the_a @ A ) @ ( set_option_a3 @ A ) ) ) ).

% option.set_sel
thf(fact_337_elem__set,axiom,
    ! [X2: produc5278197477302038359_a_c_d,Xo: option6736415434736681245_a_c_d] :
      ( ( member1180172933830803072_a_c_d @ X2 @ ( set_op1238738513205398757_a_c_d @ Xo ) )
      = ( Xo
        = ( some_P7564561949935391804_a_c_d @ X2 ) ) ) ).

% elem_set
thf(fact_338_elem__set,axiom,
    ! [X2: option_a,Xo: option_option_a] :
      ( ( member_option_a @ X2 @ ( set_option_option_a2 @ Xo ) )
      = ( Xo
        = ( some_option_a @ X2 ) ) ) ).

% elem_set
thf(fact_339_elem__set,axiom,
    ! [X2: product_prod_a_c_d,Xo: option6413918287372586467_a_c_d] :
      ( ( member8846983874477766826_a_c_d @ X2 @ ( set_op7017632568034097669_a_c_d @ Xo ) )
      = ( Xo
        = ( some_P1084500821511757806_a_c_d @ X2 ) ) ) ).

% elem_set
thf(fact_340_elem__set,axiom,
    ! [X2: a,Xo: option_a] :
      ( ( member_a @ X2 @ ( set_option_a3 @ Xo ) )
      = ( Xo
        = ( some_a @ X2 ) ) ) ).

% elem_set
thf(fact_341_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_342_set__empty__eq,axiom,
    ! [Xo: option_a] :
      ( ( ( set_option_a3 @ Xo )
        = bot_bot_set_a )
      = ( Xo = none_a ) ) ).

% set_empty_eq
thf(fact_343_option_Osimps_I14_J,axiom,
    ( ( set_option_option_a2 @ none_option_a )
    = bot_bot_set_option_a ) ).

% option.simps(14)
thf(fact_344_option_Osimps_I14_J,axiom,
    ( ( set_option_a3 @ none_a )
    = bot_bot_set_a ) ).

% option.simps(14)
thf(fact_345_option_Oset__cases,axiom,
    ! [E: produc5278197477302038359_a_c_d,A: option6736415434736681245_a_c_d] :
      ( ( member1180172933830803072_a_c_d @ E @ ( set_op1238738513205398757_a_c_d @ A ) )
     => ( A
        = ( some_P7564561949935391804_a_c_d @ E ) ) ) ).

% option.set_cases
thf(fact_346_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_347_option_Oset__cases,axiom,
    ! [E: product_prod_a_c_d,A: option6413918287372586467_a_c_d] :
      ( ( member8846983874477766826_a_c_d @ E @ ( set_op7017632568034097669_a_c_d @ A ) )
     => ( A
        = ( some_P1084500821511757806_a_c_d @ E ) ) ) ).

% option.set_cases
thf(fact_348_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_349_option_Oset__intros,axiom,
    ! [X22: produc5278197477302038359_a_c_d] : ( member1180172933830803072_a_c_d @ X22 @ ( set_op1238738513205398757_a_c_d @ ( some_P7564561949935391804_a_c_d @ X22 ) ) ) ).

% option.set_intros
thf(fact_350_option_Oset__intros,axiom,
    ! [X22: option_a] : ( member_option_a @ X22 @ ( set_option_option_a2 @ ( some_option_a @ X22 ) ) ) ).

% option.set_intros
thf(fact_351_option_Oset__intros,axiom,
    ! [X22: product_prod_a_c_d] : ( member8846983874477766826_a_c_d @ X22 @ ( set_op7017632568034097669_a_c_d @ ( some_P1084500821511757806_a_c_d @ X22 ) ) ) ).

% option.set_intros
thf(fact_352_option_Oset__intros,axiom,
    ! [X22: a] : ( member_a @ X22 @ ( set_option_a3 @ ( some_a @ X22 ) ) ) ).

% option.set_intros
thf(fact_353_ospec,axiom,
    ! [A6: option6413918287372586467_a_c_d,P2: product_prod_a_c_d > $o,X2: product_prod_a_c_d] :
      ( ! [X4: product_prod_a_c_d] :
          ( ( member8846983874477766826_a_c_d @ X4 @ ( set_op7017632568034097669_a_c_d @ A6 ) )
         => ( P2 @ X4 ) )
     => ( ( A6
          = ( some_P1084500821511757806_a_c_d @ X2 ) )
       => ( P2 @ X2 ) ) ) ).

% ospec
thf(fact_354_ospec,axiom,
    ! [A6: option_a,P2: a > $o,X2: a] :
      ( ! [X4: a] :
          ( ( member_a @ X4 @ ( set_option_a3 @ A6 ) )
         => ( P2 @ X4 ) )
     => ( ( A6
          = ( some_a @ X2 ) )
       => ( P2 @ X2 ) ) ) ).

% ospec
thf(fact_355_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_356_option_Osimps_I4_J,axiom,
    ! [F1: $o,F22: a > $o] :
      ( ( case_option_o_a @ F1 @ F22 @ none_a )
      = F1 ) ).

% option.simps(4)
thf(fact_357_option_Osimps_I15_J,axiom,
    ! [X22: product_prod_a_c_d] :
      ( ( set_op7017632568034097669_a_c_d @ ( some_P1084500821511757806_a_c_d @ X22 ) )
      = ( insert2748062992969704259_a_c_d @ X22 @ bot_bo110114171885311197_a_c_d ) ) ).

% option.simps(15)
thf(fact_358_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_359_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_360_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_361_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_362_these__empty,axiom,
    ( ( these_option_a @ bot_bo4163488203964334806tion_a )
    = bot_bot_set_option_a ) ).

% these_empty
thf(fact_363_these__empty,axiom,
    ( ( these_a @ bot_bot_set_option_a )
    = bot_bot_set_a ) ).

% these_empty
thf(fact_364_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_365_bind__runit,axiom,
    ! [X2: option6413918287372586467_a_c_d] :
      ( ( bind_P5476543181312113638_a_c_d @ X2 @ some_P1084500821511757806_a_c_d )
      = X2 ) ).

% bind_runit
thf(fact_366_bind__runit,axiom,
    ! [X2: option_a] :
      ( ( bind_a_a @ X2 @ some_a )
      = X2 ) ).

% bind_runit
thf(fact_367_restrict__out,axiom,
    ! [X2: a,A6: set_a,M: a > option_a] :
      ( ~ ( member_a @ X2 @ A6 )
     => ( ( restrict_map_a_a @ M @ A6 @ X2 )
        = none_a ) ) ).

% restrict_out
thf(fact_368_restrict__out,axiom,
    ! [X2: produc5278197477302038359_a_c_d,A6: set_Pr1275464188344874039_a_c_d,M: produc5278197477302038359_a_c_d > option_a] :
      ( ~ ( member1180172933830803072_a_c_d @ X2 @ A6 )
     => ( ( restri6481929051272950551_c_d_a @ M @ A6 @ X2 )
        = none_a ) ) ).

% restrict_out
thf(fact_369_restrict__out,axiom,
    ! [X2: option_a,A6: set_option_a,M: option_a > option_a] :
      ( ~ ( member_option_a @ X2 @ A6 )
     => ( ( restri3984065703976872170on_a_a @ M @ A6 @ X2 )
        = none_a ) ) ).

% restrict_out
thf(fact_370_these__insert__None,axiom,
    ! [A6: set_option_a] :
      ( ( these_a @ ( insert_option_a @ none_a @ A6 ) )
      = ( these_a @ A6 ) ) ).

% these_insert_None
thf(fact_371_these__insert__Some,axiom,
    ! [X2: option_a,A6: set_option_option_a] :
      ( ( these_option_a @ ( insert605063979879581146tion_a @ ( some_option_a @ X2 ) @ A6 ) )
      = ( insert_option_a @ X2 @ ( these_option_a @ A6 ) ) ) ).

% these_insert_Some
thf(fact_372_these__insert__Some,axiom,
    ! [X2: product_prod_a_c_d,A6: set_op3383557829362473625_a_c_d] :
      ( ( these_5134989223866396566_a_c_d @ ( insert6629875870486991891_a_c_d @ ( some_P1084500821511757806_a_c_d @ X2 ) @ A6 ) )
      = ( insert2748062992969704259_a_c_d @ X2 @ ( these_5134989223866396566_a_c_d @ A6 ) ) ) ).

% these_insert_Some
thf(fact_373_these__insert__Some,axiom,
    ! [X2: a,A6: set_option_a] :
      ( ( these_a @ ( insert_option_a @ ( some_a @ X2 ) @ A6 ) )
      = ( insert_a @ X2 @ ( these_a @ A6 ) ) ) ).

% these_insert_Some
thf(fact_374_override__on__insert_H,axiom,
    ! [F2: c > d,G: c > d,X2: c,X6: set_c] :
      ( ( override_on_c_d @ F2 @ G @ ( insert_c @ X2 @ X6 ) )
      = ( override_on_c_d @ ( fun_upd_c_d @ F2 @ X2 @ ( G @ X2 ) ) @ G @ X6 ) ) ).

% override_on_insert'
thf(fact_375_override__on__insert,axiom,
    ! [F2: c > d,G: c > d,X2: c,X6: set_c] :
      ( ( override_on_c_d @ F2 @ G @ ( insert_c @ X2 @ X6 ) )
      = ( fun_upd_c_d @ ( override_on_c_d @ F2 @ G @ X6 ) @ X2 @ ( G @ X2 ) ) ) ).

% override_on_insert
thf(fact_376_bind__option__cong__code,axiom,
    ! [X2: option_a,Y: option_a,F2: a > option_a] :
      ( ( X2 = Y )
     => ( ( bind_a_a @ X2 @ F2 )
        = ( bind_a_a @ Y @ F2 ) ) ) ).

% bind_option_cong_code
thf(fact_377_these__not__empty__eq,axiom,
    ! [B5: set_option_option_a] :
      ( ( ( these_option_a @ B5 )
       != bot_bot_set_option_a )
      = ( ( B5 != bot_bo4163488203964334806tion_a )
        & ( B5
         != ( insert605063979879581146tion_a @ none_option_a @ bot_bo4163488203964334806tion_a ) ) ) ) ).

% these_not_empty_eq
thf(fact_378_these__not__empty__eq,axiom,
    ! [B5: set_option_a] :
      ( ( ( these_a @ B5 )
       != bot_bot_set_a )
      = ( ( B5 != bot_bot_set_option_a )
        & ( B5
         != ( insert_option_a @ none_a @ bot_bot_set_option_a ) ) ) ) ).

% these_not_empty_eq
thf(fact_379_these__empty__eq,axiom,
    ! [B5: set_option_option_a] :
      ( ( ( these_option_a @ B5 )
        = bot_bot_set_option_a )
      = ( ( B5 = bot_bo4163488203964334806tion_a )
        | ( B5
          = ( insert605063979879581146tion_a @ none_option_a @ bot_bo4163488203964334806tion_a ) ) ) ) ).

% these_empty_eq
thf(fact_380_these__empty__eq,axiom,
    ! [B5: set_option_a] :
      ( ( ( these_a @ B5 )
        = bot_bot_set_a )
      = ( ( B5 = bot_bot_set_option_a )
        | ( B5
          = ( insert_option_a @ none_a @ bot_bot_set_option_a ) ) ) ) ).

% these_empty_eq
thf(fact_381_restrict__map__def,axiom,
    ( restrict_map_a_a
    = ( ^ [M2: a > option_a,A8: set_a,X: a] : ( if_option_a @ ( member_a @ X @ A8 ) @ ( M2 @ X ) @ none_a ) ) ) ).

% restrict_map_def
thf(fact_382_restrict__map__def,axiom,
    ( restri6481929051272950551_c_d_a
    = ( ^ [M2: produc5278197477302038359_a_c_d > option_a,A8: set_Pr1275464188344874039_a_c_d,X: produc5278197477302038359_a_c_d] : ( if_option_a @ ( member1180172933830803072_a_c_d @ X @ A8 ) @ ( M2 @ X ) @ none_a ) ) ) ).

% restrict_map_def
thf(fact_383_restrict__map__def,axiom,
    ( restri3984065703976872170on_a_a
    = ( ^ [M2: option_a > option_a,A8: set_option_a,X: option_a] : ( if_option_a @ ( member_option_a @ X @ A8 ) @ ( M2 @ X ) @ none_a ) ) ) ).

% restrict_map_def
thf(fact_384_bind__eq__Some__conv,axiom,
    ! [F2: option6413918287372586467_a_c_d,G: product_prod_a_c_d > option_a,X2: a] :
      ( ( ( bind_P2911691133002111467_c_d_a @ F2 @ G )
        = ( some_a @ X2 ) )
      = ( ? [Y3: product_prod_a_c_d] :
            ( ( F2
              = ( some_P1084500821511757806_a_c_d @ Y3 ) )
            & ( ( G @ Y3 )
              = ( some_a @ X2 ) ) ) ) ) ).

% bind_eq_Some_conv
thf(fact_385_bind__eq__Some__conv,axiom,
    ! [F2: option_a,G: a > option6413918287372586467_a_c_d,X2: product_prod_a_c_d] :
      ( ( ( bind_a4598251245030301523_a_c_d @ F2 @ G )
        = ( some_P1084500821511757806_a_c_d @ X2 ) )
      = ( ? [Y3: a] :
            ( ( F2
              = ( some_a @ Y3 ) )
            & ( ( G @ Y3 )
              = ( some_P1084500821511757806_a_c_d @ X2 ) ) ) ) ) ).

% bind_eq_Some_conv
thf(fact_386_bind__eq__Some__conv,axiom,
    ! [F2: option6413918287372586467_a_c_d,G: product_prod_a_c_d > option6413918287372586467_a_c_d,X2: product_prod_a_c_d] :
      ( ( ( bind_P5476543181312113638_a_c_d @ F2 @ G )
        = ( some_P1084500821511757806_a_c_d @ X2 ) )
      = ( ? [Y3: product_prod_a_c_d] :
            ( ( F2
              = ( some_P1084500821511757806_a_c_d @ Y3 ) )
            & ( ( G @ Y3 )
              = ( some_P1084500821511757806_a_c_d @ X2 ) ) ) ) ) ).

% bind_eq_Some_conv
thf(fact_387_bind__eq__Some__conv,axiom,
    ! [F2: option_a,G: a > option_a,X2: a] :
      ( ( ( bind_a_a @ F2 @ G )
        = ( some_a @ X2 ) )
      = ( ? [Y3: a] :
            ( ( F2
              = ( some_a @ Y3 ) )
            & ( ( G @ Y3 )
              = ( some_a @ X2 ) ) ) ) ) ).

% bind_eq_Some_conv
thf(fact_388_Option_Obind__cong,axiom,
    ! [X2: option_a,Y: option_a,F2: a > option_a,G: a > option_a] :
      ( ( X2 = Y )
     => ( ! [A7: a] :
            ( ( Y
              = ( some_a @ A7 ) )
           => ( ( F2 @ A7 )
              = ( G @ A7 ) ) )
       => ( ( bind_a_a @ X2 @ F2 )
          = ( bind_a_a @ Y @ G ) ) ) ) ).

% Option.bind_cong
thf(fact_389_bind_Obind__lunit,axiom,
    ! [X2: a,F2: a > option_a] :
      ( ( bind_a_a @ ( some_a @ X2 ) @ F2 )
      = ( F2 @ X2 ) ) ).

% bind.bind_lunit
thf(fact_390_bind_Obind__lzero,axiom,
    ! [F2: a > option_a] :
      ( ( bind_a_a @ none_a @ F2 )
      = none_a ) ).

% bind.bind_lzero
thf(fact_391_bind__option__cong,axiom,
    ! [X2: option_a,Y: option_a,F2: a > option_a,G: a > option_a] :
      ( ( X2 = Y )
     => ( ! [Z3: a] :
            ( ( member_a @ Z3 @ ( set_option_a3 @ Y ) )
           => ( ( F2 @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( bind_a_a @ X2 @ F2 )
          = ( bind_a_a @ Y @ G ) ) ) ) ).

% bind_option_cong
thf(fact_392_in__these__eq,axiom,
    ! [X2: produc5278197477302038359_a_c_d,A6: set_op7411283011066580861_a_c_d] :
      ( ( member1180172933830803072_a_c_d @ X2 @ ( these_5855049757122979476_a_c_d @ A6 ) )
      = ( member7124857669314633414_a_c_d @ ( some_P7564561949935391804_a_c_d @ X2 ) @ A6 ) ) ).

% in_these_eq
thf(fact_393_in__these__eq,axiom,
    ! [X2: option_a,A6: set_option_option_a] :
      ( ( member_option_a @ X2 @ ( these_option_a @ A6 ) )
      = ( member5113800082084363315tion_a @ ( some_option_a @ X2 ) @ A6 ) ) ).

% in_these_eq
thf(fact_394_in__these__eq,axiom,
    ! [X2: product_prod_a_c_d,A6: set_op3383557829362473625_a_c_d] :
      ( ( member8846983874477766826_a_c_d @ X2 @ ( these_5134989223866396566_a_c_d @ A6 ) )
      = ( member5926479794665358074_a_c_d @ ( some_P1084500821511757806_a_c_d @ X2 ) @ A6 ) ) ).

% in_these_eq
thf(fact_395_in__these__eq,axiom,
    ! [X2: a,A6: set_option_a] :
      ( ( member_a @ X2 @ ( these_a @ A6 ) )
      = ( member_option_a @ ( some_a @ X2 ) @ A6 ) ) ).

% in_these_eq
thf(fact_396_insert__dom,axiom,
    ! [F2: a > option_a,X2: a,Y: a] :
      ( ( ( F2 @ X2 )
        = ( some_a @ Y ) )
     => ( ( insert_a @ X2 @ ( dom_a_a @ F2 ) )
        = ( dom_a_a @ F2 ) ) ) ).

% insert_dom
thf(fact_397_insert__dom,axiom,
    ! [F2: option_a > option_a,X2: option_a,Y: a] :
      ( ( ( F2 @ X2 )
        = ( some_a @ Y ) )
     => ( ( insert_option_a @ X2 @ ( dom_option_a_a @ F2 ) )
        = ( dom_option_a_a @ F2 ) ) ) ).

% insert_dom
thf(fact_398_insert__dom,axiom,
    ! [F2: a > option6413918287372586467_a_c_d,X2: a,Y: product_prod_a_c_d] :
      ( ( ( F2 @ X2 )
        = ( some_P1084500821511757806_a_c_d @ Y ) )
     => ( ( insert_a @ X2 @ ( dom_a_3529858286862933695_a_c_d @ F2 ) )
        = ( dom_a_3529858286862933695_a_c_d @ F2 ) ) ) ).

% insert_dom
thf(fact_399_insert__dom,axiom,
    ! [F2: option_a > option6413918287372586467_a_c_d,X2: option_a,Y: product_prod_a_c_d] :
      ( ( ( F2 @ X2 )
        = ( some_P1084500821511757806_a_c_d @ Y ) )
     => ( ( insert_option_a @ X2 @ ( dom_op6156779764657312645_a_c_d @ F2 ) )
        = ( dom_op6156779764657312645_a_c_d @ F2 ) ) ) ).

% insert_dom
thf(fact_400_bind__split,axiom,
    ! [P2: option_a > $o,M: option6413918287372586467_a_c_d,F2: product_prod_a_c_d > option_a] :
      ( ( P2 @ ( bind_P2911691133002111467_c_d_a @ M @ F2 ) )
      = ( ( ( M = none_P2239999354720063602_a_c_d )
         => ( P2 @ none_a ) )
        & ! [V: product_prod_a_c_d] :
            ( ( M
              = ( some_P1084500821511757806_a_c_d @ V ) )
           => ( P2 @ ( F2 @ V ) ) ) ) ) ).

% bind_split
thf(fact_401_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_402_bind__split__asm,axiom,
    ! [P2: option_a > $o,M: option6413918287372586467_a_c_d,F2: product_prod_a_c_d > option_a] :
      ( ( P2 @ ( bind_P2911691133002111467_c_d_a @ M @ F2 ) )
      = ( ~ ( ( ( M = none_P2239999354720063602_a_c_d )
              & ~ ( P2 @ none_a ) )
            | ? [X: product_prod_a_c_d] :
                ( ( M
                  = ( some_P1084500821511757806_a_c_d @ X ) )
                & ~ ( P2 @ ( F2 @ X ) ) ) ) ) ) ).

% bind_split_asm
thf(fact_403_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_404_graph__map__upd,axiom,
    ! [M: product_prod_a_c_d > option6212624336325788467_a_c_d,K: product_prod_a_c_d,V3: option6413918287372586467_a_c_d] :
      ( ( graph_7244981646656495094_a_c_d @ ( fun_up3914643275129436357_a_c_d @ M @ K @ ( some_o8886552288663794750_a_c_d @ V3 ) ) )
      = ( insert4072338954160766247_a_c_d @ ( produc8093790510458973071_a_c_d @ K @ V3 ) @ ( graph_7244981646656495094_a_c_d @ ( fun_up3914643275129436357_a_c_d @ M @ K @ none_o2262236386069974722_a_c_d ) ) ) ) ).

% graph_map_upd
thf(fact_405_graph__map__upd,axiom,
    ! [M: a > option_c_d,K: a,V3: c > d] :
      ( ( graph_a_c_d @ ( fun_upd_a_option_c_d @ M @ K @ ( some_c_d @ V3 ) ) )
      = ( insert2748062992969704259_a_c_d @ ( product_Pair_a_c_d @ K @ V3 ) @ ( graph_a_c_d @ ( fun_upd_a_option_c_d @ M @ K @ none_c_d ) ) ) ) ).

% graph_map_upd
thf(fact_406_restrict__upd__same,axiom,
    ! [M: a > option_a,X2: a,Y: a] :
      ( ( restrict_map_a_a @ ( fun_upd_a_option_a @ M @ X2 @ ( some_a @ Y ) ) @ ( uminus_uminus_set_a @ ( insert_a @ X2 @ bot_bot_set_a ) ) )
      = ( restrict_map_a_a @ M @ ( uminus_uminus_set_a @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).

% restrict_upd_same
thf(fact_407_restrict__upd__same,axiom,
    ! [M: a > option6413918287372586467_a_c_d,X2: a,Y: product_prod_a_c_d] :
      ( ( restri776808000187872865_a_c_d @ ( fun_up6642239869823773348_a_c_d @ M @ X2 @ ( some_P1084500821511757806_a_c_d @ Y ) ) @ ( uminus_uminus_set_a @ ( insert_a @ X2 @ bot_bot_set_a ) ) )
      = ( restri776808000187872865_a_c_d @ M @ ( uminus_uminus_set_a @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).

% restrict_upd_same
thf(fact_408_restrict__upd__same,axiom,
    ! [M: option_a > option_a,X2: option_a,Y: a] :
      ( ( restri3984065703976872170on_a_a @ ( fun_up1079276522633388797tion_a @ M @ X2 @ ( some_a @ Y ) ) @ ( uminus6205308855922866075tion_a @ ( insert_option_a @ X2 @ bot_bot_set_option_a ) ) )
      = ( restri3984065703976872170on_a_a @ M @ ( uminus6205308855922866075tion_a @ ( insert_option_a @ X2 @ bot_bot_set_option_a ) ) ) ) ).

% restrict_upd_same
thf(fact_409_restrict__upd__same,axiom,
    ! [M: option_a > option6413918287372586467_a_c_d,X2: option_a,Y: product_prod_a_c_d] :
      ( ( restri9088203652661596199_a_c_d @ ( fun_up5969202740897331946_a_c_d @ M @ X2 @ ( some_P1084500821511757806_a_c_d @ Y ) ) @ ( uminus6205308855922866075tion_a @ ( insert_option_a @ X2 @ bot_bot_set_option_a ) ) )
      = ( restri9088203652661596199_a_c_d @ M @ ( uminus6205308855922866075tion_a @ ( insert_option_a @ X2 @ bot_bot_set_option_a ) ) ) ) ).

% restrict_upd_same
thf(fact_410_fun__upd__None__restrict,axiom,
    ! [X2: produc5278197477302038359_a_c_d,D2: set_Pr1275464188344874039_a_c_d,M: produc5278197477302038359_a_c_d > option_a] :
      ( ( ( member1180172933830803072_a_c_d @ X2 @ D2 )
       => ( ( fun_up7463025390225626410tion_a @ ( restri6481929051272950551_c_d_a @ M @ D2 ) @ X2 @ none_a )
          = ( restri6481929051272950551_c_d_a @ M @ ( minus_4271488753454470558_a_c_d @ D2 @ ( insert4072338954160766247_a_c_d @ X2 @ bot_bo4027597737525728931_a_c_d ) ) ) ) )
      & ( ~ ( member1180172933830803072_a_c_d @ X2 @ D2 )
       => ( ( fun_up7463025390225626410tion_a @ ( restri6481929051272950551_c_d_a @ M @ D2 ) @ X2 @ none_a )
          = ( restri6481929051272950551_c_d_a @ M @ D2 ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_411_fun__upd__None__restrict,axiom,
    ! [X2: a,D2: set_a,M: a > option_a] :
      ( ( ( member_a @ X2 @ D2 )
       => ( ( fun_upd_a_option_a @ ( restrict_map_a_a @ M @ D2 ) @ X2 @ none_a )
          = ( restrict_map_a_a @ M @ ( minus_minus_set_a @ D2 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) )
      & ( ~ ( member_a @ X2 @ D2 )
       => ( ( fun_upd_a_option_a @ ( restrict_map_a_a @ M @ D2 ) @ X2 @ none_a )
          = ( restrict_map_a_a @ M @ D2 ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_412_fun__upd__None__restrict,axiom,
    ! [X2: option_a,D2: set_option_a,M: option_a > option_a] :
      ( ( ( member_option_a @ X2 @ D2 )
       => ( ( fun_up1079276522633388797tion_a @ ( restri3984065703976872170on_a_a @ M @ D2 ) @ X2 @ none_a )
          = ( restri3984065703976872170on_a_a @ M @ ( minus_1574173051537231627tion_a @ D2 @ ( insert_option_a @ X2 @ bot_bot_set_option_a ) ) ) ) )
      & ( ~ ( member_option_a @ X2 @ D2 )
       => ( ( fun_up1079276522633388797tion_a @ ( restri3984065703976872170on_a_a @ M @ D2 ) @ X2 @ none_a )
          = ( restri3984065703976872170on_a_a @ M @ D2 ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_413_dom__fun__upd,axiom,
    ! [Y: option_a,F2: a > option_a,X2: a] :
      ( ( ( Y = none_a )
       => ( ( dom_a_a @ ( fun_upd_a_option_a @ F2 @ X2 @ Y ) )
          = ( minus_minus_set_a @ ( dom_a_a @ F2 ) @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) )
      & ( ( Y != none_a )
       => ( ( dom_a_a @ ( fun_upd_a_option_a @ F2 @ X2 @ Y ) )
          = ( insert_a @ X2 @ ( dom_a_a @ F2 ) ) ) ) ) ).

% dom_fun_upd
thf(fact_414_dom__fun__upd,axiom,
    ! [Y: option_a,F2: option_a > option_a,X2: option_a] :
      ( ( ( Y = none_a )
       => ( ( dom_option_a_a @ ( fun_up1079276522633388797tion_a @ F2 @ X2 @ Y ) )
          = ( minus_1574173051537231627tion_a @ ( dom_option_a_a @ F2 ) @ ( insert_option_a @ X2 @ bot_bot_set_option_a ) ) ) )
      & ( ( Y != none_a )
       => ( ( dom_option_a_a @ ( fun_up1079276522633388797tion_a @ F2 @ X2 @ Y ) )
          = ( insert_option_a @ X2 @ ( dom_option_a_a @ F2 ) ) ) ) ) ).

% dom_fun_upd
thf(fact_415_in__graphD,axiom,
    ! [K: product_prod_a_c_d,V3: option6413918287372586467_a_c_d,M: product_prod_a_c_d > option6212624336325788467_a_c_d] :
      ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ K @ V3 ) @ ( graph_7244981646656495094_a_c_d @ M ) )
     => ( ( M @ K )
        = ( some_o8886552288663794750_a_c_d @ V3 ) ) ) ).

% in_graphD
thf(fact_416_in__graphD,axiom,
    ! [K: a,V3: c > d,M: a > option_c_d] :
      ( ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ K @ V3 ) @ ( graph_a_c_d @ M ) )
     => ( ( M @ K )
        = ( some_c_d @ V3 ) ) ) ).

% in_graphD
thf(fact_417_in__graphI,axiom,
    ! [M: product_prod_a_c_d > option6212624336325788467_a_c_d,K: product_prod_a_c_d,V3: option6413918287372586467_a_c_d] :
      ( ( ( M @ K )
        = ( some_o8886552288663794750_a_c_d @ V3 ) )
     => ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ K @ V3 ) @ ( graph_7244981646656495094_a_c_d @ M ) ) ) ).

% in_graphI
thf(fact_418_in__graphI,axiom,
    ! [M: a > option_c_d,K: a,V3: c > d] :
      ( ( ( M @ K )
        = ( some_c_d @ V3 ) )
     => ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ K @ V3 ) @ ( graph_a_c_d @ M ) ) ) ).

% in_graphI
thf(fact_419_graph__restrictD_I1_J,axiom,
    ! [K: product_prod_a_c_d,V3: option6413918287372586467_a_c_d,M: product_prod_a_c_d > option6212624336325788467_a_c_d,A6: set_Pr334309860168054345_a_c_d] :
      ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ K @ V3 ) @ ( graph_7244981646656495094_a_c_d @ ( restri3000517089388389544_a_c_d @ M @ A6 ) ) )
     => ( member8846983874477766826_a_c_d @ K @ A6 ) ) ).

% graph_restrictD(1)
thf(fact_420_graph__restrictD_I1_J,axiom,
    ! [K: a,V3: c > d,M: a > option_c_d,A6: set_a] :
      ( ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ K @ V3 ) @ ( graph_a_c_d @ ( restrict_map_a_c_d @ M @ A6 ) ) )
     => ( member_a @ K @ A6 ) ) ).

% graph_restrictD(1)
thf(fact_421_dom__minus,axiom,
    ! [F2: a > option_a,X2: a,A6: set_a] :
      ( ( ( F2 @ X2 )
        = none_a )
     => ( ( minus_minus_set_a @ ( dom_a_a @ F2 ) @ ( insert_a @ X2 @ A6 ) )
        = ( minus_minus_set_a @ ( dom_a_a @ F2 ) @ A6 ) ) ) ).

% dom_minus
thf(fact_422_dom__minus,axiom,
    ! [F2: option_a > option_a,X2: option_a,A6: set_option_a] :
      ( ( ( F2 @ X2 )
        = none_a )
     => ( ( minus_1574173051537231627tion_a @ ( dom_option_a_a @ F2 ) @ ( insert_option_a @ X2 @ A6 ) )
        = ( minus_1574173051537231627tion_a @ ( dom_option_a_a @ F2 ) @ A6 ) ) ) ).

% dom_minus
thf(fact_423_graph__restrictD_I2_J,axiom,
    ! [K: product_prod_a_c_d,V3: option6413918287372586467_a_c_d,M: product_prod_a_c_d > option6212624336325788467_a_c_d,A6: set_Pr334309860168054345_a_c_d] :
      ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ K @ V3 ) @ ( graph_7244981646656495094_a_c_d @ ( restri3000517089388389544_a_c_d @ M @ A6 ) ) )
     => ( ( M @ K )
        = ( some_o8886552288663794750_a_c_d @ V3 ) ) ) ).

% graph_restrictD(2)
thf(fact_424_graph__restrictD_I2_J,axiom,
    ! [K: a,V3: c > d,M: a > option_c_d,A6: set_a] :
      ( ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ K @ V3 ) @ ( graph_a_c_d @ ( restrict_map_a_c_d @ M @ A6 ) ) )
     => ( ( M @ K )
        = ( some_c_d @ V3 ) ) ) ).

% graph_restrictD(2)
thf(fact_425_restrict__complement__singleton__eq,axiom,
    ! [F2: a > option_a,X2: a] :
      ( ( restrict_map_a_a @ F2 @ ( uminus_uminus_set_a @ ( insert_a @ X2 @ bot_bot_set_a ) ) )
      = ( fun_upd_a_option_a @ F2 @ X2 @ none_a ) ) ).

% restrict_complement_singleton_eq
thf(fact_426_restrict__complement__singleton__eq,axiom,
    ! [F2: option_a > option_a,X2: option_a] :
      ( ( restri3984065703976872170on_a_a @ F2 @ ( uminus6205308855922866075tion_a @ ( insert_option_a @ X2 @ bot_bot_set_option_a ) ) )
      = ( fun_up1079276522633388797tion_a @ F2 @ X2 @ none_a ) ) ).

% restrict_complement_singleton_eq
thf(fact_427_ran__map__upd__Some,axiom,
    ! [M: a > option_a,X2: a,Y: a,Z2: a] :
      ( ( ( M @ X2 )
        = ( 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 @ X2 @ ( 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_428_Field__insert,axiom,
    ! [A: a,B: a,R: set_Product_prod_a_a] :
      ( ( field_a @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ A @ B ) @ R ) )
      = ( sup_sup_set_a @ ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( field_a @ R ) ) ) ).

% Field_insert
thf(fact_429_Field__insert,axiom,
    ! [A: option_a,B: option_a,R: set_Pr7585778909603769095tion_a] :
      ( ( field_option_a @ ( insert1246254401036548087tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R ) )
      = ( sup_sup_set_option_a @ ( insert_option_a @ A @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( field_option_a @ R ) ) ) ).

% Field_insert
thf(fact_430_fun__upd__image,axiom,
    ! [X2: produc5278197477302038359_a_c_d,A6: set_Pr1275464188344874039_a_c_d,F2: produc5278197477302038359_a_c_d > a,Y: a] :
      ( ( ( member1180172933830803072_a_c_d @ X2 @ A6 )
       => ( ( image_8636529855994272620_c_d_a @ ( fun_up2227350886784581604_c_d_a @ F2 @ X2 @ Y ) @ A6 )
          = ( insert_a @ Y @ ( image_8636529855994272620_c_d_a @ F2 @ ( minus_4271488753454470558_a_c_d @ A6 @ ( insert4072338954160766247_a_c_d @ X2 @ bot_bo4027597737525728931_a_c_d ) ) ) ) ) )
      & ( ~ ( member1180172933830803072_a_c_d @ X2 @ A6 )
       => ( ( image_8636529855994272620_c_d_a @ ( fun_up2227350886784581604_c_d_a @ F2 @ X2 @ Y ) @ A6 )
          = ( image_8636529855994272620_c_d_a @ F2 @ A6 ) ) ) ) ).

% fun_upd_image
thf(fact_431_fun__upd__image,axiom,
    ! [X2: produc5278197477302038359_a_c_d,A6: set_Pr1275464188344874039_a_c_d,F2: produc5278197477302038359_a_c_d > option_a,Y: option_a] :
      ( ( ( member1180172933830803072_a_c_d @ X2 @ A6 )
       => ( ( image_158748266322232498tion_a @ ( fun_up7463025390225626410tion_a @ F2 @ X2 @ Y ) @ A6 )
          = ( insert_option_a @ Y @ ( image_158748266322232498tion_a @ F2 @ ( minus_4271488753454470558_a_c_d @ A6 @ ( insert4072338954160766247_a_c_d @ X2 @ bot_bo4027597737525728931_a_c_d ) ) ) ) ) )
      & ( ~ ( member1180172933830803072_a_c_d @ X2 @ A6 )
       => ( ( image_158748266322232498tion_a @ ( fun_up7463025390225626410tion_a @ F2 @ X2 @ Y ) @ A6 )
          = ( image_158748266322232498tion_a @ F2 @ A6 ) ) ) ) ).

% fun_upd_image
thf(fact_432_fun__upd__image,axiom,
    ! [X2: c,A6: set_c,F2: c > d,Y: d] :
      ( ( ( member_c @ X2 @ A6 )
       => ( ( image_c_d @ ( fun_upd_c_d @ F2 @ X2 @ Y ) @ A6 )
          = ( insert_d @ Y @ ( image_c_d @ F2 @ ( minus_minus_set_c @ A6 @ ( insert_c @ X2 @ bot_bot_set_c ) ) ) ) ) )
      & ( ~ ( member_c @ X2 @ A6 )
       => ( ( image_c_d @ ( fun_upd_c_d @ F2 @ X2 @ Y ) @ A6 )
          = ( image_c_d @ F2 @ A6 ) ) ) ) ).

% fun_upd_image
thf(fact_433_fun__upd__image,axiom,
    ! [X2: a,A6: set_a,F2: a > a,Y: a] :
      ( ( ( member_a @ X2 @ A6 )
       => ( ( image_a_a2 @ ( fun_upd_a_a @ F2 @ X2 @ Y ) @ A6 )
          = ( insert_a @ Y @ ( image_a_a2 @ F2 @ ( minus_minus_set_a @ A6 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) )
      & ( ~ ( member_a @ X2 @ A6 )
       => ( ( image_a_a2 @ ( fun_upd_a_a @ F2 @ X2 @ Y ) @ A6 )
          = ( image_a_a2 @ F2 @ A6 ) ) ) ) ).

% fun_upd_image
thf(fact_434_fun__upd__image,axiom,
    ! [X2: a,A6: set_a,F2: a > option_a,Y: option_a] :
      ( ( ( member_a @ X2 @ A6 )
       => ( ( image_a_option_a2 @ ( fun_upd_a_option_a @ F2 @ X2 @ Y ) @ A6 )
          = ( insert_option_a @ Y @ ( image_a_option_a2 @ F2 @ ( minus_minus_set_a @ A6 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) )
      & ( ~ ( member_a @ X2 @ A6 )
       => ( ( image_a_option_a2 @ ( fun_upd_a_option_a @ F2 @ X2 @ Y ) @ A6 )
          = ( image_a_option_a2 @ F2 @ A6 ) ) ) ) ).

% fun_upd_image
thf(fact_435_fun__upd__image,axiom,
    ! [X2: option_a,A6: set_option_a,F2: option_a > a,Y: a] :
      ( ( ( member_option_a @ X2 @ A6 )
       => ( ( image_option_a_a2 @ ( fun_upd_option_a_a @ F2 @ X2 @ Y ) @ A6 )
          = ( insert_a @ Y @ ( image_option_a_a2 @ F2 @ ( minus_1574173051537231627tion_a @ A6 @ ( insert_option_a @ X2 @ bot_bot_set_option_a ) ) ) ) ) )
      & ( ~ ( member_option_a @ X2 @ A6 )
       => ( ( image_option_a_a2 @ ( fun_upd_option_a_a @ F2 @ X2 @ Y ) @ A6 )
          = ( image_option_a_a2 @ F2 @ A6 ) ) ) ) ).

% fun_upd_image
thf(fact_436_fun__upd__image,axiom,
    ! [X2: option_a,A6: set_option_a,F2: option_a > option_a,Y: option_a] :
      ( ( ( member_option_a @ X2 @ A6 )
       => ( ( image_7439109396645324421tion_a @ ( fun_up1079276522633388797tion_a @ F2 @ X2 @ Y ) @ A6 )
          = ( insert_option_a @ Y @ ( image_7439109396645324421tion_a @ F2 @ ( minus_1574173051537231627tion_a @ A6 @ ( insert_option_a @ X2 @ bot_bot_set_option_a ) ) ) ) ) )
      & ( ~ ( member_option_a @ X2 @ A6 )
       => ( ( image_7439109396645324421tion_a @ ( fun_up1079276522633388797tion_a @ F2 @ X2 @ Y ) @ A6 )
          = ( image_7439109396645324421tion_a @ F2 @ A6 ) ) ) ) ).

% fun_upd_image
thf(fact_437_linear__order__on__singleton,axiom,
    ! [X2: a] : ( order_8768733634509060147r_on_a @ ( insert_a @ X2 @ bot_bot_set_a ) @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ X2 @ X2 ) @ bot_bo3357376287454694259od_a_a ) ) ).

% linear_order_on_singleton
thf(fact_438_linear__order__on__singleton,axiom,
    ! [X2: option_a] : ( order_7850372301378808569tion_a @ ( insert_option_a @ X2 @ bot_bot_set_option_a ) @ ( insert1246254401036548087tion_a @ ( produc9011544418120257559tion_a @ X2 @ X2 ) @ bot_bo235252021745139059tion_a ) ) ).

% linear_order_on_singleton
thf(fact_439_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_440_inj__on__empty,axiom,
    ! [F2: a > option_a] : ( inj_on_a_option_a @ F2 @ bot_bot_set_a ) ).

% inj_on_empty
thf(fact_441_image__map__upd,axiom,
    ! [X2: a,A6: set_a,M: a > option_a,Y: a] :
      ( ~ ( member_a @ X2 @ A6 )
     => ( ( image_a_option_a2 @ ( fun_upd_a_option_a @ M @ X2 @ ( some_a @ Y ) ) @ A6 )
        = ( image_a_option_a2 @ M @ A6 ) ) ) ).

% image_map_upd
thf(fact_442_image__map__upd,axiom,
    ! [X2: produc5278197477302038359_a_c_d,A6: set_Pr1275464188344874039_a_c_d,M: produc5278197477302038359_a_c_d > option_a,Y: a] :
      ( ~ ( member1180172933830803072_a_c_d @ X2 @ A6 )
     => ( ( image_158748266322232498tion_a @ ( fun_up7463025390225626410tion_a @ M @ X2 @ ( some_a @ Y ) ) @ A6 )
        = ( image_158748266322232498tion_a @ M @ A6 ) ) ) ).

% image_map_upd
thf(fact_443_image__map__upd,axiom,
    ! [X2: option_a,A6: set_option_a,M: option_a > option_a,Y: a] :
      ( ~ ( member_option_a @ X2 @ A6 )
     => ( ( image_7439109396645324421tion_a @ ( fun_up1079276522633388797tion_a @ M @ X2 @ ( some_a @ Y ) ) @ A6 )
        = ( image_7439109396645324421tion_a @ M @ A6 ) ) ) ).

% image_map_upd
thf(fact_444_image__map__upd,axiom,
    ! [X2: a,A6: set_a,M: a > option6413918287372586467_a_c_d,Y: product_prod_a_c_d] :
      ( ~ ( member_a @ X2 @ A6 )
     => ( ( image_3786525410991915292_a_c_d @ ( fun_up6642239869823773348_a_c_d @ M @ X2 @ ( some_P1084500821511757806_a_c_d @ Y ) ) @ A6 )
        = ( image_3786525410991915292_a_c_d @ M @ A6 ) ) ) ).

% image_map_upd
thf(fact_445_image__map__upd,axiom,
    ! [X2: produc5278197477302038359_a_c_d,A6: set_Pr1275464188344874039_a_c_d,M: produc5278197477302038359_a_c_d > option6413918287372586467_a_c_d,Y: product_prod_a_c_d] :
      ( ~ ( member1180172933830803072_a_c_d @ X2 @ A6 )
     => ( ( image_1524162552056343797_a_c_d @ ( fun_up4007221454163596925_a_c_d @ M @ X2 @ ( some_P1084500821511757806_a_c_d @ Y ) ) @ A6 )
        = ( image_1524162552056343797_a_c_d @ M @ A6 ) ) ) ).

% image_map_upd
thf(fact_446_image__map__upd,axiom,
    ! [X2: option_a,A6: set_option_a,M: option_a > option6413918287372586467_a_c_d,Y: product_prod_a_c_d] :
      ( ~ ( member_option_a @ X2 @ A6 )
     => ( ( image_3697009332183072098_a_c_d @ ( fun_up5969202740897331946_a_c_d @ M @ X2 @ ( some_P1084500821511757806_a_c_d @ Y ) ) @ A6 )
        = ( image_3697009332183072098_a_c_d @ M @ A6 ) ) ) ).

% image_map_upd
thf(fact_447_is__none__code_I2_J,axiom,
    ! [X2: product_prod_a_c_d] :
      ~ ( is_non1375147668126617655_a_c_d @ ( some_P1084500821511757806_a_c_d @ X2 ) ) ).

% is_none_code(2)
thf(fact_448_is__none__code_I2_J,axiom,
    ! [X2: a] :
      ~ ( is_none_a @ ( some_a @ X2 ) ) ).

% is_none_code(2)
thf(fact_449_is__none__code_I1_J,axiom,
    is_none_a @ none_a ).

% is_none_code(1)
thf(fact_450_these__image__Some__eq,axiom,
    ! [A6: set_Pr334309860168054345_a_c_d] :
      ( ( these_5134989223866396566_a_c_d @ ( image_5741706613785127421_a_c_d @ some_P1084500821511757806_a_c_d @ A6 ) )
      = A6 ) ).

% these_image_Some_eq
thf(fact_451_these__image__Some__eq,axiom,
    ! [A6: set_a] :
      ( ( these_a @ ( image_a_option_a2 @ some_a @ A6 ) )
      = A6 ) ).

% these_image_Some_eq
thf(fact_452_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_453_inj__on__insert,axiom,
    ! [F2: a > a,A: a,A6: set_a] :
      ( ( inj_on_a_a @ F2 @ ( insert_a @ A @ A6 ) )
      = ( ( inj_on_a_a @ F2 @ A6 )
        & ~ ( member_a @ ( F2 @ A ) @ ( image_a_a2 @ F2 @ ( minus_minus_set_a @ A6 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_454_inj__on__insert,axiom,
    ! [F2: a > produc5278197477302038359_a_c_d,A: a,A6: set_a] :
      ( ( inj_on2836679365399032650_a_c_d @ F2 @ ( insert_a @ A @ A6 ) )
      = ( ( inj_on2836679365399032650_a_c_d @ F2 @ A6 )
        & ~ ( member1180172933830803072_a_c_d @ ( F2 @ A ) @ ( image_6011757624558158814_a_c_d @ F2 @ ( minus_minus_set_a @ A6 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_455_inj__on__insert,axiom,
    ! [F2: a > option_a,A: a,A6: set_a] :
      ( ( inj_on_a_option_a @ F2 @ ( insert_a @ A @ A6 ) )
      = ( ( inj_on_a_option_a @ F2 @ A6 )
        & ~ ( member_option_a @ ( F2 @ A ) @ ( image_a_option_a2 @ F2 @ ( minus_minus_set_a @ A6 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_456_inj__on__insert,axiom,
    ! [F2: option_a > a,A: option_a,A6: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ ( insert_option_a @ A @ A6 ) )
      = ( ( inj_on_option_a_a @ F2 @ A6 )
        & ~ ( member_a @ ( F2 @ A ) @ ( image_option_a_a2 @ F2 @ ( minus_1574173051537231627tion_a @ A6 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_457_inj__on__insert,axiom,
    ! [F2: option_a > produc5278197477302038359_a_c_d,A: option_a,A6: set_option_a] :
      ( ( inj_on6212699291050112900_a_c_d @ F2 @ ( insert_option_a @ A @ A6 ) )
      = ( ( inj_on6212699291050112900_a_c_d @ F2 @ A6 )
        & ~ ( member1180172933830803072_a_c_d @ ( F2 @ A ) @ ( image_8542254255036258840_a_c_d @ F2 @ ( minus_1574173051537231627tion_a @ A6 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_458_inj__on__insert,axiom,
    ! [F2: option_a > option_a,A: option_a,A6: set_option_a] :
      ( ( inj_on8559383841115902449tion_a @ F2 @ ( insert_option_a @ A @ A6 ) )
      = ( ( inj_on8559383841115902449tion_a @ F2 @ A6 )
        & ~ ( member_option_a @ ( F2 @ A ) @ ( image_7439109396645324421tion_a @ F2 @ ( minus_1574173051537231627tion_a @ A6 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_459_inj__img__insertE,axiom,
    ! [F2: a > a,A6: set_a,X2: a,B5: set_a] :
      ( ( inj_on_a_a @ F2 @ A6 )
     => ( ~ ( member_a @ X2 @ B5 )
       => ( ( ( insert_a @ X2 @ B5 )
            = ( image_a_a2 @ F2 @ A6 ) )
         => ~ ! [X7: a,A9: set_a] :
                ( ~ ( member_a @ X7 @ A9 )
               => ( ( A6
                    = ( insert_a @ X7 @ A9 ) )
                 => ( ( X2
                      = ( F2 @ X7 ) )
                   => ( B5
                     != ( image_a_a2 @ F2 @ A9 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_460_inj__img__insertE,axiom,
    ! [F2: produc5278197477302038359_a_c_d > a,A6: set_Pr1275464188344874039_a_c_d,X2: a,B5: set_a] :
      ( ( inj_on5461451596835146456_c_d_a @ F2 @ A6 )
     => ( ~ ( member_a @ X2 @ B5 )
       => ( ( ( insert_a @ X2 @ B5 )
            = ( image_8636529855994272620_c_d_a @ F2 @ A6 ) )
         => ~ ! [X7: produc5278197477302038359_a_c_d,A9: set_Pr1275464188344874039_a_c_d] :
                ( ~ ( member1180172933830803072_a_c_d @ X7 @ A9 )
               => ( ( A6
                    = ( insert4072338954160766247_a_c_d @ X7 @ A9 ) )
                 => ( ( X2
                      = ( F2 @ X7 ) )
                   => ( B5
                     != ( image_8636529855994272620_c_d_a @ F2 @ A9 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_461_inj__img__insertE,axiom,
    ! [F2: option_a > a,A6: set_option_a,X2: a,B5: set_a] :
      ( ( inj_on_option_a_a @ F2 @ A6 )
     => ( ~ ( member_a @ X2 @ B5 )
       => ( ( ( insert_a @ X2 @ B5 )
            = ( image_option_a_a2 @ F2 @ A6 ) )
         => ~ ! [X7: option_a,A9: set_option_a] :
                ( ~ ( member_option_a @ X7 @ A9 )
               => ( ( A6
                    = ( insert_option_a @ X7 @ A9 ) )
                 => ( ( X2
                      = ( F2 @ X7 ) )
                   => ( B5
                     != ( image_option_a_a2 @ F2 @ A9 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_462_inj__img__insertE,axiom,
    ! [F2: a > produc5278197477302038359_a_c_d,A6: set_a,X2: produc5278197477302038359_a_c_d,B5: set_Pr1275464188344874039_a_c_d] :
      ( ( inj_on2836679365399032650_a_c_d @ F2 @ A6 )
     => ( ~ ( member1180172933830803072_a_c_d @ X2 @ B5 )
       => ( ( ( insert4072338954160766247_a_c_d @ X2 @ B5 )
            = ( image_6011757624558158814_a_c_d @ F2 @ A6 ) )
         => ~ ! [X7: a,A9: set_a] :
                ( ~ ( member_a @ X7 @ A9 )
               => ( ( A6
                    = ( insert_a @ X7 @ A9 ) )
                 => ( ( X2
                      = ( F2 @ X7 ) )
                   => ( B5
                     != ( image_6011757624558158814_a_c_d @ F2 @ A9 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_463_inj__img__insertE,axiom,
    ! [F2: produc5278197477302038359_a_c_d > produc5278197477302038359_a_c_d,A6: set_Pr1275464188344874039_a_c_d,X2: produc5278197477302038359_a_c_d,B5: set_Pr1275464188344874039_a_c_d] :
      ( ( inj_on7596037564690925361_a_c_d @ F2 @ A6 )
     => ( ~ ( member1180172933830803072_a_c_d @ X2 @ B5 )
       => ( ( ( insert4072338954160766247_a_c_d @ X2 @ B5 )
            = ( image_3091440356067162053_a_c_d @ F2 @ A6 ) )
         => ~ ! [X7: produc5278197477302038359_a_c_d,A9: set_Pr1275464188344874039_a_c_d] :
                ( ~ ( member1180172933830803072_a_c_d @ X7 @ A9 )
               => ( ( A6
                    = ( insert4072338954160766247_a_c_d @ X7 @ A9 ) )
                 => ( ( X2
                      = ( F2 @ X7 ) )
                   => ( B5
                     != ( image_3091440356067162053_a_c_d @ F2 @ A9 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_464_inj__img__insertE,axiom,
    ! [F2: option_a > produc5278197477302038359_a_c_d,A6: set_option_a,X2: produc5278197477302038359_a_c_d,B5: set_Pr1275464188344874039_a_c_d] :
      ( ( inj_on6212699291050112900_a_c_d @ F2 @ A6 )
     => ( ~ ( member1180172933830803072_a_c_d @ X2 @ B5 )
       => ( ( ( insert4072338954160766247_a_c_d @ X2 @ B5 )
            = ( image_8542254255036258840_a_c_d @ F2 @ A6 ) )
         => ~ ! [X7: option_a,A9: set_option_a] :
                ( ~ ( member_option_a @ X7 @ A9 )
               => ( ( A6
                    = ( insert_option_a @ X7 @ A9 ) )
                 => ( ( X2
                      = ( F2 @ X7 ) )
                   => ( B5
                     != ( image_8542254255036258840_a_c_d @ F2 @ A9 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_465_inj__img__insertE,axiom,
    ! [F2: a > option_a,A6: set_a,X2: option_a,B5: set_option_a] :
      ( ( inj_on_a_option_a @ F2 @ A6 )
     => ( ~ ( member_option_a @ X2 @ B5 )
       => ( ( ( insert_option_a @ X2 @ B5 )
            = ( image_a_option_a2 @ F2 @ A6 ) )
         => ~ ! [X7: a,A9: set_a] :
                ( ~ ( member_a @ X7 @ A9 )
               => ( ( A6
                    = ( insert_a @ X7 @ A9 ) )
                 => ( ( X2
                      = ( F2 @ X7 ) )
                   => ( B5
                     != ( image_a_option_a2 @ F2 @ A9 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_466_inj__img__insertE,axiom,
    ! [F2: produc5278197477302038359_a_c_d > option_a,A6: set_Pr1275464188344874039_a_c_d,X2: option_a,B5: set_option_a] :
      ( ( inj_on7052565339190862366tion_a @ F2 @ A6 )
     => ( ~ ( member_option_a @ X2 @ B5 )
       => ( ( ( insert_option_a @ X2 @ B5 )
            = ( image_158748266322232498tion_a @ F2 @ A6 ) )
         => ~ ! [X7: produc5278197477302038359_a_c_d,A9: set_Pr1275464188344874039_a_c_d] :
                ( ~ ( member1180172933830803072_a_c_d @ X7 @ A9 )
               => ( ( A6
                    = ( insert4072338954160766247_a_c_d @ X7 @ A9 ) )
                 => ( ( X2
                      = ( F2 @ X7 ) )
                   => ( B5
                     != ( image_158748266322232498tion_a @ F2 @ A9 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_467_inj__img__insertE,axiom,
    ! [F2: option_a > option_a,A6: set_option_a,X2: option_a,B5: set_option_a] :
      ( ( inj_on8559383841115902449tion_a @ F2 @ A6 )
     => ( ~ ( member_option_a @ X2 @ B5 )
       => ( ( ( insert_option_a @ X2 @ B5 )
            = ( image_7439109396645324421tion_a @ F2 @ A6 ) )
         => ~ ! [X7: option_a,A9: set_option_a] :
                ( ~ ( member_option_a @ X7 @ A9 )
               => ( ( A6
                    = ( insert_option_a @ X7 @ A9 ) )
                 => ( ( X2
                      = ( F2 @ X7 ) )
                   => ( B5
                     != ( image_7439109396645324421tion_a @ F2 @ A9 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_468_inj__on__diff,axiom,
    ! [F2: a > option_a,A6: set_a,B5: set_a] :
      ( ( inj_on_a_option_a @ F2 @ A6 )
     => ( inj_on_a_option_a @ F2 @ ( minus_minus_set_a @ A6 @ B5 ) ) ) ).

% inj_on_diff
thf(fact_469_inj__on__Int,axiom,
    ! [F2: a > option_a,A6: set_a,B5: set_a] :
      ( ( ( inj_on_a_option_a @ F2 @ A6 )
        | ( inj_on_a_option_a @ F2 @ B5 ) )
     => ( inj_on_a_option_a @ F2 @ ( inf_inf_set_a @ A6 @ B5 ) ) ) ).

% inj_on_Int
thf(fact_470_inj__Some,axiom,
    ! [A6: set_Pr334309860168054345_a_c_d] : ( inj_on7837763817969280361_a_c_d @ some_P1084500821511757806_a_c_d @ A6 ) ).

% inj_Some
thf(fact_471_inj__Some,axiom,
    ! [A6: set_a] : ( inj_on_a_option_a @ some_a @ A6 ) ).

% inj_Some
thf(fact_472_inj__on__Un__image__eq__iff,axiom,
    ! [F2: option_a > a,A6: set_option_a,B5: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ ( sup_sup_set_option_a @ A6 @ B5 ) )
     => ( ( ( image_option_a_a2 @ F2 @ A6 )
          = ( image_option_a_a2 @ F2 @ B5 ) )
        = ( A6 = B5 ) ) ) ).

% inj_on_Un_image_eq_iff
thf(fact_473_inj__on__Un__image__eq__iff,axiom,
    ! [F2: a > option_a,A6: set_a,B5: set_a] :
      ( ( inj_on_a_option_a @ F2 @ ( sup_sup_set_a @ A6 @ B5 ) )
     => ( ( ( image_a_option_a2 @ F2 @ A6 )
          = ( image_a_option_a2 @ F2 @ B5 ) )
        = ( A6 = B5 ) ) ) ).

% inj_on_Un_image_eq_iff
thf(fact_474_inj__on__fun__updI,axiom,
    ! [F2: option_a > a,A6: set_option_a,Y: a,X2: option_a] :
      ( ( inj_on_option_a_a @ F2 @ A6 )
     => ( ~ ( member_a @ Y @ ( image_option_a_a2 @ F2 @ A6 ) )
       => ( inj_on_option_a_a @ ( fun_upd_option_a_a @ F2 @ X2 @ Y ) @ A6 ) ) ) ).

% inj_on_fun_updI
thf(fact_475_inj__on__fun__updI,axiom,
    ! [F2: a > option_a,A6: set_a,Y: option_a,X2: a] :
      ( ( inj_on_a_option_a @ F2 @ A6 )
     => ( ~ ( member_option_a @ Y @ ( image_a_option_a2 @ F2 @ A6 ) )
       => ( inj_on_a_option_a @ ( fun_upd_a_option_a @ F2 @ X2 @ Y ) @ A6 ) ) ) ).

% inj_on_fun_updI
thf(fact_476_inj__on__fun__updI,axiom,
    ! [F2: c > d,A6: set_c,Y: d,X2: c] :
      ( ( inj_on_c_d @ F2 @ A6 )
     => ( ~ ( member_d @ Y @ ( image_c_d @ F2 @ A6 ) )
       => ( inj_on_c_d @ ( fun_upd_c_d @ F2 @ X2 @ Y ) @ A6 ) ) ) ).

% inj_on_fun_updI
thf(fact_477_inj__onD,axiom,
    ! [F2: a > option_a,A6: set_a,X2: a,Y: a] :
      ( ( inj_on_a_option_a @ F2 @ A6 )
     => ( ( ( F2 @ X2 )
          = ( F2 @ Y ) )
       => ( ( member_a @ X2 @ A6 )
         => ( ( member_a @ Y @ A6 )
           => ( X2 = Y ) ) ) ) ) ).

% inj_onD
thf(fact_478_inj__onI,axiom,
    ! [A6: set_a,F2: a > option_a] :
      ( ! [X4: a,Y4: a] :
          ( ( member_a @ X4 @ A6 )
         => ( ( member_a @ Y4 @ A6 )
           => ( ( ( F2 @ X4 )
                = ( F2 @ Y4 ) )
             => ( X4 = Y4 ) ) ) )
     => ( inj_on_a_option_a @ F2 @ A6 ) ) ).

% inj_onI
thf(fact_479_inj__on__def,axiom,
    ( inj_on_a_option_a
    = ( ^ [F: a > option_a,A8: set_a] :
        ! [X: a] :
          ( ( member_a @ X @ A8 )
         => ! [Y3: a] :
              ( ( member_a @ Y3 @ A8 )
             => ( ( ( F @ X )
                  = ( F @ Y3 ) )
               => ( X = Y3 ) ) ) ) ) ) ).

% inj_on_def
thf(fact_480_inj__on__cong,axiom,
    ! [A6: set_a,F2: a > option_a,G: a > option_a] :
      ( ! [A7: a] :
          ( ( member_a @ A7 @ A6 )
         => ( ( F2 @ A7 )
            = ( G @ A7 ) ) )
     => ( ( inj_on_a_option_a @ F2 @ A6 )
        = ( inj_on_a_option_a @ G @ A6 ) ) ) ).

% inj_on_cong
thf(fact_481_inj__on__eq__iff,axiom,
    ! [F2: a > option_a,A6: set_a,X2: a,Y: a] :
      ( ( inj_on_a_option_a @ F2 @ A6 )
     => ( ( member_a @ X2 @ A6 )
       => ( ( member_a @ Y @ A6 )
         => ( ( ( F2 @ X2 )
              = ( F2 @ Y ) )
            = ( X2 = Y ) ) ) ) ) ).

% inj_on_eq_iff
thf(fact_482_inj__on__contraD,axiom,
    ! [F2: a > option_a,A6: set_a,X2: a,Y: a] :
      ( ( inj_on_a_option_a @ F2 @ A6 )
     => ( ( X2 != Y )
       => ( ( member_a @ X2 @ A6 )
         => ( ( member_a @ Y @ A6 )
           => ( ( F2 @ X2 )
             != ( F2 @ Y ) ) ) ) ) ) ).

% inj_on_contraD
thf(fact_483_inj__on__inverseI,axiom,
    ! [A6: set_a,G: option_a > a,F2: a > option_a] :
      ( ! [X4: a] :
          ( ( member_a @ X4 @ A6 )
         => ( ( G @ ( F2 @ X4 ) )
            = X4 ) )
     => ( inj_on_a_option_a @ F2 @ A6 ) ) ).

% inj_on_inverseI
thf(fact_484_inj__on__image__iff,axiom,
    ! [A6: set_a,G: a > option_a,F2: a > a] :
      ( ! [X4: a] :
          ( ( member_a @ X4 @ A6 )
         => ! [Xa: a] :
              ( ( member_a @ Xa @ A6 )
             => ( ( ( G @ ( F2 @ X4 ) )
                  = ( G @ ( F2 @ Xa ) ) )
                = ( ( G @ X4 )
                  = ( G @ Xa ) ) ) ) )
     => ( ( inj_on_a_a @ F2 @ A6 )
       => ( ( inj_on_a_option_a @ G @ ( image_a_a2 @ F2 @ A6 ) )
          = ( inj_on_a_option_a @ G @ A6 ) ) ) ) ).

% inj_on_image_iff
thf(fact_485_FieldI2,axiom,
    ! [I: a,J: a,R3: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R3 )
     => ( member_a @ J @ ( field_a @ R3 ) ) ) ).

% FieldI2
thf(fact_486_FieldI2,axiom,
    ! [I: produc5278197477302038359_a_c_d,J: produc5278197477302038359_a_c_d,R3: set_Pr3774076249145843911_a_c_d] :
      ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ I @ J ) @ R3 )
     => ( member1180172933830803072_a_c_d @ J @ ( field_5664454651862650532_a_c_d @ R3 ) ) ) ).

% FieldI2
thf(fact_487_FieldI2,axiom,
    ! [I: option_a,J: option_a,R3: set_Pr7585778909603769095tion_a] :
      ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ I @ J ) @ R3 )
     => ( member_option_a @ J @ ( field_option_a @ R3 ) ) ) ).

% FieldI2
thf(fact_488_FieldI1,axiom,
    ! [I: a,J: a,R3: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R3 )
     => ( member_a @ I @ ( field_a @ R3 ) ) ) ).

% FieldI1
thf(fact_489_FieldI1,axiom,
    ! [I: produc5278197477302038359_a_c_d,J: produc5278197477302038359_a_c_d,R3: set_Pr3774076249145843911_a_c_d] :
      ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ I @ J ) @ R3 )
     => ( member1180172933830803072_a_c_d @ I @ ( field_5664454651862650532_a_c_d @ R3 ) ) ) ).

% FieldI1
thf(fact_490_FieldI1,axiom,
    ! [I: option_a,J: option_a,R3: set_Pr7585778909603769095tion_a] :
      ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ I @ J ) @ R3 )
     => ( member_option_a @ I @ ( field_option_a @ R3 ) ) ) ).

% FieldI1
thf(fact_491_is__none__simps_I2_J,axiom,
    ! [X2: product_prod_a_c_d] :
      ~ ( is_non1375147668126617655_a_c_d @ ( some_P1084500821511757806_a_c_d @ X2 ) ) ).

% is_none_simps(2)
thf(fact_492_is__none__simps_I2_J,axiom,
    ! [X2: a] :
      ~ ( is_none_a @ ( some_a @ X2 ) ) ).

% is_none_simps(2)
thf(fact_493_Option_Ois__none__def,axiom,
    ( is_none_a
    = ( ^ [X: option_a] : ( X = none_a ) ) ) ).

% Option.is_none_def
thf(fact_494_is__none__simps_I1_J,axiom,
    is_none_a @ none_a ).

% is_none_simps(1)
thf(fact_495_inj__on__Un,axiom,
    ! [F2: option_a > a,A6: set_option_a,B5: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ ( sup_sup_set_option_a @ A6 @ B5 ) )
      = ( ( inj_on_option_a_a @ F2 @ A6 )
        & ( inj_on_option_a_a @ F2 @ B5 )
        & ( ( inf_inf_set_a @ ( image_option_a_a2 @ F2 @ ( minus_1574173051537231627tion_a @ A6 @ B5 ) ) @ ( image_option_a_a2 @ F2 @ ( minus_1574173051537231627tion_a @ B5 @ A6 ) ) )
          = bot_bot_set_a ) ) ) ).

% inj_on_Un
thf(fact_496_inj__on__Un,axiom,
    ! [F2: a > option_a,A6: set_a,B5: set_a] :
      ( ( inj_on_a_option_a @ F2 @ ( sup_sup_set_a @ A6 @ B5 ) )
      = ( ( inj_on_a_option_a @ F2 @ A6 )
        & ( inj_on_a_option_a @ F2 @ B5 )
        & ( ( inf_inf_set_option_a @ ( image_a_option_a2 @ F2 @ ( minus_minus_set_a @ A6 @ B5 ) ) @ ( image_a_option_a2 @ F2 @ ( minus_minus_set_a @ B5 @ A6 ) ) )
          = bot_bot_set_option_a ) ) ) ).

% inj_on_Un
thf(fact_497_None__notin__image__Some,axiom,
    ! [A6: set_Pr334309860168054345_a_c_d] :
      ~ ( member5926479794665358074_a_c_d @ none_P2239999354720063602_a_c_d @ ( image_5741706613785127421_a_c_d @ some_P1084500821511757806_a_c_d @ A6 ) ) ).

% None_notin_image_Some
thf(fact_498_None__notin__image__Some,axiom,
    ! [A6: set_a] :
      ~ ( member_option_a @ none_a @ ( image_a_option_a2 @ some_a @ A6 ) ) ).

% None_notin_image_Some
thf(fact_499_refl__on__singleton,axiom,
    ! [X2: a] : ( refl_on_a @ ( insert_a @ X2 @ bot_bot_set_a ) @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ X2 @ X2 ) @ bot_bo3357376287454694259od_a_a ) ) ).

% refl_on_singleton
thf(fact_500_refl__on__singleton,axiom,
    ! [X2: option_a] : ( refl_on_option_a @ ( insert_option_a @ X2 @ bot_bot_set_option_a ) @ ( insert1246254401036548087tion_a @ ( produc9011544418120257559tion_a @ X2 @ X2 ) @ bot_bo235252021745139059tion_a ) ) ).

% refl_on_singleton
thf(fact_501_Range__insert,axiom,
    ! [A: product_prod_a_c_d,B: option6413918287372586467_a_c_d,R: set_Pr1275464188344874039_a_c_d] :
      ( ( range_1858866086472732781_a_c_d @ ( insert4072338954160766247_a_c_d @ ( produc8093790510458973071_a_c_d @ A @ B ) @ R ) )
      = ( insert6629875870486991891_a_c_d @ B @ ( range_1858866086472732781_a_c_d @ R ) ) ) ).

% Range_insert
thf(fact_502_Range__insert,axiom,
    ! [A: a,B: c > d,R: set_Pr334309860168054345_a_c_d] :
      ( ( range_a_c_d @ ( insert2748062992969704259_a_c_d @ ( product_Pair_a_c_d @ A @ B ) @ R ) )
      = ( insert_c_d @ B @ ( range_a_c_d @ R ) ) ) ).

% Range_insert
thf(fact_503_Domain__insert,axiom,
    ! [A: product_prod_a_c_d,B: option6413918287372586467_a_c_d,R: set_Pr1275464188344874039_a_c_d] :
      ( ( domain7659442726769224452_a_c_d @ ( insert4072338954160766247_a_c_d @ ( produc8093790510458973071_a_c_d @ A @ B ) @ R ) )
      = ( insert2748062992969704259_a_c_d @ A @ ( domain7659442726769224452_a_c_d @ R ) ) ) ).

% Domain_insert
thf(fact_504_Domain__insert,axiom,
    ! [A: a,B: c > d,R: set_Pr334309860168054345_a_c_d] :
      ( ( domain_a_c_d @ ( insert2748062992969704259_a_c_d @ ( product_Pair_a_c_d @ A @ B ) @ R ) )
      = ( insert_a @ A @ ( domain_a_c_d @ R ) ) ) ).

% Domain_insert
thf(fact_505_the__inv__into__onto,axiom,
    ! [F2: option_a > a,A6: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ A6 )
     => ( ( image_a_option_a2 @ ( the_in1757154643552616557on_a_a @ A6 @ F2 ) @ ( image_option_a_a2 @ F2 @ A6 ) )
        = A6 ) ) ).

% the_inv_into_onto
thf(fact_506_the__inv__into__onto,axiom,
    ! [F2: a > option_a,A6: set_a] :
      ( ( inj_on_a_option_a @ F2 @ A6 )
     => ( ( image_option_a_a2 @ ( the_in8758012798868597241tion_a @ A6 @ F2 ) @ ( image_a_option_a2 @ F2 @ A6 ) )
        = A6 ) ) ).

% the_inv_into_onto
thf(fact_507_Linear__order__in__diff__Id,axiom,
    ! [R: set_Product_prod_a_a,A: a,B: a] :
      ( ( order_8768733634509060147r_on_a @ ( field_a @ R ) @ R )
     => ( ( member_a @ A @ ( field_a @ R ) )
       => ( ( member_a @ B @ ( field_a @ R ) )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
            = ( ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ A ) @ ( minus_6817036919807184750od_a_a @ R @ id_a2 ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_508_Linear__order__in__diff__Id,axiom,
    ! [R: set_Pr3774076249145843911_a_c_d,A: produc5278197477302038359_a_c_d,B: produc5278197477302038359_a_c_d] :
      ( ( order_1589598524371451020_a_c_d @ ( field_5664454651862650532_a_c_d @ R ) @ R )
     => ( ( member1180172933830803072_a_c_d @ A @ ( field_5664454651862650532_a_c_d @ R ) )
       => ( ( member1180172933830803072_a_c_d @ B @ ( field_5664454651862650532_a_c_d @ R ) )
         => ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ A @ B ) @ R )
            = ( ~ ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ B @ A ) @ ( minus_6105496285253675822_a_c_d @ R @ id_Pro8620185220256667827_a_c_d ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_509_Linear__order__in__diff__Id,axiom,
    ! [R: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
      ( ( order_7850372301378808569tion_a @ ( field_option_a @ R ) @ R )
     => ( ( member_option_a @ A @ ( field_option_a @ R ) )
       => ( ( member_option_a @ B @ ( field_option_a @ R ) )
         => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R )
            = ( ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ A ) @ ( minus_6512073291116468334tion_a @ R @ id_option_a2 ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_510_refl__onD,axiom,
    ! [A6: set_a,R: set_Product_prod_a_a,A: a] :
      ( ( refl_on_a @ A6 @ R )
     => ( ( member_a @ A @ A6 )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_511_refl__onD,axiom,
    ! [A6: set_Pr1275464188344874039_a_c_d,R: set_Pr3774076249145843911_a_c_d,A: produc5278197477302038359_a_c_d] :
      ( ( refl_o726447250937964886_a_c_d @ A6 @ R )
     => ( ( member1180172933830803072_a_c_d @ A @ A6 )
       => ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_512_refl__onD,axiom,
    ! [A6: set_option_a,R: set_Pr7585778909603769095tion_a,A: option_a] :
      ( ( refl_on_option_a @ A6 @ R )
     => ( ( member_option_a @ A @ A6 )
       => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_513_refl__onD1,axiom,
    ! [A6: set_a,R: set_Product_prod_a_a,X2: a,Y: a] :
      ( ( refl_on_a @ A6 @ R )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
       => ( member_a @ X2 @ A6 ) ) ) ).

% refl_onD1
thf(fact_514_refl__onD1,axiom,
    ! [A6: set_Pr1275464188344874039_a_c_d,R: set_Pr3774076249145843911_a_c_d,X2: produc5278197477302038359_a_c_d,Y: produc5278197477302038359_a_c_d] :
      ( ( refl_o726447250937964886_a_c_d @ A6 @ R )
     => ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ X2 @ Y ) @ R )
       => ( member1180172933830803072_a_c_d @ X2 @ A6 ) ) ) ).

% refl_onD1
thf(fact_515_refl__onD1,axiom,
    ! [A6: set_option_a,R: set_Pr7585778909603769095tion_a,X2: option_a,Y: option_a] :
      ( ( refl_on_option_a @ A6 @ R )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y ) @ R )
       => ( member_option_a @ X2 @ A6 ) ) ) ).

% refl_onD1
thf(fact_516_refl__onD2,axiom,
    ! [A6: set_a,R: set_Product_prod_a_a,X2: a,Y: a] :
      ( ( refl_on_a @ A6 @ R )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
       => ( member_a @ Y @ A6 ) ) ) ).

% refl_onD2
thf(fact_517_refl__onD2,axiom,
    ! [A6: set_Pr1275464188344874039_a_c_d,R: set_Pr3774076249145843911_a_c_d,X2: produc5278197477302038359_a_c_d,Y: produc5278197477302038359_a_c_d] :
      ( ( refl_o726447250937964886_a_c_d @ A6 @ R )
     => ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ X2 @ Y ) @ R )
       => ( member1180172933830803072_a_c_d @ Y @ A6 ) ) ) ).

% refl_onD2
thf(fact_518_refl__onD2,axiom,
    ! [A6: set_option_a,R: set_Pr7585778909603769095tion_a,X2: option_a,Y: option_a] :
      ( ( refl_on_option_a @ A6 @ R )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y ) @ R )
       => ( member_option_a @ Y @ A6 ) ) ) ).

% refl_onD2
thf(fact_519_refl__on__domain,axiom,
    ! [A6: set_a,R: set_Product_prod_a_a,A: a,B: a] :
      ( ( refl_on_a @ A6 @ R )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
       => ( ( member_a @ A @ A6 )
          & ( member_a @ B @ A6 ) ) ) ) ).

% refl_on_domain
thf(fact_520_refl__on__domain,axiom,
    ! [A6: set_Pr1275464188344874039_a_c_d,R: set_Pr3774076249145843911_a_c_d,A: produc5278197477302038359_a_c_d,B: produc5278197477302038359_a_c_d] :
      ( ( refl_o726447250937964886_a_c_d @ A6 @ R )
     => ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ A @ B ) @ R )
       => ( ( member1180172933830803072_a_c_d @ A @ A6 )
          & ( member1180172933830803072_a_c_d @ B @ A6 ) ) ) ) ).

% refl_on_domain
thf(fact_521_refl__on__domain,axiom,
    ! [A6: set_option_a,R: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
      ( ( refl_on_option_a @ A6 @ R )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R )
       => ( ( member_option_a @ A @ A6 )
          & ( member_option_a @ B @ A6 ) ) ) ) ).

% refl_on_domain
thf(fact_522_Domain_Ocases,axiom,
    ! [A: product_prod_a_c_d,R: set_Pr1275464188344874039_a_c_d] :
      ( ( member8846983874477766826_a_c_d @ A @ ( domain7659442726769224452_a_c_d @ R ) )
     => ~ ! [B6: option6413918287372586467_a_c_d] :
            ~ ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ A @ B6 ) @ R ) ) ).

% Domain.cases
thf(fact_523_Domain_Ocases,axiom,
    ! [A: a,R: set_Pr334309860168054345_a_c_d] :
      ( ( member_a @ A @ ( domain_a_c_d @ R ) )
     => ~ ! [B6: c > d] :
            ~ ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ A @ B6 ) @ R ) ) ).

% Domain.cases
thf(fact_524_Domain_Osimps,axiom,
    ! [A: product_prod_a_c_d,R: set_Pr1275464188344874039_a_c_d] :
      ( ( member8846983874477766826_a_c_d @ A @ ( domain7659442726769224452_a_c_d @ R ) )
      = ( ? [A5: product_prod_a_c_d,B4: option6413918287372586467_a_c_d] :
            ( ( A = A5 )
            & ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ A5 @ B4 ) @ R ) ) ) ) ).

% Domain.simps
thf(fact_525_Domain_Osimps,axiom,
    ! [A: a,R: set_Pr334309860168054345_a_c_d] :
      ( ( member_a @ A @ ( domain_a_c_d @ R ) )
      = ( ? [A5: a,B4: c > d] :
            ( ( A = A5 )
            & ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ A5 @ B4 ) @ R ) ) ) ) ).

% Domain.simps
thf(fact_526_Domain_ODomainI,axiom,
    ! [A: product_prod_a_c_d,B: option6413918287372586467_a_c_d,R: set_Pr1275464188344874039_a_c_d] :
      ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ A @ B ) @ R )
     => ( member8846983874477766826_a_c_d @ A @ ( domain7659442726769224452_a_c_d @ R ) ) ) ).

% Domain.DomainI
thf(fact_527_Domain_ODomainI,axiom,
    ! [A: a,B: c > d,R: set_Pr334309860168054345_a_c_d] :
      ( ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ A @ B ) @ R )
     => ( member_a @ A @ ( domain_a_c_d @ R ) ) ) ).

% Domain.DomainI
thf(fact_528_DomainE,axiom,
    ! [A: product_prod_a_c_d,R: set_Pr1275464188344874039_a_c_d] :
      ( ( member8846983874477766826_a_c_d @ A @ ( domain7659442726769224452_a_c_d @ R ) )
     => ~ ! [B6: option6413918287372586467_a_c_d] :
            ~ ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ A @ B6 ) @ R ) ) ).

% DomainE
thf(fact_529_DomainE,axiom,
    ! [A: a,R: set_Pr334309860168054345_a_c_d] :
      ( ( member_a @ A @ ( domain_a_c_d @ R ) )
     => ~ ! [B6: c > d] :
            ~ ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ A @ B6 ) @ R ) ) ).

% DomainE
thf(fact_530_Domain__iff,axiom,
    ! [A: product_prod_a_c_d,R: set_Pr1275464188344874039_a_c_d] :
      ( ( member8846983874477766826_a_c_d @ A @ ( domain7659442726769224452_a_c_d @ R ) )
      = ( ? [Y3: option6413918287372586467_a_c_d] : ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ A @ Y3 ) @ R ) ) ) ).

% Domain_iff
thf(fact_531_Domain__iff,axiom,
    ! [A: a,R: set_Pr334309860168054345_a_c_d] :
      ( ( member_a @ A @ ( domain_a_c_d @ R ) )
      = ( ? [Y3: c > d] : ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ A @ Y3 ) @ R ) ) ) ).

% Domain_iff
thf(fact_532_Range_Ocases,axiom,
    ! [A: option6413918287372586467_a_c_d,R: set_Pr1275464188344874039_a_c_d] :
      ( ( member5926479794665358074_a_c_d @ A @ ( range_1858866086472732781_a_c_d @ R ) )
     => ~ ! [A7: product_prod_a_c_d] :
            ~ ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ A7 @ A ) @ R ) ) ).

% Range.cases
thf(fact_533_Range_Ocases,axiom,
    ! [A: c > d,R: set_Pr334309860168054345_a_c_d] :
      ( ( member_c_d @ A @ ( range_a_c_d @ R ) )
     => ~ ! [A7: a] :
            ~ ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ A7 @ A ) @ R ) ) ).

% Range.cases
thf(fact_534_Range_Osimps,axiom,
    ! [A: option6413918287372586467_a_c_d,R: set_Pr1275464188344874039_a_c_d] :
      ( ( member5926479794665358074_a_c_d @ A @ ( range_1858866086472732781_a_c_d @ R ) )
      = ( ? [A5: product_prod_a_c_d,B4: option6413918287372586467_a_c_d] :
            ( ( A = B4 )
            & ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ A5 @ B4 ) @ R ) ) ) ) ).

% Range.simps
thf(fact_535_Range_Osimps,axiom,
    ! [A: c > d,R: set_Pr334309860168054345_a_c_d] :
      ( ( member_c_d @ A @ ( range_a_c_d @ R ) )
      = ( ? [A5: a,B4: c > d] :
            ( ( A = B4 )
            & ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ A5 @ B4 ) @ R ) ) ) ) ).

% Range.simps
thf(fact_536_Range_Ointros,axiom,
    ! [A: product_prod_a_c_d,B: option6413918287372586467_a_c_d,R: set_Pr1275464188344874039_a_c_d] :
      ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ A @ B ) @ R )
     => ( member5926479794665358074_a_c_d @ B @ ( range_1858866086472732781_a_c_d @ R ) ) ) ).

% Range.intros
thf(fact_537_Range_Ointros,axiom,
    ! [A: a,B: c > d,R: set_Pr334309860168054345_a_c_d] :
      ( ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ A @ B ) @ R )
     => ( member_c_d @ B @ ( range_a_c_d @ R ) ) ) ).

% Range.intros
thf(fact_538_RangeE,axiom,
    ! [B: option6413918287372586467_a_c_d,R: set_Pr1275464188344874039_a_c_d] :
      ( ( member5926479794665358074_a_c_d @ B @ ( range_1858866086472732781_a_c_d @ R ) )
     => ~ ! [A7: product_prod_a_c_d] :
            ~ ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ A7 @ B ) @ R ) ) ).

% RangeE
thf(fact_539_RangeE,axiom,
    ! [B: c > d,R: set_Pr334309860168054345_a_c_d] :
      ( ( member_c_d @ B @ ( range_a_c_d @ R ) )
     => ~ ! [A7: a] :
            ~ ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ A7 @ B ) @ R ) ) ).

% RangeE
thf(fact_540_Range__iff,axiom,
    ! [A: option6413918287372586467_a_c_d,R: set_Pr1275464188344874039_a_c_d] :
      ( ( member5926479794665358074_a_c_d @ A @ ( range_1858866086472732781_a_c_d @ R ) )
      = ( ? [Y3: product_prod_a_c_d] : ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ Y3 @ A ) @ R ) ) ) ).

% Range_iff
thf(fact_541_Range__iff,axiom,
    ! [A: c > d,R: set_Pr334309860168054345_a_c_d] :
      ( ( member_c_d @ A @ ( range_a_c_d @ R ) )
      = ( ? [Y3: a] : ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ Y3 @ A ) @ R ) ) ) ).

% Range_iff
thf(fact_542_the__inv__into__f__eq,axiom,
    ! [F2: a > option_a,A6: set_a,X2: a,Y: option_a] :
      ( ( inj_on_a_option_a @ F2 @ A6 )
     => ( ( ( F2 @ X2 )
          = Y )
       => ( ( member_a @ X2 @ A6 )
         => ( ( the_in8758012798868597241tion_a @ A6 @ F2 @ Y )
            = X2 ) ) ) ) ).

% the_inv_into_f_eq
thf(fact_543_the__inv__into__f__f,axiom,
    ! [F2: a > option_a,A6: set_a,X2: a] :
      ( ( inj_on_a_option_a @ F2 @ A6 )
     => ( ( member_a @ X2 @ A6 )
       => ( ( the_in8758012798868597241tion_a @ A6 @ F2 @ ( F2 @ X2 ) )
          = X2 ) ) ) ).

% the_inv_into_f_f
thf(fact_544_inj__on__the__inv__into,axiom,
    ! [F2: option_a > a,A6: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ A6 )
     => ( inj_on_a_option_a @ ( the_in1757154643552616557on_a_a @ A6 @ F2 ) @ ( image_option_a_a2 @ F2 @ A6 ) ) ) ).

% inj_on_the_inv_into
thf(fact_545_inj__on__the__inv__into,axiom,
    ! [F2: a > option_a,A6: set_a] :
      ( ( inj_on_a_option_a @ F2 @ A6 )
     => ( inj_on_option_a_a @ ( the_in8758012798868597241tion_a @ A6 @ F2 ) @ ( image_a_option_a2 @ F2 @ A6 ) ) ) ).

% inj_on_the_inv_into
thf(fact_546_f__the__inv__into__f,axiom,
    ! [F2: option_a > a,A6: set_option_a,Y: a] :
      ( ( inj_on_option_a_a @ F2 @ A6 )
     => ( ( member_a @ Y @ ( image_option_a_a2 @ F2 @ A6 ) )
       => ( ( F2 @ ( the_in1757154643552616557on_a_a @ A6 @ F2 @ Y ) )
          = Y ) ) ) ).

% f_the_inv_into_f
thf(fact_547_f__the__inv__into__f,axiom,
    ! [F2: a > option_a,A6: set_a,Y: option_a] :
      ( ( inj_on_a_option_a @ F2 @ A6 )
     => ( ( member_option_a @ Y @ ( image_a_option_a2 @ F2 @ A6 ) )
       => ( ( F2 @ ( the_in8758012798868597241tion_a @ A6 @ F2 @ Y ) )
          = Y ) ) ) ).

% f_the_inv_into_f
thf(fact_548_notin__range__Some,axiom,
    ! [X2: option6413918287372586467_a_c_d] :
      ( ( ~ ( member5926479794665358074_a_c_d @ X2 @ ( image_5741706613785127421_a_c_d @ some_P1084500821511757806_a_c_d @ top_to5543414004138509561_a_c_d ) ) )
      = ( X2 = none_P2239999354720063602_a_c_d ) ) ).

% notin_range_Some
thf(fact_549_notin__range__Some,axiom,
    ! [X2: option_option_a] :
      ( ( ~ ( member5113800082084363315tion_a @ X2 @ ( image_2132136900116418507tion_a @ some_option_a @ top_top_set_option_a ) ) )
      = ( X2 = none_option_a ) ) ).

% notin_range_Some
thf(fact_550_notin__range__Some,axiom,
    ! [X2: option_a] :
      ( ( ~ ( member_option_a @ X2 @ ( image_a_option_a2 @ some_a @ top_top_set_a ) ) )
      = ( X2 = none_a ) ) ).

% notin_range_Some
thf(fact_551_the__inv__into__into,axiom,
    ! [F2: a > a,A6: set_a,X2: a,B5: set_a] :
      ( ( inj_on_a_a @ F2 @ A6 )
     => ( ( member_a @ X2 @ ( image_a_a2 @ F2 @ A6 ) )
       => ( ( ord_less_eq_set_a @ A6 @ B5 )
         => ( member_a @ ( the_inv_into_a_a @ A6 @ F2 @ X2 ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_552_the__inv__into__into,axiom,
    ! [F2: produc5278197477302038359_a_c_d > a,A6: set_Pr1275464188344874039_a_c_d,X2: a,B5: set_Pr1275464188344874039_a_c_d] :
      ( ( inj_on5461451596835146456_c_d_a @ F2 @ A6 )
     => ( ( member_a @ X2 @ ( image_8636529855994272620_c_d_a @ F2 @ A6 ) )
       => ( ( ord_le6392319565089278935_a_c_d @ A6 @ B5 )
         => ( member1180172933830803072_a_c_d @ ( the_in1140205326877463322_c_d_a @ A6 @ F2 @ X2 ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_553_the__inv__into__into,axiom,
    ! [F2: option_a > a,A6: set_option_a,X2: a,B5: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ A6 )
     => ( ( member_a @ X2 @ ( image_option_a_a2 @ F2 @ A6 ) )
       => ( ( ord_le1955136853071979460tion_a @ A6 @ B5 )
         => ( member_option_a @ ( the_in1757154643552616557on_a_a @ A6 @ F2 @ X2 ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_554_the__inv__into__into,axiom,
    ! [F2: a > produc5278197477302038359_a_c_d,A6: set_a,X2: produc5278197477302038359_a_c_d,B5: set_a] :
      ( ( inj_on2836679365399032650_a_c_d @ F2 @ A6 )
     => ( ( member1180172933830803072_a_c_d @ X2 @ ( image_6011757624558158814_a_c_d @ F2 @ A6 ) )
       => ( ( ord_less_eq_set_a @ A6 @ B5 )
         => ( member_a @ ( the_in7738805132296125324_a_c_d @ A6 @ F2 @ X2 ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_555_the__inv__into__into,axiom,
    ! [F2: produc5278197477302038359_a_c_d > produc5278197477302038359_a_c_d,A6: set_Pr1275464188344874039_a_c_d,X2: produc5278197477302038359_a_c_d,B5: set_Pr1275464188344874039_a_c_d] :
      ( ( inj_on7596037564690925361_a_c_d @ F2 @ A6 )
     => ( ( member1180172933830803072_a_c_d @ X2 @ ( image_3091440356067162053_a_c_d @ F2 @ A6 ) )
       => ( ( ord_le6392319565089278935_a_c_d @ A6 @ B5 )
         => ( member1180172933830803072_a_c_d @ ( the_in282633113862504307_a_c_d @ A6 @ F2 @ X2 ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_556_the__inv__into__into,axiom,
    ! [F2: option_a > produc5278197477302038359_a_c_d,A6: set_option_a,X2: produc5278197477302038359_a_c_d,B5: set_option_a] :
      ( ( inj_on6212699291050112900_a_c_d @ F2 @ A6 )
     => ( ( member1180172933830803072_a_c_d @ X2 @ ( image_8542254255036258840_a_c_d @ F2 @ A6 ) )
       => ( ( ord_le1955136853071979460tion_a @ A6 @ B5 )
         => ( member_option_a @ ( the_in4155257485301462214_a_c_d @ A6 @ F2 @ X2 ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_557_the__inv__into__into,axiom,
    ! [F2: a > option_a,A6: set_a,X2: option_a,B5: set_a] :
      ( ( inj_on_a_option_a @ F2 @ A6 )
     => ( ( member_option_a @ X2 @ ( image_a_option_a2 @ F2 @ A6 ) )
       => ( ( ord_less_eq_set_a @ A6 @ B5 )
         => ( member_a @ ( the_in8758012798868597241tion_a @ A6 @ F2 @ X2 ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_558_the__inv__into__into,axiom,
    ! [F2: produc5278197477302038359_a_c_d > option_a,A6: set_Pr1275464188344874039_a_c_d,X2: option_a,B5: set_Pr1275464188344874039_a_c_d] :
      ( ( inj_on7052565339190862366tion_a @ F2 @ A6 )
     => ( ( member_option_a @ X2 @ ( image_158748266322232498tion_a @ F2 @ A6 ) )
       => ( ( ord_le6392319565089278935_a_c_d @ A6 @ B5 )
         => ( member1180172933830803072_a_c_d @ ( the_in4995123533442211680tion_a @ A6 @ F2 @ X2 ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_559_the__inv__into__into,axiom,
    ! [F2: option_a > option_a,A6: set_option_a,X2: option_a,B5: set_option_a] :
      ( ( inj_on8559383841115902449tion_a @ F2 @ A6 )
     => ( ( member_option_a @ X2 @ ( image_7439109396645324421tion_a @ F2 @ A6 ) )
       => ( ( ord_le1955136853071979460tion_a @ A6 @ B5 )
         => ( member_option_a @ ( the_in2538339130118444083tion_a @ A6 @ F2 @ X2 ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_560_image__set__diff,axiom,
    ! [F2: a > option_a,A6: set_a,B5: set_a] :
      ( ( inj_on_a_option_a @ F2 @ top_top_set_a )
     => ( ( image_a_option_a2 @ F2 @ ( minus_minus_set_a @ A6 @ B5 ) )
        = ( minus_1574173051537231627tion_a @ ( image_a_option_a2 @ F2 @ A6 ) @ ( image_a_option_a2 @ F2 @ B5 ) ) ) ) ).

% image_set_diff
thf(fact_561_image__set__diff,axiom,
    ! [F2: option_a > a,A6: set_option_a,B5: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ top_top_set_option_a )
     => ( ( image_option_a_a2 @ F2 @ ( minus_1574173051537231627tion_a @ A6 @ B5 ) )
        = ( minus_minus_set_a @ ( image_option_a_a2 @ F2 @ A6 ) @ ( image_option_a_a2 @ F2 @ B5 ) ) ) ) ).

% image_set_diff
thf(fact_562_surj__Compl__image__subset,axiom,
    ! [F2: a > a,A6: 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 @ A6 ) ) @ ( image_a_a2 @ F2 @ ( uminus_uminus_set_a @ A6 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_563_surj__Compl__image__subset,axiom,
    ! [F2: a > option_a,A6: 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 @ A6 ) ) @ ( image_a_option_a2 @ F2 @ ( uminus_uminus_set_a @ A6 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_564_surj__Compl__image__subset,axiom,
    ! [F2: option_a > a,A6: 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 @ A6 ) ) @ ( image_option_a_a2 @ F2 @ ( uminus6205308855922866075tion_a @ A6 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_565_surj__Compl__image__subset,axiom,
    ! [F2: option_a > option_a,A6: 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 @ A6 ) ) @ ( image_7439109396645324421tion_a @ F2 @ ( uminus6205308855922866075tion_a @ A6 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_566_inj__image__subset__iff,axiom,
    ! [F2: a > option_a,A6: set_a,B5: set_a] :
      ( ( inj_on_a_option_a @ F2 @ top_top_set_a )
     => ( ( ord_le1955136853071979460tion_a @ ( image_a_option_a2 @ F2 @ A6 ) @ ( image_a_option_a2 @ F2 @ B5 ) )
        = ( ord_less_eq_set_a @ A6 @ B5 ) ) ) ).

% inj_image_subset_iff
thf(fact_567_inj__image__subset__iff,axiom,
    ! [F2: option_a > a,A6: set_option_a,B5: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ top_top_set_option_a )
     => ( ( ord_less_eq_set_a @ ( image_option_a_a2 @ F2 @ A6 ) @ ( image_option_a_a2 @ F2 @ B5 ) )
        = ( ord_le1955136853071979460tion_a @ A6 @ B5 ) ) ) ).

% inj_image_subset_iff
thf(fact_568_subrelI,axiom,
    ! [R: set_Pr1275464188344874039_a_c_d,S4: set_Pr1275464188344874039_a_c_d] :
      ( ! [X4: product_prod_a_c_d,Y4: option6413918287372586467_a_c_d] :
          ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ X4 @ Y4 ) @ R )
         => ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ X4 @ Y4 ) @ S4 ) )
     => ( ord_le6392319565089278935_a_c_d @ R @ S4 ) ) ).

% subrelI
thf(fact_569_subrelI,axiom,
    ! [R: set_Pr334309860168054345_a_c_d,S4: set_Pr334309860168054345_a_c_d] :
      ( ! [X4: a,Y4: c > d] :
          ( ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ X4 @ Y4 ) @ R )
         => ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ X4 @ Y4 ) @ S4 ) )
     => ( ord_le5099265691656296105_a_c_d @ R @ S4 ) ) ).

% subrelI
thf(fact_570_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_571_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_572_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_573_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_574_surjI,axiom,
    ! [G: a > a,F2: a > a] :
      ( ! [X4: a] :
          ( ( G @ ( F2 @ X4 ) )
          = X4 )
     => ( ( image_a_a2 @ G @ top_top_set_a )
        = top_top_set_a ) ) ).

% surjI
thf(fact_575_surjI,axiom,
    ! [G: a > option_a,F2: option_a > a] :
      ( ! [X4: option_a] :
          ( ( G @ ( F2 @ X4 ) )
          = X4 )
     => ( ( image_a_option_a2 @ G @ top_top_set_a )
        = top_top_set_option_a ) ) ).

% surjI
thf(fact_576_surjI,axiom,
    ! [G: option_a > a,F2: a > option_a] :
      ( ! [X4: a] :
          ( ( G @ ( F2 @ X4 ) )
          = X4 )
     => ( ( image_option_a_a2 @ G @ top_top_set_option_a )
        = top_top_set_a ) ) ).

% surjI
thf(fact_577_surjI,axiom,
    ! [G: option_a > option_a,F2: option_a > option_a] :
      ( ! [X4: option_a] :
          ( ( G @ ( F2 @ X4 ) )
          = X4 )
     => ( ( image_7439109396645324421tion_a @ G @ top_top_set_option_a )
        = top_top_set_option_a ) ) ).

% surjI
thf(fact_578_surjE,axiom,
    ! [F2: a > a,Y: a] :
      ( ( ( image_a_a2 @ F2 @ top_top_set_a )
        = top_top_set_a )
     => ~ ! [X4: a] :
            ( Y
           != ( F2 @ X4 ) ) ) ).

% surjE
thf(fact_579_surjE,axiom,
    ! [F2: a > option_a,Y: option_a] :
      ( ( ( image_a_option_a2 @ F2 @ top_top_set_a )
        = top_top_set_option_a )
     => ~ ! [X4: a] :
            ( Y
           != ( F2 @ X4 ) ) ) ).

% surjE
thf(fact_580_surjE,axiom,
    ! [F2: option_a > a,Y: a] :
      ( ( ( image_option_a_a2 @ F2 @ top_top_set_option_a )
        = top_top_set_a )
     => ~ ! [X4: option_a] :
            ( Y
           != ( F2 @ X4 ) ) ) ).

% surjE
thf(fact_581_surjE,axiom,
    ! [F2: option_a > option_a,Y: option_a] :
      ( ( ( image_7439109396645324421tion_a @ F2 @ top_top_set_option_a )
        = top_top_set_option_a )
     => ~ ! [X4: option_a] :
            ( Y
           != ( F2 @ X4 ) ) ) ).

% surjE
thf(fact_582_surjD,axiom,
    ! [F2: a > a,Y: a] :
      ( ( ( image_a_a2 @ F2 @ top_top_set_a )
        = top_top_set_a )
     => ? [X4: a] :
          ( Y
          = ( F2 @ X4 ) ) ) ).

% surjD
thf(fact_583_surjD,axiom,
    ! [F2: a > option_a,Y: option_a] :
      ( ( ( image_a_option_a2 @ F2 @ top_top_set_a )
        = top_top_set_option_a )
     => ? [X4: a] :
          ( Y
          = ( F2 @ X4 ) ) ) ).

% surjD
thf(fact_584_surjD,axiom,
    ! [F2: option_a > a,Y: a] :
      ( ( ( image_option_a_a2 @ F2 @ top_top_set_option_a )
        = top_top_set_a )
     => ? [X4: option_a] :
          ( Y
          = ( F2 @ X4 ) ) ) ).

% surjD
thf(fact_585_surjD,axiom,
    ! [F2: option_a > option_a,Y: option_a] :
      ( ( ( image_7439109396645324421tion_a @ F2 @ top_top_set_option_a )
        = top_top_set_option_a )
     => ? [X4: option_a] :
          ( Y
          = ( F2 @ X4 ) ) ) ).

% surjD
thf(fact_586_subset__inj__on,axiom,
    ! [F2: a > option_a,B5: set_a,A6: set_a] :
      ( ( inj_on_a_option_a @ F2 @ B5 )
     => ( ( ord_less_eq_set_a @ A6 @ B5 )
       => ( inj_on_a_option_a @ F2 @ A6 ) ) ) ).

% subset_inj_on
thf(fact_587_inj__on__subset,axiom,
    ! [F2: a > option_a,A6: set_a,B5: set_a] :
      ( ( inj_on_a_option_a @ F2 @ A6 )
     => ( ( ord_less_eq_set_a @ B5 @ A6 )
       => ( inj_on_a_option_a @ F2 @ B5 ) ) ) ).

% inj_on_subset
thf(fact_588_injD,axiom,
    ! [F2: a > option_a,X2: a,Y: a] :
      ( ( inj_on_a_option_a @ F2 @ top_top_set_a )
     => ( ( ( F2 @ X2 )
          = ( F2 @ Y ) )
       => ( X2 = Y ) ) ) ).

% injD
thf(fact_589_injI,axiom,
    ! [F2: a > option_a] :
      ( ! [X4: a,Y4: a] :
          ( ( ( F2 @ X4 )
            = ( F2 @ Y4 ) )
         => ( X4 = Y4 ) )
     => ( inj_on_a_option_a @ F2 @ top_top_set_a ) ) ).

% injI
thf(fact_590_inj__eq,axiom,
    ! [F2: a > option_a,X2: a,Y: a] :
      ( ( inj_on_a_option_a @ F2 @ top_top_set_a )
     => ( ( ( F2 @ X2 )
          = ( F2 @ Y ) )
        = ( X2 = Y ) ) ) ).

% inj_eq
thf(fact_591_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_592_inj__image__Compl__subset,axiom,
    ! [F2: a > option_a,A6: set_a] :
      ( ( inj_on_a_option_a @ F2 @ top_top_set_a )
     => ( ord_le1955136853071979460tion_a @ ( image_a_option_a2 @ F2 @ ( uminus_uminus_set_a @ A6 ) ) @ ( uminus6205308855922866075tion_a @ ( image_a_option_a2 @ F2 @ A6 ) ) ) ) ).

% inj_image_Compl_subset
thf(fact_593_inj__image__Compl__subset,axiom,
    ! [F2: option_a > a,A6: 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 @ A6 ) ) @ ( uminus_uminus_set_a @ ( image_option_a_a2 @ F2 @ A6 ) ) ) ) ).

% inj_image_Compl_subset
thf(fact_594_inj__on__image__eq__iff,axiom,
    ! [F2: option_a > a,C3: set_option_a,A6: set_option_a,B5: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ C3 )
     => ( ( ord_le1955136853071979460tion_a @ A6 @ C3 )
       => ( ( ord_le1955136853071979460tion_a @ B5 @ C3 )
         => ( ( ( image_option_a_a2 @ F2 @ A6 )
              = ( image_option_a_a2 @ F2 @ B5 ) )
            = ( A6 = B5 ) ) ) ) ) ).

% inj_on_image_eq_iff
thf(fact_595_inj__on__image__eq__iff,axiom,
    ! [F2: a > option_a,C3: set_a,A6: set_a,B5: set_a] :
      ( ( inj_on_a_option_a @ F2 @ C3 )
     => ( ( ord_less_eq_set_a @ A6 @ C3 )
       => ( ( ord_less_eq_set_a @ B5 @ C3 )
         => ( ( ( image_a_option_a2 @ F2 @ A6 )
              = ( image_a_option_a2 @ F2 @ B5 ) )
            = ( A6 = B5 ) ) ) ) ) ).

% inj_on_image_eq_iff
thf(fact_596_inj__on__image__mem__iff,axiom,
    ! [F2: a > a,B5: set_a,A: a,A6: set_a] :
      ( ( inj_on_a_a @ F2 @ B5 )
     => ( ( member_a @ A @ B5 )
       => ( ( ord_less_eq_set_a @ A6 @ B5 )
         => ( ( member_a @ ( F2 @ A ) @ ( image_a_a2 @ F2 @ A6 ) )
            = ( member_a @ A @ A6 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_597_inj__on__image__mem__iff,axiom,
    ! [F2: a > produc5278197477302038359_a_c_d,B5: set_a,A: a,A6: set_a] :
      ( ( inj_on2836679365399032650_a_c_d @ F2 @ B5 )
     => ( ( member_a @ A @ B5 )
       => ( ( ord_less_eq_set_a @ A6 @ B5 )
         => ( ( member1180172933830803072_a_c_d @ ( F2 @ A ) @ ( image_6011757624558158814_a_c_d @ F2 @ A6 ) )
            = ( member_a @ A @ A6 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_598_inj__on__image__mem__iff,axiom,
    ! [F2: a > option_a,B5: set_a,A: a,A6: set_a] :
      ( ( inj_on_a_option_a @ F2 @ B5 )
     => ( ( member_a @ A @ B5 )
       => ( ( ord_less_eq_set_a @ A6 @ B5 )
         => ( ( member_option_a @ ( F2 @ A ) @ ( image_a_option_a2 @ F2 @ A6 ) )
            = ( member_a @ A @ A6 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_599_inj__on__image__mem__iff,axiom,
    ! [F2: produc5278197477302038359_a_c_d > a,B5: set_Pr1275464188344874039_a_c_d,A: produc5278197477302038359_a_c_d,A6: set_Pr1275464188344874039_a_c_d] :
      ( ( inj_on5461451596835146456_c_d_a @ F2 @ B5 )
     => ( ( member1180172933830803072_a_c_d @ A @ B5 )
       => ( ( ord_le6392319565089278935_a_c_d @ A6 @ B5 )
         => ( ( member_a @ ( F2 @ A ) @ ( image_8636529855994272620_c_d_a @ F2 @ A6 ) )
            = ( member1180172933830803072_a_c_d @ A @ A6 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_600_inj__on__image__mem__iff,axiom,
    ! [F2: produc5278197477302038359_a_c_d > produc5278197477302038359_a_c_d,B5: set_Pr1275464188344874039_a_c_d,A: produc5278197477302038359_a_c_d,A6: set_Pr1275464188344874039_a_c_d] :
      ( ( inj_on7596037564690925361_a_c_d @ F2 @ B5 )
     => ( ( member1180172933830803072_a_c_d @ A @ B5 )
       => ( ( ord_le6392319565089278935_a_c_d @ A6 @ B5 )
         => ( ( member1180172933830803072_a_c_d @ ( F2 @ A ) @ ( image_3091440356067162053_a_c_d @ F2 @ A6 ) )
            = ( member1180172933830803072_a_c_d @ A @ A6 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_601_inj__on__image__mem__iff,axiom,
    ! [F2: produc5278197477302038359_a_c_d > option_a,B5: set_Pr1275464188344874039_a_c_d,A: produc5278197477302038359_a_c_d,A6: set_Pr1275464188344874039_a_c_d] :
      ( ( inj_on7052565339190862366tion_a @ F2 @ B5 )
     => ( ( member1180172933830803072_a_c_d @ A @ B5 )
       => ( ( ord_le6392319565089278935_a_c_d @ A6 @ B5 )
         => ( ( member_option_a @ ( F2 @ A ) @ ( image_158748266322232498tion_a @ F2 @ A6 ) )
            = ( member1180172933830803072_a_c_d @ A @ A6 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_602_inj__on__image__mem__iff,axiom,
    ! [F2: option_a > a,B5: set_option_a,A: option_a,A6: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ B5 )
     => ( ( member_option_a @ A @ B5 )
       => ( ( ord_le1955136853071979460tion_a @ A6 @ B5 )
         => ( ( member_a @ ( F2 @ A ) @ ( image_option_a_a2 @ F2 @ A6 ) )
            = ( member_option_a @ A @ A6 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_603_inj__on__image__mem__iff,axiom,
    ! [F2: option_a > produc5278197477302038359_a_c_d,B5: set_option_a,A: option_a,A6: set_option_a] :
      ( ( inj_on6212699291050112900_a_c_d @ F2 @ B5 )
     => ( ( member_option_a @ A @ B5 )
       => ( ( ord_le1955136853071979460tion_a @ A6 @ B5 )
         => ( ( member1180172933830803072_a_c_d @ ( F2 @ A ) @ ( image_8542254255036258840_a_c_d @ F2 @ A6 ) )
            = ( member_option_a @ A @ A6 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_604_inj__on__image__mem__iff,axiom,
    ! [F2: option_a > option_a,B5: set_option_a,A: option_a,A6: set_option_a] :
      ( ( inj_on8559383841115902449tion_a @ F2 @ B5 )
     => ( ( member_option_a @ A @ B5 )
       => ( ( ord_le1955136853071979460tion_a @ A6 @ B5 )
         => ( ( member_option_a @ ( F2 @ A ) @ ( image_7439109396645324421tion_a @ F2 @ A6 ) )
            = ( member_option_a @ A @ A6 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_605_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_606_range__ex1__eq,axiom,
    ! [F2: a > produc5278197477302038359_a_c_d,B: produc5278197477302038359_a_c_d] :
      ( ( inj_on2836679365399032650_a_c_d @ F2 @ top_top_set_a )
     => ( ( member1180172933830803072_a_c_d @ B @ ( image_6011757624558158814_a_c_d @ F2 @ top_top_set_a ) )
        = ( ? [X: a] :
              ( ( B
                = ( F2 @ X ) )
              & ! [Y3: a] :
                  ( ( B
                    = ( F2 @ Y3 ) )
                 => ( Y3 = X ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_607_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_608_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_609_range__ex1__eq,axiom,
    ! [F2: option_a > produc5278197477302038359_a_c_d,B: produc5278197477302038359_a_c_d] :
      ( ( inj_on6212699291050112900_a_c_d @ F2 @ top_top_set_option_a )
     => ( ( member1180172933830803072_a_c_d @ B @ ( image_8542254255036258840_a_c_d @ 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_610_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_611_inj__image__eq__iff,axiom,
    ! [F2: a > option_a,A6: set_a,B5: set_a] :
      ( ( inj_on_a_option_a @ F2 @ top_top_set_a )
     => ( ( ( image_a_option_a2 @ F2 @ A6 )
          = ( image_a_option_a2 @ F2 @ B5 ) )
        = ( A6 = B5 ) ) ) ).

% inj_image_eq_iff
thf(fact_612_inj__image__eq__iff,axiom,
    ! [F2: option_a > a,A6: set_option_a,B5: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ top_top_set_option_a )
     => ( ( ( image_option_a_a2 @ F2 @ A6 )
          = ( image_option_a_a2 @ F2 @ B5 ) )
        = ( A6 = B5 ) ) ) ).

% inj_image_eq_iff
thf(fact_613_inj__image__mem__iff,axiom,
    ! [F2: produc5278197477302038359_a_c_d > a,A: produc5278197477302038359_a_c_d,A6: set_Pr1275464188344874039_a_c_d] :
      ( ( inj_on5461451596835146456_c_d_a @ F2 @ top_to2771551084318925191_a_c_d )
     => ( ( member_a @ ( F2 @ A ) @ ( image_8636529855994272620_c_d_a @ F2 @ A6 ) )
        = ( member1180172933830803072_a_c_d @ A @ A6 ) ) ) ).

% inj_image_mem_iff
thf(fact_614_inj__image__mem__iff,axiom,
    ! [F2: produc5278197477302038359_a_c_d > produc5278197477302038359_a_c_d,A: produc5278197477302038359_a_c_d,A6: set_Pr1275464188344874039_a_c_d] :
      ( ( inj_on7596037564690925361_a_c_d @ F2 @ top_to2771551084318925191_a_c_d )
     => ( ( member1180172933830803072_a_c_d @ ( F2 @ A ) @ ( image_3091440356067162053_a_c_d @ F2 @ A6 ) )
        = ( member1180172933830803072_a_c_d @ A @ A6 ) ) ) ).

% inj_image_mem_iff
thf(fact_615_inj__image__mem__iff,axiom,
    ! [F2: produc5278197477302038359_a_c_d > option_a,A: produc5278197477302038359_a_c_d,A6: set_Pr1275464188344874039_a_c_d] :
      ( ( inj_on7052565339190862366tion_a @ F2 @ top_to2771551084318925191_a_c_d )
     => ( ( member_option_a @ ( F2 @ A ) @ ( image_158748266322232498tion_a @ F2 @ A6 ) )
        = ( member1180172933830803072_a_c_d @ A @ A6 ) ) ) ).

% inj_image_mem_iff
thf(fact_616_inj__image__mem__iff,axiom,
    ! [F2: a > a,A: a,A6: set_a] :
      ( ( inj_on_a_a @ F2 @ top_top_set_a )
     => ( ( member_a @ ( F2 @ A ) @ ( image_a_a2 @ F2 @ A6 ) )
        = ( member_a @ A @ A6 ) ) ) ).

% inj_image_mem_iff
thf(fact_617_inj__image__mem__iff,axiom,
    ! [F2: a > produc5278197477302038359_a_c_d,A: a,A6: set_a] :
      ( ( inj_on2836679365399032650_a_c_d @ F2 @ top_top_set_a )
     => ( ( member1180172933830803072_a_c_d @ ( F2 @ A ) @ ( image_6011757624558158814_a_c_d @ F2 @ A6 ) )
        = ( member_a @ A @ A6 ) ) ) ).

% inj_image_mem_iff
thf(fact_618_inj__image__mem__iff,axiom,
    ! [F2: a > option_a,A: a,A6: set_a] :
      ( ( inj_on_a_option_a @ F2 @ top_top_set_a )
     => ( ( member_option_a @ ( F2 @ A ) @ ( image_a_option_a2 @ F2 @ A6 ) )
        = ( member_a @ A @ A6 ) ) ) ).

% inj_image_mem_iff
thf(fact_619_inj__image__mem__iff,axiom,
    ! [F2: option_a > a,A: option_a,A6: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ top_top_set_option_a )
     => ( ( member_a @ ( F2 @ A ) @ ( image_option_a_a2 @ F2 @ A6 ) )
        = ( member_option_a @ A @ A6 ) ) ) ).

% inj_image_mem_iff
thf(fact_620_inj__image__mem__iff,axiom,
    ! [F2: option_a > produc5278197477302038359_a_c_d,A: option_a,A6: set_option_a] :
      ( ( inj_on6212699291050112900_a_c_d @ F2 @ top_top_set_option_a )
     => ( ( member1180172933830803072_a_c_d @ ( F2 @ A ) @ ( image_8542254255036258840_a_c_d @ F2 @ A6 ) )
        = ( member_option_a @ A @ A6 ) ) ) ).

% inj_image_mem_iff
thf(fact_621_inj__image__mem__iff,axiom,
    ! [F2: option_a > option_a,A: option_a,A6: set_option_a] :
      ( ( inj_on8559383841115902449tion_a @ F2 @ top_top_set_option_a )
     => ( ( member_option_a @ ( F2 @ A ) @ ( image_7439109396645324421tion_a @ F2 @ A6 ) )
        = ( member_option_a @ A @ A6 ) ) ) ).

% inj_image_mem_iff
thf(fact_622_reflD,axiom,
    ! [R: set_Product_prod_a_a,A: a] :
      ( ( refl_on_a @ top_top_set_a @ R )
     => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A ) @ R ) ) ).

% reflD
thf(fact_623_reflD,axiom,
    ! [R: set_Pr7585778909603769095tion_a,A: option_a] :
      ( ( refl_on_option_a @ top_top_set_option_a @ R )
     => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ A ) @ R ) ) ).

% reflD
thf(fact_624_reflI,axiom,
    ! [R: set_Product_prod_a_a] :
      ( ! [X4: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ X4 ) @ R )
     => ( refl_on_a @ top_top_set_a @ R ) ) ).

% reflI
thf(fact_625_reflI,axiom,
    ! [R: set_Pr7585778909603769095tion_a] :
      ( ! [X4: option_a] : ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X4 @ X4 ) @ R )
     => ( refl_on_option_a @ top_top_set_option_a @ R ) ) ).

% reflI
thf(fact_626_UNIV__option__conv,axiom,
    ( top_to2846071696153141577_a_c_d
    = ( insert6629875870486991891_a_c_d @ none_P2239999354720063602_a_c_d @ ( image_5741706613785127421_a_c_d @ some_P1084500821511757806_a_c_d @ top_to5543414004138509561_a_c_d ) ) ) ).

% UNIV_option_conv
thf(fact_627_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_628_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_629_the__inv__f__f,axiom,
    ! [F2: a > option_a,X2: a] :
      ( ( inj_on_a_option_a @ F2 @ top_top_set_a )
     => ( ( the_in8758012798868597241tion_a @ top_top_set_a @ F2 @ ( F2 @ X2 ) )
        = X2 ) ) ).

% the_inv_f_f
thf(fact_630_inj__on__image__Int,axiom,
    ! [F2: option_a > a,C3: set_option_a,A6: set_option_a,B5: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ C3 )
     => ( ( ord_le1955136853071979460tion_a @ A6 @ C3 )
       => ( ( ord_le1955136853071979460tion_a @ B5 @ C3 )
         => ( ( image_option_a_a2 @ F2 @ ( inf_inf_set_option_a @ A6 @ B5 ) )
            = ( inf_inf_set_a @ ( image_option_a_a2 @ F2 @ A6 ) @ ( image_option_a_a2 @ F2 @ B5 ) ) ) ) ) ) ).

% inj_on_image_Int
thf(fact_631_inj__on__image__Int,axiom,
    ! [F2: a > option_a,C3: set_a,A6: set_a,B5: set_a] :
      ( ( inj_on_a_option_a @ F2 @ C3 )
     => ( ( ord_less_eq_set_a @ A6 @ C3 )
       => ( ( ord_less_eq_set_a @ B5 @ C3 )
         => ( ( image_a_option_a2 @ F2 @ ( inf_inf_set_a @ A6 @ B5 ) )
            = ( inf_inf_set_option_a @ ( image_a_option_a2 @ F2 @ A6 ) @ ( image_a_option_a2 @ F2 @ B5 ) ) ) ) ) ) ).

% inj_on_image_Int
thf(fact_632_inj__on__image__set__diff,axiom,
    ! [F2: option_a > a,C3: set_option_a,A6: set_option_a,B5: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ C3 )
     => ( ( ord_le1955136853071979460tion_a @ ( minus_1574173051537231627tion_a @ A6 @ B5 ) @ C3 )
       => ( ( ord_le1955136853071979460tion_a @ B5 @ C3 )
         => ( ( image_option_a_a2 @ F2 @ ( minus_1574173051537231627tion_a @ A6 @ B5 ) )
            = ( minus_minus_set_a @ ( image_option_a_a2 @ F2 @ A6 ) @ ( image_option_a_a2 @ F2 @ B5 ) ) ) ) ) ) ).

% inj_on_image_set_diff
thf(fact_633_inj__on__image__set__diff,axiom,
    ! [F2: a > option_a,C3: set_a,A6: set_a,B5: set_a] :
      ( ( inj_on_a_option_a @ F2 @ C3 )
     => ( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A6 @ B5 ) @ C3 )
       => ( ( ord_less_eq_set_a @ B5 @ C3 )
         => ( ( image_a_option_a2 @ F2 @ ( minus_minus_set_a @ A6 @ B5 ) )
            = ( minus_1574173051537231627tion_a @ ( image_a_option_a2 @ F2 @ A6 ) @ ( image_a_option_a2 @ F2 @ B5 ) ) ) ) ) ) ).

% inj_on_image_set_diff
thf(fact_634_image__Int,axiom,
    ! [F2: a > option_a,A6: set_a,B5: set_a] :
      ( ( inj_on_a_option_a @ F2 @ top_top_set_a )
     => ( ( image_a_option_a2 @ F2 @ ( inf_inf_set_a @ A6 @ B5 ) )
        = ( inf_inf_set_option_a @ ( image_a_option_a2 @ F2 @ A6 ) @ ( image_a_option_a2 @ F2 @ B5 ) ) ) ) ).

% image_Int
thf(fact_635_image__Int,axiom,
    ! [F2: option_a > a,A6: set_option_a,B5: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ top_top_set_option_a )
     => ( ( image_option_a_a2 @ F2 @ ( inf_inf_set_option_a @ A6 @ B5 ) )
        = ( inf_inf_set_a @ ( image_option_a_a2 @ F2 @ A6 ) @ ( image_option_a_a2 @ F2 @ B5 ) ) ) ) ).

% image_Int
thf(fact_636_Linear__order__wf__diff__Id,axiom,
    ! [R: set_Product_prod_a_a] :
      ( ( order_8768733634509060147r_on_a @ ( field_a @ R ) @ R )
     => ( ( wf_a @ ( minus_6817036919807184750od_a_a @ R @ id_a2 ) )
        = ( ! [A8: set_a] :
              ( ( ord_less_eq_set_a @ A8 @ ( field_a @ R ) )
             => ( ( A8 != bot_bot_set_a )
               => ? [X: a] :
                    ( ( member_a @ X @ A8 )
                    & ! [Y3: a] :
                        ( ( member_a @ Y3 @ A8 )
                       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_637_Linear__order__wf__diff__Id,axiom,
    ! [R: set_Pr7585778909603769095tion_a] :
      ( ( order_7850372301378808569tion_a @ ( field_option_a @ R ) @ R )
     => ( ( wf_option_a @ ( minus_6512073291116468334tion_a @ R @ id_option_a2 ) )
        = ( ! [A8: set_option_a] :
              ( ( ord_le1955136853071979460tion_a @ A8 @ ( field_option_a @ R ) )
             => ( ( A8 != bot_bot_set_option_a )
               => ? [X: option_a] :
                    ( ( member_option_a @ X @ A8 )
                    & ! [Y3: option_a] :
                        ( ( member_option_a @ Y3 @ A8 )
                       => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y3 ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_638_wo__rel_Ocases__Total3,axiom,
    ! [R: set_Product_prod_a_a,A: a,B: a,Phi: a > a > $o] :
      ( ( bNF_We1162827675446709994_rel_a @ R )
     => ( ( ord_less_eq_set_a @ ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( field_a @ R ) )
       => ( ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( minus_6817036919807184750od_a_a @ R @ id_a2 ) )
              | ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ A ) @ ( minus_6817036919807184750od_a_a @ R @ id_a2 ) ) )
           => ( Phi @ A @ B ) )
         => ( ( ( A = B )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_639_wo__rel_Ocases__Total3,axiom,
    ! [R: set_Pr7585778909603769095tion_a,A: option_a,B: option_a,Phi: option_a > option_a > $o] :
      ( ( bNF_We8432232079604507440tion_a @ R )
     => ( ( ord_le1955136853071979460tion_a @ ( insert_option_a @ A @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( field_option_a @ R ) )
       => ( ( ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ ( minus_6512073291116468334tion_a @ R @ id_option_a2 ) )
              | ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ A ) @ ( minus_6512073291116468334tion_a @ R @ id_option_a2 ) ) )
           => ( Phi @ A @ B ) )
         => ( ( ( A = B )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_640_total__onI,axiom,
    ! [A6: set_a,R: set_Product_prod_a_a] :
      ( ! [X4: a,Y4: a] :
          ( ( member_a @ X4 @ A6 )
         => ( ( member_a @ Y4 @ A6 )
           => ( ( X4 != Y4 )
             => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R )
                | ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X4 ) @ R ) ) ) ) )
     => ( total_on_a @ A6 @ R ) ) ).

% total_onI
thf(fact_641_total__onI,axiom,
    ! [A6: set_Pr1275464188344874039_a_c_d,R: set_Pr3774076249145843911_a_c_d] :
      ( ! [X4: produc5278197477302038359_a_c_d,Y4: produc5278197477302038359_a_c_d] :
          ( ( member1180172933830803072_a_c_d @ X4 @ A6 )
         => ( ( member1180172933830803072_a_c_d @ Y4 @ A6 )
           => ( ( X4 != Y4 )
             => ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ X4 @ Y4 ) @ R )
                | ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ Y4 @ X4 ) @ R ) ) ) ) )
     => ( total_7546330196025013979_a_c_d @ A6 @ R ) ) ).

% total_onI
thf(fact_642_total__onI,axiom,
    ! [A6: set_option_a,R: set_Pr7585778909603769095tion_a] :
      ( ! [X4: option_a,Y4: option_a] :
          ( ( member_option_a @ X4 @ A6 )
         => ( ( member_option_a @ Y4 @ A6 )
           => ( ( X4 != Y4 )
             => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X4 @ Y4 ) @ R )
                | ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y4 @ X4 ) @ R ) ) ) ) )
     => ( total_on_option_a @ A6 @ R ) ) ).

% total_onI
thf(fact_643_wo__rel_Oin__notinI,axiom,
    ! [R: set_Product_prod_a_a,J: a,I: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R )
     => ( ( ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ J @ I ) @ R )
          | ( J = I ) )
       => ( ( member_a @ I @ ( field_a @ R ) )
         => ( ( member_a @ J @ ( field_a @ R ) )
           => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_644_wo__rel_Oin__notinI,axiom,
    ! [R: set_Pr3774076249145843911_a_c_d,J: produc5278197477302038359_a_c_d,I: produc5278197477302038359_a_c_d] :
      ( ( bNF_We47639529709206339_a_c_d @ R )
     => ( ( ~ ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ J @ I ) @ R )
          | ( J = I ) )
       => ( ( member1180172933830803072_a_c_d @ I @ ( field_5664454651862650532_a_c_d @ R ) )
         => ( ( member1180172933830803072_a_c_d @ J @ ( field_5664454651862650532_a_c_d @ R ) )
           => ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ I @ J ) @ R ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_645_wo__rel_Oin__notinI,axiom,
    ! [R: set_Pr7585778909603769095tion_a,J: option_a,I: option_a] :
      ( ( bNF_We8432232079604507440tion_a @ R )
     => ( ( ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ J @ I ) @ R )
          | ( J = I ) )
       => ( ( member_option_a @ I @ ( field_option_a @ R ) )
         => ( ( member_option_a @ J @ ( field_option_a @ R ) )
           => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ I @ J ) @ R ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_646_well__order__induct__imp,axiom,
    ! [R: set_Product_prod_a_a,P2: a > $o,A: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R )
     => ( ! [X4: a] :
            ( ! [Y5: a] :
                ( ( ( Y5 != X4 )
                  & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ X4 ) @ R ) )
               => ( ( member_a @ Y5 @ ( field_a @ R ) )
                 => ( P2 @ Y5 ) ) )
           => ( ( member_a @ X4 @ ( field_a @ R ) )
             => ( P2 @ X4 ) ) )
       => ( ( member_a @ A @ ( field_a @ R ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_647_well__order__induct__imp,axiom,
    ! [R: set_Pr3774076249145843911_a_c_d,P2: produc5278197477302038359_a_c_d > $o,A: produc5278197477302038359_a_c_d] :
      ( ( bNF_We47639529709206339_a_c_d @ R )
     => ( ! [X4: produc5278197477302038359_a_c_d] :
            ( ! [Y5: produc5278197477302038359_a_c_d] :
                ( ( ( Y5 != X4 )
                  & ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ Y5 @ X4 ) @ R ) )
               => ( ( member1180172933830803072_a_c_d @ Y5 @ ( field_5664454651862650532_a_c_d @ R ) )
                 => ( P2 @ Y5 ) ) )
           => ( ( member1180172933830803072_a_c_d @ X4 @ ( field_5664454651862650532_a_c_d @ R ) )
             => ( P2 @ X4 ) ) )
       => ( ( member1180172933830803072_a_c_d @ A @ ( field_5664454651862650532_a_c_d @ R ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_648_well__order__induct__imp,axiom,
    ! [R: set_Pr7585778909603769095tion_a,P2: option_a > $o,A: option_a] :
      ( ( bNF_We8432232079604507440tion_a @ R )
     => ( ! [X4: option_a] :
            ( ! [Y5: option_a] :
                ( ( ( Y5 != X4 )
                  & ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y5 @ X4 ) @ R ) )
               => ( ( member_option_a @ Y5 @ ( field_option_a @ R ) )
                 => ( P2 @ Y5 ) ) )
           => ( ( member_option_a @ X4 @ ( field_option_a @ R ) )
             => ( P2 @ X4 ) ) )
       => ( ( member_option_a @ A @ ( field_option_a @ R ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_649_totalI,axiom,
    ! [R: set_Product_prod_a_a] :
      ( ! [X4: a,Y4: a] :
          ( ( X4 != Y4 )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R )
            | ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X4 ) @ R ) ) )
     => ( total_on_a @ top_top_set_a @ R ) ) ).

% totalI
thf(fact_650_totalI,axiom,
    ! [R: set_Pr7585778909603769095tion_a] :
      ( ! [X4: option_a,Y4: option_a] :
          ( ( X4 != Y4 )
         => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X4 @ Y4 ) @ R )
            | ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y4 @ X4 ) @ R ) ) )
     => ( total_on_option_a @ top_top_set_option_a @ R ) ) ).

% totalI
thf(fact_651_wf__eq__minimal2,axiom,
    ( wf_a
    = ( ^ [R2: set_Product_prod_a_a] :
        ! [A8: set_a] :
          ( ( ( ord_less_eq_set_a @ A8 @ ( field_a @ R2 ) )
            & ( A8 != bot_bot_set_a ) )
         => ? [X: a] :
              ( ( member_a @ X @ A8 )
              & ! [Y3: a] :
                  ( ( member_a @ Y3 @ A8 )
                 => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X ) @ R2 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_652_wf__eq__minimal2,axiom,
    ( wf_option_a
    = ( ^ [R2: set_Pr7585778909603769095tion_a] :
        ! [A8: set_option_a] :
          ( ( ( ord_le1955136853071979460tion_a @ A8 @ ( field_option_a @ R2 ) )
            & ( A8 != bot_bot_set_option_a ) )
         => ? [X: option_a] :
              ( ( member_option_a @ X @ A8 )
              & ! [Y3: option_a] :
                  ( ( member_option_a @ Y3 @ A8 )
                 => ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y3 @ X ) @ R2 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_653_wo__rel_Ocases__Total,axiom,
    ! [R: set_Product_prod_a_a,A: a,B: a,Phi: a > a > $o] :
      ( ( bNF_We1162827675446709994_rel_a @ R )
     => ( ( ord_less_eq_set_a @ ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( field_a @ R ) )
       => ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
           => ( Phi @ A @ B ) )
         => ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ A ) @ R )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_654_wo__rel_Ocases__Total,axiom,
    ! [R: set_Pr7585778909603769095tion_a,A: option_a,B: option_a,Phi: option_a > option_a > $o] :
      ( ( bNF_We8432232079604507440tion_a @ R )
     => ( ( ord_le1955136853071979460tion_a @ ( insert_option_a @ A @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( field_option_a @ R ) )
       => ( ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R )
           => ( Phi @ A @ B ) )
         => ( ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ A ) @ R )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_655_wfE__min_H,axiom,
    ! [R3: set_Pr3774076249145843911_a_c_d,Q2: set_Pr1275464188344874039_a_c_d] :
      ( ( wf_Pro840557295966663232_a_c_d @ R3 )
     => ( ( Q2 != bot_bo4027597737525728931_a_c_d )
       => ~ ! [Z3: produc5278197477302038359_a_c_d] :
              ( ( member1180172933830803072_a_c_d @ Z3 @ Q2 )
             => ~ ! [Y5: produc5278197477302038359_a_c_d] :
                    ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ Y5 @ Z3 ) @ R3 )
                   => ~ ( member1180172933830803072_a_c_d @ Y5 @ Q2 ) ) ) ) ) ).

% wfE_min'
thf(fact_656_wfE__min_H,axiom,
    ! [R3: set_Product_prod_a_a,Q2: set_a] :
      ( ( wf_a @ R3 )
     => ( ( Q2 != bot_bot_set_a )
       => ~ ! [Z3: a] :
              ( ( member_a @ Z3 @ Q2 )
             => ~ ! [Y5: a] :
                    ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ Z3 ) @ R3 )
                   => ~ ( member_a @ Y5 @ Q2 ) ) ) ) ) ).

% wfE_min'
thf(fact_657_wfE__min_H,axiom,
    ! [R3: set_Pr7585778909603769095tion_a,Q2: set_option_a] :
      ( ( wf_option_a @ R3 )
     => ( ( Q2 != bot_bot_set_option_a )
       => ~ ! [Z3: option_a] :
              ( ( member_option_a @ Z3 @ Q2 )
             => ~ ! [Y5: option_a] :
                    ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y5 @ Z3 ) @ R3 )
                   => ~ ( member_option_a @ Y5 @ Q2 ) ) ) ) ) ).

% wfE_min'
thf(fact_658_wo__rel_Oequals__minim,axiom,
    ! [R: set_Product_prod_a_a,B5: set_a,A: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R )
     => ( ( ord_less_eq_set_a @ B5 @ ( field_a @ R ) )
       => ( ( member_a @ A @ B5 )
         => ( ! [B6: a] :
                ( ( member_a @ B6 @ B5 )
               => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B6 ) @ R ) )
           => ( A
              = ( bNF_We5615626441682584778inim_a @ R @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_659_wo__rel_Oequals__minim,axiom,
    ! [R: set_Pr3774076249145843911_a_c_d,B5: set_Pr1275464188344874039_a_c_d,A: produc5278197477302038359_a_c_d] :
      ( ( bNF_We47639529709206339_a_c_d @ R )
     => ( ( ord_le6392319565089278935_a_c_d @ B5 @ ( field_5664454651862650532_a_c_d @ R ) )
       => ( ( member1180172933830803072_a_c_d @ A @ B5 )
         => ( ! [B6: produc5278197477302038359_a_c_d] :
                ( ( member1180172933830803072_a_c_d @ B6 @ B5 )
               => ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ A @ B6 ) @ R ) )
           => ( A
              = ( bNF_We9171917446092943651_a_c_d @ R @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_660_wo__rel_Oequals__minim,axiom,
    ! [R: set_Pr7585778909603769095tion_a,B5: set_option_a,A: option_a] :
      ( ( bNF_We8432232079604507440tion_a @ R )
     => ( ( ord_le1955136853071979460tion_a @ B5 @ ( field_option_a @ R ) )
       => ( ( member_option_a @ A @ B5 )
         => ( ! [B6: option_a] :
                ( ( member_option_a @ B6 @ B5 )
               => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B6 ) @ R ) )
           => ( A
              = ( bNF_We6579146059749918992tion_a @ R @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_661_wo__rel_Ominim__least,axiom,
    ! [R: set_Product_prod_a_a,B5: set_a,B: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R )
     => ( ( ord_less_eq_set_a @ B5 @ ( field_a @ R ) )
       => ( ( member_a @ B @ B5 )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( bNF_We5615626441682584778inim_a @ R @ B5 ) @ B ) @ R ) ) ) ) ).

% wo_rel.minim_least
thf(fact_662_wo__rel_Ominim__least,axiom,
    ! [R: set_Pr3774076249145843911_a_c_d,B5: set_Pr1275464188344874039_a_c_d,B: produc5278197477302038359_a_c_d] :
      ( ( bNF_We47639529709206339_a_c_d @ R )
     => ( ( ord_le6392319565089278935_a_c_d @ B5 @ ( field_5664454651862650532_a_c_d @ R ) )
       => ( ( member1180172933830803072_a_c_d @ B @ B5 )
         => ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ ( bNF_We9171917446092943651_a_c_d @ R @ B5 ) @ B ) @ R ) ) ) ) ).

% wo_rel.minim_least
thf(fact_663_wo__rel_Ominim__least,axiom,
    ! [R: set_Pr7585778909603769095tion_a,B5: set_option_a,B: option_a] :
      ( ( bNF_We8432232079604507440tion_a @ R )
     => ( ( ord_le1955136853071979460tion_a @ B5 @ ( field_option_a @ R ) )
       => ( ( member_option_a @ B @ B5 )
         => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ ( bNF_We6579146059749918992tion_a @ R @ B5 ) @ B ) @ R ) ) ) ) ).

% wo_rel.minim_least
thf(fact_664_wo__rel_Omax2__greater__among,axiom,
    ! [R: set_Pr3774076249145843911_a_c_d,A: produc5278197477302038359_a_c_d,B: produc5278197477302038359_a_c_d] :
      ( ( bNF_We47639529709206339_a_c_d @ R )
     => ( ( member1180172933830803072_a_c_d @ A @ ( field_5664454651862650532_a_c_d @ R ) )
       => ( ( member1180172933830803072_a_c_d @ B @ ( field_5664454651862650532_a_c_d @ R ) )
         => ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ A @ ( bNF_We3665937542762464421_a_c_d @ R @ A @ B ) ) @ R )
            & ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ B @ ( bNF_We3665937542762464421_a_c_d @ R @ A @ B ) ) @ R )
            & ( member1180172933830803072_a_c_d @ ( bNF_We3665937542762464421_a_c_d @ R @ A @ B ) @ ( insert4072338954160766247_a_c_d @ A @ ( insert4072338954160766247_a_c_d @ B @ bot_bo4027597737525728931_a_c_d ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_665_wo__rel_Omax2__greater__among,axiom,
    ! [R: set_Product_prod_a_a,A: a,B: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R )
     => ( ( member_a @ A @ ( field_a @ R ) )
       => ( ( member_a @ B @ ( field_a @ R ) )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ ( bNF_We3763454674811381836max2_a @ R @ A @ B ) ) @ R )
            & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ ( bNF_We3763454674811381836max2_a @ R @ A @ B ) ) @ R )
            & ( member_a @ ( bNF_We3763454674811381836max2_a @ R @ A @ B ) @ ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_666_wo__rel_Omax2__greater__among,axiom,
    ! [R: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
      ( ( bNF_We8432232079604507440tion_a @ R )
     => ( ( member_option_a @ A @ ( field_option_a @ R ) )
       => ( ( member_option_a @ B @ ( field_option_a @ R ) )
         => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ ( bNF_We4567742444881707410tion_a @ R @ A @ B ) ) @ R )
            & ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ ( bNF_We4567742444881707410tion_a @ R @ A @ B ) ) @ R )
            & ( member_option_a @ ( bNF_We4567742444881707410tion_a @ R @ A @ B ) @ ( insert_option_a @ A @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_667_wo__rel_Omax2__equals1,axiom,
    ! [R: set_Product_prod_a_a,A: a,B: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R )
     => ( ( member_a @ A @ ( field_a @ R ) )
       => ( ( member_a @ B @ ( field_a @ R ) )
         => ( ( ( bNF_We3763454674811381836max2_a @ R @ A @ B )
              = A )
            = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ A ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_668_wo__rel_Omax2__equals1,axiom,
    ! [R: set_Pr3774076249145843911_a_c_d,A: produc5278197477302038359_a_c_d,B: produc5278197477302038359_a_c_d] :
      ( ( bNF_We47639529709206339_a_c_d @ R )
     => ( ( member1180172933830803072_a_c_d @ A @ ( field_5664454651862650532_a_c_d @ R ) )
       => ( ( member1180172933830803072_a_c_d @ B @ ( field_5664454651862650532_a_c_d @ R ) )
         => ( ( ( bNF_We3665937542762464421_a_c_d @ R @ A @ B )
              = A )
            = ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ B @ A ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_669_wo__rel_Omax2__equals1,axiom,
    ! [R: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
      ( ( bNF_We8432232079604507440tion_a @ R )
     => ( ( member_option_a @ A @ ( field_option_a @ R ) )
       => ( ( member_option_a @ B @ ( field_option_a @ R ) )
         => ( ( ( bNF_We4567742444881707410tion_a @ R @ A @ B )
              = A )
            = ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ A ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_670_wo__rel_Omax2__equals2,axiom,
    ! [R: set_Product_prod_a_a,A: a,B: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R )
     => ( ( member_a @ A @ ( field_a @ R ) )
       => ( ( member_a @ B @ ( field_a @ R ) )
         => ( ( ( bNF_We3763454674811381836max2_a @ R @ A @ B )
              = B )
            = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_671_wo__rel_Omax2__equals2,axiom,
    ! [R: set_Pr3774076249145843911_a_c_d,A: produc5278197477302038359_a_c_d,B: produc5278197477302038359_a_c_d] :
      ( ( bNF_We47639529709206339_a_c_d @ R )
     => ( ( member1180172933830803072_a_c_d @ A @ ( field_5664454651862650532_a_c_d @ R ) )
       => ( ( member1180172933830803072_a_c_d @ B @ ( field_5664454651862650532_a_c_d @ R ) )
         => ( ( ( bNF_We3665937542762464421_a_c_d @ R @ A @ B )
              = B )
            = ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ A @ B ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_672_wo__rel_Omax2__equals2,axiom,
    ! [R: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
      ( ( bNF_We8432232079604507440tion_a @ R )
     => ( ( member_option_a @ A @ ( field_option_a @ R ) )
       => ( ( member_option_a @ B @ ( field_option_a @ R ) )
         => ( ( ( bNF_We4567742444881707410tion_a @ R @ A @ B )
              = B )
            = ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_673_wo__rel_Omax2__greater,axiom,
    ! [R: set_Product_prod_a_a,A: a,B: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R )
     => ( ( member_a @ A @ ( field_a @ R ) )
       => ( ( member_a @ B @ ( field_a @ R ) )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ ( bNF_We3763454674811381836max2_a @ R @ A @ B ) ) @ R )
            & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ ( bNF_We3763454674811381836max2_a @ R @ A @ B ) ) @ R ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_674_wo__rel_Omax2__greater,axiom,
    ! [R: set_Pr3774076249145843911_a_c_d,A: produc5278197477302038359_a_c_d,B: produc5278197477302038359_a_c_d] :
      ( ( bNF_We47639529709206339_a_c_d @ R )
     => ( ( member1180172933830803072_a_c_d @ A @ ( field_5664454651862650532_a_c_d @ R ) )
       => ( ( member1180172933830803072_a_c_d @ B @ ( field_5664454651862650532_a_c_d @ R ) )
         => ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ A @ ( bNF_We3665937542762464421_a_c_d @ R @ A @ B ) ) @ R )
            & ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ B @ ( bNF_We3665937542762464421_a_c_d @ R @ A @ B ) ) @ R ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_675_wo__rel_Omax2__greater,axiom,
    ! [R: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
      ( ( bNF_We8432232079604507440tion_a @ R )
     => ( ( member_option_a @ A @ ( field_option_a @ R ) )
       => ( ( member_option_a @ B @ ( field_option_a @ R ) )
         => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ ( bNF_We4567742444881707410tion_a @ R @ A @ B ) ) @ R )
            & ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ ( bNF_We4567742444881707410tion_a @ R @ A @ B ) ) @ R ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_676_wf__eq__minimal,axiom,
    ( wf_a
    = ( ^ [R2: 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 ) @ R2 )
                 => ~ ( member_a @ Y3 @ Q5 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_677_wf__eq__minimal,axiom,
    ( wf_Pro840557295966663232_a_c_d
    = ( ^ [R2: set_Pr3774076249145843911_a_c_d] :
        ! [Q5: set_Pr1275464188344874039_a_c_d] :
          ( ? [X: produc5278197477302038359_a_c_d] : ( member1180172933830803072_a_c_d @ X @ Q5 )
         => ? [X: produc5278197477302038359_a_c_d] :
              ( ( member1180172933830803072_a_c_d @ X @ Q5 )
              & ! [Y3: produc5278197477302038359_a_c_d] :
                  ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ Y3 @ X ) @ R2 )
                 => ~ ( member1180172933830803072_a_c_d @ Y3 @ Q5 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_678_wf__eq__minimal,axiom,
    ( wf_option_a
    = ( ^ [R2: 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 ) @ R2 )
                 => ~ ( member_option_a @ Y3 @ Q5 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_679_wfI__min,axiom,
    ! [R3: set_Product_prod_a_a] :
      ( ! [X4: a,Q6: set_a] :
          ( ( member_a @ X4 @ Q6 )
         => ? [Xa2: a] :
              ( ( member_a @ Xa2 @ Q6 )
              & ! [Y4: a] :
                  ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ Xa2 ) @ R3 )
                 => ~ ( member_a @ Y4 @ Q6 ) ) ) )
     => ( wf_a @ R3 ) ) ).

% wfI_min
thf(fact_680_wfI__min,axiom,
    ! [R3: set_Pr3774076249145843911_a_c_d] :
      ( ! [X4: produc5278197477302038359_a_c_d,Q6: set_Pr1275464188344874039_a_c_d] :
          ( ( member1180172933830803072_a_c_d @ X4 @ Q6 )
         => ? [Xa2: produc5278197477302038359_a_c_d] :
              ( ( member1180172933830803072_a_c_d @ Xa2 @ Q6 )
              & ! [Y4: produc5278197477302038359_a_c_d] :
                  ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ Y4 @ Xa2 ) @ R3 )
                 => ~ ( member1180172933830803072_a_c_d @ Y4 @ Q6 ) ) ) )
     => ( wf_Pro840557295966663232_a_c_d @ R3 ) ) ).

% wfI_min
thf(fact_681_wfI__min,axiom,
    ! [R3: set_Pr7585778909603769095tion_a] :
      ( ! [X4: option_a,Q6: set_option_a] :
          ( ( member_option_a @ X4 @ Q6 )
         => ? [Xa2: option_a] :
              ( ( member_option_a @ Xa2 @ Q6 )
              & ! [Y4: option_a] :
                  ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y4 @ Xa2 ) @ R3 )
                 => ~ ( member_option_a @ Y4 @ Q6 ) ) ) )
     => ( wf_option_a @ R3 ) ) ).

% wfI_min
thf(fact_682_wfE__min,axiom,
    ! [R3: set_Product_prod_a_a,X2: a,Q2: set_a] :
      ( ( wf_a @ R3 )
     => ( ( member_a @ X2 @ Q2 )
       => ~ ! [Z3: a] :
              ( ( member_a @ Z3 @ Q2 )
             => ~ ! [Y5: a] :
                    ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ Z3 ) @ R3 )
                   => ~ ( member_a @ Y5 @ Q2 ) ) ) ) ) ).

% wfE_min
thf(fact_683_wfE__min,axiom,
    ! [R3: set_Pr3774076249145843911_a_c_d,X2: produc5278197477302038359_a_c_d,Q2: set_Pr1275464188344874039_a_c_d] :
      ( ( wf_Pro840557295966663232_a_c_d @ R3 )
     => ( ( member1180172933830803072_a_c_d @ X2 @ Q2 )
       => ~ ! [Z3: produc5278197477302038359_a_c_d] :
              ( ( member1180172933830803072_a_c_d @ Z3 @ Q2 )
             => ~ ! [Y5: produc5278197477302038359_a_c_d] :
                    ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ Y5 @ Z3 ) @ R3 )
                   => ~ ( member1180172933830803072_a_c_d @ Y5 @ Q2 ) ) ) ) ) ).

% wfE_min
thf(fact_684_wfE__min,axiom,
    ! [R3: set_Pr7585778909603769095tion_a,X2: option_a,Q2: set_option_a] :
      ( ( wf_option_a @ R3 )
     => ( ( member_option_a @ X2 @ Q2 )
       => ~ ! [Z3: option_a] :
              ( ( member_option_a @ Z3 @ Q2 )
             => ~ ! [Y5: option_a] :
                    ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y5 @ Z3 ) @ R3 )
                   => ~ ( member_option_a @ Y5 @ Q2 ) ) ) ) ) ).

% wfE_min
thf(fact_685_Linear__order__Well__order__iff,axiom,
    ! [R: set_Product_prod_a_a] :
      ( ( order_8768733634509060147r_on_a @ ( field_a @ R ) @ R )
     => ( ( order_6972113574731384220r_on_a @ ( field_a @ R ) @ R )
        = ( ! [A8: set_a] :
              ( ( ord_less_eq_set_a @ A8 @ ( field_a @ R ) )
             => ( ( A8 != bot_bot_set_a )
               => ? [X: a] :
                    ( ( member_a @ X @ A8 )
                    & ! [Y3: a] :
                        ( ( member_a @ Y3 @ A8 )
                       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_686_Linear__order__Well__order__iff,axiom,
    ! [R: set_Pr7585778909603769095tion_a] :
      ( ( order_7850372301378808569tion_a @ ( field_option_a @ R ) @ R )
     => ( ( order_4821795997958563554tion_a @ ( field_option_a @ R ) @ R )
        = ( ! [A8: set_option_a] :
              ( ( ord_le1955136853071979460tion_a @ A8 @ ( field_option_a @ R ) )
             => ( ( A8 != bot_bot_set_option_a )
               => ? [X: option_a] :
                    ( ( member_option_a @ X @ A8 )
                    & ! [Y3: option_a] :
                        ( ( member_option_a @ Y3 @ A8 )
                       => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y3 ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_687_underS__incl__iff,axiom,
    ! [R: set_Product_prod_a_a,A: a,B: a] :
      ( ( order_8768733634509060147r_on_a @ ( field_a @ R ) @ R )
     => ( ( member_a @ A @ ( field_a @ R ) )
       => ( ( member_a @ B @ ( field_a @ R ) )
         => ( ( ord_less_eq_set_a @ ( order_underS_a @ R @ A ) @ ( order_underS_a @ R @ B ) )
            = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R ) ) ) ) ) ).

% underS_incl_iff
thf(fact_688_underS__incl__iff,axiom,
    ! [R: set_Pr3774076249145843911_a_c_d,A: produc5278197477302038359_a_c_d,B: produc5278197477302038359_a_c_d] :
      ( ( order_1589598524371451020_a_c_d @ ( field_5664454651862650532_a_c_d @ R ) @ R )
     => ( ( member1180172933830803072_a_c_d @ A @ ( field_5664454651862650532_a_c_d @ R ) )
       => ( ( member1180172933830803072_a_c_d @ B @ ( field_5664454651862650532_a_c_d @ R ) )
         => ( ( ord_le6392319565089278935_a_c_d @ ( order_3579777048248667421_a_c_d @ R @ A ) @ ( order_3579777048248667421_a_c_d @ R @ B ) )
            = ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ A @ B ) @ R ) ) ) ) ) ).

% underS_incl_iff
thf(fact_689_underS__incl__iff,axiom,
    ! [R: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
      ( ( order_7850372301378808569tion_a @ ( field_option_a @ R ) @ R )
     => ( ( member_option_a @ A @ ( field_option_a @ R ) )
       => ( ( member_option_a @ B @ ( field_option_a @ R ) )
         => ( ( ord_le1955136853071979460tion_a @ ( order_8525669848891258378tion_a @ R @ A ) @ ( order_8525669848891258378tion_a @ R @ B ) )
            = ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R ) ) ) ) ) ).

% underS_incl_iff
thf(fact_690_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_691_finite__range__updI,axiom,
    ! [F2: a > option6413918287372586467_a_c_d,A: a,B: product_prod_a_c_d] :
      ( ( finite3027439597799462906_a_c_d @ ( image_3786525410991915292_a_c_d @ F2 @ top_top_set_a ) )
     => ( finite3027439597799462906_a_c_d @ ( image_3786525410991915292_a_c_d @ ( fun_up6642239869823773348_a_c_d @ F2 @ A @ ( some_P1084500821511757806_a_c_d @ B ) ) @ top_top_set_a ) ) ) ).

% finite_range_updI
thf(fact_692_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_693_finite__range__updI,axiom,
    ! [F2: option_a > option6413918287372586467_a_c_d,A: option_a,B: product_prod_a_c_d] :
      ( ( finite3027439597799462906_a_c_d @ ( image_3697009332183072098_a_c_d @ F2 @ top_top_set_option_a ) )
     => ( finite3027439597799462906_a_c_d @ ( image_3697009332183072098_a_c_d @ ( fun_up5969202740897331946_a_c_d @ F2 @ A @ ( some_P1084500821511757806_a_c_d @ B ) ) @ top_top_set_option_a ) ) ) ).

% finite_range_updI
thf(fact_694_finite__option__UNIV,axiom,
    ( ( finite8114217219359860531tion_a @ top_to1659475022456381882tion_a )
    = ( finite1674126218327898605tion_a @ top_top_set_option_a ) ) ).

% finite_option_UNIV
thf(fact_695_finite__option__UNIV,axiom,
    ( ( finite1674126218327898605tion_a @ top_top_set_option_a )
    = ( finite_finite_a @ top_top_set_a ) ) ).

% finite_option_UNIV
thf(fact_696_well__order__on__domain,axiom,
    ! [A6: set_a,R: set_Product_prod_a_a,A: a,B: a] :
      ( ( order_6972113574731384220r_on_a @ A6 @ R )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
       => ( ( member_a @ A @ A6 )
          & ( member_a @ B @ A6 ) ) ) ) ).

% well_order_on_domain
thf(fact_697_well__order__on__domain,axiom,
    ! [A6: set_Pr1275464188344874039_a_c_d,R: set_Pr3774076249145843911_a_c_d,A: produc5278197477302038359_a_c_d,B: produc5278197477302038359_a_c_d] :
      ( ( order_244995887285936629_a_c_d @ A6 @ R )
     => ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ A @ B ) @ R )
       => ( ( member1180172933830803072_a_c_d @ A @ A6 )
          & ( member1180172933830803072_a_c_d @ B @ A6 ) ) ) ) ).

% well_order_on_domain
thf(fact_698_well__order__on__domain,axiom,
    ! [A6: set_option_a,R: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
      ( ( order_4821795997958563554tion_a @ A6 @ R )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R )
       => ( ( member_option_a @ A @ A6 )
          & ( member_option_a @ B @ A6 ) ) ) ) ).

% well_order_on_domain
thf(fact_699_underS__I,axiom,
    ! [I: a,J: a,R3: set_Product_prod_a_a] :
      ( ( I != J )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R3 )
       => ( member_a @ I @ ( order_underS_a @ R3 @ J ) ) ) ) ).

% underS_I
thf(fact_700_underS__I,axiom,
    ! [I: produc5278197477302038359_a_c_d,J: produc5278197477302038359_a_c_d,R3: set_Pr3774076249145843911_a_c_d] :
      ( ( I != J )
     => ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ I @ J ) @ R3 )
       => ( member1180172933830803072_a_c_d @ I @ ( order_3579777048248667421_a_c_d @ R3 @ J ) ) ) ) ).

% underS_I
thf(fact_701_underS__I,axiom,
    ! [I: option_a,J: option_a,R3: set_Pr7585778909603769095tion_a] :
      ( ( I != J )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ I @ J ) @ R3 )
       => ( member_option_a @ I @ ( order_8525669848891258378tion_a @ R3 @ J ) ) ) ) ).

% underS_I
thf(fact_702_underS__E,axiom,
    ! [I: a,R3: set_Product_prod_a_a,J: a] :
      ( ( member_a @ I @ ( order_underS_a @ R3 @ J ) )
     => ( ( I != J )
        & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R3 ) ) ) ).

% underS_E
thf(fact_703_underS__E,axiom,
    ! [I: produc5278197477302038359_a_c_d,R3: set_Pr3774076249145843911_a_c_d,J: produc5278197477302038359_a_c_d] :
      ( ( member1180172933830803072_a_c_d @ I @ ( order_3579777048248667421_a_c_d @ R3 @ J ) )
     => ( ( I != J )
        & ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ I @ J ) @ R3 ) ) ) ).

% underS_E
thf(fact_704_underS__E,axiom,
    ! [I: option_a,R3: set_Pr7585778909603769095tion_a,J: option_a] :
      ( ( member_option_a @ I @ ( order_8525669848891258378tion_a @ R3 @ J ) )
     => ( ( I != J )
        & ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ I @ J ) @ R3 ) ) ) ).

% underS_E
thf(fact_705_finite__range__Some,axiom,
    ( ( finite3027439597799462906_a_c_d @ ( image_5741706613785127421_a_c_d @ some_P1084500821511757806_a_c_d @ top_to5543414004138509561_a_c_d ) )
    = ( finite6487500827408305578_a_c_d @ top_to5543414004138509561_a_c_d ) ) ).

% finite_range_Some
thf(fact_706_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_707_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_708_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_709_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_710_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_711_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_712_finite__map__freshness,axiom,
    ! [F2: a > option_a] :
      ( ( finite_finite_a @ ( dom_a_a @ F2 ) )
     => ( ~ ( finite_finite_a @ top_top_set_a )
       => ? [X4: a] :
            ( ( F2 @ X4 )
            = none_a ) ) ) ).

% finite_map_freshness
thf(fact_713_finite__map__freshness,axiom,
    ! [F2: option_a > option_a] :
      ( ( finite1674126218327898605tion_a @ ( dom_option_a_a @ F2 ) )
     => ( ~ ( finite1674126218327898605tion_a @ top_top_set_option_a )
       => ? [X4: option_a] :
            ( ( F2 @ X4 )
            = none_a ) ) ) ).

% finite_map_freshness
thf(fact_714_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_715_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_716_max__ext_Ocases,axiom,
    ! [A12: set_Pr1275464188344874039_a_c_d,A22: set_Pr1275464188344874039_a_c_d,R3: set_Pr3774076249145843911_a_c_d] :
      ( ( member6005089157543085840_a_c_d @ ( produc4179448859695865687_a_c_d @ A12 @ A22 ) @ ( max_ex7813175997721881362_a_c_d @ R3 ) )
     => ~ ( ( finite5665696198532655488_a_c_d @ A12 )
         => ( ( finite5665696198532655488_a_c_d @ A22 )
           => ( ( A22 != bot_bo4027597737525728931_a_c_d )
             => ~ ! [X8: produc5278197477302038359_a_c_d] :
                    ( ( member1180172933830803072_a_c_d @ X8 @ A12 )
                   => ? [Xa: produc5278197477302038359_a_c_d] :
                        ( ( member1180172933830803072_a_c_d @ Xa @ A22 )
                        & ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ X8 @ Xa ) @ R3 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_717_max__ext_Ocases,axiom,
    ! [A12: set_a,A22: set_a,R3: set_Product_prod_a_a] :
      ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ A12 @ A22 ) @ ( max_ext_a @ R3 ) )
     => ~ ( ( finite_finite_a @ A12 )
         => ( ( finite_finite_a @ A22 )
           => ( ( A22 != bot_bot_set_a )
             => ~ ! [X8: a] :
                    ( ( member_a @ X8 @ A12 )
                   => ? [Xa: a] :
                        ( ( member_a @ Xa @ A22 )
                        & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X8 @ Xa ) @ R3 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_718_max__ext_Ocases,axiom,
    ! [A12: set_option_a,A22: set_option_a,R3: set_Pr7585778909603769095tion_a] :
      ( ( member5358692782348450128tion_a @ ( produc8179951581375851543tion_a @ A12 @ A22 ) @ ( max_ext_option_a @ R3 ) )
     => ~ ( ( finite1674126218327898605tion_a @ A12 )
         => ( ( finite1674126218327898605tion_a @ A22 )
           => ( ( A22 != bot_bot_set_option_a )
             => ~ ! [X8: option_a] :
                    ( ( member_option_a @ X8 @ A12 )
                   => ? [Xa: option_a] :
                        ( ( member_option_a @ Xa @ A22 )
                        & ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X8 @ Xa ) @ R3 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_719_max__ext_Osimps,axiom,
    ! [A12: set_Pr1275464188344874039_a_c_d,A22: set_Pr1275464188344874039_a_c_d,R3: set_Pr3774076249145843911_a_c_d] :
      ( ( member6005089157543085840_a_c_d @ ( produc4179448859695865687_a_c_d @ A12 @ A22 ) @ ( max_ex7813175997721881362_a_c_d @ R3 ) )
      = ( ( finite5665696198532655488_a_c_d @ A12 )
        & ( finite5665696198532655488_a_c_d @ A22 )
        & ( A22 != bot_bo4027597737525728931_a_c_d )
        & ! [X: produc5278197477302038359_a_c_d] :
            ( ( member1180172933830803072_a_c_d @ X @ A12 )
           => ? [Y3: produc5278197477302038359_a_c_d] :
                ( ( member1180172933830803072_a_c_d @ Y3 @ A22 )
                & ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ X @ Y3 ) @ R3 ) ) ) ) ) ).

% max_ext.simps
thf(fact_720_max__ext_Osimps,axiom,
    ! [A12: set_a,A22: set_a,R3: set_Product_prod_a_a] :
      ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ A12 @ A22 ) @ ( max_ext_a @ R3 ) )
      = ( ( finite_finite_a @ A12 )
        & ( finite_finite_a @ A22 )
        & ( A22 != bot_bot_set_a )
        & ! [X: a] :
            ( ( member_a @ X @ A12 )
           => ? [Y3: a] :
                ( ( member_a @ Y3 @ A22 )
                & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R3 ) ) ) ) ) ).

% max_ext.simps
thf(fact_721_max__ext_Osimps,axiom,
    ! [A12: set_option_a,A22: set_option_a,R3: set_Pr7585778909603769095tion_a] :
      ( ( member5358692782348450128tion_a @ ( produc8179951581375851543tion_a @ A12 @ A22 ) @ ( max_ext_option_a @ R3 ) )
      = ( ( finite1674126218327898605tion_a @ A12 )
        & ( finite1674126218327898605tion_a @ A22 )
        & ( A22 != bot_bot_set_option_a )
        & ! [X: option_a] :
            ( ( member_option_a @ X @ A12 )
           => ? [Y3: option_a] :
                ( ( member_option_a @ Y3 @ A22 )
                & ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y3 ) @ R3 ) ) ) ) ) ).

% max_ext.simps
thf(fact_722_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_723_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_724_max__ext_Omax__extI,axiom,
    ! [X6: set_Pr1275464188344874039_a_c_d,Y6: set_Pr1275464188344874039_a_c_d,R3: set_Pr3774076249145843911_a_c_d] :
      ( ( finite5665696198532655488_a_c_d @ X6 )
     => ( ( finite5665696198532655488_a_c_d @ Y6 )
       => ( ( Y6 != bot_bo4027597737525728931_a_c_d )
         => ( ! [X4: produc5278197477302038359_a_c_d] :
                ( ( member1180172933830803072_a_c_d @ X4 @ X6 )
               => ? [Xa2: produc5278197477302038359_a_c_d] :
                    ( ( member1180172933830803072_a_c_d @ Xa2 @ Y6 )
                    & ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ X4 @ Xa2 ) @ R3 ) ) )
           => ( member6005089157543085840_a_c_d @ ( produc4179448859695865687_a_c_d @ X6 @ Y6 ) @ ( max_ex7813175997721881362_a_c_d @ R3 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_725_max__ext_Omax__extI,axiom,
    ! [X6: set_a,Y6: set_a,R3: set_Product_prod_a_a] :
      ( ( finite_finite_a @ X6 )
     => ( ( finite_finite_a @ Y6 )
       => ( ( Y6 != bot_bot_set_a )
         => ( ! [X4: a] :
                ( ( member_a @ X4 @ X6 )
               => ? [Xa2: a] :
                    ( ( member_a @ Xa2 @ Y6 )
                    & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Xa2 ) @ R3 ) ) )
           => ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X6 @ Y6 ) @ ( max_ext_a @ R3 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_726_max__ext_Omax__extI,axiom,
    ! [X6: set_option_a,Y6: set_option_a,R3: set_Pr7585778909603769095tion_a] :
      ( ( finite1674126218327898605tion_a @ X6 )
     => ( ( finite1674126218327898605tion_a @ Y6 )
       => ( ( Y6 != bot_bot_set_option_a )
         => ( ! [X4: option_a] :
                ( ( member_option_a @ X4 @ X6 )
               => ? [Xa2: option_a] :
                    ( ( member_option_a @ Xa2 @ Y6 )
                    & ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X4 @ Xa2 ) @ R3 ) ) )
           => ( member5358692782348450128tion_a @ ( produc8179951581375851543tion_a @ X6 @ Y6 ) @ ( max_ext_option_a @ R3 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_727_wo__rel_OisMinim__def,axiom,
    ! [R: set_Product_prod_a_a,A6: set_a,B: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R )
     => ( ( bNF_We6697304935525757620inim_a @ R @ A6 @ B )
        = ( ( member_a @ B @ A6 )
          & ! [X: a] :
              ( ( member_a @ X @ A6 )
             => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ X ) @ R ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_728_wo__rel_OisMinim__def,axiom,
    ! [R: set_Pr3774076249145843911_a_c_d,A6: set_Pr1275464188344874039_a_c_d,B: produc5278197477302038359_a_c_d] :
      ( ( bNF_We47639529709206339_a_c_d @ R )
     => ( ( bNF_We3608594581950107405_a_c_d @ R @ A6 @ B )
        = ( ( member1180172933830803072_a_c_d @ B @ A6 )
          & ! [X: produc5278197477302038359_a_c_d] :
              ( ( member1180172933830803072_a_c_d @ X @ A6 )
             => ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ B @ X ) @ R ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_729_wo__rel_OisMinim__def,axiom,
    ! [R: set_Pr7585778909603769095tion_a,A6: set_option_a,B: option_a] :
      ( ( bNF_We8432232079604507440tion_a @ R )
     => ( ( bNF_We2467337426749329402tion_a @ R @ A6 @ B )
        = ( ( member_option_a @ B @ A6 )
          & ! [X: option_a] :
              ( ( member_option_a @ X @ A6 )
             => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ X ) @ R ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_730_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_731_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_732_trans__onD,axiom,
    ! [A6: set_a,R: set_Product_prod_a_a,X2: a,Y: a,Z2: a] :
      ( ( trans_on_a @ A6 @ R )
     => ( ( member_a @ X2 @ A6 )
       => ( ( member_a @ Y @ A6 )
         => ( ( member_a @ Z2 @ A6 )
           => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
             => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z2 ) @ R )
               => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Z2 ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_733_trans__onD,axiom,
    ! [A6: set_Pr1275464188344874039_a_c_d,R: set_Pr3774076249145843911_a_c_d,X2: produc5278197477302038359_a_c_d,Y: produc5278197477302038359_a_c_d,Z2: produc5278197477302038359_a_c_d] :
      ( ( trans_3107571255393791223_a_c_d @ A6 @ R )
     => ( ( member1180172933830803072_a_c_d @ X2 @ A6 )
       => ( ( member1180172933830803072_a_c_d @ Y @ A6 )
         => ( ( member1180172933830803072_a_c_d @ Z2 @ A6 )
           => ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ X2 @ Y ) @ R )
             => ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ Y @ Z2 ) @ R )
               => ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ X2 @ Z2 ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_734_trans__onD,axiom,
    ! [A6: set_option_a,R: set_Pr7585778909603769095tion_a,X2: option_a,Y: option_a,Z2: option_a] :
      ( ( trans_on_option_a @ A6 @ R )
     => ( ( member_option_a @ X2 @ A6 )
       => ( ( member_option_a @ Y @ A6 )
         => ( ( member_option_a @ Z2 @ A6 )
           => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y ) @ R )
             => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y @ Z2 ) @ R )
               => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Z2 ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_735_trans__onI,axiom,
    ! [A6: set_a,R: set_Product_prod_a_a] :
      ( ! [X4: a,Y4: a,Z3: a] :
          ( ( member_a @ X4 @ A6 )
         => ( ( member_a @ Y4 @ A6 )
           => ( ( member_a @ Z3 @ A6 )
             => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R )
               => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ Z3 ) @ R )
                 => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Z3 ) @ R ) ) ) ) ) )
     => ( trans_on_a @ A6 @ R ) ) ).

% trans_onI
thf(fact_736_trans__onI,axiom,
    ! [A6: set_Pr1275464188344874039_a_c_d,R: set_Pr3774076249145843911_a_c_d] :
      ( ! [X4: produc5278197477302038359_a_c_d,Y4: produc5278197477302038359_a_c_d,Z3: produc5278197477302038359_a_c_d] :
          ( ( member1180172933830803072_a_c_d @ X4 @ A6 )
         => ( ( member1180172933830803072_a_c_d @ Y4 @ A6 )
           => ( ( member1180172933830803072_a_c_d @ Z3 @ A6 )
             => ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ X4 @ Y4 ) @ R )
               => ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ Y4 @ Z3 ) @ R )
                 => ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ X4 @ Z3 ) @ R ) ) ) ) ) )
     => ( trans_3107571255393791223_a_c_d @ A6 @ R ) ) ).

% trans_onI
thf(fact_737_trans__onI,axiom,
    ! [A6: set_option_a,R: set_Pr7585778909603769095tion_a] :
      ( ! [X4: option_a,Y4: option_a,Z3: option_a] :
          ( ( member_option_a @ X4 @ A6 )
         => ( ( member_option_a @ Y4 @ A6 )
           => ( ( member_option_a @ Z3 @ A6 )
             => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X4 @ Y4 ) @ R )
               => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y4 @ Z3 ) @ R )
                 => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X4 @ Z3 ) @ R ) ) ) ) ) )
     => ( trans_on_option_a @ A6 @ R ) ) ).

% trans_onI
thf(fact_738_transI,axiom,
    ! [R: set_Product_prod_a_a] :
      ( ! [X4: a,Y4: a,Z3: a] :
          ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ Z3 ) @ R )
           => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Z3 ) @ R ) ) )
     => ( trans_on_a @ top_top_set_a @ R ) ) ).

% transI
thf(fact_739_transI,axiom,
    ! [R: set_Pr7585778909603769095tion_a] :
      ( ! [X4: option_a,Y4: option_a,Z3: option_a] :
          ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X4 @ Y4 ) @ R )
         => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y4 @ Z3 ) @ R )
           => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X4 @ Z3 ) @ R ) ) )
     => ( trans_on_option_a @ top_top_set_option_a @ R ) ) ).

% transI
thf(fact_740_transE,axiom,
    ! [R: set_Product_prod_a_a,X2: a,Y: a,Z2: a] :
      ( ( trans_on_a @ top_top_set_a @ R )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
       => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z2 ) @ R )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Z2 ) @ R ) ) ) ) ).

% transE
thf(fact_741_transE,axiom,
    ! [R: set_Pr7585778909603769095tion_a,X2: option_a,Y: option_a,Z2: option_a] :
      ( ( trans_on_option_a @ top_top_set_option_a @ R )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y ) @ R )
       => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y @ Z2 ) @ R )
         => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Z2 ) @ R ) ) ) ) ).

% transE
thf(fact_742_transD,axiom,
    ! [R: set_Product_prod_a_a,X2: a,Y: a,Z2: a] :
      ( ( trans_on_a @ top_top_set_a @ R )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
       => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z2 ) @ R )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Z2 ) @ R ) ) ) ) ).

% transD
thf(fact_743_transD,axiom,
    ! [R: set_Pr7585778909603769095tion_a,X2: option_a,Y: option_a,Z2: option_a] :
      ( ( trans_on_option_a @ top_top_set_option_a @ R )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y ) @ R )
       => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y @ Z2 ) @ R )
         => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Z2 ) @ R ) ) ) ) ).

% transD
thf(fact_744_image__strict__mono,axiom,
    ! [F2: option_a > a,B5: set_option_a,A6: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ B5 )
     => ( ( ord_le5631237216984945872tion_a @ A6 @ B5 )
       => ( ord_less_set_a @ ( image_option_a_a2 @ F2 @ A6 ) @ ( image_option_a_a2 @ F2 @ B5 ) ) ) ) ).

% image_strict_mono
thf(fact_745_image__strict__mono,axiom,
    ! [F2: a > option_a,B5: set_a,A6: set_a] :
      ( ( inj_on_a_option_a @ F2 @ B5 )
     => ( ( ord_less_set_a @ A6 @ B5 )
       => ( ord_le5631237216984945872tion_a @ ( image_a_option_a2 @ F2 @ A6 ) @ ( image_a_option_a2 @ F2 @ B5 ) ) ) ) ).

% image_strict_mono
thf(fact_746_in__finite__psubset,axiom,
    ! [A6: set_option_a,B5: set_option_a] :
      ( ( member5358692782348450128tion_a @ ( produc8179951581375851543tion_a @ A6 @ B5 ) @ finite4966134214920407047tion_a )
      = ( ( ord_le5631237216984945872tion_a @ A6 @ B5 )
        & ( finite1674126218327898605tion_a @ B5 ) ) ) ).

% in_finite_psubset
thf(fact_747_in__finite__psubset,axiom,
    ! [A6: set_a,B5: set_a] :
      ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ A6 @ B5 ) @ finite_psubset_a )
      = ( ( ord_less_set_a @ A6 @ B5 )
        & ( finite_finite_a @ B5 ) ) ) ).

% in_finite_psubset
thf(fact_748_underS__incr,axiom,
    ! [R: set_Product_prod_a_a,A: a,B: a] :
      ( ( trans_on_a @ top_top_set_a @ R )
     => ( ( antisym_on_a @ top_top_set_a @ R )
       => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
         => ( ord_less_eq_set_a @ ( order_underS_a @ R @ A ) @ ( order_underS_a @ R @ B ) ) ) ) ) ).

% underS_incr
thf(fact_749_underS__incr,axiom,
    ! [R: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
      ( ( trans_on_option_a @ top_top_set_option_a @ R )
     => ( ( antisym_on_option_a @ top_top_set_option_a @ R )
       => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R )
         => ( ord_le1955136853071979460tion_a @ ( order_8525669848891258378tion_a @ R @ A ) @ ( order_8525669848891258378tion_a @ R @ B ) ) ) ) ) ).

% underS_incr
thf(fact_750_in__lex__prod,axiom,
    ! [A: product_prod_a_c_d,B: option6413918287372586467_a_c_d,A4: product_prod_a_c_d,B3: option6413918287372586467_a_c_d,R: set_Pr5254629501202997351_a_c_d,S4: set_Pr3188740713198616583_a_c_d] :
      ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ ( produc8093790510458973071_a_c_d @ A @ B ) @ ( produc8093790510458973071_a_c_d @ A4 @ B3 ) ) @ ( lex_pr675406489056450471_a_c_d @ R @ S4 ) )
      = ( ( member4024245474877636784_a_c_d @ ( produc2275631196151162047_a_c_d @ A @ A4 ) @ R )
        | ( ( A = A4 )
          & ( member9059391734154409040_a_c_d @ ( produc4503948553971640415_a_c_d @ B @ B3 ) @ S4 ) ) ) ) ).

% in_lex_prod
thf(fact_751_in__lex__prod,axiom,
    ! [A: a,B: c > d,A4: a,B3: c > d,R: set_Product_prod_a_a,S4: set_Pr7973059830462150285_d_c_d] :
      ( ( member4024245474877636784_a_c_d @ ( produc2275631196151162047_a_c_d @ ( product_Pair_a_c_d @ A @ B ) @ ( product_Pair_a_c_d @ A4 @ B3 ) ) @ ( lex_prod_a_c_d @ R @ S4 ) )
      = ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A4 ) @ R )
        | ( ( A = A4 )
          & ( member1278719481969914326_d_c_d @ ( product_Pair_c_d_c_d @ B @ B3 ) @ S4 ) ) ) ) ).

% in_lex_prod
thf(fact_752_under__incr,axiom,
    ! [R: set_Product_prod_a_a,A: a,B: a] :
      ( ( trans_on_a @ top_top_set_a @ R )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
       => ( ord_less_eq_set_a @ ( order_under_a @ R @ A ) @ ( order_under_a @ R @ B ) ) ) ) ).

% under_incr
thf(fact_753_under__incr,axiom,
    ! [R: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
      ( ( trans_on_option_a @ top_top_set_option_a @ R )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R )
       => ( ord_le1955136853071979460tion_a @ ( order_under_option_a @ R @ A ) @ ( order_under_option_a @ R @ B ) ) ) ) ).

% under_incr
thf(fact_754_antisym__onD,axiom,
    ! [A6: set_a,R: set_Product_prod_a_a,X2: a,Y: a] :
      ( ( antisym_on_a @ A6 @ R )
     => ( ( member_a @ X2 @ A6 )
       => ( ( member_a @ Y @ A6 )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
           => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ R )
             => ( X2 = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_755_antisym__onD,axiom,
    ! [A6: set_Pr1275464188344874039_a_c_d,R: set_Pr3774076249145843911_a_c_d,X2: produc5278197477302038359_a_c_d,Y: produc5278197477302038359_a_c_d] :
      ( ( antisy2793470566104255546_a_c_d @ A6 @ R )
     => ( ( member1180172933830803072_a_c_d @ X2 @ A6 )
       => ( ( member1180172933830803072_a_c_d @ Y @ A6 )
         => ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ X2 @ Y ) @ R )
           => ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ Y @ X2 ) @ R )
             => ( X2 = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_756_antisym__onD,axiom,
    ! [A6: set_option_a,R: set_Pr7585778909603769095tion_a,X2: option_a,Y: option_a] :
      ( ( antisym_on_option_a @ A6 @ R )
     => ( ( member_option_a @ X2 @ A6 )
       => ( ( member_option_a @ Y @ A6 )
         => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y ) @ R )
           => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y @ X2 ) @ R )
             => ( X2 = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_757_antisym__onI,axiom,
    ! [A6: set_a,R: set_Product_prod_a_a] :
      ( ! [X4: a,Y4: a] :
          ( ( member_a @ X4 @ A6 )
         => ( ( member_a @ Y4 @ A6 )
           => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R )
             => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X4 ) @ R )
               => ( X4 = Y4 ) ) ) ) )
     => ( antisym_on_a @ A6 @ R ) ) ).

% antisym_onI
thf(fact_758_antisym__onI,axiom,
    ! [A6: set_Pr1275464188344874039_a_c_d,R: set_Pr3774076249145843911_a_c_d] :
      ( ! [X4: produc5278197477302038359_a_c_d,Y4: produc5278197477302038359_a_c_d] :
          ( ( member1180172933830803072_a_c_d @ X4 @ A6 )
         => ( ( member1180172933830803072_a_c_d @ Y4 @ A6 )
           => ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ X4 @ Y4 ) @ R )
             => ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ Y4 @ X4 ) @ R )
               => ( X4 = Y4 ) ) ) ) )
     => ( antisy2793470566104255546_a_c_d @ A6 @ R ) ) ).

% antisym_onI
thf(fact_759_antisym__onI,axiom,
    ! [A6: set_option_a,R: set_Pr7585778909603769095tion_a] :
      ( ! [X4: option_a,Y4: option_a] :
          ( ( member_option_a @ X4 @ A6 )
         => ( ( member_option_a @ Y4 @ A6 )
           => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X4 @ Y4 ) @ R )
             => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y4 @ X4 ) @ R )
               => ( X4 = Y4 ) ) ) ) )
     => ( antisym_on_option_a @ A6 @ R ) ) ).

% antisym_onI
thf(fact_760_antisymD,axiom,
    ! [R: set_Product_prod_a_a,X2: a,Y: a] :
      ( ( antisym_on_a @ top_top_set_a @ R )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
       => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ R )
         => ( X2 = Y ) ) ) ) ).

% antisymD
thf(fact_761_antisymD,axiom,
    ! [R: set_Pr7585778909603769095tion_a,X2: option_a,Y: option_a] :
      ( ( antisym_on_option_a @ top_top_set_option_a @ R )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y ) @ R )
       => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y @ X2 ) @ R )
         => ( X2 = Y ) ) ) ) ).

% antisymD
thf(fact_762_antisymI,axiom,
    ! [R: set_Product_prod_a_a] :
      ( ! [X4: a,Y4: a] :
          ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X4 ) @ R )
           => ( X4 = Y4 ) ) )
     => ( antisym_on_a @ top_top_set_a @ R ) ) ).

% antisymI
thf(fact_763_antisymI,axiom,
    ! [R: set_Pr7585778909603769095tion_a] :
      ( ! [X4: option_a,Y4: option_a] :
          ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X4 @ Y4 ) @ R )
         => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y4 @ X4 ) @ R )
           => ( X4 = Y4 ) ) )
     => ( antisym_on_option_a @ top_top_set_option_a @ R ) ) ).

% antisymI
thf(fact_764_same__fstI,axiom,
    ! [P2: product_prod_a_c_d > $o,X2: product_prod_a_c_d,Y7: option6413918287372586467_a_c_d,Y: option6413918287372586467_a_c_d,R3: product_prod_a_c_d > set_Pr3188740713198616583_a_c_d] :
      ( ( P2 @ X2 )
     => ( ( member9059391734154409040_a_c_d @ ( produc4503948553971640415_a_c_d @ Y7 @ Y ) @ ( R3 @ X2 ) )
       => ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ ( produc8093790510458973071_a_c_d @ X2 @ Y7 ) @ ( produc8093790510458973071_a_c_d @ X2 @ Y ) ) @ ( same_f7651772208441708388_a_c_d @ P2 @ R3 ) ) ) ) ).

% same_fstI
thf(fact_765_same__fstI,axiom,
    ! [P2: a > $o,X2: a,Y7: c > d,Y: c > d,R3: a > set_Pr7973059830462150285_d_c_d] :
      ( ( P2 @ X2 )
     => ( ( member1278719481969914326_d_c_d @ ( product_Pair_c_d_c_d @ Y7 @ Y ) @ ( R3 @ X2 ) )
       => ( member4024245474877636784_a_c_d @ ( produc2275631196151162047_a_c_d @ ( product_Pair_a_c_d @ X2 @ Y7 ) @ ( product_Pair_a_c_d @ X2 @ Y ) ) @ ( same_fst_a_c_d @ P2 @ R3 ) ) ) ) ).

% same_fstI
thf(fact_766_wo__rel_Osuc__greater,axiom,
    ! [R: set_Pr3774076249145843911_a_c_d,B5: set_Pr1275464188344874039_a_c_d,B: produc5278197477302038359_a_c_d] :
      ( ( bNF_We47639529709206339_a_c_d @ R )
     => ( ( ord_le6392319565089278935_a_c_d @ B5 @ ( field_5664454651862650532_a_c_d @ R ) )
       => ( ( ( order_689020630899515938_a_c_d @ R @ B5 )
           != bot_bo4027597737525728931_a_c_d )
         => ( ( member1180172933830803072_a_c_d @ B @ B5 )
           => ( ( ( bNF_We673253246704241848_a_c_d @ R @ B5 )
               != B )
              & ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ B @ ( bNF_We673253246704241848_a_c_d @ R @ B5 ) ) @ R ) ) ) ) ) ) ).

% wo_rel.suc_greater
thf(fact_767_wo__rel_Osuc__greater,axiom,
    ! [R: set_Product_prod_a_a,B5: set_a,B: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R )
     => ( ( ord_less_eq_set_a @ B5 @ ( field_a @ R ) )
       => ( ( ( order_AboveS_a @ R @ B5 )
           != bot_bot_set_a )
         => ( ( member_a @ B @ B5 )
           => ( ( ( bNF_We6154283375207884895_suc_a @ R @ B5 )
               != B )
              & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ ( bNF_We6154283375207884895_suc_a @ R @ B5 ) ) @ R ) ) ) ) ) ) ).

% wo_rel.suc_greater
thf(fact_768_wo__rel_Osuc__greater,axiom,
    ! [R: set_Pr7585778909603769095tion_a,B5: set_option_a,B: option_a] :
      ( ( bNF_We8432232079604507440tion_a @ R )
     => ( ( ord_le1955136853071979460tion_a @ B5 @ ( field_option_a @ R ) )
       => ( ( ( order_6500638856667293583tion_a @ R @ B5 )
           != bot_bot_set_option_a )
         => ( ( member_option_a @ B @ B5 )
           => ( ( ( bNF_We5356091070762920229tion_a @ R @ B5 )
               != B )
              & ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ ( bNF_We5356091070762920229tion_a @ R @ B5 ) ) @ R ) ) ) ) ) ) ).

% wo_rel.suc_greater
thf(fact_769_wo__rel_Osuc__least__AboveS,axiom,
    ! [R: set_Product_prod_a_a,A: a,B5: set_a] :
      ( ( bNF_We1162827675446709994_rel_a @ R )
     => ( ( member_a @ A @ ( order_AboveS_a @ R @ B5 ) )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( bNF_We6154283375207884895_suc_a @ R @ B5 ) @ A ) @ R ) ) ) ).

% wo_rel.suc_least_AboveS
thf(fact_770_wo__rel_Osuc__least__AboveS,axiom,
    ! [R: set_Pr3774076249145843911_a_c_d,A: produc5278197477302038359_a_c_d,B5: set_Pr1275464188344874039_a_c_d] :
      ( ( bNF_We47639529709206339_a_c_d @ R )
     => ( ( member1180172933830803072_a_c_d @ A @ ( order_689020630899515938_a_c_d @ R @ B5 ) )
       => ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ ( bNF_We673253246704241848_a_c_d @ R @ B5 ) @ A ) @ R ) ) ) ).

% wo_rel.suc_least_AboveS
thf(fact_771_wo__rel_Osuc__least__AboveS,axiom,
    ! [R: set_Pr7585778909603769095tion_a,A: option_a,B5: set_option_a] :
      ( ( bNF_We8432232079604507440tion_a @ R )
     => ( ( member_option_a @ A @ ( order_6500638856667293583tion_a @ R @ B5 ) )
       => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ ( bNF_We5356091070762920229tion_a @ R @ B5 ) @ A ) @ R ) ) ) ).

% wo_rel.suc_least_AboveS
thf(fact_772_wo__rel_Oequals__suc__AboveS,axiom,
    ! [R: set_Product_prod_a_a,B5: set_a,A: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R )
     => ( ( ord_less_eq_set_a @ B5 @ ( field_a @ R ) )
       => ( ( member_a @ A @ ( order_AboveS_a @ R @ B5 ) )
         => ( ! [A3: a] :
                ( ( member_a @ A3 @ ( order_AboveS_a @ R @ B5 ) )
               => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A3 ) @ R ) )
           => ( A
              = ( bNF_We6154283375207884895_suc_a @ R @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
thf(fact_773_wo__rel_Oequals__suc__AboveS,axiom,
    ! [R: set_Pr3774076249145843911_a_c_d,B5: set_Pr1275464188344874039_a_c_d,A: produc5278197477302038359_a_c_d] :
      ( ( bNF_We47639529709206339_a_c_d @ R )
     => ( ( ord_le6392319565089278935_a_c_d @ B5 @ ( field_5664454651862650532_a_c_d @ R ) )
       => ( ( member1180172933830803072_a_c_d @ A @ ( order_689020630899515938_a_c_d @ R @ B5 ) )
         => ( ! [A3: produc5278197477302038359_a_c_d] :
                ( ( member1180172933830803072_a_c_d @ A3 @ ( order_689020630899515938_a_c_d @ R @ B5 ) )
               => ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ A @ A3 ) @ R ) )
           => ( A
              = ( bNF_We673253246704241848_a_c_d @ R @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
thf(fact_774_wo__rel_Oequals__suc__AboveS,axiom,
    ! [R: set_Pr7585778909603769095tion_a,B5: set_option_a,A: option_a] :
      ( ( bNF_We8432232079604507440tion_a @ R )
     => ( ( ord_le1955136853071979460tion_a @ B5 @ ( field_option_a @ R ) )
       => ( ( member_option_a @ A @ ( order_6500638856667293583tion_a @ R @ B5 ) )
         => ( ! [A3: option_a] :
                ( ( member_option_a @ A3 @ ( order_6500638856667293583tion_a @ R @ B5 ) )
               => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ A3 ) @ R ) )
           => ( A
              = ( bNF_We5356091070762920229tion_a @ R @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
thf(fact_775_Not__Domain__rtrancl,axiom,
    ! [X2: a,R3: set_Product_prod_a_a,Y: a] :
      ( ~ ( member_a @ X2 @ ( domain_a_a @ R3 ) )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ ( transitive_rtrancl_a @ R3 ) )
        = ( X2 = Y ) ) ) ).

% Not_Domain_rtrancl
thf(fact_776_Not__Domain__rtrancl,axiom,
    ! [X2: produc5278197477302038359_a_c_d,R3: set_Pr3774076249145843911_a_c_d,Y: produc5278197477302038359_a_c_d] :
      ( ~ ( member1180172933830803072_a_c_d @ X2 @ ( domain6468958756806197708_a_c_d @ R3 ) )
     => ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ X2 @ Y ) @ ( transi5408070501777088858_a_c_d @ R3 ) )
        = ( X2 = Y ) ) ) ).

% Not_Domain_rtrancl
thf(fact_777_Not__Domain__rtrancl,axiom,
    ! [X2: option_a,R3: set_Pr7585778909603769095tion_a,Y: option_a] :
      ( ~ ( member_option_a @ X2 @ ( domain5649462347324568460tion_a @ R3 ) )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y ) @ ( transi330218190764880583tion_a @ R3 ) )
        = ( X2 = Y ) ) ) ).

% Not_Domain_rtrancl
thf(fact_778_converse__rtrancl__induct2,axiom,
    ! [Ax: product_prod_a_c_d,Ay: option6413918287372586467_a_c_d,Bx: product_prod_a_c_d,By: option6413918287372586467_a_c_d,R: set_Pr3774076249145843911_a_c_d,P2: product_prod_a_c_d > option6413918287372586467_a_c_d > $o] :
      ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ ( produc8093790510458973071_a_c_d @ Ax @ Ay ) @ ( produc8093790510458973071_a_c_d @ Bx @ By ) ) @ ( transi5408070501777088858_a_c_d @ R ) )
     => ( ( P2 @ Bx @ By )
       => ( ! [A7: product_prod_a_c_d,B6: option6413918287372586467_a_c_d,Aa: product_prod_a_c_d,Ba: option6413918287372586467_a_c_d] :
              ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ ( produc8093790510458973071_a_c_d @ A7 @ B6 ) @ ( produc8093790510458973071_a_c_d @ Aa @ Ba ) ) @ R )
             => ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ ( produc8093790510458973071_a_c_d @ Aa @ Ba ) @ ( produc8093790510458973071_a_c_d @ Bx @ By ) ) @ ( transi5408070501777088858_a_c_d @ R ) )
               => ( ( P2 @ Aa @ Ba )
                 => ( P2 @ A7 @ B6 ) ) ) )
         => ( P2 @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_779_converse__rtrancl__induct2,axiom,
    ! [Ax: a,Ay: c > d,Bx: a,By: c > d,R: set_Pr5254629501202997351_a_c_d,P2: a > ( c > d ) > $o] :
      ( ( member4024245474877636784_a_c_d @ ( produc2275631196151162047_a_c_d @ ( product_Pair_a_c_d @ Ax @ Ay ) @ ( product_Pair_a_c_d @ Bx @ By ) ) @ ( transi1691159240759869776_a_c_d @ R ) )
     => ( ( P2 @ Bx @ By )
       => ( ! [A7: a,B6: c > d,Aa: a,Ba: c > d] :
              ( ( member4024245474877636784_a_c_d @ ( produc2275631196151162047_a_c_d @ ( product_Pair_a_c_d @ A7 @ B6 ) @ ( product_Pair_a_c_d @ Aa @ Ba ) ) @ R )
             => ( ( member4024245474877636784_a_c_d @ ( produc2275631196151162047_a_c_d @ ( product_Pair_a_c_d @ Aa @ Ba ) @ ( product_Pair_a_c_d @ Bx @ By ) ) @ ( transi1691159240759869776_a_c_d @ R ) )
               => ( ( P2 @ Aa @ Ba )
                 => ( P2 @ A7 @ B6 ) ) ) )
         => ( P2 @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_780_converse__rtranclE2,axiom,
    ! [Xa3: product_prod_a_c_d,Xb: option6413918287372586467_a_c_d,Za: product_prod_a_c_d,Zb: option6413918287372586467_a_c_d,R: set_Pr3774076249145843911_a_c_d] :
      ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ ( produc8093790510458973071_a_c_d @ Xa3 @ Xb ) @ ( produc8093790510458973071_a_c_d @ Za @ Zb ) ) @ ( transi5408070501777088858_a_c_d @ R ) )
     => ( ( ( produc8093790510458973071_a_c_d @ Xa3 @ Xb )
         != ( produc8093790510458973071_a_c_d @ Za @ Zb ) )
       => ~ ! [A7: product_prod_a_c_d,B6: option6413918287372586467_a_c_d] :
              ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ ( produc8093790510458973071_a_c_d @ Xa3 @ Xb ) @ ( produc8093790510458973071_a_c_d @ A7 @ B6 ) ) @ R )
             => ~ ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ ( produc8093790510458973071_a_c_d @ A7 @ B6 ) @ ( produc8093790510458973071_a_c_d @ Za @ Zb ) ) @ ( transi5408070501777088858_a_c_d @ R ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_781_converse__rtranclE2,axiom,
    ! [Xa3: a,Xb: c > d,Za: a,Zb: c > d,R: set_Pr5254629501202997351_a_c_d] :
      ( ( member4024245474877636784_a_c_d @ ( produc2275631196151162047_a_c_d @ ( product_Pair_a_c_d @ Xa3 @ Xb ) @ ( product_Pair_a_c_d @ Za @ Zb ) ) @ ( transi1691159240759869776_a_c_d @ R ) )
     => ( ( ( product_Pair_a_c_d @ Xa3 @ Xb )
         != ( product_Pair_a_c_d @ Za @ Zb ) )
       => ~ ! [A7: a,B6: c > d] :
              ( ( member4024245474877636784_a_c_d @ ( produc2275631196151162047_a_c_d @ ( product_Pair_a_c_d @ Xa3 @ Xb ) @ ( product_Pair_a_c_d @ A7 @ B6 ) ) @ R )
             => ~ ( member4024245474877636784_a_c_d @ ( produc2275631196151162047_a_c_d @ ( product_Pair_a_c_d @ A7 @ B6 ) @ ( product_Pair_a_c_d @ Za @ Zb ) ) @ ( transi1691159240759869776_a_c_d @ R ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_782_rtrancl__induct2,axiom,
    ! [Ax: product_prod_a_c_d,Ay: option6413918287372586467_a_c_d,Bx: product_prod_a_c_d,By: option6413918287372586467_a_c_d,R: set_Pr3774076249145843911_a_c_d,P2: product_prod_a_c_d > option6413918287372586467_a_c_d > $o] :
      ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ ( produc8093790510458973071_a_c_d @ Ax @ Ay ) @ ( produc8093790510458973071_a_c_d @ Bx @ By ) ) @ ( transi5408070501777088858_a_c_d @ R ) )
     => ( ( P2 @ Ax @ Ay )
       => ( ! [A7: product_prod_a_c_d,B6: option6413918287372586467_a_c_d,Aa: product_prod_a_c_d,Ba: option6413918287372586467_a_c_d] :
              ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ ( produc8093790510458973071_a_c_d @ Ax @ Ay ) @ ( produc8093790510458973071_a_c_d @ A7 @ B6 ) ) @ ( transi5408070501777088858_a_c_d @ R ) )
             => ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ ( produc8093790510458973071_a_c_d @ A7 @ B6 ) @ ( produc8093790510458973071_a_c_d @ Aa @ Ba ) ) @ R )
               => ( ( P2 @ A7 @ B6 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_783_rtrancl__induct2,axiom,
    ! [Ax: a,Ay: c > d,Bx: a,By: c > d,R: set_Pr5254629501202997351_a_c_d,P2: a > ( c > d ) > $o] :
      ( ( member4024245474877636784_a_c_d @ ( produc2275631196151162047_a_c_d @ ( product_Pair_a_c_d @ Ax @ Ay ) @ ( product_Pair_a_c_d @ Bx @ By ) ) @ ( transi1691159240759869776_a_c_d @ R ) )
     => ( ( P2 @ Ax @ Ay )
       => ( ! [A7: a,B6: c > d,Aa: a,Ba: c > d] :
              ( ( member4024245474877636784_a_c_d @ ( produc2275631196151162047_a_c_d @ ( product_Pair_a_c_d @ Ax @ Ay ) @ ( product_Pair_a_c_d @ A7 @ B6 ) ) @ ( transi1691159240759869776_a_c_d @ R ) )
             => ( ( member4024245474877636784_a_c_d @ ( produc2275631196151162047_a_c_d @ ( product_Pair_a_c_d @ A7 @ B6 ) @ ( product_Pair_a_c_d @ Aa @ Ba ) ) @ R )
               => ( ( P2 @ A7 @ B6 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_784_wo__rel_Osuc__ofilter__in,axiom,
    ! [R: set_Pr3774076249145843911_a_c_d,A6: set_Pr1275464188344874039_a_c_d,B: produc5278197477302038359_a_c_d] :
      ( ( bNF_We47639529709206339_a_c_d @ R )
     => ( ( order_6254496387943360493_a_c_d @ R @ A6 )
       => ( ( ( order_689020630899515938_a_c_d @ R @ A6 )
           != bot_bo4027597737525728931_a_c_d )
         => ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ B @ ( bNF_We673253246704241848_a_c_d @ R @ A6 ) ) @ R )
           => ( ( B
               != ( bNF_We673253246704241848_a_c_d @ R @ A6 ) )
             => ( member1180172933830803072_a_c_d @ B @ A6 ) ) ) ) ) ) ).

% wo_rel.suc_ofilter_in
thf(fact_785_wo__rel_Osuc__ofilter__in,axiom,
    ! [R: set_Product_prod_a_a,A6: set_a,B: a] :
      ( ( bNF_We1162827675446709994_rel_a @ R )
     => ( ( order_ofilter_a @ R @ A6 )
       => ( ( ( order_AboveS_a @ R @ A6 )
           != bot_bot_set_a )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ ( bNF_We6154283375207884895_suc_a @ R @ A6 ) ) @ R )
           => ( ( B
               != ( bNF_We6154283375207884895_suc_a @ R @ A6 ) )
             => ( member_a @ B @ A6 ) ) ) ) ) ) ).

% wo_rel.suc_ofilter_in
thf(fact_786_wo__rel_Osuc__ofilter__in,axiom,
    ! [R: set_Pr7585778909603769095tion_a,A6: set_option_a,B: option_a] :
      ( ( bNF_We8432232079604507440tion_a @ R )
     => ( ( order_6420974439381506266tion_a @ R @ A6 )
       => ( ( ( order_6500638856667293583tion_a @ R @ A6 )
           != bot_bot_set_option_a )
         => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ ( bNF_We5356091070762920229tion_a @ R @ A6 ) ) @ R )
           => ( ( B
               != ( bNF_We5356091070762920229tion_a @ R @ A6 ) )
             => ( member_option_a @ B @ A6 ) ) ) ) ) ) ).

% wo_rel.suc_ofilter_in
thf(fact_787_vimage__subsetI,axiom,
    ! [F2: a > option_a,B5: set_option_a,A6: set_a] :
      ( ( inj_on_a_option_a @ F2 @ top_top_set_a )
     => ( ( ord_le1955136853071979460tion_a @ B5 @ ( image_a_option_a2 @ F2 @ A6 ) )
       => ( ord_less_eq_set_a @ ( vimage_a_option_a @ F2 @ B5 ) @ A6 ) ) ) ).

% vimage_subsetI
thf(fact_788_vimage__subsetI,axiom,
    ! [F2: option_a > a,B5: set_a,A6: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ top_top_set_option_a )
     => ( ( ord_less_eq_set_a @ B5 @ ( image_option_a_a2 @ F2 @ A6 ) )
       => ( ord_le1955136853071979460tion_a @ ( vimage_option_a_a @ F2 @ B5 ) @ A6 ) ) ) ).

% vimage_subsetI
thf(fact_789_image2__eqI,axiom,
    ! [B: product_prod_a_c_d,F2: a > product_prod_a_c_d,X2: a,C: option6413918287372586467_a_c_d,G: a > option6413918287372586467_a_c_d,A6: set_a] :
      ( ( B
        = ( F2 @ X2 ) )
     => ( ( C
          = ( G @ X2 ) )
       => ( ( member_a @ X2 @ A6 )
         => ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ B @ C ) @ ( bNF_Gr4981583572320052764_a_c_d @ A6 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_790_image2__eqI,axiom,
    ! [B: product_prod_a_c_d,F2: produc5278197477302038359_a_c_d > product_prod_a_c_d,X2: produc5278197477302038359_a_c_d,C: option6413918287372586467_a_c_d,G: produc5278197477302038359_a_c_d > option6413918287372586467_a_c_d,A6: set_Pr1275464188344874039_a_c_d] :
      ( ( B
        = ( F2 @ X2 ) )
     => ( ( C
          = ( G @ X2 ) )
       => ( ( member1180172933830803072_a_c_d @ X2 @ A6 )
         => ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ B @ C ) @ ( bNF_Gr589831376474905333_a_c_d @ A6 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_791_image2__eqI,axiom,
    ! [B: product_prod_a_c_d,F2: option_a > product_prod_a_c_d,X2: option_a,C: option6413918287372586467_a_c_d,G: option_a > option6413918287372586467_a_c_d,A6: set_option_a] :
      ( ( B
        = ( F2 @ X2 ) )
     => ( ( C
          = ( G @ X2 ) )
       => ( ( member_option_a @ X2 @ A6 )
         => ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ B @ C ) @ ( bNF_Gr2659421419784889954_a_c_d @ A6 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_792_image2__eqI,axiom,
    ! [B: a,F2: a > a,X2: a,C: c > d,G: a > c > d,A6: set_a] :
      ( ( B
        = ( F2 @ X2 ) )
     => ( ( C
          = ( G @ X2 ) )
       => ( ( member_a @ X2 @ A6 )
         => ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ B @ C ) @ ( bNF_Gr7437496851395112512_a_c_d @ A6 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_793_image2__eqI,axiom,
    ! [B: a,F2: produc5278197477302038359_a_c_d > a,X2: produc5278197477302038359_a_c_d,C: c > d,G: produc5278197477302038359_a_c_d > c > d,A6: set_Pr1275464188344874039_a_c_d] :
      ( ( B
        = ( F2 @ X2 ) )
     => ( ( C
          = ( G @ X2 ) )
       => ( ( member1180172933830803072_a_c_d @ X2 @ A6 )
         => ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ B @ C ) @ ( bNF_Gr4265814288748638247_a_c_d @ A6 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_794_image2__eqI,axiom,
    ! [B: a,F2: option_a > a,X2: option_a,C: c > d,G: option_a > c > d,A6: set_option_a] :
      ( ( B
        = ( F2 @ X2 ) )
     => ( ( C
          = ( G @ X2 ) )
       => ( ( member_option_a @ X2 @ A6 )
         => ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ B @ C ) @ ( bNF_Gr1391775451562859130_a_c_d @ A6 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_795_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_796_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_797_map__option__eq__Some,axiom,
    ! [F2: product_prod_a_c_d > a,Xo: option6413918287372586467_a_c_d,Y: a] :
      ( ( ( map_op5913999614919751774_c_d_a @ F2 @ Xo )
        = ( some_a @ Y ) )
      = ( ? [Z: product_prod_a_c_d] :
            ( ( Xo
              = ( some_P1084500821511757806_a_c_d @ Z ) )
            & ( ( F2 @ Z )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_798_map__option__eq__Some,axiom,
    ! [F2: a > product_prod_a_c_d,Xo: option_a,Y: product_prod_a_c_d] :
      ( ( ( map_op7600559726947941830_a_c_d @ F2 @ Xo )
        = ( some_P1084500821511757806_a_c_d @ Y ) )
      = ( ? [Z: a] :
            ( ( Xo
              = ( some_a @ Z ) )
            & ( ( F2 @ Z )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_799_map__option__eq__Some,axiom,
    ! [F2: product_prod_a_c_d > product_prod_a_c_d,Xo: option6413918287372586467_a_c_d,Y: product_prod_a_c_d] :
      ( ( ( map_op989909284770252723_a_c_d @ F2 @ Xo )
        = ( some_P1084500821511757806_a_c_d @ Y ) )
      = ( ? [Z: product_prod_a_c_d] :
            ( ( Xo
              = ( some_P1084500821511757806_a_c_d @ Z ) )
            & ( ( F2 @ Z )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_800_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_801_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_802_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_803_None__eq__map__option__iff,axiom,
    ! [F2: a > a,X2: option_a] :
      ( ( none_a
        = ( map_option_a_a @ F2 @ X2 ) )
      = ( X2 = none_a ) ) ).

% None_eq_map_option_iff
thf(fact_804_is__none__map__option,axiom,
    ! [F2: a > a,X2: option_a] :
      ( ( is_none_a @ ( map_option_a_a @ F2 @ X2 ) )
      = ( is_none_a @ X2 ) ) ).

% is_none_map_option
thf(fact_805_option_Osimps_I8_J,axiom,
    ! [F2: a > a] :
      ( ( map_option_a_a @ F2 @ none_a )
      = none_a ) ).

% option.simps(8)
thf(fact_806_option_Osimps_I9_J,axiom,
    ! [F2: a > product_prod_a_c_d,X22: a] :
      ( ( map_op7600559726947941830_a_c_d @ F2 @ ( some_a @ X22 ) )
      = ( some_P1084500821511757806_a_c_d @ ( F2 @ X22 ) ) ) ).

% option.simps(9)
thf(fact_807_option_Osimps_I9_J,axiom,
    ! [F2: product_prod_a_c_d > a,X22: product_prod_a_c_d] :
      ( ( map_op5913999614919751774_c_d_a @ F2 @ ( some_P1084500821511757806_a_c_d @ X22 ) )
      = ( some_a @ ( F2 @ X22 ) ) ) ).

% option.simps(9)
thf(fact_808_option_Osimps_I9_J,axiom,
    ! [F2: product_prod_a_c_d > product_prod_a_c_d,X22: product_prod_a_c_d] :
      ( ( map_op989909284770252723_a_c_d @ F2 @ ( some_P1084500821511757806_a_c_d @ X22 ) )
      = ( some_P1084500821511757806_a_c_d @ ( F2 @ X22 ) ) ) ).

% option.simps(9)
thf(fact_809_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_810_map__option__cong,axiom,
    ! [X2: option_a,Y: option_a,F2: a > a,G: a > a] :
      ( ( X2 = Y )
     => ( ! [A7: a] :
            ( ( Y
              = ( some_a @ A7 ) )
           => ( ( F2 @ A7 )
              = ( G @ A7 ) ) )
       => ( ( map_option_a_a @ F2 @ X2 )
          = ( map_option_a_a @ G @ Y ) ) ) ) ).

% map_option_cong
thf(fact_811_map__option__idI,axiom,
    ! [X2: option6736415434736681245_a_c_d,F2: produc5278197477302038359_a_c_d > produc5278197477302038359_a_c_d] :
      ( ! [Y4: produc5278197477302038359_a_c_d] :
          ( ( member1180172933830803072_a_c_d @ Y4 @ ( set_op1238738513205398757_a_c_d @ X2 ) )
         => ( ( F2 @ Y4 )
            = Y4 ) )
     => ( ( map_op4319857742139324491_a_c_d @ F2 @ X2 )
        = X2 ) ) ).

% map_option_idI
thf(fact_812_map__option__idI,axiom,
    ! [X2: option_option_a,F2: option_a > option_a] :
      ( ! [Y4: option_a] :
          ( ( member_option_a @ Y4 @ ( set_option_option_a2 @ X2 ) )
         => ( ( F2 @ Y4 )
            = Y4 ) )
     => ( ( map_op788413144570152203tion_a @ F2 @ X2 )
        = X2 ) ) ).

% map_option_idI
thf(fact_813_map__option__idI,axiom,
    ! [X2: option_a,F2: a > a] :
      ( ! [Y4: a] :
          ( ( member_a @ Y4 @ ( set_option_a3 @ X2 ) )
         => ( ( F2 @ Y4 )
            = Y4 ) )
     => ( ( map_option_a_a @ F2 @ X2 )
        = X2 ) ) ).

% map_option_idI
thf(fact_814_option_Omap__ident__strong,axiom,
    ! [T: option6736415434736681245_a_c_d,F2: produc5278197477302038359_a_c_d > produc5278197477302038359_a_c_d] :
      ( ! [Z3: produc5278197477302038359_a_c_d] :
          ( ( member1180172933830803072_a_c_d @ Z3 @ ( set_op1238738513205398757_a_c_d @ T ) )
         => ( ( F2 @ Z3 )
            = Z3 ) )
     => ( ( map_op4319857742139324491_a_c_d @ F2 @ T )
        = T ) ) ).

% option.map_ident_strong
thf(fact_815_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_816_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_817_option_Oinj__map__strong,axiom,
    ! [X2: option_a,Xa3: option_a,F2: a > a,Fa: a > a] :
      ( ! [Z3: a,Za2: a] :
          ( ( member_a @ Z3 @ ( set_option_a3 @ X2 ) )
         => ( ( member_a @ Za2 @ ( set_option_a3 @ Xa3 ) )
           => ( ( ( F2 @ Z3 )
                = ( Fa @ Za2 ) )
             => ( Z3 = Za2 ) ) ) )
     => ( ( ( map_option_a_a @ F2 @ X2 )
          = ( map_option_a_a @ Fa @ Xa3 ) )
       => ( X2 = Xa3 ) ) ) ).

% option.inj_map_strong
thf(fact_818_option_Omap__cong0,axiom,
    ! [X2: option_a,F2: a > a,G: a > a] :
      ( ! [Z3: a] :
          ( ( member_a @ Z3 @ ( set_option_a3 @ X2 ) )
         => ( ( F2 @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( map_option_a_a @ F2 @ X2 )
        = ( map_option_a_a @ G @ X2 ) ) ) ).

% option.map_cong0
thf(fact_819_option_Omap__cong,axiom,
    ! [X2: option_a,Ya: option_a,F2: a > a,G: a > a] :
      ( ( X2 = Ya )
     => ( ! [Z3: a] :
            ( ( member_a @ Z3 @ ( set_option_a3 @ Ya ) )
           => ( ( F2 @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( map_option_a_a @ F2 @ X2 )
          = ( map_option_a_a @ G @ Ya ) ) ) ) ).

% option.map_cong
thf(fact_820_surj__image__vimage__eq,axiom,
    ! [F2: a > a,A6: set_a] :
      ( ( ( image_a_a2 @ F2 @ top_top_set_a )
        = top_top_set_a )
     => ( ( image_a_a2 @ F2 @ ( vimage_a_a @ F2 @ A6 ) )
        = A6 ) ) ).

% surj_image_vimage_eq
thf(fact_821_surj__image__vimage__eq,axiom,
    ! [F2: a > option_a,A6: 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 @ A6 ) )
        = A6 ) ) ).

% surj_image_vimage_eq
thf(fact_822_surj__image__vimage__eq,axiom,
    ! [F2: option_a > a,A6: 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 @ A6 ) )
        = A6 ) ) ).

% surj_image_vimage_eq
thf(fact_823_surj__image__vimage__eq,axiom,
    ! [F2: option_a > option_a,A6: set_option_a] :
      ( ( ( image_7439109396645324421tion_a @ F2 @ top_top_set_option_a )
        = top_top_set_option_a )
     => ( ( image_7439109396645324421tion_a @ F2 @ ( vimage1562710927270423099tion_a @ F2 @ A6 ) )
        = A6 ) ) ).

% surj_image_vimage_eq
thf(fact_824_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_825_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_826_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_827_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_828_the__map__option,axiom,
    ! [X2: option_a,F2: a > a] :
      ( ~ ( is_none_a @ X2 )
     => ( ( the_a @ ( map_option_a_a @ F2 @ X2 ) )
        = ( F2 @ ( the_a @ X2 ) ) ) ) ).

% the_map_option
thf(fact_829_surj__vimage__empty,axiom,
    ! [F2: a > a,A6: set_a] :
      ( ( ( image_a_a2 @ F2 @ top_top_set_a )
        = top_top_set_a )
     => ( ( ( vimage_a_a @ F2 @ A6 )
          = bot_bot_set_a )
        = ( A6 = bot_bot_set_a ) ) ) ).

% surj_vimage_empty
thf(fact_830_surj__vimage__empty,axiom,
    ! [F2: a > option_a,A6: set_option_a] :
      ( ( ( image_a_option_a2 @ F2 @ top_top_set_a )
        = top_top_set_option_a )
     => ( ( ( vimage_a_option_a @ F2 @ A6 )
          = bot_bot_set_a )
        = ( A6 = bot_bot_set_option_a ) ) ) ).

% surj_vimage_empty
thf(fact_831_surj__vimage__empty,axiom,
    ! [F2: option_a > a,A6: set_a] :
      ( ( ( image_option_a_a2 @ F2 @ top_top_set_option_a )
        = top_top_set_a )
     => ( ( ( vimage_option_a_a @ F2 @ A6 )
          = bot_bot_set_option_a )
        = ( A6 = bot_bot_set_a ) ) ) ).

% surj_vimage_empty
thf(fact_832_surj__vimage__empty,axiom,
    ! [F2: option_a > option_a,A6: set_option_a] :
      ( ( ( image_7439109396645324421tion_a @ F2 @ top_top_set_option_a )
        = top_top_set_option_a )
     => ( ( ( vimage1562710927270423099tion_a @ F2 @ A6 )
          = bot_bot_set_option_a )
        = ( A6 = bot_bot_set_option_a ) ) ) ).

% surj_vimage_empty
thf(fact_833_vimage__subsetD,axiom,
    ! [F2: a > a,B5: set_a,A6: set_a] :
      ( ( ( image_a_a2 @ F2 @ top_top_set_a )
        = top_top_set_a )
     => ( ( ord_less_eq_set_a @ ( vimage_a_a @ F2 @ B5 ) @ A6 )
       => ( ord_less_eq_set_a @ B5 @ ( image_a_a2 @ F2 @ A6 ) ) ) ) ).

% vimage_subsetD
thf(fact_834_vimage__subsetD,axiom,
    ! [F2: a > option_a,B5: set_option_a,A6: 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 @ B5 ) @ A6 )
       => ( ord_le1955136853071979460tion_a @ B5 @ ( image_a_option_a2 @ F2 @ A6 ) ) ) ) ).

% vimage_subsetD
thf(fact_835_vimage__subsetD,axiom,
    ! [F2: option_a > a,B5: set_a,A6: 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 @ B5 ) @ A6 )
       => ( ord_less_eq_set_a @ B5 @ ( image_option_a_a2 @ F2 @ A6 ) ) ) ) ).

% vimage_subsetD
thf(fact_836_vimage__subsetD,axiom,
    ! [F2: option_a > option_a,B5: set_option_a,A6: set_option_a] :
      ( ( ( image_7439109396645324421tion_a @ F2 @ top_top_set_option_a )
        = top_top_set_option_a )
     => ( ( ord_le1955136853071979460tion_a @ ( vimage1562710927270423099tion_a @ F2 @ B5 ) @ A6 )
       => ( ord_le1955136853071979460tion_a @ B5 @ ( image_7439109396645324421tion_a @ F2 @ A6 ) ) ) ) ).

% vimage_subsetD
thf(fact_837_inj__vimage__image__eq,axiom,
    ! [F2: a > option_a,A6: set_a] :
      ( ( inj_on_a_option_a @ F2 @ top_top_set_a )
     => ( ( vimage_a_option_a @ F2 @ ( image_a_option_a2 @ F2 @ A6 ) )
        = A6 ) ) ).

% inj_vimage_image_eq
thf(fact_838_inj__vimage__image__eq,axiom,
    ! [F2: option_a > a,A6: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ top_top_set_option_a )
     => ( ( vimage_option_a_a @ F2 @ ( image_option_a_a2 @ F2 @ A6 ) )
        = A6 ) ) ).

% inj_vimage_image_eq
thf(fact_839_vimage__subset__eq,axiom,
    ! [F2: a > a,B5: set_a,A6: set_a] :
      ( ( bij_betw_a_a @ F2 @ top_top_set_a @ top_top_set_a )
     => ( ( ord_less_eq_set_a @ ( vimage_a_a @ F2 @ B5 ) @ A6 )
        = ( ord_less_eq_set_a @ B5 @ ( image_a_a2 @ F2 @ A6 ) ) ) ) ).

% vimage_subset_eq
thf(fact_840_vimage__subset__eq,axiom,
    ! [F2: a > option_a,B5: set_option_a,A6: 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 @ B5 ) @ A6 )
        = ( ord_le1955136853071979460tion_a @ B5 @ ( image_a_option_a2 @ F2 @ A6 ) ) ) ) ).

% vimage_subset_eq
thf(fact_841_vimage__subset__eq,axiom,
    ! [F2: option_a > a,B5: set_a,A6: 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 @ B5 ) @ A6 )
        = ( ord_less_eq_set_a @ B5 @ ( image_option_a_a2 @ F2 @ A6 ) ) ) ) ).

% vimage_subset_eq
thf(fact_842_vimage__subset__eq,axiom,
    ! [F2: option_a > option_a,B5: set_option_a,A6: set_option_a] :
      ( ( bij_be5431266891817924854tion_a @ F2 @ top_top_set_option_a @ top_top_set_option_a )
     => ( ( ord_le1955136853071979460tion_a @ ( vimage1562710927270423099tion_a @ F2 @ B5 ) @ A6 )
        = ( ord_le1955136853071979460tion_a @ B5 @ ( image_7439109396645324421tion_a @ F2 @ A6 ) ) ) ) ).

% vimage_subset_eq
thf(fact_843_asymI,axiom,
    ! [R: set_Product_prod_a_a] :
      ( ! [X4: a,Y4: a] :
          ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R )
         => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X4 ) @ R ) )
     => ( asym_on_a @ top_top_set_a @ R ) ) ).

% asymI
thf(fact_844_asymI,axiom,
    ! [R: set_Pr7585778909603769095tion_a] :
      ( ! [X4: option_a,Y4: option_a] :
          ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X4 @ Y4 ) @ R )
         => ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y4 @ X4 ) @ R ) )
     => ( asym_on_option_a @ top_top_set_option_a @ R ) ) ).

% asymI
thf(fact_845_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_846_asym__onI,axiom,
    ! [A6: set_a,R: set_Product_prod_a_a] :
      ( ! [X4: a,Y4: a] :
          ( ( member_a @ X4 @ A6 )
         => ( ( member_a @ Y4 @ A6 )
           => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R )
             => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X4 ) @ R ) ) ) )
     => ( asym_on_a @ A6 @ R ) ) ).

% asym_onI
thf(fact_847_asym__onI,axiom,
    ! [A6: set_Pr1275464188344874039_a_c_d,R: set_Pr3774076249145843911_a_c_d] :
      ( ! [X4: produc5278197477302038359_a_c_d,Y4: produc5278197477302038359_a_c_d] :
          ( ( member1180172933830803072_a_c_d @ X4 @ A6 )
         => ( ( member1180172933830803072_a_c_d @ Y4 @ A6 )
           => ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ X4 @ Y4 ) @ R )
             => ~ ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ Y4 @ X4 ) @ R ) ) ) )
     => ( asym_o6600848050852821641_a_c_d @ A6 @ R ) ) ).

% asym_onI
thf(fact_848_asym__onI,axiom,
    ! [A6: set_option_a,R: set_Pr7585778909603769095tion_a] :
      ( ! [X4: option_a,Y4: option_a] :
          ( ( member_option_a @ X4 @ A6 )
         => ( ( member_option_a @ Y4 @ A6 )
           => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X4 @ Y4 ) @ R )
             => ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y4 @ X4 ) @ R ) ) ) )
     => ( asym_on_option_a @ A6 @ R ) ) ).

% asym_onI
thf(fact_849_case__map__option,axiom,
    ! [G: a,H: option_a > a,F2: a > option_a,X2: option_a] :
      ( ( case_o926465512965637841tion_a @ G @ H @ ( map_op2340691886215429841tion_a @ F2 @ X2 ) )
      = ( case_option_a_a @ G @ ( comp_option_a_a_a @ H @ F2 ) @ X2 ) ) ).

% case_map_option
thf(fact_850_case__map__option,axiom,
    ! [G: $o,H: a > $o,F2: a > a,X2: option_a] :
      ( ( case_option_o_a @ G @ H @ ( map_option_a_a @ F2 @ X2 ) )
      = ( case_option_o_a @ G @ ( comp_a_o_a @ H @ F2 ) @ X2 ) ) ).

% case_map_option
thf(fact_851_inj__on__imageI2,axiom,
    ! [F3: option_a > a,F2: a > option_a,A6: set_a] :
      ( ( inj_on_a_a @ ( comp_option_a_a_a @ F3 @ F2 ) @ A6 )
     => ( inj_on_a_option_a @ F2 @ A6 ) ) ).

% inj_on_imageI2
thf(fact_852_inj__on__imageI2,axiom,
    ! [F3: option_a > option_a,F2: a > option_a,A6: set_a] :
      ( ( inj_on_a_option_a @ ( comp_o6087033147929006299on_a_a @ F3 @ F2 ) @ A6 )
     => ( inj_on_a_option_a @ F2 @ A6 ) ) ).

% inj_on_imageI2
thf(fact_853_bij__betw__imp__inj__on,axiom,
    ! [F2: a > option_a,A6: set_a,B5: set_option_a] :
      ( ( bij_betw_a_option_a @ F2 @ A6 @ B5 )
     => ( inj_on_a_option_a @ F2 @ A6 ) ) ).

% bij_betw_imp_inj_on
thf(fact_854_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_855_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_856_vimage__comp,axiom,
    ! [F2: a > option_a,G: option_a > a,X2: set_a] :
      ( ( vimage_a_option_a @ F2 @ ( vimage_option_a_a @ G @ X2 ) )
      = ( vimage_a_a @ ( comp_option_a_a_a @ G @ F2 ) @ X2 ) ) ).

% vimage_comp
thf(fact_857_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_858_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_859_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_860_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_861_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_862_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_863_bij__betw__empty1,axiom,
    ! [F2: a > a,A6: set_a] :
      ( ( bij_betw_a_a @ F2 @ bot_bot_set_a @ A6 )
     => ( A6 = bot_bot_set_a ) ) ).

% bij_betw_empty1
thf(fact_864_bij__betw__empty1,axiom,
    ! [F2: a > option_a,A6: set_option_a] :
      ( ( bij_betw_a_option_a @ F2 @ bot_bot_set_a @ A6 )
     => ( A6 = bot_bot_set_option_a ) ) ).

% bij_betw_empty1
thf(fact_865_bij__betw__empty1,axiom,
    ! [F2: option_a > a,A6: set_a] :
      ( ( bij_betw_option_a_a @ F2 @ bot_bot_set_option_a @ A6 )
     => ( A6 = bot_bot_set_a ) ) ).

% bij_betw_empty1
thf(fact_866_bij__betw__empty1,axiom,
    ! [F2: option_a > option_a,A6: set_option_a] :
      ( ( bij_be5431266891817924854tion_a @ F2 @ bot_bot_set_option_a @ A6 )
     => ( A6 = bot_bot_set_option_a ) ) ).

% bij_betw_empty1
thf(fact_867_bij__betw__empty2,axiom,
    ! [F2: a > a,A6: set_a] :
      ( ( bij_betw_a_a @ F2 @ A6 @ bot_bot_set_a )
     => ( A6 = bot_bot_set_a ) ) ).

% bij_betw_empty2
thf(fact_868_bij__betw__empty2,axiom,
    ! [F2: option_a > a,A6: set_option_a] :
      ( ( bij_betw_option_a_a @ F2 @ A6 @ bot_bot_set_a )
     => ( A6 = bot_bot_set_option_a ) ) ).

% bij_betw_empty2
thf(fact_869_bij__betw__empty2,axiom,
    ! [F2: a > option_a,A6: set_a] :
      ( ( bij_betw_a_option_a @ F2 @ A6 @ bot_bot_set_option_a )
     => ( A6 = bot_bot_set_a ) ) ).

% bij_betw_empty2
thf(fact_870_bij__betw__empty2,axiom,
    ! [F2: option_a > option_a,A6: set_option_a] :
      ( ( bij_be5431266891817924854tion_a @ F2 @ A6 @ bot_bot_set_option_a )
     => ( A6 = bot_bot_set_option_a ) ) ).

% bij_betw_empty2
thf(fact_871_fun__upd__comp,axiom,
    ! [F2: option_a > a,G: a > option_a,X2: a,Y: option_a] :
      ( ( comp_option_a_a_a @ F2 @ ( fun_upd_a_option_a @ G @ X2 @ Y ) )
      = ( fun_upd_a_a @ ( comp_option_a_a_a @ F2 @ G ) @ X2 @ ( F2 @ Y ) ) ) ).

% fun_upd_comp
thf(fact_872_fun__upd__comp,axiom,
    ! [F2: d > d,G: c > d,X2: c,Y: d] :
      ( ( comp_d_d_c @ F2 @ ( fun_upd_c_d @ G @ X2 @ Y ) )
      = ( fun_upd_c_d @ ( comp_d_d_c @ F2 @ G ) @ X2 @ ( F2 @ Y ) ) ) ).

% fun_upd_comp
thf(fact_873_bij__betw__iff__bijections,axiom,
    ( bij_betw_a_a
    = ( ^ [F: a > a,A8: set_a,B7: set_a] :
        ? [G2: a > a] :
          ( ! [X: a] :
              ( ( member_a @ X @ A8 )
             => ( ( member_a @ ( F @ X ) @ B7 )
                & ( ( G2 @ ( F @ X ) )
                  = X ) ) )
          & ! [X: a] :
              ( ( member_a @ X @ B7 )
             => ( ( member_a @ ( G2 @ X ) @ A8 )
                & ( ( F @ ( G2 @ X ) )
                  = X ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_874_bij__betw__iff__bijections,axiom,
    ( bij_be5244696088125194461_c_d_a
    = ( ^ [F: produc5278197477302038359_a_c_d > a,A8: set_Pr1275464188344874039_a_c_d,B7: set_a] :
        ? [G2: a > produc5278197477302038359_a_c_d] :
          ( ! [X: produc5278197477302038359_a_c_d] :
              ( ( member1180172933830803072_a_c_d @ X @ A8 )
             => ( ( member_a @ ( F @ X ) @ B7 )
                & ( ( G2 @ ( F @ X ) )
                  = X ) ) )
          & ! [X: a] :
              ( ( member_a @ X @ B7 )
             => ( ( member1180172933830803072_a_c_d @ ( G2 @ X ) @ A8 )
                & ( ( F @ ( G2 @ X ) )
                  = X ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_875_bij__betw__iff__bijections,axiom,
    ( bij_betw_option_a_a
    = ( ^ [F: option_a > a,A8: set_option_a,B7: set_a] :
        ? [G2: a > option_a] :
          ( ! [X: option_a] :
              ( ( member_option_a @ X @ A8 )
             => ( ( member_a @ ( F @ X ) @ B7 )
                & ( ( G2 @ ( F @ X ) )
                  = X ) ) )
          & ! [X: a] :
              ( ( member_a @ X @ B7 )
             => ( ( member_option_a @ ( G2 @ X ) @ A8 )
                & ( ( F @ ( G2 @ X ) )
                  = X ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_876_bij__betw__iff__bijections,axiom,
    ( bij_be2619923856689080655_a_c_d
    = ( ^ [F: a > produc5278197477302038359_a_c_d,A8: set_a,B7: set_Pr1275464188344874039_a_c_d] :
        ? [G2: produc5278197477302038359_a_c_d > a] :
          ( ! [X: a] :
              ( ( member_a @ X @ A8 )
             => ( ( member1180172933830803072_a_c_d @ ( F @ X ) @ B7 )
                & ( ( G2 @ ( F @ X ) )
                  = X ) ) )
          & ! [X: produc5278197477302038359_a_c_d] :
              ( ( member1180172933830803072_a_c_d @ X @ B7 )
             => ( ( member_a @ ( G2 @ X ) @ A8 )
                & ( ( F @ ( G2 @ X ) )
                  = X ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_877_bij__betw__iff__bijections,axiom,
    ( bij_be3277052638744491318_a_c_d
    = ( ^ [F: produc5278197477302038359_a_c_d > produc5278197477302038359_a_c_d,A8: set_Pr1275464188344874039_a_c_d,B7: set_Pr1275464188344874039_a_c_d] :
        ? [G2: produc5278197477302038359_a_c_d > produc5278197477302038359_a_c_d] :
          ( ! [X: produc5278197477302038359_a_c_d] :
              ( ( member1180172933830803072_a_c_d @ X @ A8 )
             => ( ( member1180172933830803072_a_c_d @ ( F @ X ) @ B7 )
                & ( ( G2 @ ( F @ X ) )
                  = X ) ) )
          & ! [X: produc5278197477302038359_a_c_d] :
              ( ( member1180172933830803072_a_c_d @ X @ B7 )
             => ( ( member1180172933830803072_a_c_d @ ( G2 @ X ) @ A8 )
                & ( ( F @ ( G2 @ X ) )
                  = X ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_878_bij__betw__iff__bijections,axiom,
    ( bij_be7402712960780075529_a_c_d
    = ( ^ [F: option_a > produc5278197477302038359_a_c_d,A8: set_option_a,B7: set_Pr1275464188344874039_a_c_d] :
        ? [G2: produc5278197477302038359_a_c_d > option_a] :
          ( ! [X: option_a] :
              ( ( member_option_a @ X @ A8 )
             => ( ( member1180172933830803072_a_c_d @ ( F @ X ) @ B7 )
                & ( ( G2 @ ( F @ X ) )
                  = X ) ) )
          & ! [X: produc5278197477302038359_a_c_d] :
              ( ( member1180172933830803072_a_c_d @ X @ B7 )
             => ( ( member_option_a @ ( G2 @ X ) @ A8 )
                & ( ( F @ ( G2 @ X ) )
                  = X ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_879_bij__betw__iff__bijections,axiom,
    ( bij_betw_a_option_a
    = ( ^ [F: a > option_a,A8: set_a,B7: set_option_a] :
        ? [G2: option_a > a] :
          ( ! [X: a] :
              ( ( member_a @ X @ A8 )
             => ( ( member_option_a @ ( F @ X ) @ B7 )
                & ( ( G2 @ ( F @ X ) )
                  = X ) ) )
          & ! [X: option_a] :
              ( ( member_option_a @ X @ B7 )
             => ( ( member_a @ ( G2 @ X ) @ A8 )
                & ( ( F @ ( G2 @ X ) )
                  = X ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_880_bij__betw__iff__bijections,axiom,
    ( bij_be8242579008920824995tion_a
    = ( ^ [F: produc5278197477302038359_a_c_d > option_a,A8: set_Pr1275464188344874039_a_c_d,B7: set_option_a] :
        ? [G2: option_a > produc5278197477302038359_a_c_d] :
          ( ! [X: produc5278197477302038359_a_c_d] :
              ( ( member1180172933830803072_a_c_d @ X @ A8 )
             => ( ( member_option_a @ ( F @ X ) @ B7 )
                & ( ( G2 @ ( F @ X ) )
                  = X ) ) )
          & ! [X: option_a] :
              ( ( member_option_a @ X @ B7 )
             => ( ( member1180172933830803072_a_c_d @ ( G2 @ X ) @ A8 )
                & ( ( F @ ( G2 @ X ) )
                  = X ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_881_bij__betw__iff__bijections,axiom,
    ( bij_be5431266891817924854tion_a
    = ( ^ [F: option_a > option_a,A8: set_option_a,B7: set_option_a] :
        ? [G2: option_a > option_a] :
          ( ! [X: option_a] :
              ( ( member_option_a @ X @ A8 )
             => ( ( member_option_a @ ( F @ X ) @ B7 )
                & ( ( G2 @ ( F @ X ) )
                  = X ) ) )
          & ! [X: option_a] :
              ( ( member_option_a @ X @ B7 )
             => ( ( member_option_a @ ( G2 @ X ) @ A8 )
                & ( ( F @ ( G2 @ X ) )
                  = X ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_882_bij__betw__comp__iff,axiom,
    ! [F2: a > option_a,A6: set_a,A10: set_option_a,F3: option_a > a,A11: set_a] :
      ( ( bij_betw_a_option_a @ F2 @ A6 @ A10 )
     => ( ( bij_betw_option_a_a @ F3 @ A10 @ A11 )
        = ( bij_betw_a_a @ ( comp_option_a_a_a @ F3 @ F2 ) @ A6 @ A11 ) ) ) ).

% bij_betw_comp_iff
thf(fact_883_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_884_bij__betw__trans,axiom,
    ! [F2: a > option_a,A6: set_a,B5: set_option_a,G: option_a > a,C3: set_a] :
      ( ( bij_betw_a_option_a @ F2 @ A6 @ B5 )
     => ( ( bij_betw_option_a_a @ G @ B5 @ C3 )
       => ( bij_betw_a_a @ ( comp_option_a_a_a @ G @ F2 ) @ A6 @ C3 ) ) ) ).

% bij_betw_trans
thf(fact_885_bij__betw__apply,axiom,
    ! [F2: a > a,A6: set_a,B5: set_a,A: a] :
      ( ( bij_betw_a_a @ F2 @ A6 @ B5 )
     => ( ( member_a @ A @ A6 )
       => ( member_a @ ( F2 @ A ) @ B5 ) ) ) ).

% bij_betw_apply
thf(fact_886_bij__betw__apply,axiom,
    ! [F2: a > produc5278197477302038359_a_c_d,A6: set_a,B5: set_Pr1275464188344874039_a_c_d,A: a] :
      ( ( bij_be2619923856689080655_a_c_d @ F2 @ A6 @ B5 )
     => ( ( member_a @ A @ A6 )
       => ( member1180172933830803072_a_c_d @ ( F2 @ A ) @ B5 ) ) ) ).

% bij_betw_apply
thf(fact_887_bij__betw__apply,axiom,
    ! [F2: a > option_a,A6: set_a,B5: set_option_a,A: a] :
      ( ( bij_betw_a_option_a @ F2 @ A6 @ B5 )
     => ( ( member_a @ A @ A6 )
       => ( member_option_a @ ( F2 @ A ) @ B5 ) ) ) ).

% bij_betw_apply
thf(fact_888_bij__betw__apply,axiom,
    ! [F2: produc5278197477302038359_a_c_d > a,A6: set_Pr1275464188344874039_a_c_d,B5: set_a,A: produc5278197477302038359_a_c_d] :
      ( ( bij_be5244696088125194461_c_d_a @ F2 @ A6 @ B5 )
     => ( ( member1180172933830803072_a_c_d @ A @ A6 )
       => ( member_a @ ( F2 @ A ) @ B5 ) ) ) ).

% bij_betw_apply
thf(fact_889_bij__betw__apply,axiom,
    ! [F2: produc5278197477302038359_a_c_d > produc5278197477302038359_a_c_d,A6: set_Pr1275464188344874039_a_c_d,B5: set_Pr1275464188344874039_a_c_d,A: produc5278197477302038359_a_c_d] :
      ( ( bij_be3277052638744491318_a_c_d @ F2 @ A6 @ B5 )
     => ( ( member1180172933830803072_a_c_d @ A @ A6 )
       => ( member1180172933830803072_a_c_d @ ( F2 @ A ) @ B5 ) ) ) ).

% bij_betw_apply
thf(fact_890_bij__betw__apply,axiom,
    ! [F2: produc5278197477302038359_a_c_d > option_a,A6: set_Pr1275464188344874039_a_c_d,B5: set_option_a,A: produc5278197477302038359_a_c_d] :
      ( ( bij_be8242579008920824995tion_a @ F2 @ A6 @ B5 )
     => ( ( member1180172933830803072_a_c_d @ A @ A6 )
       => ( member_option_a @ ( F2 @ A ) @ B5 ) ) ) ).

% bij_betw_apply
thf(fact_891_bij__betw__apply,axiom,
    ! [F2: option_a > a,A6: set_option_a,B5: set_a,A: option_a] :
      ( ( bij_betw_option_a_a @ F2 @ A6 @ B5 )
     => ( ( member_option_a @ A @ A6 )
       => ( member_a @ ( F2 @ A ) @ B5 ) ) ) ).

% bij_betw_apply
thf(fact_892_bij__betw__apply,axiom,
    ! [F2: option_a > produc5278197477302038359_a_c_d,A6: set_option_a,B5: set_Pr1275464188344874039_a_c_d,A: option_a] :
      ( ( bij_be7402712960780075529_a_c_d @ F2 @ A6 @ B5 )
     => ( ( member_option_a @ A @ A6 )
       => ( member1180172933830803072_a_c_d @ ( F2 @ A ) @ B5 ) ) ) ).

% bij_betw_apply
thf(fact_893_bij__betw__apply,axiom,
    ! [F2: option_a > option_a,A6: set_option_a,B5: set_option_a,A: option_a] :
      ( ( bij_be5431266891817924854tion_a @ F2 @ A6 @ B5 )
     => ( ( member_option_a @ A @ A6 )
       => ( member_option_a @ ( F2 @ A ) @ B5 ) ) ) ).

% bij_betw_apply
thf(fact_894_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_895_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_896_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_897_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_898_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_899_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_900_bij__betw__imp__surj__on,axiom,
    ! [F2: a > option_a,A6: set_a,B5: set_option_a] :
      ( ( bij_betw_a_option_a @ F2 @ A6 @ B5 )
     => ( ( image_a_option_a2 @ F2 @ A6 )
        = B5 ) ) ).

% bij_betw_imp_surj_on
thf(fact_901_bij__betw__imp__surj__on,axiom,
    ! [F2: option_a > a,A6: set_option_a,B5: set_a] :
      ( ( bij_betw_option_a_a @ F2 @ A6 @ B5 )
     => ( ( image_option_a_a2 @ F2 @ A6 )
        = B5 ) ) ).

% bij_betw_imp_surj_on
thf(fact_902_image__eq__imp__comp,axiom,
    ! [F2: option_a > option_a,A6: set_option_a,G: a > option_a,B5: set_a,H: option_a > a] :
      ( ( ( image_7439109396645324421tion_a @ F2 @ A6 )
        = ( image_a_option_a2 @ G @ B5 ) )
     => ( ( image_option_a_a2 @ ( comp_o3864519266390211175tion_a @ H @ F2 ) @ A6 )
        = ( image_a_a2 @ ( comp_option_a_a_a @ H @ G ) @ B5 ) ) ) ).

% image_eq_imp_comp
thf(fact_903_image__eq__imp__comp,axiom,
    ! [F2: a > a,A6: set_a,G: option_a > a,B5: set_option_a,H: a > option_a] :
      ( ( ( image_a_a2 @ F2 @ A6 )
        = ( image_option_a_a2 @ G @ B5 ) )
     => ( ( image_a_option_a2 @ ( comp_a_option_a_a @ H @ F2 ) @ A6 )
        = ( image_7439109396645324421tion_a @ ( comp_a6249931511552232923tion_a @ H @ G ) @ B5 ) ) ) ).

% image_eq_imp_comp
thf(fact_904_image__eq__imp__comp,axiom,
    ! [F2: a > option_a,A6: set_a,G: option_a > option_a,B5: set_option_a,H: option_a > a] :
      ( ( ( image_a_option_a2 @ F2 @ A6 )
        = ( image_7439109396645324421tion_a @ G @ B5 ) )
     => ( ( image_a_a2 @ ( comp_option_a_a_a @ H @ F2 ) @ A6 )
        = ( image_option_a_a2 @ ( comp_o3864519266390211175tion_a @ H @ G ) @ B5 ) ) ) ).

% image_eq_imp_comp
thf(fact_905_image__eq__imp__comp,axiom,
    ! [F2: a > option_a,A6: set_a,G: a > option_a,B5: set_a,H: option_a > a] :
      ( ( ( image_a_option_a2 @ F2 @ A6 )
        = ( image_a_option_a2 @ G @ B5 ) )
     => ( ( image_a_a2 @ ( comp_option_a_a_a @ H @ F2 ) @ A6 )
        = ( image_a_a2 @ ( comp_option_a_a_a @ H @ G ) @ B5 ) ) ) ).

% image_eq_imp_comp
thf(fact_906_image__eq__imp__comp,axiom,
    ! [F2: a > option_a,A6: set_a,G: a > option_a,B5: set_a,H: option_a > option_a] :
      ( ( ( image_a_option_a2 @ F2 @ A6 )
        = ( image_a_option_a2 @ G @ B5 ) )
     => ( ( image_a_option_a2 @ ( comp_o6087033147929006299on_a_a @ H @ F2 ) @ A6 )
        = ( image_a_option_a2 @ ( comp_o6087033147929006299on_a_a @ H @ G ) @ B5 ) ) ) ).

% image_eq_imp_comp
thf(fact_907_image__eq__imp__comp,axiom,
    ! [F2: option_a > a,A6: set_option_a,G: a > a,B5: set_a,H: a > option_a] :
      ( ( ( image_option_a_a2 @ F2 @ A6 )
        = ( image_a_a2 @ G @ B5 ) )
     => ( ( image_7439109396645324421tion_a @ ( comp_a6249931511552232923tion_a @ H @ F2 ) @ A6 )
        = ( image_a_option_a2 @ ( comp_a_option_a_a @ H @ G ) @ B5 ) ) ) ).

% image_eq_imp_comp
thf(fact_908_image__eq__imp__comp,axiom,
    ! [F2: option_a > a,A6: set_option_a,G: option_a > a,B5: set_option_a,H: a > a] :
      ( ( ( image_option_a_a2 @ F2 @ A6 )
        = ( image_option_a_a2 @ G @ B5 ) )
     => ( ( image_option_a_a2 @ ( comp_a_a_option_a @ H @ F2 ) @ A6 )
        = ( image_option_a_a2 @ ( comp_a_a_option_a @ H @ G ) @ B5 ) ) ) ).

% image_eq_imp_comp
thf(fact_909_image__comp,axiom,
    ! [F2: option_a > option_a,G: a > option_a,R: set_a] :
      ( ( image_7439109396645324421tion_a @ F2 @ ( image_a_option_a2 @ G @ R ) )
      = ( image_a_option_a2 @ ( comp_o6087033147929006299on_a_a @ F2 @ G ) @ R ) ) ).

% image_comp
thf(fact_910_image__comp,axiom,
    ! [F2: a > a,G: option_a > a,R: set_option_a] :
      ( ( image_a_a2 @ F2 @ ( image_option_a_a2 @ G @ R ) )
      = ( image_option_a_a2 @ ( comp_a_a_option_a @ F2 @ G ) @ R ) ) ).

% image_comp
thf(fact_911_image__comp,axiom,
    ! [F2: a > option_a,G: a > a,R: set_a] :
      ( ( image_a_option_a2 @ F2 @ ( image_a_a2 @ G @ R ) )
      = ( image_a_option_a2 @ ( comp_a_option_a_a @ F2 @ G ) @ R ) ) ).

% image_comp
thf(fact_912_image__comp,axiom,
    ! [F2: a > option_a,G: option_a > a,R: set_option_a] :
      ( ( image_a_option_a2 @ F2 @ ( image_option_a_a2 @ G @ R ) )
      = ( image_7439109396645324421tion_a @ ( comp_a6249931511552232923tion_a @ F2 @ G ) @ R ) ) ).

% image_comp
thf(fact_913_image__comp,axiom,
    ! [F2: option_a > a,G: option_a > option_a,R: set_option_a] :
      ( ( image_option_a_a2 @ F2 @ ( image_7439109396645324421tion_a @ G @ R ) )
      = ( image_option_a_a2 @ ( comp_o3864519266390211175tion_a @ F2 @ G ) @ R ) ) ).

% image_comp
thf(fact_914_image__comp,axiom,
    ! [F2: option_a > a,G: a > option_a,R: set_a] :
      ( ( image_option_a_a2 @ F2 @ ( image_a_option_a2 @ G @ R ) )
      = ( image_a_a2 @ ( comp_option_a_a_a @ F2 @ G ) @ R ) ) ).

% image_comp
thf(fact_915_involuntory__imp__bij,axiom,
    ! [F2: a > a] :
      ( ! [X4: a] :
          ( ( F2 @ ( F2 @ X4 ) )
          = X4 )
     => ( bij_betw_a_a @ F2 @ top_top_set_a @ top_top_set_a ) ) ).

% involuntory_imp_bij
thf(fact_916_involuntory__imp__bij,axiom,
    ! [F2: option_a > option_a] :
      ( ! [X4: option_a] :
          ( ( F2 @ ( F2 @ X4 ) )
          = X4 )
     => ( bij_be5431266891817924854tion_a @ F2 @ top_top_set_option_a @ top_top_set_option_a ) ) ).

% involuntory_imp_bij
thf(fact_917_bij__pointE,axiom,
    ! [F2: a > a,Y: a] :
      ( ( bij_betw_a_a @ F2 @ top_top_set_a @ top_top_set_a )
     => ~ ! [X4: a] :
            ( ( Y
              = ( F2 @ X4 ) )
           => ~ ! [X9: a] :
                  ( ( Y
                    = ( F2 @ X9 ) )
                 => ( X9 = X4 ) ) ) ) ).

% bij_pointE
thf(fact_918_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 )
     => ~ ! [X4: a] :
            ( ( Y
              = ( F2 @ X4 ) )
           => ~ ! [X9: a] :
                  ( ( Y
                    = ( F2 @ X9 ) )
                 => ( X9 = X4 ) ) ) ) ).

% bij_pointE
thf(fact_919_bij__pointE,axiom,
    ! [F2: option_a > a,Y: a] :
      ( ( bij_betw_option_a_a @ F2 @ top_top_set_option_a @ top_top_set_a )
     => ~ ! [X4: option_a] :
            ( ( Y
              = ( F2 @ X4 ) )
           => ~ ! [X9: option_a] :
                  ( ( Y
                    = ( F2 @ X9 ) )
                 => ( X9 = X4 ) ) ) ) ).

% bij_pointE
thf(fact_920_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 )
     => ~ ! [X4: option_a] :
            ( ( Y
              = ( F2 @ X4 ) )
           => ~ ! [X9: option_a] :
                  ( ( Y
                    = ( F2 @ X9 ) )
                 => ( X9 = X4 ) ) ) ) ).

% bij_pointE
thf(fact_921_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_922_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_923_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_924_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_925_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_926_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_927_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_928_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_929_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_930_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_931_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_932_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_933_asym__onD,axiom,
    ! [A6: set_a,R: set_Product_prod_a_a,X2: a,Y: a] :
      ( ( asym_on_a @ A6 @ R )
     => ( ( member_a @ X2 @ A6 )
       => ( ( member_a @ Y @ A6 )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
           => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ R ) ) ) ) ) ).

% asym_onD
thf(fact_934_asym__onD,axiom,
    ! [A6: set_Pr1275464188344874039_a_c_d,R: set_Pr3774076249145843911_a_c_d,X2: produc5278197477302038359_a_c_d,Y: produc5278197477302038359_a_c_d] :
      ( ( asym_o6600848050852821641_a_c_d @ A6 @ R )
     => ( ( member1180172933830803072_a_c_d @ X2 @ A6 )
       => ( ( member1180172933830803072_a_c_d @ Y @ A6 )
         => ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ X2 @ Y ) @ R )
           => ~ ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ Y @ X2 ) @ R ) ) ) ) ) ).

% asym_onD
thf(fact_935_asym__onD,axiom,
    ! [A6: set_option_a,R: set_Pr7585778909603769095tion_a,X2: option_a,Y: option_a] :
      ( ( asym_on_option_a @ A6 @ R )
     => ( ( member_option_a @ X2 @ A6 )
       => ( ( member_option_a @ Y @ A6 )
         => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y ) @ R )
           => ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y @ X2 ) @ R ) ) ) ) ) ).

% asym_onD
thf(fact_936_bij__betw__comp__iff2,axiom,
    ! [F3: option_a > a,A10: set_option_a,A11: set_a,F2: a > option_a,A6: set_a] :
      ( ( bij_betw_option_a_a @ F3 @ A10 @ A11 )
     => ( ( ord_le1955136853071979460tion_a @ ( image_a_option_a2 @ F2 @ A6 ) @ A10 )
       => ( ( bij_betw_a_option_a @ F2 @ A6 @ A10 )
          = ( bij_betw_a_a @ ( comp_option_a_a_a @ F3 @ F2 ) @ A6 @ A11 ) ) ) ) ).

% bij_betw_comp_iff2
thf(fact_937_bij__betw__imp__surj,axiom,
    ! [F2: a > a,A6: set_a] :
      ( ( bij_betw_a_a @ F2 @ A6 @ top_top_set_a )
     => ( ( image_a_a2 @ F2 @ top_top_set_a )
        = top_top_set_a ) ) ).

% bij_betw_imp_surj
thf(fact_938_bij__betw__imp__surj,axiom,
    ! [F2: option_a > a,A6: set_option_a] :
      ( ( bij_betw_option_a_a @ F2 @ A6 @ top_top_set_a )
     => ( ( image_option_a_a2 @ F2 @ top_top_set_option_a )
        = top_top_set_a ) ) ).

% bij_betw_imp_surj
thf(fact_939_bij__betw__imp__surj,axiom,
    ! [F2: a > option_a,A6: set_a] :
      ( ( bij_betw_a_option_a @ F2 @ A6 @ 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_940_bij__betw__imp__surj,axiom,
    ! [F2: option_a > option_a,A6: set_option_a] :
      ( ( bij_be5431266891817924854tion_a @ F2 @ A6 @ 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_941_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_942_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_943_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_944_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_945_bij__betw__byWitness,axiom,
    ! [A6: set_a,F3: option_a > a,F2: a > option_a,A10: set_option_a] :
      ( ! [X4: a] :
          ( ( member_a @ X4 @ A6 )
         => ( ( F3 @ ( F2 @ X4 ) )
            = X4 ) )
     => ( ! [X4: option_a] :
            ( ( member_option_a @ X4 @ A10 )
           => ( ( F2 @ ( F3 @ X4 ) )
              = X4 ) )
       => ( ( ord_le1955136853071979460tion_a @ ( image_a_option_a2 @ F2 @ A6 ) @ A10 )
         => ( ( ord_less_eq_set_a @ ( image_option_a_a2 @ F3 @ A10 ) @ A6 )
           => ( bij_betw_a_option_a @ F2 @ A6 @ A10 ) ) ) ) ) ).

% bij_betw_byWitness
thf(fact_946_bij__betw__byWitness,axiom,
    ! [A6: set_option_a,F3: a > option_a,F2: option_a > a,A10: set_a] :
      ( ! [X4: option_a] :
          ( ( member_option_a @ X4 @ A6 )
         => ( ( F3 @ ( F2 @ X4 ) )
            = X4 ) )
     => ( ! [X4: a] :
            ( ( member_a @ X4 @ A10 )
           => ( ( F2 @ ( F3 @ X4 ) )
              = X4 ) )
       => ( ( ord_less_eq_set_a @ ( image_option_a_a2 @ F2 @ A6 ) @ A10 )
         => ( ( ord_le1955136853071979460tion_a @ ( image_a_option_a2 @ F3 @ A10 ) @ A6 )
           => ( bij_betw_option_a_a @ F2 @ A6 @ A10 ) ) ) ) ) ).

% bij_betw_byWitness
thf(fact_947_bij__betw__subset,axiom,
    ! [F2: a > option_a,A6: set_a,A10: set_option_a,B5: set_a,B8: set_option_a] :
      ( ( bij_betw_a_option_a @ F2 @ A6 @ A10 )
     => ( ( ord_less_eq_set_a @ B5 @ A6 )
       => ( ( ( image_a_option_a2 @ F2 @ B5 )
            = B8 )
         => ( bij_betw_a_option_a @ F2 @ B5 @ B8 ) ) ) ) ).

% bij_betw_subset
thf(fact_948_bij__betw__subset,axiom,
    ! [F2: option_a > a,A6: set_option_a,A10: set_a,B5: set_option_a,B8: set_a] :
      ( ( bij_betw_option_a_a @ F2 @ A6 @ A10 )
     => ( ( ord_le1955136853071979460tion_a @ B5 @ A6 )
       => ( ( ( image_option_a_a2 @ F2 @ B5 )
            = B8 )
         => ( bij_betw_option_a_a @ F2 @ B5 @ B8 ) ) ) ) ).

% bij_betw_subset
thf(fact_949_inj__on__imp__bij__betw,axiom,
    ! [F2: option_a > a,A6: set_option_a] :
      ( ( inj_on_option_a_a @ F2 @ A6 )
     => ( bij_betw_option_a_a @ F2 @ A6 @ ( image_option_a_a2 @ F2 @ A6 ) ) ) ).

% inj_on_imp_bij_betw
thf(fact_950_inj__on__imp__bij__betw,axiom,
    ! [F2: a > option_a,A6: set_a] :
      ( ( inj_on_a_option_a @ F2 @ A6 )
     => ( bij_betw_a_option_a @ F2 @ A6 @ ( image_a_option_a2 @ F2 @ A6 ) ) ) ).

% inj_on_imp_bij_betw
thf(fact_951_bij__betw__imageI,axiom,
    ! [F2: option_a > a,A6: set_option_a,B5: set_a] :
      ( ( inj_on_option_a_a @ F2 @ A6 )
     => ( ( ( image_option_a_a2 @ F2 @ A6 )
          = B5 )
       => ( bij_betw_option_a_a @ F2 @ A6 @ B5 ) ) ) ).

% bij_betw_imageI
thf(fact_952_bij__betw__imageI,axiom,
    ! [F2: a > option_a,A6: set_a,B5: set_option_a] :
      ( ( inj_on_a_option_a @ F2 @ A6 )
     => ( ( ( image_a_option_a2 @ F2 @ A6 )
          = B5 )
       => ( bij_betw_a_option_a @ F2 @ A6 @ B5 ) ) ) ).

% bij_betw_imageI
thf(fact_953_bij__betw__def,axiom,
    ( bij_betw_option_a_a
    = ( ^ [F: option_a > a,A8: set_option_a,B7: set_a] :
          ( ( inj_on_option_a_a @ F @ A8 )
          & ( ( image_option_a_a2 @ F @ A8 )
            = B7 ) ) ) ) ).

% bij_betw_def
thf(fact_954_bij__betw__def,axiom,
    ( bij_betw_a_option_a
    = ( ^ [F: a > option_a,A8: set_a,B7: set_option_a] :
          ( ( inj_on_a_option_a @ F @ A8 )
          & ( ( image_a_option_a2 @ F @ A8 )
            = B7 ) ) ) ) ).

% bij_betw_def
thf(fact_955_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_956_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_957_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_958_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_959_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_960_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_961_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_962_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_963_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_964_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_965_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_966_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_967_comp__inj__on__iff,axiom,
    ! [F2: option_a > a,A6: set_option_a,F3: a > option_a] :
      ( ( inj_on_option_a_a @ F2 @ A6 )
     => ( ( inj_on_a_option_a @ F3 @ ( image_option_a_a2 @ F2 @ A6 ) )
        = ( inj_on8559383841115902449tion_a @ ( comp_a6249931511552232923tion_a @ F3 @ F2 ) @ A6 ) ) ) ).

% comp_inj_on_iff
thf(fact_968_comp__inj__on__iff,axiom,
    ! [F2: a > a,A6: set_a,F3: a > option_a] :
      ( ( inj_on_a_a @ F2 @ A6 )
     => ( ( inj_on_a_option_a @ F3 @ ( image_a_a2 @ F2 @ A6 ) )
        = ( inj_on_a_option_a @ ( comp_a_option_a_a @ F3 @ F2 ) @ A6 ) ) ) ).

% comp_inj_on_iff
thf(fact_969_comp__inj__on__iff,axiom,
    ! [F2: a > option_a,A6: set_a,F3: option_a > a] :
      ( ( inj_on_a_option_a @ F2 @ A6 )
     => ( ( inj_on_option_a_a @ F3 @ ( image_a_option_a2 @ F2 @ A6 ) )
        = ( inj_on_a_a @ ( comp_option_a_a_a @ F3 @ F2 ) @ A6 ) ) ) ).

% comp_inj_on_iff
thf(fact_970_comp__inj__on__iff,axiom,
    ! [F2: a > option_a,A6: set_a,F3: option_a > option_a] :
      ( ( inj_on_a_option_a @ F2 @ A6 )
     => ( ( inj_on8559383841115902449tion_a @ F3 @ ( image_a_option_a2 @ F2 @ A6 ) )
        = ( inj_on_a_option_a @ ( comp_o6087033147929006299on_a_a @ F3 @ F2 ) @ A6 ) ) ) ).

% comp_inj_on_iff
thf(fact_971_inj__on__imageI,axiom,
    ! [G: option_a > a,F2: a > option_a,A6: set_a] :
      ( ( inj_on_a_a @ ( comp_option_a_a_a @ G @ F2 ) @ A6 )
     => ( inj_on_option_a_a @ G @ ( image_a_option_a2 @ F2 @ A6 ) ) ) ).

% inj_on_imageI
thf(fact_972_inj__on__imageI,axiom,
    ! [G: a > option_a,F2: option_a > a,A6: set_option_a] :
      ( ( inj_on8559383841115902449tion_a @ ( comp_a6249931511552232923tion_a @ G @ F2 ) @ A6 )
     => ( inj_on_a_option_a @ G @ ( image_option_a_a2 @ F2 @ A6 ) ) ) ).

% inj_on_imageI
thf(fact_973_inj__on__imageI,axiom,
    ! [G: option_a > option_a,F2: a > option_a,A6: set_a] :
      ( ( inj_on_a_option_a @ ( comp_o6087033147929006299on_a_a @ G @ F2 ) @ A6 )
     => ( inj_on8559383841115902449tion_a @ G @ ( image_a_option_a2 @ F2 @ A6 ) ) ) ).

% inj_on_imageI
thf(fact_974_inj__on__imageI,axiom,
    ! [G: a > option_a,F2: a > a,A6: set_a] :
      ( ( inj_on_a_option_a @ ( comp_a_option_a_a @ G @ F2 ) @ A6 )
     => ( inj_on_a_option_a @ G @ ( image_a_a2 @ F2 @ A6 ) ) ) ).

% inj_on_imageI
thf(fact_975_comp__inj__on,axiom,
    ! [F2: option_a > a,A6: set_option_a,G: a > option_a] :
      ( ( inj_on_option_a_a @ F2 @ A6 )
     => ( ( inj_on_a_option_a @ G @ ( image_option_a_a2 @ F2 @ A6 ) )
       => ( inj_on8559383841115902449tion_a @ ( comp_a6249931511552232923tion_a @ G @ F2 ) @ A6 ) ) ) ).

% comp_inj_on
thf(fact_976_comp__inj__on,axiom,
    ! [F2: a > a,A6: set_a,G: a > option_a] :
      ( ( inj_on_a_a @ F2 @ A6 )
     => ( ( inj_on_a_option_a @ G @ ( image_a_a2 @ F2 @ A6 ) )
       => ( inj_on_a_option_a @ ( comp_a_option_a_a @ G @ F2 ) @ A6 ) ) ) ).

% comp_inj_on
thf(fact_977_comp__inj__on,axiom,
    ! [F2: a > option_a,A6: set_a,G: option_a > a] :
      ( ( inj_on_a_option_a @ F2 @ A6 )
     => ( ( inj_on_option_a_a @ G @ ( image_a_option_a2 @ F2 @ A6 ) )
       => ( inj_on_a_a @ ( comp_option_a_a_a @ G @ F2 ) @ A6 ) ) ) ).

% comp_inj_on
thf(fact_978_comp__inj__on,axiom,
    ! [F2: a > option_a,A6: set_a,G: option_a > option_a] :
      ( ( inj_on_a_option_a @ F2 @ A6 )
     => ( ( inj_on8559383841115902449tion_a @ G @ ( image_a_option_a2 @ F2 @ A6 ) )
       => ( inj_on_a_option_a @ ( comp_o6087033147929006299on_a_a @ G @ F2 ) @ A6 ) ) ) ).

% comp_inj_on
thf(fact_979_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_980_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_981_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_982_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_983_bind__map__option,axiom,
    ! [F2: a > a,X2: option_a,G: a > option_a] :
      ( ( bind_a_a @ ( map_option_a_a @ F2 @ X2 ) @ G )
      = ( bind_a_a @ X2 @ ( comp_a_option_a_a @ G @ F2 ) ) ) ).

% bind_map_option
thf(fact_984_map__option__bind,axiom,
    ! [F2: a > a,X2: option_a,G: a > option_a] :
      ( ( map_option_a_a @ F2 @ ( bind_a_a @ X2 @ G ) )
      = ( bind_a_a @ X2 @ ( comp_o6087033147929006299on_a_a @ ( map_option_a_a @ F2 ) @ G ) ) ) ).

% map_option_bind
thf(fact_985_asym__iff,axiom,
    ! [R: set_Product_prod_a_a] :
      ( ( asym_on_a @ top_top_set_a @ R )
      = ( ! [X: a,Y3: a] :
            ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R )
           => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X ) @ R ) ) ) ) ).

% asym_iff
thf(fact_986_asym__iff,axiom,
    ! [R: set_Pr7585778909603769095tion_a] :
      ( ( asym_on_option_a @ top_top_set_option_a @ R )
      = ( ! [X: option_a,Y3: option_a] :
            ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y3 ) @ R )
           => ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y3 @ X ) @ R ) ) ) ) ).

% asym_iff
thf(fact_987_asymD,axiom,
    ! [R: set_Product_prod_a_a,X2: a,Y: a] :
      ( ( asym_on_a @ top_top_set_a @ R )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
       => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ R ) ) ) ).

% asymD
thf(fact_988_asymD,axiom,
    ! [R: set_Pr7585778909603769095tion_a,X2: option_a,Y: option_a] :
      ( ( asym_on_option_a @ top_top_set_option_a @ R )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y ) @ R )
       => ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y @ X2 ) @ R ) ) ) ).

% asymD
thf(fact_989_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_990_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_991_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_992_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_993_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_994_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_995_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_996_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_997_notIn__Un__bij__betw,axiom,
    ! [B: produc5278197477302038359_a_c_d,A6: set_Pr1275464188344874039_a_c_d,F2: produc5278197477302038359_a_c_d > produc5278197477302038359_a_c_d,A10: set_Pr1275464188344874039_a_c_d] :
      ( ~ ( member1180172933830803072_a_c_d @ B @ A6 )
     => ( ~ ( member1180172933830803072_a_c_d @ ( F2 @ B ) @ A10 )
       => ( ( bij_be3277052638744491318_a_c_d @ F2 @ A6 @ A10 )
         => ( bij_be3277052638744491318_a_c_d @ F2 @ ( sup_su3456763302433169931_a_c_d @ A6 @ ( insert4072338954160766247_a_c_d @ B @ bot_bo4027597737525728931_a_c_d ) ) @ ( sup_su3456763302433169931_a_c_d @ A10 @ ( insert4072338954160766247_a_c_d @ ( F2 @ B ) @ bot_bo4027597737525728931_a_c_d ) ) ) ) ) ) ).

% notIn_Un_bij_betw
thf(fact_998_notIn__Un__bij__betw,axiom,
    ! [B: produc5278197477302038359_a_c_d,A6: set_Pr1275464188344874039_a_c_d,F2: produc5278197477302038359_a_c_d > a,A10: set_a] :
      ( ~ ( member1180172933830803072_a_c_d @ B @ A6 )
     => ( ~ ( member_a @ ( F2 @ B ) @ A10 )
       => ( ( bij_be5244696088125194461_c_d_a @ F2 @ A6 @ A10 )
         => ( bij_be5244696088125194461_c_d_a @ F2 @ ( sup_su3456763302433169931_a_c_d @ A6 @ ( insert4072338954160766247_a_c_d @ B @ bot_bo4027597737525728931_a_c_d ) ) @ ( sup_sup_set_a @ A10 @ ( insert_a @ ( F2 @ B ) @ bot_bot_set_a ) ) ) ) ) ) ).

% notIn_Un_bij_betw
thf(fact_999_notIn__Un__bij__betw,axiom,
    ! [B: produc5278197477302038359_a_c_d,A6: set_Pr1275464188344874039_a_c_d,F2: produc5278197477302038359_a_c_d > option_a,A10: set_option_a] :
      ( ~ ( member1180172933830803072_a_c_d @ B @ A6 )
     => ( ~ ( member_option_a @ ( F2 @ B ) @ A10 )
       => ( ( bij_be8242579008920824995tion_a @ F2 @ A6 @ A10 )
         => ( bij_be8242579008920824995tion_a @ F2 @ ( sup_su3456763302433169931_a_c_d @ A6 @ ( insert4072338954160766247_a_c_d @ B @ bot_bo4027597737525728931_a_c_d ) ) @ ( sup_sup_set_option_a @ A10 @ ( insert_option_a @ ( F2 @ B ) @ bot_bot_set_option_a ) ) ) ) ) ) ).

% notIn_Un_bij_betw
thf(fact_1000_notIn__Un__bij__betw,axiom,
    ! [B: a,A6: set_a,F2: a > produc5278197477302038359_a_c_d,A10: set_Pr1275464188344874039_a_c_d] :
      ( ~ ( member_a @ B @ A6 )
     => ( ~ ( member1180172933830803072_a_c_d @ ( F2 @ B ) @ A10 )
       => ( ( bij_be2619923856689080655_a_c_d @ F2 @ A6 @ A10 )
         => ( bij_be2619923856689080655_a_c_d @ F2 @ ( sup_sup_set_a @ A6 @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( sup_su3456763302433169931_a_c_d @ A10 @ ( insert4072338954160766247_a_c_d @ ( F2 @ B ) @ bot_bo4027597737525728931_a_c_d ) ) ) ) ) ) ).

% notIn_Un_bij_betw
thf(fact_1001_notIn__Un__bij__betw,axiom,
    ! [B: a,A6: set_a,F2: a > a,A10: set_a] :
      ( ~ ( member_a @ B @ A6 )
     => ( ~ ( member_a @ ( F2 @ B ) @ A10 )
       => ( ( bij_betw_a_a @ F2 @ A6 @ A10 )
         => ( bij_betw_a_a @ F2 @ ( sup_sup_set_a @ A6 @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( sup_sup_set_a @ A10 @ ( insert_a @ ( F2 @ B ) @ bot_bot_set_a ) ) ) ) ) ) ).

% notIn_Un_bij_betw
thf(fact_1002_notIn__Un__bij__betw,axiom,
    ! [B: a,A6: set_a,F2: a > option_a,A10: set_option_a] :
      ( ~ ( member_a @ B @ A6 )
     => ( ~ ( member_option_a @ ( F2 @ B ) @ A10 )
       => ( ( bij_betw_a_option_a @ F2 @ A6 @ A10 )
         => ( bij_betw_a_option_a @ F2 @ ( sup_sup_set_a @ A6 @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( sup_sup_set_option_a @ A10 @ ( insert_option_a @ ( F2 @ B ) @ bot_bot_set_option_a ) ) ) ) ) ) ).

% notIn_Un_bij_betw
thf(fact_1003_notIn__Un__bij__betw,axiom,
    ! [B: option_a,A6: set_option_a,F2: option_a > produc5278197477302038359_a_c_d,A10: set_Pr1275464188344874039_a_c_d] :
      ( ~ ( member_option_a @ B @ A6 )
     => ( ~ ( member1180172933830803072_a_c_d @ ( F2 @ B ) @ A10 )
       => ( ( bij_be7402712960780075529_a_c_d @ F2 @ A6 @ A10 )
         => ( bij_be7402712960780075529_a_c_d @ F2 @ ( sup_sup_set_option_a @ A6 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( sup_su3456763302433169931_a_c_d @ A10 @ ( insert4072338954160766247_a_c_d @ ( F2 @ B ) @ bot_bo4027597737525728931_a_c_d ) ) ) ) ) ) ).

% notIn_Un_bij_betw
thf(fact_1004_notIn__Un__bij__betw,axiom,
    ! [B: option_a,A6: set_option_a,F2: option_a > a,A10: set_a] :
      ( ~ ( member_option_a @ B @ A6 )
     => ( ~ ( member_a @ ( F2 @ B ) @ A10 )
       => ( ( bij_betw_option_a_a @ F2 @ A6 @ A10 )
         => ( bij_betw_option_a_a @ F2 @ ( sup_sup_set_option_a @ A6 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( sup_sup_set_a @ A10 @ ( insert_a @ ( F2 @ B ) @ bot_bot_set_a ) ) ) ) ) ) ).

% notIn_Un_bij_betw
thf(fact_1005_notIn__Un__bij__betw,axiom,
    ! [B: option_a,A6: set_option_a,F2: option_a > option_a,A10: set_option_a] :
      ( ~ ( member_option_a @ B @ A6 )
     => ( ~ ( member_option_a @ ( F2 @ B ) @ A10 )
       => ( ( bij_be5431266891817924854tion_a @ F2 @ A6 @ A10 )
         => ( bij_be5431266891817924854tion_a @ F2 @ ( sup_sup_set_option_a @ A6 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( sup_sup_set_option_a @ A10 @ ( insert_option_a @ ( F2 @ B ) @ bot_bot_set_option_a ) ) ) ) ) ) ).

% notIn_Un_bij_betw
thf(fact_1006_notIn__Un__bij__betw3,axiom,
    ! [B: produc5278197477302038359_a_c_d,A6: set_Pr1275464188344874039_a_c_d,F2: produc5278197477302038359_a_c_d > produc5278197477302038359_a_c_d,A10: set_Pr1275464188344874039_a_c_d] :
      ( ~ ( member1180172933830803072_a_c_d @ B @ A6 )
     => ( ~ ( member1180172933830803072_a_c_d @ ( F2 @ B ) @ A10 )
       => ( ( bij_be3277052638744491318_a_c_d @ F2 @ A6 @ A10 )
          = ( bij_be3277052638744491318_a_c_d @ F2 @ ( sup_su3456763302433169931_a_c_d @ A6 @ ( insert4072338954160766247_a_c_d @ B @ bot_bo4027597737525728931_a_c_d ) ) @ ( sup_su3456763302433169931_a_c_d @ A10 @ ( insert4072338954160766247_a_c_d @ ( F2 @ B ) @ bot_bo4027597737525728931_a_c_d ) ) ) ) ) ) ).

% notIn_Un_bij_betw3
thf(fact_1007_notIn__Un__bij__betw3,axiom,
    ! [B: produc5278197477302038359_a_c_d,A6: set_Pr1275464188344874039_a_c_d,F2: produc5278197477302038359_a_c_d > a,A10: set_a] :
      ( ~ ( member1180172933830803072_a_c_d @ B @ A6 )
     => ( ~ ( member_a @ ( F2 @ B ) @ A10 )
       => ( ( bij_be5244696088125194461_c_d_a @ F2 @ A6 @ A10 )
          = ( bij_be5244696088125194461_c_d_a @ F2 @ ( sup_su3456763302433169931_a_c_d @ A6 @ ( insert4072338954160766247_a_c_d @ B @ bot_bo4027597737525728931_a_c_d ) ) @ ( sup_sup_set_a @ A10 @ ( insert_a @ ( F2 @ B ) @ bot_bot_set_a ) ) ) ) ) ) ).

% notIn_Un_bij_betw3
thf(fact_1008_notIn__Un__bij__betw3,axiom,
    ! [B: produc5278197477302038359_a_c_d,A6: set_Pr1275464188344874039_a_c_d,F2: produc5278197477302038359_a_c_d > option_a,A10: set_option_a] :
      ( ~ ( member1180172933830803072_a_c_d @ B @ A6 )
     => ( ~ ( member_option_a @ ( F2 @ B ) @ A10 )
       => ( ( bij_be8242579008920824995tion_a @ F2 @ A6 @ A10 )
          = ( bij_be8242579008920824995tion_a @ F2 @ ( sup_su3456763302433169931_a_c_d @ A6 @ ( insert4072338954160766247_a_c_d @ B @ bot_bo4027597737525728931_a_c_d ) ) @ ( sup_sup_set_option_a @ A10 @ ( insert_option_a @ ( F2 @ B ) @ bot_bot_set_option_a ) ) ) ) ) ) ).

% notIn_Un_bij_betw3
thf(fact_1009_notIn__Un__bij__betw3,axiom,
    ! [B: a,A6: set_a,F2: a > produc5278197477302038359_a_c_d,A10: set_Pr1275464188344874039_a_c_d] :
      ( ~ ( member_a @ B @ A6 )
     => ( ~ ( member1180172933830803072_a_c_d @ ( F2 @ B ) @ A10 )
       => ( ( bij_be2619923856689080655_a_c_d @ F2 @ A6 @ A10 )
          = ( bij_be2619923856689080655_a_c_d @ F2 @ ( sup_sup_set_a @ A6 @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( sup_su3456763302433169931_a_c_d @ A10 @ ( insert4072338954160766247_a_c_d @ ( F2 @ B ) @ bot_bo4027597737525728931_a_c_d ) ) ) ) ) ) ).

% notIn_Un_bij_betw3
thf(fact_1010_notIn__Un__bij__betw3,axiom,
    ! [B: a,A6: set_a,F2: a > a,A10: set_a] :
      ( ~ ( member_a @ B @ A6 )
     => ( ~ ( member_a @ ( F2 @ B ) @ A10 )
       => ( ( bij_betw_a_a @ F2 @ A6 @ A10 )
          = ( bij_betw_a_a @ F2 @ ( sup_sup_set_a @ A6 @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( sup_sup_set_a @ A10 @ ( insert_a @ ( F2 @ B ) @ bot_bot_set_a ) ) ) ) ) ) ).

% notIn_Un_bij_betw3
thf(fact_1011_notIn__Un__bij__betw3,axiom,
    ! [B: a,A6: set_a,F2: a > option_a,A10: set_option_a] :
      ( ~ ( member_a @ B @ A6 )
     => ( ~ ( member_option_a @ ( F2 @ B ) @ A10 )
       => ( ( bij_betw_a_option_a @ F2 @ A6 @ A10 )
          = ( bij_betw_a_option_a @ F2 @ ( sup_sup_set_a @ A6 @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( sup_sup_set_option_a @ A10 @ ( insert_option_a @ ( F2 @ B ) @ bot_bot_set_option_a ) ) ) ) ) ) ).

% notIn_Un_bij_betw3
thf(fact_1012_notIn__Un__bij__betw3,axiom,
    ! [B: option_a,A6: set_option_a,F2: option_a > produc5278197477302038359_a_c_d,A10: set_Pr1275464188344874039_a_c_d] :
      ( ~ ( member_option_a @ B @ A6 )
     => ( ~ ( member1180172933830803072_a_c_d @ ( F2 @ B ) @ A10 )
       => ( ( bij_be7402712960780075529_a_c_d @ F2 @ A6 @ A10 )
          = ( bij_be7402712960780075529_a_c_d @ F2 @ ( sup_sup_set_option_a @ A6 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( sup_su3456763302433169931_a_c_d @ A10 @ ( insert4072338954160766247_a_c_d @ ( F2 @ B ) @ bot_bo4027597737525728931_a_c_d ) ) ) ) ) ) ).

% notIn_Un_bij_betw3
thf(fact_1013_notIn__Un__bij__betw3,axiom,
    ! [B: option_a,A6: set_option_a,F2: option_a > a,A10: set_a] :
      ( ~ ( member_option_a @ B @ A6 )
     => ( ~ ( member_a @ ( F2 @ B ) @ A10 )
       => ( ( bij_betw_option_a_a @ F2 @ A6 @ A10 )
          = ( bij_betw_option_a_a @ F2 @ ( sup_sup_set_option_a @ A6 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( sup_sup_set_a @ A10 @ ( insert_a @ ( F2 @ B ) @ bot_bot_set_a ) ) ) ) ) ) ).

% notIn_Un_bij_betw3
thf(fact_1014_notIn__Un__bij__betw3,axiom,
    ! [B: option_a,A6: set_option_a,F2: option_a > option_a,A10: set_option_a] :
      ( ~ ( member_option_a @ B @ A6 )
     => ( ~ ( member_option_a @ ( F2 @ B ) @ A10 )
       => ( ( bij_be5431266891817924854tion_a @ F2 @ A6 @ A10 )
          = ( bij_be5431266891817924854tion_a @ F2 @ ( sup_sup_set_option_a @ A6 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( sup_sup_set_option_a @ A10 @ ( insert_option_a @ ( F2 @ B ) @ bot_bot_set_option_a ) ) ) ) ) ) ).

% notIn_Un_bij_betw3
thf(fact_1015_bij__betw__partition,axiom,
    ! [F2: a > a,A6: set_a,C3: set_a,B5: set_a,D2: set_a] :
      ( ( bij_betw_a_a @ F2 @ ( sup_sup_set_a @ A6 @ C3 ) @ ( sup_sup_set_a @ B5 @ D2 ) )
     => ( ( bij_betw_a_a @ F2 @ C3 @ D2 )
       => ( ( ( inf_inf_set_a @ A6 @ C3 )
            = bot_bot_set_a )
         => ( ( ( inf_inf_set_a @ B5 @ D2 )
              = bot_bot_set_a )
           => ( bij_betw_a_a @ F2 @ A6 @ B5 ) ) ) ) ) ).

% bij_betw_partition
thf(fact_1016_bij__betw__partition,axiom,
    ! [F2: a > option_a,A6: set_a,C3: set_a,B5: set_option_a,D2: set_option_a] :
      ( ( bij_betw_a_option_a @ F2 @ ( sup_sup_set_a @ A6 @ C3 ) @ ( sup_sup_set_option_a @ B5 @ D2 ) )
     => ( ( bij_betw_a_option_a @ F2 @ C3 @ D2 )
       => ( ( ( inf_inf_set_a @ A6 @ C3 )
            = bot_bot_set_a )
         => ( ( ( inf_inf_set_option_a @ B5 @ D2 )
              = bot_bot_set_option_a )
           => ( bij_betw_a_option_a @ F2 @ A6 @ B5 ) ) ) ) ) ).

% bij_betw_partition
thf(fact_1017_bij__betw__partition,axiom,
    ! [F2: option_a > a,A6: set_option_a,C3: set_option_a,B5: set_a,D2: set_a] :
      ( ( bij_betw_option_a_a @ F2 @ ( sup_sup_set_option_a @ A6 @ C3 ) @ ( sup_sup_set_a @ B5 @ D2 ) )
     => ( ( bij_betw_option_a_a @ F2 @ C3 @ D2 )
       => ( ( ( inf_inf_set_option_a @ A6 @ C3 )
            = bot_bot_set_option_a )
         => ( ( ( inf_inf_set_a @ B5 @ D2 )
              = bot_bot_set_a )
           => ( bij_betw_option_a_a @ F2 @ A6 @ B5 ) ) ) ) ) ).

% bij_betw_partition
thf(fact_1018_bij__betw__partition,axiom,
    ! [F2: option_a > option_a,A6: set_option_a,C3: set_option_a,B5: set_option_a,D2: set_option_a] :
      ( ( bij_be5431266891817924854tion_a @ F2 @ ( sup_sup_set_option_a @ A6 @ C3 ) @ ( sup_sup_set_option_a @ B5 @ D2 ) )
     => ( ( bij_be5431266891817924854tion_a @ F2 @ C3 @ D2 )
       => ( ( ( inf_inf_set_option_a @ A6 @ C3 )
            = bot_bot_set_option_a )
         => ( ( ( inf_inf_set_option_a @ B5 @ D2 )
              = bot_bot_set_option_a )
           => ( bij_be5431266891817924854tion_a @ F2 @ A6 @ B5 ) ) ) ) ) ).

% bij_betw_partition
thf(fact_1019_bij__image__Compl__eq,axiom,
    ! [F2: a > a,A6: set_a] :
      ( ( bij_betw_a_a @ F2 @ top_top_set_a @ top_top_set_a )
     => ( ( image_a_a2 @ F2 @ ( uminus_uminus_set_a @ A6 ) )
        = ( uminus_uminus_set_a @ ( image_a_a2 @ F2 @ A6 ) ) ) ) ).

% bij_image_Compl_eq
thf(fact_1020_bij__image__Compl__eq,axiom,
    ! [F2: a > option_a,A6: 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 @ A6 ) )
        = ( uminus6205308855922866075tion_a @ ( image_a_option_a2 @ F2 @ A6 ) ) ) ) ).

% bij_image_Compl_eq
thf(fact_1021_bij__image__Compl__eq,axiom,
    ! [F2: option_a > a,A6: 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 @ A6 ) )
        = ( uminus_uminus_set_a @ ( image_option_a_a2 @ F2 @ A6 ) ) ) ) ).

% bij_image_Compl_eq
thf(fact_1022_bij__image__Compl__eq,axiom,
    ! [F2: option_a > option_a,A6: set_option_a] :
      ( ( bij_be5431266891817924854tion_a @ F2 @ top_top_set_option_a @ top_top_set_option_a )
     => ( ( image_7439109396645324421tion_a @ F2 @ ( uminus6205308855922866075tion_a @ A6 ) )
        = ( uminus6205308855922866075tion_a @ ( image_7439109396645324421tion_a @ F2 @ A6 ) ) ) ) ).

% bij_image_Compl_eq
thf(fact_1023_the__inv__into__comp,axiom,
    ! [F2: a > a,G: option_a > a,A6: set_option_a,X2: a] :
      ( ( inj_on_a_a @ F2 @ ( image_option_a_a2 @ G @ A6 ) )
     => ( ( inj_on_option_a_a @ G @ A6 )
       => ( ( member_a @ X2 @ ( image_a_a2 @ F2 @ ( image_option_a_a2 @ G @ A6 ) ) )
         => ( ( the_in1757154643552616557on_a_a @ A6 @ ( comp_a_a_option_a @ F2 @ G ) @ X2 )
            = ( comp_a_option_a_a @ ( the_in1757154643552616557on_a_a @ A6 @ G ) @ ( the_inv_into_a_a @ ( image_option_a_a2 @ G @ A6 ) @ F2 ) @ X2 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1024_the__inv__into__comp,axiom,
    ! [F2: option_a > a,G: a > option_a,A6: set_a,X2: a] :
      ( ( inj_on_option_a_a @ F2 @ ( image_a_option_a2 @ G @ A6 ) )
     => ( ( inj_on_a_option_a @ G @ A6 )
       => ( ( member_a @ X2 @ ( image_option_a_a2 @ F2 @ ( image_a_option_a2 @ G @ A6 ) ) )
         => ( ( the_inv_into_a_a @ A6 @ ( comp_option_a_a_a @ F2 @ G ) @ X2 )
            = ( comp_option_a_a_a @ ( the_in8758012798868597241tion_a @ A6 @ G ) @ ( the_in1757154643552616557on_a_a @ ( image_a_option_a2 @ G @ A6 ) @ F2 ) @ X2 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1025_the__inv__into__comp,axiom,
    ! [F2: a > produc5278197477302038359_a_c_d,G: option_a > a,A6: set_option_a,X2: produc5278197477302038359_a_c_d] :
      ( ( inj_on2836679365399032650_a_c_d @ F2 @ ( image_option_a_a2 @ G @ A6 ) )
     => ( ( inj_on_option_a_a @ G @ A6 )
       => ( ( member1180172933830803072_a_c_d @ X2 @ ( image_6011757624558158814_a_c_d @ F2 @ ( image_option_a_a2 @ G @ A6 ) ) )
         => ( ( the_in4155257485301462214_a_c_d @ A6 @ ( comp_a2437979404838630664tion_a @ F2 @ G ) @ X2 )
            = ( comp_a1598113356697881198_a_c_d @ ( the_in1757154643552616557on_a_a @ A6 @ G ) @ ( the_in7738805132296125324_a_c_d @ ( image_option_a_a2 @ G @ A6 ) @ F2 ) @ X2 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1026_the__inv__into__comp,axiom,
    ! [F2: option_a > produc5278197477302038359_a_c_d,G: a > option_a,A6: set_a,X2: produc5278197477302038359_a_c_d] :
      ( ( inj_on6212699291050112900_a_c_d @ F2 @ ( image_a_option_a2 @ G @ A6 ) )
     => ( ( inj_on_a_option_a @ G @ A6 )
       => ( ( member1180172933830803072_a_c_d @ X2 @ ( image_8542254255036258840_a_c_d @ F2 @ ( image_a_option_a2 @ G @ A6 ) ) )
         => ( ( the_in7738805132296125324_a_c_d @ A6 @ ( comp_o6118957220454002824_c_d_a @ F2 @ G ) @ X2 )
            = ( comp_o3494184989017889018_a_c_d @ ( the_in8758012798868597241tion_a @ A6 @ G ) @ ( the_in4155257485301462214_a_c_d @ ( image_a_option_a2 @ G @ A6 ) @ F2 ) @ X2 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1027_the__inv__into__comp,axiom,
    ! [F2: option_a > option_a,G: a > option_a,A6: set_a,X2: option_a] :
      ( ( inj_on8559383841115902449tion_a @ F2 @ ( image_a_option_a2 @ G @ A6 ) )
     => ( ( inj_on_a_option_a @ G @ A6 )
       => ( ( member_option_a @ X2 @ ( image_7439109396645324421tion_a @ F2 @ ( image_a_option_a2 @ G @ A6 ) ) )
         => ( ( the_in8758012798868597241tion_a @ A6 @ ( comp_o6087033147929006299on_a_a @ F2 @ G ) @ X2 )
            = ( comp_o3864519266390211175tion_a @ ( the_in8758012798868597241tion_a @ A6 @ G ) @ ( the_in2538339130118444083tion_a @ ( image_a_option_a2 @ G @ A6 ) @ F2 ) @ X2 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1028_the__inv__into__comp,axiom,
    ! [F2: a > option_a,G: option_a > a,A6: set_option_a,X2: option_a] :
      ( ( inj_on_a_option_a @ F2 @ ( image_option_a_a2 @ G @ A6 ) )
     => ( ( inj_on_option_a_a @ G @ A6 )
       => ( ( member_option_a @ X2 @ ( image_a_option_a2 @ F2 @ ( image_option_a_a2 @ G @ A6 ) ) )
         => ( ( the_in2538339130118444083tion_a @ A6 @ ( comp_a6249931511552232923tion_a @ F2 @ G ) @ X2 )
            = ( comp_a6249931511552232923tion_a @ ( the_in1757154643552616557on_a_a @ A6 @ G ) @ ( the_in8758012798868597241tion_a @ ( image_option_a_a2 @ G @ A6 ) @ F2 ) @ X2 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1029_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_1030_comp__the__Some,axiom,
    ( ( comp_o6230065071311576686_a_c_d @ the_Pr7316212213771180927_a_c_d @ some_P1084500821511757806_a_c_d )
    = id_Pro4866527663230331490_a_c_d ) ).

% comp_the_Some
thf(fact_1031_comp__the__Some,axiom,
    ( ( comp_option_a_a_a @ the_a @ some_a )
    = id_a ) ).

% comp_the_Some
thf(fact_1032_Image__singleton__iff,axiom,
    ! [B: option6413918287372586467_a_c_d,R: set_Pr1275464188344874039_a_c_d,A: product_prod_a_c_d] :
      ( ( member5926479794665358074_a_c_d @ B @ ( image_218652559420133323_a_c_d @ R @ ( insert2748062992969704259_a_c_d @ A @ bot_bo110114171885311197_a_c_d ) ) )
      = ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ A @ B ) @ R ) ) ).

% Image_singleton_iff
thf(fact_1033_Image__singleton__iff,axiom,
    ! [B: a,R: set_Product_prod_a_a,A: a] :
      ( ( member_a @ B @ ( image_a_a @ R @ ( insert_a @ A @ bot_bot_set_a ) ) )
      = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R ) ) ).

% Image_singleton_iff
thf(fact_1034_Image__singleton__iff,axiom,
    ! [B: produc5278197477302038359_a_c_d,R: set_Pr6557577845950427232_a_c_d,A: a] :
      ( ( member1180172933830803072_a_c_d @ B @ ( image_5830428142920477612_a_c_d @ R @ ( insert_a @ A @ bot_bot_set_a ) ) )
      = ( member6653950053022509865_a_c_d @ ( produc909605494754173808_a_c_d @ A @ B ) @ R ) ) ).

% Image_singleton_iff
thf(fact_1035_Image__singleton__iff,axiom,
    ! [B: option_a,R: set_Pr3411724424142761165tion_a,A: a] :
      ( ( member_option_a @ B @ ( image_a_option_a @ R @ ( insert_a @ A @ bot_bot_set_a ) ) )
      = ( member6937434987665551382tion_a @ ( produc1224194096085666781tion_a @ A @ B ) @ R ) ) ).

% Image_singleton_iff
thf(fact_1036_Image__singleton__iff,axiom,
    ! [B: c > d,R: set_Pr334309860168054345_a_c_d,A: a] :
      ( ( member_c_d @ B @ ( image_a_c_d @ R @ ( insert_a @ A @ bot_bot_set_a ) ) )
      = ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ A @ B ) @ R ) ) ).

% Image_singleton_iff
thf(fact_1037_Image__singleton__iff,axiom,
    ! [B: a,R: set_Pr6308966090954093121on_a_a,A: option_a] :
      ( ( member_a @ B @ ( image_option_a_a @ R @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) )
      = ( member6056235002698166154on_a_a @ ( produc3446707977624461905on_a_a @ A @ B ) @ R ) ) ).

% Image_singleton_iff
thf(fact_1038_Image__singleton__iff,axiom,
    ! [B: produc5278197477302038359_a_c_d,R: set_Pr2239412299151736218_a_c_d,A: option_a] :
      ( ( member1180172933830803072_a_c_d @ B @ ( image_221249642492245222_a_c_d @ R @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) )
      = ( member8792505911481038691_a_c_d @ ( produc3880585002510677674_a_c_d @ A @ B ) @ R ) ) ).

% Image_singleton_iff
thf(fact_1039_Image__singleton__iff,axiom,
    ! [B: option_a,R: set_Pr7585778909603769095tion_a,A: option_a] :
      ( ( member_option_a @ B @ ( image_4442594622209975379tion_a @ R @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) )
      = ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R ) ) ).

% Image_singleton_iff
thf(fact_1040_id__apply,axiom,
    ( id_a
    = ( ^ [X: a] : X ) ) ).

% id_apply
thf(fact_1041_image__id,axiom,
    ( ( image_a_a2 @ id_a )
    = id_set_a ) ).

% image_id
thf(fact_1042_ImageI,axiom,
    ! [A: a,B: a,R: set_Product_prod_a_a,A6: set_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
     => ( ( member_a @ A @ A6 )
       => ( member_a @ B @ ( image_a_a @ R @ A6 ) ) ) ) ).

% ImageI
thf(fact_1043_ImageI,axiom,
    ! [A: a,B: option_a,R: set_Pr3411724424142761165tion_a,A6: set_a] :
      ( ( member6937434987665551382tion_a @ ( produc1224194096085666781tion_a @ A @ B ) @ R )
     => ( ( member_a @ A @ A6 )
       => ( member_option_a @ B @ ( image_a_option_a @ R @ A6 ) ) ) ) ).

% ImageI
thf(fact_1044_ImageI,axiom,
    ! [A: option_a,B: a,R: set_Pr6308966090954093121on_a_a,A6: set_option_a] :
      ( ( member6056235002698166154on_a_a @ ( produc3446707977624461905on_a_a @ A @ B ) @ R )
     => ( ( member_option_a @ A @ A6 )
       => ( member_a @ B @ ( image_option_a_a @ R @ A6 ) ) ) ) ).

% ImageI
thf(fact_1045_ImageI,axiom,
    ! [A: option_a,B: option_a,R: set_Pr7585778909603769095tion_a,A6: set_option_a] :
      ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R )
     => ( ( member_option_a @ A @ A6 )
       => ( member_option_a @ B @ ( image_4442594622209975379tion_a @ R @ A6 ) ) ) ) ).

% ImageI
thf(fact_1046_ImageI,axiom,
    ! [A: a,B: c > d,R: set_Pr334309860168054345_a_c_d,A6: set_a] :
      ( ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ A @ B ) @ R )
     => ( ( member_a @ A @ A6 )
       => ( member_c_d @ B @ ( image_a_c_d @ R @ A6 ) ) ) ) ).

% ImageI
thf(fact_1047_ImageI,axiom,
    ! [A: product_prod_a_c_d,B: option6413918287372586467_a_c_d,R: set_Pr1275464188344874039_a_c_d,A6: set_Pr334309860168054345_a_c_d] :
      ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ A @ B ) @ R )
     => ( ( member8846983874477766826_a_c_d @ A @ A6 )
       => ( member5926479794665358074_a_c_d @ B @ ( image_218652559420133323_a_c_d @ R @ A6 ) ) ) ) ).

% ImageI
thf(fact_1048_ImageI,axiom,
    ! [A: a,B: produc5278197477302038359_a_c_d,R: set_Pr6557577845950427232_a_c_d,A6: set_a] :
      ( ( member6653950053022509865_a_c_d @ ( produc909605494754173808_a_c_d @ A @ B ) @ R )
     => ( ( member_a @ A @ A6 )
       => ( member1180172933830803072_a_c_d @ B @ ( image_5830428142920477612_a_c_d @ R @ A6 ) ) ) ) ).

% ImageI
thf(fact_1049_ImageI,axiom,
    ! [A: produc5278197477302038359_a_c_d,B: a,R: set_Pr5483263022462721390_c_d_a,A6: set_Pr1275464188344874039_a_c_d] :
      ( ( member1010981217928118071_c_d_a @ ( produc3534377726190287614_c_d_a @ A @ B ) @ R )
     => ( ( member1180172933830803072_a_c_d @ A @ A6 )
       => ( member_a @ B @ ( image_8455200374356591418_c_d_a @ R @ A6 ) ) ) ) ).

% ImageI
thf(fact_1050_ImageI,axiom,
    ! [A: produc5278197477302038359_a_c_d,B: option_a,R: set_Pr3014883941228383924tion_a,A6: set_Pr1275464188344874039_a_c_d] :
      ( ( member4468344759597462909tion_a @ ( produc4720451050651427140tion_a @ A @ B ) @ R )
     => ( ( member1180172933830803072_a_c_d @ A @ A6 )
       => ( member_option_a @ B @ ( image_1061115690632994688tion_a @ R @ A6 ) ) ) ) ).

% ImageI
thf(fact_1051_ImageI,axiom,
    ! [A: option_a,B: produc5278197477302038359_a_c_d,R: set_Pr2239412299151736218_a_c_d,A6: set_option_a] :
      ( ( member8792505911481038691_a_c_d @ ( produc3880585002510677674_a_c_d @ A @ B ) @ R )
     => ( ( member_option_a @ A @ A6 )
       => ( member1180172933830803072_a_c_d @ B @ ( image_221249642492245222_a_c_d @ R @ A6 ) ) ) ) ).

% ImageI
thf(fact_1052_bij__betw__id,axiom,
    ! [A6: set_a] : ( bij_betw_a_a @ id_a @ A6 @ A6 ) ).

% bij_betw_id
thf(fact_1053_vimage__id,axiom,
    ( ( vimage_a_a @ id_a )
    = id_set_a ) ).

% vimage_id
thf(fact_1054_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_1055_bij__betw__id__iff,axiom,
    ! [A6: set_a,B5: set_a] :
      ( ( bij_betw_a_a @ id_a @ A6 @ B5 )
      = ( A6 = B5 ) ) ).

% bij_betw_id_iff
thf(fact_1056_ImageE,axiom,
    ! [B: a,R: set_Product_prod_a_a,A6: set_a] :
      ( ( member_a @ B @ ( image_a_a @ R @ A6 ) )
     => ~ ! [X4: a] :
            ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ B ) @ R )
           => ~ ( member_a @ X4 @ A6 ) ) ) ).

% ImageE
thf(fact_1057_ImageE,axiom,
    ! [B: a,R: set_Pr6308966090954093121on_a_a,A6: set_option_a] :
      ( ( member_a @ B @ ( image_option_a_a @ R @ A6 ) )
     => ~ ! [X4: option_a] :
            ( ( member6056235002698166154on_a_a @ ( produc3446707977624461905on_a_a @ X4 @ B ) @ R )
           => ~ ( member_option_a @ X4 @ A6 ) ) ) ).

% ImageE
thf(fact_1058_ImageE,axiom,
    ! [B: option_a,R: set_Pr3411724424142761165tion_a,A6: set_a] :
      ( ( member_option_a @ B @ ( image_a_option_a @ R @ A6 ) )
     => ~ ! [X4: a] :
            ( ( member6937434987665551382tion_a @ ( produc1224194096085666781tion_a @ X4 @ B ) @ R )
           => ~ ( member_a @ X4 @ A6 ) ) ) ).

% ImageE
thf(fact_1059_ImageE,axiom,
    ! [B: option_a,R: set_Pr7585778909603769095tion_a,A6: set_option_a] :
      ( ( member_option_a @ B @ ( image_4442594622209975379tion_a @ R @ A6 ) )
     => ~ ! [X4: option_a] :
            ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X4 @ B ) @ R )
           => ~ ( member_option_a @ X4 @ A6 ) ) ) ).

% ImageE
thf(fact_1060_ImageE,axiom,
    ! [B: c > d,R: set_Pr334309860168054345_a_c_d,A6: set_a] :
      ( ( member_c_d @ B @ ( image_a_c_d @ R @ A6 ) )
     => ~ ! [X4: a] :
            ( ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ X4 @ B ) @ R )
           => ~ ( member_a @ X4 @ A6 ) ) ) ).

% ImageE
thf(fact_1061_ImageE,axiom,
    ! [B: option6413918287372586467_a_c_d,R: set_Pr1275464188344874039_a_c_d,A6: set_Pr334309860168054345_a_c_d] :
      ( ( member5926479794665358074_a_c_d @ B @ ( image_218652559420133323_a_c_d @ R @ A6 ) )
     => ~ ! [X4: product_prod_a_c_d] :
            ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ X4 @ B ) @ R )
           => ~ ( member8846983874477766826_a_c_d @ X4 @ A6 ) ) ) ).

% ImageE
thf(fact_1062_ImageE,axiom,
    ! [B: a,R: set_Pr5483263022462721390_c_d_a,A6: set_Pr1275464188344874039_a_c_d] :
      ( ( member_a @ B @ ( image_8455200374356591418_c_d_a @ R @ A6 ) )
     => ~ ! [X4: produc5278197477302038359_a_c_d] :
            ( ( member1010981217928118071_c_d_a @ ( produc3534377726190287614_c_d_a @ X4 @ B ) @ R )
           => ~ ( member1180172933830803072_a_c_d @ X4 @ A6 ) ) ) ).

% ImageE
thf(fact_1063_ImageE,axiom,
    ! [B: produc5278197477302038359_a_c_d,R: set_Pr6557577845950427232_a_c_d,A6: set_a] :
      ( ( member1180172933830803072_a_c_d @ B @ ( image_5830428142920477612_a_c_d @ R @ A6 ) )
     => ~ ! [X4: a] :
            ( ( member6653950053022509865_a_c_d @ ( produc909605494754173808_a_c_d @ X4 @ B ) @ R )
           => ~ ( member_a @ X4 @ A6 ) ) ) ).

% ImageE
thf(fact_1064_ImageE,axiom,
    ! [B: produc5278197477302038359_a_c_d,R: set_Pr2239412299151736218_a_c_d,A6: set_option_a] :
      ( ( member1180172933830803072_a_c_d @ B @ ( image_221249642492245222_a_c_d @ R @ A6 ) )
     => ~ ! [X4: option_a] :
            ( ( member8792505911481038691_a_c_d @ ( produc3880585002510677674_a_c_d @ X4 @ B ) @ R )
           => ~ ( member_option_a @ X4 @ A6 ) ) ) ).

% ImageE
thf(fact_1065_ImageE,axiom,
    ! [B: option_a,R: set_Pr3014883941228383924tion_a,A6: set_Pr1275464188344874039_a_c_d] :
      ( ( member_option_a @ B @ ( image_1061115690632994688tion_a @ R @ A6 ) )
     => ~ ! [X4: produc5278197477302038359_a_c_d] :
            ( ( member4468344759597462909tion_a @ ( produc4720451050651427140tion_a @ X4 @ B ) @ R )
           => ~ ( member1180172933830803072_a_c_d @ X4 @ A6 ) ) ) ).

% ImageE
thf(fact_1066_Image__iff,axiom,
    ! [B: option6413918287372586467_a_c_d,R: set_Pr1275464188344874039_a_c_d,A6: set_Pr334309860168054345_a_c_d] :
      ( ( member5926479794665358074_a_c_d @ B @ ( image_218652559420133323_a_c_d @ R @ A6 ) )
      = ( ? [X: product_prod_a_c_d] :
            ( ( member8846983874477766826_a_c_d @ X @ A6 )
            & ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ X @ B ) @ R ) ) ) ) ).

% Image_iff
thf(fact_1067_Image__iff,axiom,
    ! [B: c > d,R: set_Pr334309860168054345_a_c_d,A6: set_a] :
      ( ( member_c_d @ B @ ( image_a_c_d @ R @ A6 ) )
      = ( ? [X: a] :
            ( ( member_a @ X @ A6 )
            & ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ X @ B ) @ R ) ) ) ) ).

% Image_iff
thf(fact_1068_rev__ImageI,axiom,
    ! [A: a,A6: set_a,B: a,R: set_Product_prod_a_a] :
      ( ( member_a @ A @ A6 )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
       => ( member_a @ B @ ( image_a_a @ R @ A6 ) ) ) ) ).

% rev_ImageI
thf(fact_1069_rev__ImageI,axiom,
    ! [A: a,A6: set_a,B: option_a,R: set_Pr3411724424142761165tion_a] :
      ( ( member_a @ A @ A6 )
     => ( ( member6937434987665551382tion_a @ ( produc1224194096085666781tion_a @ A @ B ) @ R )
       => ( member_option_a @ B @ ( image_a_option_a @ R @ A6 ) ) ) ) ).

% rev_ImageI
thf(fact_1070_rev__ImageI,axiom,
    ! [A: option_a,A6: set_option_a,B: a,R: set_Pr6308966090954093121on_a_a] :
      ( ( member_option_a @ A @ A6 )
     => ( ( member6056235002698166154on_a_a @ ( produc3446707977624461905on_a_a @ A @ B ) @ R )
       => ( member_a @ B @ ( image_option_a_a @ R @ A6 ) ) ) ) ).

% rev_ImageI
thf(fact_1071_rev__ImageI,axiom,
    ! [A: option_a,A6: set_option_a,B: option_a,R: set_Pr7585778909603769095tion_a] :
      ( ( member_option_a @ A @ A6 )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R )
       => ( member_option_a @ B @ ( image_4442594622209975379tion_a @ R @ A6 ) ) ) ) ).

% rev_ImageI
thf(fact_1072_rev__ImageI,axiom,
    ! [A: a,A6: set_a,B: c > d,R: set_Pr334309860168054345_a_c_d] :
      ( ( member_a @ A @ A6 )
     => ( ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ A @ B ) @ R )
       => ( member_c_d @ B @ ( image_a_c_d @ R @ A6 ) ) ) ) ).

% rev_ImageI
thf(fact_1073_rev__ImageI,axiom,
    ! [A: product_prod_a_c_d,A6: set_Pr334309860168054345_a_c_d,B: option6413918287372586467_a_c_d,R: set_Pr1275464188344874039_a_c_d] :
      ( ( member8846983874477766826_a_c_d @ A @ A6 )
     => ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ A @ B ) @ R )
       => ( member5926479794665358074_a_c_d @ B @ ( image_218652559420133323_a_c_d @ R @ A6 ) ) ) ) ).

% rev_ImageI
thf(fact_1074_rev__ImageI,axiom,
    ! [A: a,A6: set_a,B: produc5278197477302038359_a_c_d,R: set_Pr6557577845950427232_a_c_d] :
      ( ( member_a @ A @ A6 )
     => ( ( member6653950053022509865_a_c_d @ ( produc909605494754173808_a_c_d @ A @ B ) @ R )
       => ( member1180172933830803072_a_c_d @ B @ ( image_5830428142920477612_a_c_d @ R @ A6 ) ) ) ) ).

% rev_ImageI
thf(fact_1075_rev__ImageI,axiom,
    ! [A: produc5278197477302038359_a_c_d,A6: set_Pr1275464188344874039_a_c_d,B: a,R: set_Pr5483263022462721390_c_d_a] :
      ( ( member1180172933830803072_a_c_d @ A @ A6 )
     => ( ( member1010981217928118071_c_d_a @ ( produc3534377726190287614_c_d_a @ A @ B ) @ R )
       => ( member_a @ B @ ( image_8455200374356591418_c_d_a @ R @ A6 ) ) ) ) ).

% rev_ImageI
thf(fact_1076_rev__ImageI,axiom,
    ! [A: produc5278197477302038359_a_c_d,A6: set_Pr1275464188344874039_a_c_d,B: option_a,R: set_Pr3014883941228383924tion_a] :
      ( ( member1180172933830803072_a_c_d @ A @ A6 )
     => ( ( member4468344759597462909tion_a @ ( produc4720451050651427140tion_a @ A @ B ) @ R )
       => ( member_option_a @ B @ ( image_1061115690632994688tion_a @ R @ A6 ) ) ) ) ).

% rev_ImageI
thf(fact_1077_rev__ImageI,axiom,
    ! [A: option_a,A6: set_option_a,B: produc5278197477302038359_a_c_d,R: set_Pr2239412299151736218_a_c_d] :
      ( ( member_option_a @ A @ A6 )
     => ( ( member8792505911481038691_a_c_d @ ( produc3880585002510677674_a_c_d @ A @ B ) @ R )
       => ( member1180172933830803072_a_c_d @ B @ ( image_221249642492245222_a_c_d @ R @ A6 ) ) ) ) ).

% rev_ImageI
thf(fact_1078_id__def,axiom,
    ( id_a
    = ( ^ [X: a] : X ) ) ).

% id_def
thf(fact_1079_eq__id__iff,axiom,
    ! [F2: a > a] :
      ( ( ! [X: a] :
            ( ( F2 @ X )
            = X ) )
      = ( F2 = id_a ) ) ).

% eq_id_iff
thf(fact_1080_option_Omap__id0,axiom,
    ( ( map_option_a_a @ id_a )
    = id_option_a ) ).

% option.map_id0
thf(fact_1081_option_Omap__id,axiom,
    ! [T: option_a] :
      ( ( map_option_a_a @ id_a @ T )
      = T ) ).

% option.map_id
thf(fact_1082_inj__on__id,axiom,
    ! [A6: set_a] : ( inj_on_a_a @ id_a @ A6 ) ).

% inj_on_id
thf(fact_1083_surj__id,axiom,
    ( ( image_a_a2 @ id_a @ top_top_set_a )
    = top_top_set_a ) ).

% surj_id
thf(fact_1084_surj__id,axiom,
    ( ( image_7439109396645324421tion_a @ id_option_a @ top_top_set_option_a )
    = top_top_set_option_a ) ).

% surj_id
thf(fact_1085_bij__id,axiom,
    bij_betw_a_a @ id_a @ top_top_set_a @ top_top_set_a ).

% bij_id
thf(fact_1086_bij__id,axiom,
    bij_be5431266891817924854tion_a @ id_option_a @ top_top_set_option_a @ top_top_set_option_a ).

% bij_id
thf(fact_1087_subset__Image1__Image1__iff,axiom,
    ! [R: set_Pr3774076249145843911_a_c_d,A: produc5278197477302038359_a_c_d,B: produc5278197477302038359_a_c_d] :
      ( ( order_7509936545869068142_a_c_d @ ( field_5664454651862650532_a_c_d @ R ) @ R )
     => ( ( member1180172933830803072_a_c_d @ A @ ( field_5664454651862650532_a_c_d @ R ) )
       => ( ( member1180172933830803072_a_c_d @ B @ ( field_5664454651862650532_a_c_d @ R ) )
         => ( ( ord_le6392319565089278935_a_c_d @ ( image_5860303855050338195_a_c_d @ R @ ( insert4072338954160766247_a_c_d @ A @ bot_bo4027597737525728931_a_c_d ) ) @ ( image_5860303855050338195_a_c_d @ R @ ( insert4072338954160766247_a_c_d @ B @ bot_bo4027597737525728931_a_c_d ) ) )
            = ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ B @ A ) @ R ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_1088_subset__Image1__Image1__iff,axiom,
    ! [R: set_Product_prod_a_a,A: a,B: a] :
      ( ( order_preorder_on_a @ ( field_a @ R ) @ R )
     => ( ( member_a @ A @ ( field_a @ R ) )
       => ( ( member_a @ B @ ( field_a @ R ) )
         => ( ( ord_less_eq_set_a @ ( image_a_a @ R @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( image_a_a @ R @ ( insert_a @ B @ bot_bot_set_a ) ) )
            = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ A ) @ R ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_1089_subset__Image1__Image1__iff,axiom,
    ! [R: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
      ( ( order_4134995541221112539tion_a @ ( field_option_a @ R ) @ R )
     => ( ( member_option_a @ A @ ( field_option_a @ R ) )
       => ( ( member_option_a @ B @ ( field_option_a @ R ) )
         => ( ( ord_le1955136853071979460tion_a @ ( image_4442594622209975379tion_a @ R @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) @ ( image_4442594622209975379tion_a @ R @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) )
            = ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ A ) @ R ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_1090_equiv__class__nondisjoint,axiom,
    ! [A6: set_Pr1275464188344874039_a_c_d,R: set_Pr3774076249145843911_a_c_d,X2: produc5278197477302038359_a_c_d,A: produc5278197477302038359_a_c_d,B: produc5278197477302038359_a_c_d] :
      ( ( equiv_777496545229528845_a_c_d @ A6 @ R )
     => ( ( member1180172933830803072_a_c_d @ X2 @ ( inf_in8400640451169763877_a_c_d @ ( image_5860303855050338195_a_c_d @ R @ ( insert4072338954160766247_a_c_d @ A @ bot_bo4027597737525728931_a_c_d ) ) @ ( image_5860303855050338195_a_c_d @ R @ ( insert4072338954160766247_a_c_d @ B @ bot_bo4027597737525728931_a_c_d ) ) ) )
       => ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ A @ B ) @ R ) ) ) ).

% equiv_class_nondisjoint
thf(fact_1091_equiv__class__nondisjoint,axiom,
    ! [A6: set_a,R: set_Product_prod_a_a,X2: a,A: a,B: a] :
      ( ( equiv_equiv_a @ A6 @ R )
     => ( ( member_a @ X2 @ ( inf_inf_set_a @ ( image_a_a @ R @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( image_a_a @ R @ ( insert_a @ B @ bot_bot_set_a ) ) ) )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R ) ) ) ).

% equiv_class_nondisjoint
thf(fact_1092_equiv__class__nondisjoint,axiom,
    ! [A6: set_option_a,R: set_Pr7585778909603769095tion_a,X2: option_a,A: option_a,B: option_a] :
      ( ( equiv_equiv_option_a @ A6 @ R )
     => ( ( member_option_a @ X2 @ ( inf_inf_set_option_a @ ( image_4442594622209975379tion_a @ R @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) @ ( image_4442594622209975379tion_a @ R @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) ) )
       => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R ) ) ) ).

% equiv_class_nondisjoint
thf(fact_1093_eq__equiv__class,axiom,
    ! [R: set_Pr3774076249145843911_a_c_d,A: produc5278197477302038359_a_c_d,B: produc5278197477302038359_a_c_d,A6: set_Pr1275464188344874039_a_c_d] :
      ( ( ( image_5860303855050338195_a_c_d @ R @ ( insert4072338954160766247_a_c_d @ A @ bot_bo4027597737525728931_a_c_d ) )
        = ( image_5860303855050338195_a_c_d @ R @ ( insert4072338954160766247_a_c_d @ B @ bot_bo4027597737525728931_a_c_d ) ) )
     => ( ( equiv_777496545229528845_a_c_d @ A6 @ R )
       => ( ( member1180172933830803072_a_c_d @ B @ A6 )
         => ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ A @ B ) @ R ) ) ) ) ).

% eq_equiv_class
thf(fact_1094_eq__equiv__class,axiom,
    ! [R: set_Product_prod_a_a,A: a,B: a,A6: set_a] :
      ( ( ( image_a_a @ R @ ( insert_a @ A @ bot_bot_set_a ) )
        = ( image_a_a @ R @ ( insert_a @ B @ bot_bot_set_a ) ) )
     => ( ( equiv_equiv_a @ A6 @ R )
       => ( ( member_a @ B @ A6 )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R ) ) ) ) ).

% eq_equiv_class
thf(fact_1095_eq__equiv__class,axiom,
    ! [R: set_Pr7585778909603769095tion_a,A: option_a,B: option_a,A6: set_option_a] :
      ( ( ( image_4442594622209975379tion_a @ R @ ( insert_option_a @ A @ bot_bot_set_option_a ) )
        = ( image_4442594622209975379tion_a @ R @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) )
     => ( ( equiv_equiv_option_a @ A6 @ R )
       => ( ( member_option_a @ B @ A6 )
         => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R ) ) ) ) ).

% eq_equiv_class
thf(fact_1096_equiv__class__eq,axiom,
    ! [A6: set_a,R: set_Product_prod_a_a,A: a,B: a] :
      ( ( equiv_equiv_a @ A6 @ R )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
       => ( ( image_a_a @ R @ ( insert_a @ A @ bot_bot_set_a ) )
          = ( image_a_a @ R @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ) ).

% equiv_class_eq
thf(fact_1097_equiv__class__eq,axiom,
    ! [A6: set_option_a,R: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
      ( ( equiv_equiv_option_a @ A6 @ R )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R )
       => ( ( image_4442594622209975379tion_a @ R @ ( insert_option_a @ A @ bot_bot_set_option_a ) )
          = ( image_4442594622209975379tion_a @ R @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) ) ) ) ).

% equiv_class_eq
thf(fact_1098_eq__equiv__class__iff,axiom,
    ! [A6: set_Pr1275464188344874039_a_c_d,R: set_Pr3774076249145843911_a_c_d,X2: produc5278197477302038359_a_c_d,Y: produc5278197477302038359_a_c_d] :
      ( ( equiv_777496545229528845_a_c_d @ A6 @ R )
     => ( ( member1180172933830803072_a_c_d @ X2 @ A6 )
       => ( ( member1180172933830803072_a_c_d @ Y @ A6 )
         => ( ( ( image_5860303855050338195_a_c_d @ R @ ( insert4072338954160766247_a_c_d @ X2 @ bot_bo4027597737525728931_a_c_d ) )
              = ( image_5860303855050338195_a_c_d @ R @ ( insert4072338954160766247_a_c_d @ Y @ bot_bo4027597737525728931_a_c_d ) ) )
            = ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ X2 @ Y ) @ R ) ) ) ) ) ).

% eq_equiv_class_iff
thf(fact_1099_eq__equiv__class__iff,axiom,
    ! [A6: set_a,R: set_Product_prod_a_a,X2: a,Y: a] :
      ( ( equiv_equiv_a @ A6 @ R )
     => ( ( member_a @ X2 @ A6 )
       => ( ( member_a @ Y @ A6 )
         => ( ( ( image_a_a @ R @ ( insert_a @ X2 @ bot_bot_set_a ) )
              = ( image_a_a @ R @ ( insert_a @ Y @ bot_bot_set_a ) ) )
            = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R ) ) ) ) ) ).

% eq_equiv_class_iff
thf(fact_1100_eq__equiv__class__iff,axiom,
    ! [A6: set_option_a,R: set_Pr7585778909603769095tion_a,X2: option_a,Y: option_a] :
      ( ( equiv_equiv_option_a @ A6 @ R )
     => ( ( member_option_a @ X2 @ A6 )
       => ( ( member_option_a @ Y @ A6 )
         => ( ( ( image_4442594622209975379tion_a @ R @ ( insert_option_a @ X2 @ bot_bot_set_option_a ) )
              = ( image_4442594622209975379tion_a @ R @ ( insert_option_a @ Y @ bot_bot_set_option_a ) ) )
            = ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y ) @ R ) ) ) ) ) ).

% eq_equiv_class_iff
thf(fact_1101_equiv__class__eq__iff,axiom,
    ! [A6: set_Pr1275464188344874039_a_c_d,R: set_Pr3774076249145843911_a_c_d,X2: produc5278197477302038359_a_c_d,Y: produc5278197477302038359_a_c_d] :
      ( ( equiv_777496545229528845_a_c_d @ A6 @ R )
     => ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ X2 @ Y ) @ R )
        = ( ( ( image_5860303855050338195_a_c_d @ R @ ( insert4072338954160766247_a_c_d @ X2 @ bot_bo4027597737525728931_a_c_d ) )
            = ( image_5860303855050338195_a_c_d @ R @ ( insert4072338954160766247_a_c_d @ Y @ bot_bo4027597737525728931_a_c_d ) ) )
          & ( member1180172933830803072_a_c_d @ X2 @ A6 )
          & ( member1180172933830803072_a_c_d @ Y @ A6 ) ) ) ) ).

% equiv_class_eq_iff
thf(fact_1102_equiv__class__eq__iff,axiom,
    ! [A6: set_a,R: set_Product_prod_a_a,X2: a,Y: a] :
      ( ( equiv_equiv_a @ A6 @ R )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
        = ( ( ( image_a_a @ R @ ( insert_a @ X2 @ bot_bot_set_a ) )
            = ( image_a_a @ R @ ( insert_a @ Y @ bot_bot_set_a ) ) )
          & ( member_a @ X2 @ A6 )
          & ( member_a @ Y @ A6 ) ) ) ) ).

% equiv_class_eq_iff
thf(fact_1103_equiv__class__eq__iff,axiom,
    ! [A6: set_option_a,R: set_Pr7585778909603769095tion_a,X2: option_a,Y: option_a] :
      ( ( equiv_equiv_option_a @ A6 @ R )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y ) @ R )
        = ( ( ( image_4442594622209975379tion_a @ R @ ( insert_option_a @ X2 @ bot_bot_set_option_a ) )
            = ( image_4442594622209975379tion_a @ R @ ( insert_option_a @ Y @ bot_bot_set_option_a ) ) )
          & ( member_option_a @ X2 @ A6 )
          & ( member_option_a @ Y @ A6 ) ) ) ) ).

% equiv_class_eq_iff
thf(fact_1104_equiv__class__subset,axiom,
    ! [A6: set_a,R: set_Product_prod_a_a,A: a,B: a] :
      ( ( equiv_equiv_a @ A6 @ R )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
       => ( ord_less_eq_set_a @ ( image_a_a @ R @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( image_a_a @ R @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ) ).

% equiv_class_subset
thf(fact_1105_equiv__class__subset,axiom,
    ! [A6: set_option_a,R: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
      ( ( equiv_equiv_option_a @ A6 @ R )
     => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R )
       => ( ord_le1955136853071979460tion_a @ ( image_4442594622209975379tion_a @ R @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) @ ( image_4442594622209975379tion_a @ R @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) ) ) ) ).

% equiv_class_subset
thf(fact_1106_subset__equiv__class,axiom,
    ! [A6: set_Pr1275464188344874039_a_c_d,R: set_Pr3774076249145843911_a_c_d,B: produc5278197477302038359_a_c_d,A: produc5278197477302038359_a_c_d] :
      ( ( equiv_777496545229528845_a_c_d @ A6 @ R )
     => ( ( ord_le6392319565089278935_a_c_d @ ( image_5860303855050338195_a_c_d @ R @ ( insert4072338954160766247_a_c_d @ B @ bot_bo4027597737525728931_a_c_d ) ) @ ( image_5860303855050338195_a_c_d @ R @ ( insert4072338954160766247_a_c_d @ A @ bot_bo4027597737525728931_a_c_d ) ) )
       => ( ( member1180172933830803072_a_c_d @ B @ A6 )
         => ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ A @ B ) @ R ) ) ) ) ).

% subset_equiv_class
thf(fact_1107_subset__equiv__class,axiom,
    ! [A6: set_a,R: set_Product_prod_a_a,B: a,A: a] :
      ( ( equiv_equiv_a @ A6 @ R )
     => ( ( ord_less_eq_set_a @ ( image_a_a @ R @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( image_a_a @ R @ ( insert_a @ A @ bot_bot_set_a ) ) )
       => ( ( member_a @ B @ A6 )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R ) ) ) ) ).

% subset_equiv_class
thf(fact_1108_subset__equiv__class,axiom,
    ! [A6: set_option_a,R: set_Pr7585778909603769095tion_a,B: option_a,A: option_a] :
      ( ( equiv_equiv_option_a @ A6 @ R )
     => ( ( ord_le1955136853071979460tion_a @ ( image_4442594622209975379tion_a @ R @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( image_4442594622209975379tion_a @ R @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) )
       => ( ( member_option_a @ B @ A6 )
         => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R ) ) ) ) ).

% subset_equiv_class
thf(fact_1109_proj__iff,axiom,
    ! [A6: set_a,R: set_Product_prod_a_a,X2: a,Y: a] :
      ( ( equiv_equiv_a @ A6 @ R )
     => ( ( ord_less_eq_set_a @ ( insert_a @ X2 @ ( insert_a @ Y @ bot_bot_set_a ) ) @ A6 )
       => ( ( ( equiv_proj_a_a @ R @ X2 )
            = ( equiv_proj_a_a @ R @ Y ) )
          = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R ) ) ) ) ).

% proj_iff
thf(fact_1110_proj__iff,axiom,
    ! [A6: set_option_a,R: set_Pr7585778909603769095tion_a,X2: option_a,Y: option_a] :
      ( ( equiv_equiv_option_a @ A6 @ R )
     => ( ( ord_le1955136853071979460tion_a @ ( insert_option_a @ X2 @ ( insert_option_a @ Y @ bot_bot_set_option_a ) ) @ A6 )
       => ( ( ( equiv_6865337221296424970tion_a @ R @ X2 )
            = ( equiv_6865337221296424970tion_a @ R @ Y ) )
          = ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y ) @ R ) ) ) ) ).

% proj_iff
thf(fact_1111_disjnt__equiv__class,axiom,
    ! [A6: set_a,R: set_Product_prod_a_a,A: a,B: a] :
      ( ( equiv_equiv_a @ A6 @ R )
     => ( ( disjnt_a @ ( image_a_a @ R @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( image_a_a @ R @ ( insert_a @ B @ bot_bot_set_a ) ) )
        = ( ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R ) ) ) ) ).

% disjnt_equiv_class
thf(fact_1112_disjnt__equiv__class,axiom,
    ! [A6: set_option_a,R: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
      ( ( equiv_equiv_option_a @ A6 @ R )
     => ( ( disjnt_option_a @ ( image_4442594622209975379tion_a @ R @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) @ ( image_4442594622209975379tion_a @ R @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) )
        = ( ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R ) ) ) ) ).

% disjnt_equiv_class
thf(fact_1113_in__quotient__imp__in__rel,axiom,
    ! [A6: set_a,R: set_Product_prod_a_a,X6: set_a,X2: a,Y: a] :
      ( ( equiv_equiv_a @ A6 @ R )
     => ( ( member_set_a @ X6 @ ( equiv_quotient_a @ A6 @ R ) )
       => ( ( ord_less_eq_set_a @ ( insert_a @ X2 @ ( insert_a @ Y @ bot_bot_set_a ) ) @ X6 )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R ) ) ) ) ).

% in_quotient_imp_in_rel
thf(fact_1114_in__quotient__imp__in__rel,axiom,
    ! [A6: set_option_a,R: set_Pr7585778909603769095tion_a,X6: set_option_a,X2: option_a,Y: option_a] :
      ( ( equiv_equiv_option_a @ A6 @ R )
     => ( ( member_set_option_a @ X6 @ ( equiv_2859340374733651339tion_a @ A6 @ R ) )
       => ( ( ord_le1955136853071979460tion_a @ ( insert_option_a @ X2 @ ( insert_option_a @ Y @ bot_bot_set_option_a ) ) @ X6 )
         => ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y ) @ R ) ) ) ) ).

% in_quotient_imp_in_rel
thf(fact_1115_in__quotient__imp__closed,axiom,
    ! [A6: set_a,R: set_Product_prod_a_a,X6: set_a,X2: a,Y: a] :
      ( ( equiv_equiv_a @ A6 @ R )
     => ( ( member_set_a @ X6 @ ( equiv_quotient_a @ A6 @ R ) )
       => ( ( member_a @ X2 @ X6 )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
           => ( member_a @ Y @ X6 ) ) ) ) ) ).

% in_quotient_imp_closed
thf(fact_1116_in__quotient__imp__closed,axiom,
    ! [A6: set_Pr1275464188344874039_a_c_d,R: set_Pr3774076249145843911_a_c_d,X6: set_Pr1275464188344874039_a_c_d,X2: produc5278197477302038359_a_c_d,Y: produc5278197477302038359_a_c_d] :
      ( ( equiv_777496545229528845_a_c_d @ A6 @ R )
     => ( ( member1125768928354541408_a_c_d @ X6 @ ( equiv_3313360957152018974_a_c_d @ A6 @ R ) )
       => ( ( member1180172933830803072_a_c_d @ X2 @ X6 )
         => ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ X2 @ Y ) @ R )
           => ( member1180172933830803072_a_c_d @ Y @ X6 ) ) ) ) ) ).

% in_quotient_imp_closed
thf(fact_1117_in__quotient__imp__closed,axiom,
    ! [A6: set_option_a,R: set_Pr7585778909603769095tion_a,X6: set_option_a,X2: option_a,Y: option_a] :
      ( ( equiv_equiv_option_a @ A6 @ R )
     => ( ( member_set_option_a @ X6 @ ( equiv_2859340374733651339tion_a @ A6 @ R ) )
       => ( ( member_option_a @ X2 @ X6 )
         => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y ) @ R )
           => ( member_option_a @ Y @ X6 ) ) ) ) ) ).

% in_quotient_imp_closed
thf(fact_1118_quotient__eq__iff,axiom,
    ! [A6: set_a,R: set_Product_prod_a_a,X6: set_a,Y6: set_a,X2: a,Y: a] :
      ( ( equiv_equiv_a @ A6 @ R )
     => ( ( member_set_a @ X6 @ ( equiv_quotient_a @ A6 @ R ) )
       => ( ( member_set_a @ Y6 @ ( equiv_quotient_a @ A6 @ R ) )
         => ( ( member_a @ X2 @ X6 )
           => ( ( member_a @ Y @ Y6 )
             => ( ( X6 = Y6 )
                = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R ) ) ) ) ) ) ) ).

% quotient_eq_iff
thf(fact_1119_quotient__eq__iff,axiom,
    ! [A6: set_Pr1275464188344874039_a_c_d,R: set_Pr3774076249145843911_a_c_d,X6: set_Pr1275464188344874039_a_c_d,Y6: set_Pr1275464188344874039_a_c_d,X2: produc5278197477302038359_a_c_d,Y: produc5278197477302038359_a_c_d] :
      ( ( equiv_777496545229528845_a_c_d @ A6 @ R )
     => ( ( member1125768928354541408_a_c_d @ X6 @ ( equiv_3313360957152018974_a_c_d @ A6 @ R ) )
       => ( ( member1125768928354541408_a_c_d @ Y6 @ ( equiv_3313360957152018974_a_c_d @ A6 @ R ) )
         => ( ( member1180172933830803072_a_c_d @ X2 @ X6 )
           => ( ( member1180172933830803072_a_c_d @ Y @ Y6 )
             => ( ( X6 = Y6 )
                = ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ X2 @ Y ) @ R ) ) ) ) ) ) ) ).

% quotient_eq_iff
thf(fact_1120_quotient__eq__iff,axiom,
    ! [A6: set_option_a,R: set_Pr7585778909603769095tion_a,X6: set_option_a,Y6: set_option_a,X2: option_a,Y: option_a] :
      ( ( equiv_equiv_option_a @ A6 @ R )
     => ( ( member_set_option_a @ X6 @ ( equiv_2859340374733651339tion_a @ A6 @ R ) )
       => ( ( member_set_option_a @ Y6 @ ( equiv_2859340374733651339tion_a @ A6 @ R ) )
         => ( ( member_option_a @ X2 @ X6 )
           => ( ( member_option_a @ Y @ Y6 )
             => ( ( X6 = Y6 )
                = ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y ) @ R ) ) ) ) ) ) ) ).

% quotient_eq_iff
thf(fact_1121_quotient__eqI,axiom,
    ! [A6: set_a,R: set_Product_prod_a_a,X6: set_a,Y6: set_a,X2: a,Y: a] :
      ( ( equiv_equiv_a @ A6 @ R )
     => ( ( member_set_a @ X6 @ ( equiv_quotient_a @ A6 @ R ) )
       => ( ( member_set_a @ Y6 @ ( equiv_quotient_a @ A6 @ R ) )
         => ( ( member_a @ X2 @ X6 )
           => ( ( member_a @ Y @ Y6 )
             => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
               => ( X6 = Y6 ) ) ) ) ) ) ) ).

% quotient_eqI
thf(fact_1122_quotient__eqI,axiom,
    ! [A6: set_Pr1275464188344874039_a_c_d,R: set_Pr3774076249145843911_a_c_d,X6: set_Pr1275464188344874039_a_c_d,Y6: set_Pr1275464188344874039_a_c_d,X2: produc5278197477302038359_a_c_d,Y: produc5278197477302038359_a_c_d] :
      ( ( equiv_777496545229528845_a_c_d @ A6 @ R )
     => ( ( member1125768928354541408_a_c_d @ X6 @ ( equiv_3313360957152018974_a_c_d @ A6 @ R ) )
       => ( ( member1125768928354541408_a_c_d @ Y6 @ ( equiv_3313360957152018974_a_c_d @ A6 @ R ) )
         => ( ( member1180172933830803072_a_c_d @ X2 @ X6 )
           => ( ( member1180172933830803072_a_c_d @ Y @ Y6 )
             => ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ X2 @ Y ) @ R )
               => ( X6 = Y6 ) ) ) ) ) ) ) ).

% quotient_eqI
thf(fact_1123_quotient__eqI,axiom,
    ! [A6: set_option_a,R: set_Pr7585778909603769095tion_a,X6: set_option_a,Y6: set_option_a,X2: option_a,Y: option_a] :
      ( ( equiv_equiv_option_a @ A6 @ R )
     => ( ( member_set_option_a @ X6 @ ( equiv_2859340374733651339tion_a @ A6 @ R ) )
       => ( ( member_set_option_a @ Y6 @ ( equiv_2859340374733651339tion_a @ A6 @ R ) )
         => ( ( member_option_a @ X2 @ X6 )
           => ( ( member_option_a @ Y @ Y6 )
             => ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y ) @ R )
               => ( X6 = Y6 ) ) ) ) ) ) ) ).

% quotient_eqI
thf(fact_1124_eq__equiv__class__iff2,axiom,
    ! [A6: set_Pr1275464188344874039_a_c_d,R: set_Pr3774076249145843911_a_c_d,X2: produc5278197477302038359_a_c_d,Y: produc5278197477302038359_a_c_d] :
      ( ( equiv_777496545229528845_a_c_d @ A6 @ R )
     => ( ( member1180172933830803072_a_c_d @ X2 @ A6 )
       => ( ( member1180172933830803072_a_c_d @ Y @ A6 )
         => ( ( ( equiv_3313360957152018974_a_c_d @ ( insert4072338954160766247_a_c_d @ X2 @ bot_bo4027597737525728931_a_c_d ) @ R )
              = ( equiv_3313360957152018974_a_c_d @ ( insert4072338954160766247_a_c_d @ Y @ bot_bo4027597737525728931_a_c_d ) @ R ) )
            = ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ X2 @ Y ) @ R ) ) ) ) ) ).

% eq_equiv_class_iff2
thf(fact_1125_eq__equiv__class__iff2,axiom,
    ! [A6: set_a,R: set_Product_prod_a_a,X2: a,Y: a] :
      ( ( equiv_equiv_a @ A6 @ R )
     => ( ( member_a @ X2 @ A6 )
       => ( ( member_a @ Y @ A6 )
         => ( ( ( equiv_quotient_a @ ( insert_a @ X2 @ bot_bot_set_a ) @ R )
              = ( equiv_quotient_a @ ( insert_a @ Y @ bot_bot_set_a ) @ R ) )
            = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R ) ) ) ) ) ).

% eq_equiv_class_iff2
thf(fact_1126_eq__equiv__class__iff2,axiom,
    ! [A6: set_option_a,R: set_Pr7585778909603769095tion_a,X2: option_a,Y: option_a] :
      ( ( equiv_equiv_option_a @ A6 @ R )
     => ( ( member_option_a @ X2 @ A6 )
       => ( ( member_option_a @ Y @ A6 )
         => ( ( ( equiv_2859340374733651339tion_a @ ( insert_option_a @ X2 @ bot_bot_set_option_a ) @ R )
              = ( equiv_2859340374733651339tion_a @ ( insert_option_a @ Y @ bot_bot_set_option_a ) @ R ) )
            = ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y ) @ R ) ) ) ) ) ).

% eq_equiv_class_iff2
thf(fact_1127_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_1128_map__of__Cons__code_I2_J,axiom,
    ! [L: product_prod_a_c_d,K: product_prod_a_c_d,V3: option6413918287372586467_a_c_d,Ps: list_P1929962060191741789_a_c_d] :
      ( ( ( L = K )
       => ( ( map_of5572901299533018075_a_c_d @ ( cons_P4841026258484161293_a_c_d @ ( produc8093790510458973071_a_c_d @ L @ V3 ) @ Ps ) @ K )
          = ( some_o8886552288663794750_a_c_d @ V3 ) ) )
      & ( ( L != K )
       => ( ( map_of5572901299533018075_a_c_d @ ( cons_P4841026258484161293_a_c_d @ ( produc8093790510458973071_a_c_d @ L @ V3 ) @ Ps ) @ K )
          = ( map_of5572901299533018075_a_c_d @ Ps @ K ) ) ) ) ).

% map_of_Cons_code(2)
thf(fact_1129_map__of__Cons__code_I2_J,axiom,
    ! [L: a,K: a,V3: c > d,Ps: list_P2433834481846699939_a_c_d] :
      ( ( ( L = K )
       => ( ( map_of_a_c_d @ ( cons_P1689682164644567517_a_c_d @ ( product_Pair_a_c_d @ L @ V3 ) @ Ps ) @ K )
          = ( some_c_d @ V3 ) ) )
      & ( ( L != K )
       => ( ( map_of_a_c_d @ ( cons_P1689682164644567517_a_c_d @ ( product_Pair_a_c_d @ L @ V3 ) @ Ps ) @ K )
          = ( map_of_a_c_d @ Ps @ K ) ) ) ) ).

% map_of_Cons_code(2)
thf(fact_1130_single__valuedD,axiom,
    ! [R: set_Pr1275464188344874039_a_c_d,X2: product_prod_a_c_d,Y: option6413918287372586467_a_c_d,Z2: option6413918287372586467_a_c_d] :
      ( ( single1503972237520253819_a_c_d @ R )
     => ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ X2 @ Y ) @ R )
       => ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ X2 @ Z2 ) @ R )
         => ( Y = Z2 ) ) ) ) ).

% single_valuedD
thf(fact_1131_single__valuedD,axiom,
    ! [R: set_Pr334309860168054345_a_c_d,X2: a,Y: c > d,Z2: c > d] :
      ( ( single_valued_a_c_d @ R )
     => ( ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ X2 @ Y ) @ R )
       => ( ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ X2 @ Z2 ) @ R )
         => ( Y = Z2 ) ) ) ) ).

% single_valuedD
thf(fact_1132_single__valuedI,axiom,
    ! [R: set_Pr1275464188344874039_a_c_d] :
      ( ! [X4: product_prod_a_c_d,Y4: option6413918287372586467_a_c_d,Z3: option6413918287372586467_a_c_d] :
          ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ X4 @ Y4 ) @ R )
         => ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ X4 @ Z3 ) @ R )
           => ( Y4 = Z3 ) ) )
     => ( single1503972237520253819_a_c_d @ R ) ) ).

% single_valuedI
thf(fact_1133_single__valuedI,axiom,
    ! [R: set_Pr334309860168054345_a_c_d] :
      ( ! [X4: a,Y4: c > d,Z3: c > d] :
          ( ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ X4 @ Y4 ) @ R )
         => ( ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ X4 @ Z3 ) @ R )
           => ( Y4 = Z3 ) ) )
     => ( single_valued_a_c_d @ R ) ) ).

% single_valuedI
thf(fact_1134_single__valued__def,axiom,
    ( single1503972237520253819_a_c_d
    = ( ^ [R2: set_Pr1275464188344874039_a_c_d] :
        ! [X: product_prod_a_c_d,Y3: option6413918287372586467_a_c_d] :
          ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ X @ Y3 ) @ R2 )
         => ! [Z: option6413918287372586467_a_c_d] :
              ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ X @ Z ) @ R2 )
             => ( Y3 = Z ) ) ) ) ) ).

% single_valued_def
thf(fact_1135_single__valued__def,axiom,
    ( single_valued_a_c_d
    = ( ^ [R2: set_Pr334309860168054345_a_c_d] :
        ! [X: a,Y3: c > d] :
          ( ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ X @ Y3 ) @ R2 )
         => ! [Z: c > d] :
              ( ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ X @ Z ) @ R2 )
             => ( Y3 = Z ) ) ) ) ) ).

% single_valued_def
thf(fact_1136_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_1137_map__upds__Cons,axiom,
    ! [M: a > option6413918287372586467_a_c_d,A: a,As: list_a,B: product_prod_a_c_d,Bs: list_P2433834481846699939_a_c_d] :
      ( ( map_up6617793535811126555_a_c_d @ M @ ( cons_a @ A @ As ) @ ( cons_P1689682164644567517_a_c_d @ B @ Bs ) )
      = ( map_up6617793535811126555_a_c_d @ ( fun_up6642239869823773348_a_c_d @ M @ A @ ( some_P1084500821511757806_a_c_d @ B ) ) @ As @ Bs ) ) ).

% map_upds_Cons
thf(fact_1138_map__of__SomeD,axiom,
    ! [Xs: list_P1929962060191741789_a_c_d,K: product_prod_a_c_d,Y: option6413918287372586467_a_c_d] :
      ( ( ( map_of5572901299533018075_a_c_d @ Xs @ K )
        = ( some_o8886552288663794750_a_c_d @ Y ) )
     => ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ K @ Y ) @ ( set_Pr8461946692310444268_a_c_d @ Xs ) ) ) ).

% map_of_SomeD
thf(fact_1139_map__of__SomeD,axiom,
    ! [Xs: list_P2433834481846699939_a_c_d,K: a,Y: c > d] :
      ( ( ( map_of_a_c_d @ Xs @ K )
        = ( some_c_d @ Y ) )
     => ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ K @ Y ) @ ( set_Pr2012686464001265470_a_c_d @ Xs ) ) ) ).

% map_of_SomeD
thf(fact_1140_weak__map__of__SomeI,axiom,
    ! [K: product_prod_a_c_d,X2: option6413918287372586467_a_c_d,L: list_P1929962060191741789_a_c_d] :
      ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ K @ X2 ) @ ( set_Pr8461946692310444268_a_c_d @ L ) )
     => ? [X4: option6413918287372586467_a_c_d] :
          ( ( map_of5572901299533018075_a_c_d @ L @ K )
          = ( some_o8886552288663794750_a_c_d @ X4 ) ) ) ).

% weak_map_of_SomeI
thf(fact_1141_weak__map__of__SomeI,axiom,
    ! [K: a,X2: c > d,L: list_P2433834481846699939_a_c_d] :
      ( ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ K @ X2 ) @ ( set_Pr2012686464001265470_a_c_d @ L ) )
     => ? [X4: c > d] :
          ( ( map_of_a_c_d @ L @ K )
          = ( some_c_d @ X4 ) ) ) ).

% weak_map_of_SomeI
thf(fact_1142_map__upds__twist,axiom,
    ! [A: produc5278197477302038359_a_c_d,As: list_P1929962060191741789_a_c_d,M: produc5278197477302038359_a_c_d > option_a,B: a,Bs: list_a] :
      ( ~ ( member1180172933830803072_a_c_d @ A @ ( set_Pr8461946692310444268_a_c_d @ As ) )
     => ( ( map_up87012832410759389_c_d_a @ ( fun_up7463025390225626410tion_a @ M @ A @ ( some_a @ B ) ) @ As @ Bs )
        = ( fun_up7463025390225626410tion_a @ ( map_up87012832410759389_c_d_a @ M @ As @ Bs ) @ A @ ( some_a @ B ) ) ) ) ).

% map_upds_twist
thf(fact_1143_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_1144_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_1145_map__upds__twist,axiom,
    ! [A: produc5278197477302038359_a_c_d,As: list_P1929962060191741789_a_c_d,M: produc5278197477302038359_a_c_d > option6413918287372586467_a_c_d,B: product_prod_a_c_d,Bs: list_P2433834481846699939_a_c_d] :
      ( ~ ( member1180172933830803072_a_c_d @ A @ ( set_Pr8461946692310444268_a_c_d @ As ) )
     => ( ( map_up1324058162482958964_a_c_d @ ( fun_up4007221454163596925_a_c_d @ M @ A @ ( some_P1084500821511757806_a_c_d @ B ) ) @ As @ Bs )
        = ( fun_up4007221454163596925_a_c_d @ ( map_up1324058162482958964_a_c_d @ M @ As @ Bs ) @ A @ ( some_P1084500821511757806_a_c_d @ B ) ) ) ) ).

% map_upds_twist
thf(fact_1146_map__upds__twist,axiom,
    ! [A: option_a,As: list_option_a,M: option_a > option6413918287372586467_a_c_d,B: product_prod_a_c_d,Bs: list_P2433834481846699939_a_c_d] :
      ( ~ ( member_option_a @ A @ ( set_option_a2 @ As ) )
     => ( ( map_up2305867519919030241_a_c_d @ ( fun_up5969202740897331946_a_c_d @ M @ A @ ( some_P1084500821511757806_a_c_d @ B ) ) @ As @ Bs )
        = ( fun_up5969202740897331946_a_c_d @ ( map_up2305867519919030241_a_c_d @ M @ As @ Bs ) @ A @ ( some_P1084500821511757806_a_c_d @ B ) ) ) ) ).

% map_upds_twist
thf(fact_1147_map__upds__twist,axiom,
    ! [A: a,As: list_a,M: a > option6413918287372586467_a_c_d,B: product_prod_a_c_d,Bs: list_P2433834481846699939_a_c_d] :
      ( ~ ( member_a @ A @ ( set_a2 @ As ) )
     => ( ( map_up6617793535811126555_a_c_d @ ( fun_up6642239869823773348_a_c_d @ M @ A @ ( some_P1084500821511757806_a_c_d @ B ) ) @ As @ Bs )
        = ( fun_up6642239869823773348_a_c_d @ ( map_up6617793535811126555_a_c_d @ M @ As @ Bs ) @ A @ ( some_P1084500821511757806_a_c_d @ B ) ) ) ) ).

% map_upds_twist
thf(fact_1148_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_1149_fst__eqD,axiom,
    ! [X2: product_prod_a_c_d,Y: option6413918287372586467_a_c_d,A: product_prod_a_c_d] :
      ( ( ( produc7453127856291044731_a_c_d @ ( produc8093790510458973071_a_c_d @ X2 @ Y ) )
        = A )
     => ( X2 = A ) ) ).

% fst_eqD
thf(fact_1150_fst__eqD,axiom,
    ! [X2: a,Y: c > d,A: a] :
      ( ( ( product_fst_a_c_d @ ( product_Pair_a_c_d @ X2 @ Y ) )
        = A )
     => ( X2 = A ) ) ).

% fst_eqD
thf(fact_1151_fst__conv,axiom,
    ! [X1: product_prod_a_c_d,X22: option6413918287372586467_a_c_d] :
      ( ( produc7453127856291044731_a_c_d @ ( produc8093790510458973071_a_c_d @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_1152_fst__conv,axiom,
    ! [X1: a,X22: c > d] :
      ( ( product_fst_a_c_d @ ( product_Pair_a_c_d @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_1153_graph__domD,axiom,
    ! [X2: produc5278197477302038359_a_c_d,M: product_prod_a_c_d > option6212624336325788467_a_c_d] :
      ( ( member1180172933830803072_a_c_d @ X2 @ ( graph_7244981646656495094_a_c_d @ M ) )
     => ( member8846983874477766826_a_c_d @ ( produc7453127856291044731_a_c_d @ X2 ) @ ( dom_Pr8274521719316492234_a_c_d @ M ) ) ) ).

% graph_domD
thf(fact_1154_map__of__eq__None__iff,axiom,
    ! [Xys: list_P1396940483166286381od_a_a,X2: a] :
      ( ( ( map_of_a_a @ Xys @ X2 )
        = none_a )
      = ( ~ ( member_a @ X2 @ ( image_3437945252899457948_a_a_a @ product_fst_a_a @ ( set_Product_prod_a_a2 @ Xys ) ) ) ) ) ).

% map_of_eq_None_iff
thf(fact_1155_map__of__eq__None__iff,axiom,
    ! [Xys: list_P4266695021714950804_c_d_a,X2: produc5278197477302038359_a_c_d] :
      ( ( ( map_of5307797687691286346_c_d_a @ Xys @ X2 )
        = none_a )
      = ( ~ ( member1180172933830803072_a_c_d @ X2 @ ( image_4649473323379164110_a_c_d @ produc2935319049184305386_c_d_a @ ( set_Pr446018216176770467_c_d_a @ Xys ) ) ) ) ) ).

% map_of_eq_None_iff
thf(fact_1156_map__of__eq__None__iff,axiom,
    ! [Xys: list_P6260409590414597735on_a_a,X2: option_a] :
      ( ( ( map_of_option_a_a @ Xys @ X2 )
        = none_a )
      = ( ~ ( member_option_a @ X2 @ ( image_3098826861768462248tion_a @ produc8941638570267940413on_a_a @ ( set_Pr1233600038994746358on_a_a @ Xys ) ) ) ) ) ).

% map_of_eq_None_iff
thf(fact_1157_eq__fst__iff,axiom,
    ! [A: product_prod_a_c_d,P: produc5278197477302038359_a_c_d] :
      ( ( A
        = ( produc7453127856291044731_a_c_d @ P ) )
      = ( ? [B4: option6413918287372586467_a_c_d] :
            ( P
            = ( produc8093790510458973071_a_c_d @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_1158_eq__fst__iff,axiom,
    ! [A: a,P: product_prod_a_c_d] :
      ( ( A
        = ( product_fst_a_c_d @ P ) )
      = ( ? [B4: c > d] :
            ( P
            = ( product_Pair_a_c_d @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_1159_fstI,axiom,
    ! [X2: produc5278197477302038359_a_c_d,Y: product_prod_a_c_d,Z2: option6413918287372586467_a_c_d] :
      ( ( X2
        = ( produc8093790510458973071_a_c_d @ Y @ Z2 ) )
     => ( ( produc7453127856291044731_a_c_d @ X2 )
        = Y ) ) ).

% fstI
thf(fact_1160_fstI,axiom,
    ! [X2: product_prod_a_c_d,Y: a,Z2: c > d] :
      ( ( X2
        = ( product_Pair_a_c_d @ Y @ Z2 ) )
     => ( ( product_fst_a_c_d @ X2 )
        = Y ) ) ).

% fstI
thf(fact_1161_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_1162_prod_Ocollapse,axiom,
    ! [Prod: produc5278197477302038359_a_c_d] :
      ( ( produc8093790510458973071_a_c_d @ ( produc7453127856291044731_a_c_d @ Prod ) @ ( produc196187098351132349_a_c_d @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_1163_prod_Ocollapse,axiom,
    ! [Prod: product_prod_a_c_d] :
      ( ( product_Pair_a_c_d @ ( product_fst_a_c_d @ Prod ) @ ( product_snd_a_c_d @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_1164_sndI,axiom,
    ! [X2: produc5278197477302038359_a_c_d,Y: product_prod_a_c_d,Z2: option6413918287372586467_a_c_d] :
      ( ( X2
        = ( produc8093790510458973071_a_c_d @ Y @ Z2 ) )
     => ( ( produc196187098351132349_a_c_d @ X2 )
        = Z2 ) ) ).

% sndI
thf(fact_1165_sndI,axiom,
    ! [X2: product_prod_a_c_d,Y: a,Z2: c > d] :
      ( ( X2
        = ( product_Pair_a_c_d @ Y @ Z2 ) )
     => ( ( product_snd_a_c_d @ X2 )
        = Z2 ) ) ).

% sndI
thf(fact_1166_eq__snd__iff,axiom,
    ! [B: option6413918287372586467_a_c_d,P: produc5278197477302038359_a_c_d] :
      ( ( B
        = ( produc196187098351132349_a_c_d @ P ) )
      = ( ? [A5: product_prod_a_c_d] :
            ( P
            = ( produc8093790510458973071_a_c_d @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_1167_eq__snd__iff,axiom,
    ! [B: c > d,P: product_prod_a_c_d] :
      ( ( B
        = ( product_snd_a_c_d @ P ) )
      = ( ? [A5: a] :
            ( P
            = ( product_Pair_a_c_d @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_1168_snd__eqD,axiom,
    ! [X2: product_prod_a_c_d,Y: option6413918287372586467_a_c_d,A: option6413918287372586467_a_c_d] :
      ( ( ( produc196187098351132349_a_c_d @ ( produc8093790510458973071_a_c_d @ X2 @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_1169_snd__eqD,axiom,
    ! [X2: a,Y: c > d,A: c > d] :
      ( ( ( product_snd_a_c_d @ ( product_Pair_a_c_d @ X2 @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_1170_snd__conv,axiom,
    ! [X1: product_prod_a_c_d,X22: option6413918287372586467_a_c_d] :
      ( ( produc196187098351132349_a_c_d @ ( produc8093790510458973071_a_c_d @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_1171_snd__conv,axiom,
    ! [X1: a,X22: c > d] :
      ( ( product_snd_a_c_d @ ( product_Pair_a_c_d @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_1172_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: product_prod_a_c_d > option6413918287372586467_a_c_d > $o,X2: product_prod_a_c_d,Y: option6413918287372586467_a_c_d,A: produc5278197477302038359_a_c_d] :
      ( ( P2 @ X2 @ Y )
     => ( ( A
          = ( produc8093790510458973071_a_c_d @ X2 @ Y ) )
       => ( P2 @ ( produc7453127856291044731_a_c_d @ A ) @ ( produc196187098351132349_a_c_d @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1173_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: a > ( c > d ) > $o,X2: a,Y: c > d,A: product_prod_a_c_d] :
      ( ( P2 @ X2 @ Y )
     => ( ( A
          = ( product_Pair_a_c_d @ X2 @ Y ) )
       => ( P2 @ ( product_fst_a_c_d @ A ) @ ( product_snd_a_c_d @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1174_prod_Oexhaust__sel,axiom,
    ! [Prod: produc5278197477302038359_a_c_d] :
      ( Prod
      = ( produc8093790510458973071_a_c_d @ ( produc7453127856291044731_a_c_d @ Prod ) @ ( produc196187098351132349_a_c_d @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_1175_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_a_c_d] :
      ( Prod
      = ( product_Pair_a_c_d @ ( product_fst_a_c_d @ Prod ) @ ( product_snd_a_c_d @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_1176_surjective__pairing,axiom,
    ! [T: produc5278197477302038359_a_c_d] :
      ( T
      = ( produc8093790510458973071_a_c_d @ ( produc7453127856291044731_a_c_d @ T ) @ ( produc196187098351132349_a_c_d @ T ) ) ) ).

% surjective_pairing
thf(fact_1177_surjective__pairing,axiom,
    ! [T: product_prod_a_c_d] :
      ( T
      = ( product_Pair_a_c_d @ ( product_fst_a_c_d @ T ) @ ( product_snd_a_c_d @ T ) ) ) ).

% surjective_pairing
thf(fact_1178_graph__ranD,axiom,
    ! [X2: produc5278197477302038359_a_c_d,M: product_prod_a_c_d > option6212624336325788467_a_c_d] :
      ( ( member1180172933830803072_a_c_d @ X2 @ ( graph_7244981646656495094_a_c_d @ M ) )
     => ( member5926479794665358074_a_c_d @ ( produc196187098351132349_a_c_d @ X2 ) @ ( ran_Pr8895374234876358631_a_c_d @ M ) ) ) ).

% graph_ranD
thf(fact_1179_map__upd__upds__conv__if,axiom,
    ! [X2: produc5278197477302038359_a_c_d,Ys: list_P2433834481846699939_a_c_d,Xs: list_P1929962060191741789_a_c_d,F2: produc5278197477302038359_a_c_d > option6413918287372586467_a_c_d,Y: product_prod_a_c_d] :
      ( ( ( member1180172933830803072_a_c_d @ X2 @ ( set_Pr8461946692310444268_a_c_d @ ( take_P8537678046265997125_a_c_d @ ( size_s5659457497308758583_a_c_d @ Ys ) @ Xs ) ) )
       => ( ( map_up1324058162482958964_a_c_d @ ( fun_up4007221454163596925_a_c_d @ F2 @ X2 @ ( some_P1084500821511757806_a_c_d @ Y ) ) @ Xs @ Ys )
          = ( map_up1324058162482958964_a_c_d @ F2 @ Xs @ Ys ) ) )
      & ( ~ ( member1180172933830803072_a_c_d @ X2 @ ( set_Pr8461946692310444268_a_c_d @ ( take_P8537678046265997125_a_c_d @ ( size_s5659457497308758583_a_c_d @ Ys ) @ Xs ) ) )
       => ( ( map_up1324058162482958964_a_c_d @ ( fun_up4007221454163596925_a_c_d @ F2 @ X2 @ ( some_P1084500821511757806_a_c_d @ Y ) ) @ Xs @ Ys )
          = ( fun_up4007221454163596925_a_c_d @ ( map_up1324058162482958964_a_c_d @ F2 @ Xs @ Ys ) @ X2 @ ( some_P1084500821511757806_a_c_d @ Y ) ) ) ) ) ).

% map_upd_upds_conv_if
thf(fact_1180_map__upd__upds__conv__if,axiom,
    ! [X2: option_a,Ys: list_P2433834481846699939_a_c_d,Xs: list_option_a,F2: option_a > option6413918287372586467_a_c_d,Y: product_prod_a_c_d] :
      ( ( ( member_option_a @ X2 @ ( set_option_a2 @ ( take_option_a @ ( size_s5659457497308758583_a_c_d @ Ys ) @ Xs ) ) )
       => ( ( map_up2305867519919030241_a_c_d @ ( fun_up5969202740897331946_a_c_d @ F2 @ X2 @ ( some_P1084500821511757806_a_c_d @ Y ) ) @ Xs @ Ys )
          = ( map_up2305867519919030241_a_c_d @ F2 @ Xs @ Ys ) ) )
      & ( ~ ( member_option_a @ X2 @ ( set_option_a2 @ ( take_option_a @ ( size_s5659457497308758583_a_c_d @ Ys ) @ Xs ) ) )
       => ( ( map_up2305867519919030241_a_c_d @ ( fun_up5969202740897331946_a_c_d @ F2 @ X2 @ ( some_P1084500821511757806_a_c_d @ Y ) ) @ Xs @ Ys )
          = ( fun_up5969202740897331946_a_c_d @ ( map_up2305867519919030241_a_c_d @ F2 @ Xs @ Ys ) @ X2 @ ( some_P1084500821511757806_a_c_d @ Y ) ) ) ) ) ).

% map_upd_upds_conv_if
thf(fact_1181_map__upd__upds__conv__if,axiom,
    ! [X2: a,Ys: list_P2433834481846699939_a_c_d,Xs: list_a,F2: a > option6413918287372586467_a_c_d,Y: product_prod_a_c_d] :
      ( ( ( member_a @ X2 @ ( set_a2 @ ( take_a @ ( size_s5659457497308758583_a_c_d @ Ys ) @ Xs ) ) )
       => ( ( map_up6617793535811126555_a_c_d @ ( fun_up6642239869823773348_a_c_d @ F2 @ X2 @ ( some_P1084500821511757806_a_c_d @ Y ) ) @ Xs @ Ys )
          = ( map_up6617793535811126555_a_c_d @ F2 @ Xs @ Ys ) ) )
      & ( ~ ( member_a @ X2 @ ( set_a2 @ ( take_a @ ( size_s5659457497308758583_a_c_d @ Ys ) @ Xs ) ) )
       => ( ( map_up6617793535811126555_a_c_d @ ( fun_up6642239869823773348_a_c_d @ F2 @ X2 @ ( some_P1084500821511757806_a_c_d @ Y ) ) @ Xs @ Ys )
          = ( fun_up6642239869823773348_a_c_d @ ( map_up6617793535811126555_a_c_d @ F2 @ Xs @ Ys ) @ X2 @ ( some_P1084500821511757806_a_c_d @ Y ) ) ) ) ) ).

% map_upd_upds_conv_if
thf(fact_1182_map__upd__upds__conv__if,axiom,
    ! [X2: produc5278197477302038359_a_c_d,Ys: list_a,Xs: list_P1929962060191741789_a_c_d,F2: produc5278197477302038359_a_c_d > option_a,Y: a] :
      ( ( ( member1180172933830803072_a_c_d @ X2 @ ( set_Pr8461946692310444268_a_c_d @ ( take_P8537678046265997125_a_c_d @ ( size_size_list_a @ Ys ) @ Xs ) ) )
       => ( ( map_up87012832410759389_c_d_a @ ( fun_up7463025390225626410tion_a @ F2 @ X2 @ ( some_a @ Y ) ) @ Xs @ Ys )
          = ( map_up87012832410759389_c_d_a @ F2 @ Xs @ Ys ) ) )
      & ( ~ ( member1180172933830803072_a_c_d @ X2 @ ( set_Pr8461946692310444268_a_c_d @ ( take_P8537678046265997125_a_c_d @ ( size_size_list_a @ Ys ) @ Xs ) ) )
       => ( ( map_up87012832410759389_c_d_a @ ( fun_up7463025390225626410tion_a @ F2 @ X2 @ ( some_a @ Y ) ) @ Xs @ Ys )
          = ( fun_up7463025390225626410tion_a @ ( map_up87012832410759389_c_d_a @ F2 @ Xs @ Ys ) @ X2 @ ( some_a @ Y ) ) ) ) ) ).

% map_upd_upds_conv_if
thf(fact_1183_map__upd__upds__conv__if,axiom,
    ! [X2: option_a,Ys: list_a,Xs: list_option_a,F2: option_a > option_a,Y: a] :
      ( ( ( member_option_a @ X2 @ ( set_option_a2 @ ( take_option_a @ ( size_size_list_a @ Ys ) @ Xs ) ) )
       => ( ( map_upds_option_a_a @ ( fun_up1079276522633388797tion_a @ F2 @ X2 @ ( some_a @ Y ) ) @ Xs @ Ys )
          = ( map_upds_option_a_a @ F2 @ Xs @ Ys ) ) )
      & ( ~ ( member_option_a @ X2 @ ( set_option_a2 @ ( take_option_a @ ( size_size_list_a @ Ys ) @ Xs ) ) )
       => ( ( map_upds_option_a_a @ ( fun_up1079276522633388797tion_a @ F2 @ X2 @ ( some_a @ Y ) ) @ Xs @ Ys )
          = ( fun_up1079276522633388797tion_a @ ( map_upds_option_a_a @ F2 @ Xs @ Ys ) @ X2 @ ( some_a @ Y ) ) ) ) ) ).

% map_upd_upds_conv_if
thf(fact_1184_map__upd__upds__conv__if,axiom,
    ! [X2: a,Ys: list_a,Xs: list_a,F2: a > option_a,Y: a] :
      ( ( ( member_a @ X2 @ ( set_a2 @ ( take_a @ ( size_size_list_a @ Ys ) @ Xs ) ) )
       => ( ( map_upds_a_a @ ( fun_upd_a_option_a @ F2 @ X2 @ ( some_a @ Y ) ) @ Xs @ Ys )
          = ( map_upds_a_a @ F2 @ Xs @ Ys ) ) )
      & ( ~ ( member_a @ X2 @ ( set_a2 @ ( take_a @ ( size_size_list_a @ Ys ) @ Xs ) ) )
       => ( ( map_upds_a_a @ ( fun_upd_a_option_a @ F2 @ X2 @ ( some_a @ Y ) ) @ Xs @ Ys )
          = ( fun_upd_a_option_a @ ( map_upds_a_a @ F2 @ Xs @ Ys ) @ X2 @ ( some_a @ Y ) ) ) ) ) ).

% map_upd_upds_conv_if
thf(fact_1185_exI__realizer,axiom,
    ! [P2: option6413918287372586467_a_c_d > product_prod_a_c_d > $o,Y: option6413918287372586467_a_c_d,X2: product_prod_a_c_d] :
      ( ( P2 @ Y @ X2 )
     => ( P2 @ ( produc196187098351132349_a_c_d @ ( produc8093790510458973071_a_c_d @ X2 @ Y ) ) @ ( produc7453127856291044731_a_c_d @ ( produc8093790510458973071_a_c_d @ X2 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_1186_exI__realizer,axiom,
    ! [P2: ( c > d ) > a > $o,Y: c > d,X2: a] :
      ( ( P2 @ Y @ X2 )
     => ( P2 @ ( product_snd_a_c_d @ ( product_Pair_a_c_d @ X2 @ Y ) ) @ ( product_fst_a_c_d @ ( product_Pair_a_c_d @ X2 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_1187_conjI__realizer,axiom,
    ! [P2: product_prod_a_c_d > $o,P: product_prod_a_c_d,Q2: option6413918287372586467_a_c_d > $o,Q: option6413918287372586467_a_c_d] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( produc7453127856291044731_a_c_d @ ( produc8093790510458973071_a_c_d @ P @ Q ) ) )
          & ( Q2 @ ( produc196187098351132349_a_c_d @ ( produc8093790510458973071_a_c_d @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_1188_conjI__realizer,axiom,
    ! [P2: a > $o,P: a,Q2: ( c > d ) > $o,Q: c > d] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( product_fst_a_c_d @ ( product_Pair_a_c_d @ P @ Q ) ) )
          & ( Q2 @ ( product_snd_a_c_d @ ( product_Pair_a_c_d @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_1189_prod_Oswap__def,axiom,
    ( produc6816644282256357167_a_c_d
    = ( ^ [P5: produc1187341335561805655_a_c_d] : ( produc8093790510458973071_a_c_d @ ( produc8450192715806166205_a_c_d @ P5 ) @ ( produc6483761436891302779_a_c_d @ P5 ) ) ) ) ).

% prod.swap_def
thf(fact_1190_prod_Oswap__def,axiom,
    ( product_swap_c_d_a
    = ( ^ [P5: product_prod_c_d_a] : ( product_Pair_a_c_d @ ( product_snd_c_d_a @ P5 ) @ ( product_fst_c_d_a @ P5 ) ) ) ) ).

% prod.swap_def
thf(fact_1191_map__of__zip__upd,axiom,
    ! [Ys: list_P2433834481846699939_a_c_d,Xs: list_P1929962060191741789_a_c_d,Zs: list_P2433834481846699939_a_c_d,X2: produc5278197477302038359_a_c_d,Y: product_prod_a_c_d,Z2: product_prod_a_c_d] :
      ( ( ( size_s5659457497308758583_a_c_d @ Ys )
        = ( size_s7553148943745924553_a_c_d @ Xs ) )
     => ( ( ( size_s5659457497308758583_a_c_d @ Zs )
          = ( size_s7553148943745924553_a_c_d @ Xs ) )
       => ( ~ ( member1180172933830803072_a_c_d @ X2 @ ( set_Pr8461946692310444268_a_c_d @ Xs ) )
         => ( ( ( fun_up4007221454163596925_a_c_d @ ( map_of8696117664838035911_a_c_d @ ( zip_Pr1023814744309915123_a_c_d @ Xs @ Ys ) ) @ X2 @ ( some_P1084500821511757806_a_c_d @ Y ) )
              = ( fun_up4007221454163596925_a_c_d @ ( map_of8696117664838035911_a_c_d @ ( zip_Pr1023814744309915123_a_c_d @ Xs @ Zs ) ) @ X2 @ ( some_P1084500821511757806_a_c_d @ Z2 ) ) )
           => ( ( map_of8696117664838035911_a_c_d @ ( zip_Pr1023814744309915123_a_c_d @ Xs @ Ys ) )
              = ( map_of8696117664838035911_a_c_d @ ( zip_Pr1023814744309915123_a_c_d @ Xs @ Zs ) ) ) ) ) ) ) ).

% map_of_zip_upd
thf(fact_1192_map__of__zip__upd,axiom,
    ! [Ys: list_P2433834481846699939_a_c_d,Xs: list_option_a,Zs: list_P2433834481846699939_a_c_d,X2: option_a,Y: product_prod_a_c_d,Z2: product_prod_a_c_d] :
      ( ( ( size_s5659457497308758583_a_c_d @ Ys )
        = ( size_s3078493964004954806tion_a @ Xs ) )
     => ( ( ( size_s5659457497308758583_a_c_d @ Zs )
          = ( size_s3078493964004954806tion_a @ Xs ) )
       => ( ~ ( member_option_a @ X2 @ ( set_option_a2 @ Xs ) )
         => ( ( ( fun_up5969202740897331946_a_c_d @ ( map_of4369161196331947700_a_c_d @ ( zip_op6360952879482353888_a_c_d @ Xs @ Ys ) ) @ X2 @ ( some_P1084500821511757806_a_c_d @ Y ) )
              = ( fun_up5969202740897331946_a_c_d @ ( map_of4369161196331947700_a_c_d @ ( zip_op6360952879482353888_a_c_d @ Xs @ Zs ) ) @ X2 @ ( some_P1084500821511757806_a_c_d @ Z2 ) ) )
           => ( ( map_of4369161196331947700_a_c_d @ ( zip_op6360952879482353888_a_c_d @ Xs @ Ys ) )
              = ( map_of4369161196331947700_a_c_d @ ( zip_op6360952879482353888_a_c_d @ Xs @ Zs ) ) ) ) ) ) ) ).

% map_of_zip_upd
thf(fact_1193_map__of__zip__upd,axiom,
    ! [Ys: list_P2433834481846699939_a_c_d,Xs: list_a,Zs: list_P2433834481846699939_a_c_d,X2: a,Y: product_prod_a_c_d,Z2: product_prod_a_c_d] :
      ( ( ( size_s5659457497308758583_a_c_d @ Ys )
        = ( size_size_list_a @ Xs ) )
     => ( ( ( size_s5659457497308758583_a_c_d @ Zs )
          = ( size_size_list_a @ Xs ) )
       => ( ~ ( member_a @ X2 @ ( set_a2 @ Xs ) )
         => ( ( ( fun_up6642239869823773348_a_c_d @ ( map_of7052448288080188526_a_c_d @ ( zip_a_1300947307190184090_a_c_d @ Xs @ Ys ) ) @ X2 @ ( some_P1084500821511757806_a_c_d @ Y ) )
              = ( fun_up6642239869823773348_a_c_d @ ( map_of7052448288080188526_a_c_d @ ( zip_a_1300947307190184090_a_c_d @ Xs @ Zs ) ) @ X2 @ ( some_P1084500821511757806_a_c_d @ Z2 ) ) )
           => ( ( map_of7052448288080188526_a_c_d @ ( zip_a_1300947307190184090_a_c_d @ Xs @ Ys ) )
              = ( map_of7052448288080188526_a_c_d @ ( zip_a_1300947307190184090_a_c_d @ Xs @ Zs ) ) ) ) ) ) ) ).

% map_of_zip_upd
thf(fact_1194_map__of__zip__upd,axiom,
    ! [Ys: list_a,Xs: list_P1929962060191741789_a_c_d,Zs: list_a,X2: produc5278197477302038359_a_c_d,Y: a,Z2: a] :
      ( ( ( size_size_list_a @ Ys )
        = ( size_s7553148943745924553_a_c_d @ Xs ) )
     => ( ( ( size_size_list_a @ Zs )
          = ( size_s7553148943745924553_a_c_d @ Xs ) )
       => ( ~ ( member1180172933830803072_a_c_d @ X2 @ ( set_Pr8461946692310444268_a_c_d @ Xs ) )
         => ( ( ( fun_up7463025390225626410tion_a @ ( map_of5307797687691286346_c_d_a @ ( zip_Pr6313174362860302878_c_d_a @ Xs @ Ys ) ) @ X2 @ ( some_a @ Y ) )
              = ( fun_up7463025390225626410tion_a @ ( map_of5307797687691286346_c_d_a @ ( zip_Pr6313174362860302878_c_d_a @ Xs @ Zs ) ) @ X2 @ ( some_a @ Z2 ) ) )
           => ( ( map_of5307797687691286346_c_d_a @ ( zip_Pr6313174362860302878_c_d_a @ Xs @ Ys ) )
              = ( map_of5307797687691286346_c_d_a @ ( zip_Pr6313174362860302878_c_d_a @ Xs @ Zs ) ) ) ) ) ) ) ).

% map_of_zip_upd
thf(fact_1195_map__of__zip__upd,axiom,
    ! [Ys: list_a,Xs: list_option_a,Zs: list_a,X2: 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 @ X2 @ ( set_option_a2 @ Xs ) )
         => ( ( ( fun_up1079276522633388797tion_a @ ( map_of_option_a_a @ ( zip_option_a_a @ Xs @ Ys ) ) @ X2 @ ( some_a @ Y ) )
              = ( fun_up1079276522633388797tion_a @ ( map_of_option_a_a @ ( zip_option_a_a @ Xs @ Zs ) ) @ X2 @ ( 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_1196_map__of__zip__upd,axiom,
    ! [Ys: list_a,Xs: list_a,Zs: list_a,X2: 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 @ X2 @ ( set_a2 @ Xs ) )
         => ( ( ( fun_upd_a_option_a @ ( map_of_a_a @ ( zip_a_a @ Xs @ Ys ) ) @ X2 @ ( some_a @ Y ) )
              = ( fun_upd_a_option_a @ ( map_of_a_a @ ( zip_a_a @ Xs @ Zs ) ) @ X2 @ ( 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_1197_pair__in__swap__image,axiom,
    ! [Y: option6413918287372586467_a_c_d,X2: product_prod_a_c_d,A6: set_Pr1275464188344874039_a_c_d] :
      ( ( member6312688828945346176_a_c_d @ ( produc7124424091059231119_a_c_d @ Y @ X2 ) @ ( image_8223956251181705157_a_c_d @ produc7786010701656099119_a_c_d @ A6 ) )
      = ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ X2 @ Y ) @ A6 ) ) ).

% pair_in_swap_image
thf(fact_1198_pair__in__swap__image,axiom,
    ! [Y: c > d,X2: a,A6: set_Pr334309860168054345_a_c_d] :
      ( ( member6651178222534243620_c_d_a @ ( product_Pair_c_d_a @ Y @ X2 ) @ ( image_4148376343195491495_c_d_a @ product_swap_a_c_d @ A6 ) )
      = ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ X2 @ Y ) @ A6 ) ) ).

% pair_in_swap_image
thf(fact_1199_pair__in__swap__image,axiom,
    ! [Y: product_prod_a_c_d,X2: option6413918287372586467_a_c_d,A6: set_Pr1695729385883591735_a_c_d] :
      ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ Y @ X2 ) @ ( image_8264601463523870661_a_c_d @ produc6816644282256357167_a_c_d @ A6 ) )
      = ( member6312688828945346176_a_c_d @ ( produc7124424091059231119_a_c_d @ X2 @ Y ) @ A6 ) ) ).

% pair_in_swap_image
thf(fact_1200_pair__in__swap__image,axiom,
    ! [Y: a,X2: c > d,A6: set_Pr2254288101409822147_c_d_a] :
      ( ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ Y @ X2 ) @ ( image_6914783986489374375_a_c_d @ product_swap_c_d_a @ A6 ) )
      = ( member6651178222534243620_c_d_a @ ( product_Pair_c_d_a @ X2 @ Y ) @ A6 ) ) ).

% pair_in_swap_image
thf(fact_1201_swap__simp,axiom,
    ! [X2: option6413918287372586467_a_c_d,Y: product_prod_a_c_d] :
      ( ( produc6816644282256357167_a_c_d @ ( produc7124424091059231119_a_c_d @ X2 @ Y ) )
      = ( produc8093790510458973071_a_c_d @ Y @ X2 ) ) ).

% swap_simp
thf(fact_1202_swap__simp,axiom,
    ! [X2: c > d,Y: a] :
      ( ( product_swap_c_d_a @ ( product_Pair_c_d_a @ X2 @ Y ) )
      = ( product_Pair_a_c_d @ Y @ X2 ) ) ).

% swap_simp
thf(fact_1203_swap__simp,axiom,
    ! [X2: product_prod_a_c_d,Y: option6413918287372586467_a_c_d] :
      ( ( produc7786010701656099119_a_c_d @ ( produc8093790510458973071_a_c_d @ X2 @ Y ) )
      = ( produc7124424091059231119_a_c_d @ Y @ X2 ) ) ).

% swap_simp
thf(fact_1204_swap__simp,axiom,
    ! [X2: a,Y: c > d] :
      ( ( product_swap_a_c_d @ ( product_Pair_a_c_d @ X2 @ Y ) )
      = ( product_Pair_c_d_a @ Y @ X2 ) ) ).

% swap_simp
thf(fact_1205_map__of__zip__is__None,axiom,
    ! [Xs: list_P1929962060191741789_a_c_d,Ys: list_a,X2: produc5278197477302038359_a_c_d] :
      ( ( ( size_s7553148943745924553_a_c_d @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( ( map_of5307797687691286346_c_d_a @ ( zip_Pr6313174362860302878_c_d_a @ Xs @ Ys ) @ X2 )
          = none_a )
        = ( ~ ( member1180172933830803072_a_c_d @ X2 @ ( set_Pr8461946692310444268_a_c_d @ Xs ) ) ) ) ) ).

% map_of_zip_is_None
thf(fact_1206_map__of__zip__is__None,axiom,
    ! [Xs: list_option_a,Ys: list_a,X2: option_a] :
      ( ( ( size_s3078493964004954806tion_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( ( map_of_option_a_a @ ( zip_option_a_a @ Xs @ Ys ) @ X2 )
          = none_a )
        = ( ~ ( member_option_a @ X2 @ ( set_option_a2 @ Xs ) ) ) ) ) ).

% map_of_zip_is_None
thf(fact_1207_map__of__zip__is__None,axiom,
    ! [Xs: list_a,Ys: list_a,X2: a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( ( map_of_a_a @ ( zip_a_a @ Xs @ Ys ) @ X2 )
          = none_a )
        = ( ~ ( member_a @ X2 @ ( set_a2 @ Xs ) ) ) ) ) ).

% map_of_zip_is_None
thf(fact_1208_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_1209_map__of__zip__is__Some,axiom,
    ! [Xs: list_P1929962060191741789_a_c_d,Ys: list_P2433834481846699939_a_c_d,X2: produc5278197477302038359_a_c_d] :
      ( ( ( size_s7553148943745924553_a_c_d @ Xs )
        = ( size_s5659457497308758583_a_c_d @ Ys ) )
     => ( ( member1180172933830803072_a_c_d @ X2 @ ( set_Pr8461946692310444268_a_c_d @ Xs ) )
        = ( ? [Y3: product_prod_a_c_d] :
              ( ( map_of8696117664838035911_a_c_d @ ( zip_Pr1023814744309915123_a_c_d @ Xs @ Ys ) @ X2 )
              = ( some_P1084500821511757806_a_c_d @ Y3 ) ) ) ) ) ).

% map_of_zip_is_Some
thf(fact_1210_map__of__zip__is__Some,axiom,
    ! [Xs: list_option_a,Ys: list_P2433834481846699939_a_c_d,X2: option_a] :
      ( ( ( size_s3078493964004954806tion_a @ Xs )
        = ( size_s5659457497308758583_a_c_d @ Ys ) )
     => ( ( member_option_a @ X2 @ ( set_option_a2 @ Xs ) )
        = ( ? [Y3: product_prod_a_c_d] :
              ( ( map_of4369161196331947700_a_c_d @ ( zip_op6360952879482353888_a_c_d @ Xs @ Ys ) @ X2 )
              = ( some_P1084500821511757806_a_c_d @ Y3 ) ) ) ) ) ).

% map_of_zip_is_Some
thf(fact_1211_map__of__zip__is__Some,axiom,
    ! [Xs: list_P1929962060191741789_a_c_d,Ys: list_a,X2: produc5278197477302038359_a_c_d] :
      ( ( ( size_s7553148943745924553_a_c_d @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member1180172933830803072_a_c_d @ X2 @ ( set_Pr8461946692310444268_a_c_d @ Xs ) )
        = ( ? [Y3: a] :
              ( ( map_of5307797687691286346_c_d_a @ ( zip_Pr6313174362860302878_c_d_a @ Xs @ Ys ) @ X2 )
              = ( some_a @ Y3 ) ) ) ) ) ).

% map_of_zip_is_Some
thf(fact_1212_map__of__zip__is__Some,axiom,
    ! [Xs: list_option_a,Ys: list_a,X2: option_a] :
      ( ( ( size_s3078493964004954806tion_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member_option_a @ X2 @ ( set_option_a2 @ Xs ) )
        = ( ? [Y3: a] :
              ( ( map_of_option_a_a @ ( zip_option_a_a @ Xs @ Ys ) @ X2 )
              = ( some_a @ Y3 ) ) ) ) ) ).

% map_of_zip_is_Some
thf(fact_1213_map__of__zip__is__Some,axiom,
    ! [Xs: list_a,Ys: list_P2433834481846699939_a_c_d,X2: a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_s5659457497308758583_a_c_d @ Ys ) )
     => ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
        = ( ? [Y3: product_prod_a_c_d] :
              ( ( map_of7052448288080188526_a_c_d @ ( zip_a_1300947307190184090_a_c_d @ Xs @ Ys ) @ X2 )
              = ( some_P1084500821511757806_a_c_d @ Y3 ) ) ) ) ) ).

% map_of_zip_is_Some
thf(fact_1214_map__of__zip__is__Some,axiom,
    ! [Xs: list_a,Ys: list_a,X2: a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
        = ( ? [Y3: a] :
              ( ( map_of_a_a @ ( zip_a_a @ Xs @ Ys ) @ X2 )
              = ( some_a @ Y3 ) ) ) ) ) ).

% map_of_zip_is_Some
thf(fact_1215_zip__Cons__Cons,axiom,
    ! [X2: a,Xs: list_a,Y: a,Ys: list_a] :
      ( ( zip_a_a @ ( cons_a @ X2 @ Xs ) @ ( cons_a @ Y @ Ys ) )
      = ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ ( zip_a_a @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_1216_zip__Cons__Cons,axiom,
    ! [X2: product_prod_a_c_d,Xs: list_P2433834481846699939_a_c_d,Y: option6413918287372586467_a_c_d,Ys: list_o3515477207332150515_a_c_d] :
      ( ( zip_Pr428572426551296687_a_c_d @ ( cons_P1689682164644567517_a_c_d @ X2 @ Xs ) @ ( cons_o2716606887476812717_a_c_d @ Y @ Ys ) )
      = ( cons_P4841026258484161293_a_c_d @ ( produc8093790510458973071_a_c_d @ X2 @ Y ) @ ( zip_Pr428572426551296687_a_c_d @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_1217_zip__Cons__Cons,axiom,
    ! [X2: a,Xs: list_a,Y: c > d,Ys: list_c_d] :
      ( ( zip_a_c_d @ ( cons_a @ X2 @ Xs ) @ ( cons_c_d @ Y @ Ys ) )
      = ( cons_P1689682164644567517_a_c_d @ ( product_Pair_a_c_d @ X2 @ Y ) @ ( zip_a_c_d @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_1218_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_P2433834481846699939_a_c_d,Ys: list_o3515477207332150515_a_c_d,X2: product_prod_a_c_d] :
      ( ( ( size_s5659457497308758583_a_c_d @ Xs )
        = ( size_s4259128241386891655_a_c_d @ Ys ) )
     => ( ( member8846983874477766826_a_c_d @ X2 @ ( set_Pr2012686464001265470_a_c_d @ Xs ) )
       => ~ ! [Y4: option6413918287372586467_a_c_d] :
              ~ ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ X2 @ Y4 ) @ ( set_Pr8461946692310444268_a_c_d @ ( zip_Pr428572426551296687_a_c_d @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1219_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_P1929962060191741789_a_c_d,Ys: list_a,X2: produc5278197477302038359_a_c_d] :
      ( ( ( size_s7553148943745924553_a_c_d @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member1180172933830803072_a_c_d @ X2 @ ( set_Pr8461946692310444268_a_c_d @ Xs ) )
       => ~ ! [Y4: a] :
              ~ ( member1010981217928118071_c_d_a @ ( produc3534377726190287614_c_d_a @ X2 @ Y4 ) @ ( set_Pr446018216176770467_c_d_a @ ( zip_Pr6313174362860302878_c_d_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1220_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_option_a,Ys: list_a,X2: option_a] :
      ( ( ( size_s3078493964004954806tion_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member_option_a @ X2 @ ( set_option_a2 @ Xs ) )
       => ~ ! [Y4: a] :
              ~ ( member6056235002698166154on_a_a @ ( produc3446707977624461905on_a_a @ X2 @ Y4 ) @ ( set_Pr1233600038994746358on_a_a @ ( zip_option_a_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1221_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_a,Ys: list_c_d,X2: a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_c_d @ Ys ) )
     => ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
       => ~ ! [Y4: c > d] :
              ~ ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ X2 @ Y4 ) @ ( set_Pr2012686464001265470_a_c_d @ ( zip_a_c_d @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1222_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_a,Ys: list_a,X2: a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
       => ~ ! [Y4: a] :
              ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y4 ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1223_zip__same,axiom,
    ! [A: produc5278197477302038359_a_c_d,B: produc5278197477302038359_a_c_d,Xs: list_P1929962060191741789_a_c_d] :
      ( ( member398054542024605456_a_c_d @ ( produc4538534844009793367_a_c_d @ A @ B ) @ ( set_Pr6388181779389218172_a_c_d @ ( zip_Pr1029612570978745975_a_c_d @ Xs @ Xs ) ) )
      = ( ( member1180172933830803072_a_c_d @ A @ ( set_Pr8461946692310444268_a_c_d @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_1224_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_1225_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_1226_in__set__zipE,axiom,
    ! [X2: a,Y: a,Xs: list_a,Ys: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) )
     => ~ ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
         => ~ ( member_a @ Y @ ( set_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1227_in__set__zipE,axiom,
    ! [X2: option_a,Y: a,Xs: list_option_a,Ys: list_a] :
      ( ( member6056235002698166154on_a_a @ ( produc3446707977624461905on_a_a @ X2 @ Y ) @ ( set_Pr1233600038994746358on_a_a @ ( zip_option_a_a @ Xs @ Ys ) ) )
     => ~ ( ( member_option_a @ X2 @ ( set_option_a2 @ Xs ) )
         => ~ ( member_a @ Y @ ( set_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1228_in__set__zipE,axiom,
    ! [X2: a,Y: option_a,Xs: list_a,Ys: list_option_a] :
      ( ( member6937434987665551382tion_a @ ( produc1224194096085666781tion_a @ X2 @ Y ) @ ( set_Pr2114800023962131586tion_a @ ( zip_a_option_a @ Xs @ Ys ) ) )
     => ~ ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
         => ~ ( member_option_a @ Y @ ( set_option_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1229_in__set__zipE,axiom,
    ! [X2: option_a,Y: option_a,Xs: list_option_a,Ys: list_option_a] :
      ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y ) @ ( set_Pr948796958549772220tion_a @ ( zip_op6411647709037274935tion_a @ Xs @ Ys ) ) )
     => ~ ( ( member_option_a @ X2 @ ( set_option_a2 @ Xs ) )
         => ~ ( member_option_a @ Y @ ( set_option_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1230_in__set__zipE,axiom,
    ! [X2: a,Y: c > d,Xs: list_a,Ys: list_c_d] :
      ( ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ X2 @ Y ) @ ( set_Pr2012686464001265470_a_c_d @ ( zip_a_c_d @ Xs @ Ys ) ) )
     => ~ ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
         => ~ ( member_c_d @ Y @ ( set_c_d2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1231_in__set__zipE,axiom,
    ! [X2: product_prod_a_c_d,Y: option6413918287372586467_a_c_d,Xs: list_P2433834481846699939_a_c_d,Ys: list_o3515477207332150515_a_c_d] :
      ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ X2 @ Y ) @ ( set_Pr8461946692310444268_a_c_d @ ( zip_Pr428572426551296687_a_c_d @ Xs @ Ys ) ) )
     => ~ ( ( member8846983874477766826_a_c_d @ X2 @ ( set_Pr2012686464001265470_a_c_d @ Xs ) )
         => ~ ( member5926479794665358074_a_c_d @ Y @ ( set_op2507751132114234254_a_c_d @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1232_in__set__zipE,axiom,
    ! [X2: produc5278197477302038359_a_c_d,Y: a,Xs: list_P1929962060191741789_a_c_d,Ys: list_a] :
      ( ( member1010981217928118071_c_d_a @ ( produc3534377726190287614_c_d_a @ X2 @ Y ) @ ( set_Pr446018216176770467_c_d_a @ ( zip_Pr6313174362860302878_c_d_a @ Xs @ Ys ) ) )
     => ~ ( ( member1180172933830803072_a_c_d @ X2 @ ( set_Pr8461946692310444268_a_c_d @ Xs ) )
         => ~ ( member_a @ Y @ ( set_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1233_in__set__zipE,axiom,
    ! [X2: a,Y: produc5278197477302038359_a_c_d,Xs: list_a,Ys: list_P1929962060191741789_a_c_d] :
      ( ( member6653950053022509865_a_c_d @ ( produc909605494754173808_a_c_d @ X2 @ Y ) @ ( set_Pr6088987051271162261_a_c_d @ ( zip_a_3688402131424189072_a_c_d @ Xs @ Ys ) ) )
     => ~ ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
         => ~ ( member1180172933830803072_a_c_d @ Y @ ( set_Pr8461946692310444268_a_c_d @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1234_in__set__zipE,axiom,
    ! [X2: produc5278197477302038359_a_c_d,Y: option_a,Xs: list_P1929962060191741789_a_c_d,Ys: list_option_a] :
      ( ( member4468344759597462909tion_a @ ( produc4720451050651427140tion_a @ X2 @ Y ) @ ( set_Pr8422585129994568169tion_a @ ( zip_Pr1423292784968050788tion_a @ Xs @ Ys ) ) )
     => ~ ( ( member1180172933830803072_a_c_d @ X2 @ ( set_Pr8461946692310444268_a_c_d @ Xs ) )
         => ~ ( member_option_a @ Y @ ( set_option_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1235_in__set__zipE,axiom,
    ! [X2: option_a,Y: produc5278197477302038359_a_c_d,Xs: list_option_a,Ys: list_P1929962060191741789_a_c_d] :
      ( ( member8792505911481038691_a_c_d @ ( produc3880585002510677674_a_c_d @ X2 @ Y ) @ ( set_Pr3523374245023368143_a_c_d @ ( zip_op583426736827301322_a_c_d @ Xs @ Ys ) ) )
     => ~ ( ( member_option_a @ X2 @ ( set_option_a2 @ Xs ) )
         => ~ ( member1180172933830803072_a_c_d @ Y @ ( set_Pr8461946692310444268_a_c_d @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1236_set__zip__leftD,axiom,
    ! [X2: product_prod_a_c_d,Y: option6413918287372586467_a_c_d,Xs: list_P2433834481846699939_a_c_d,Ys: list_o3515477207332150515_a_c_d] :
      ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ X2 @ Y ) @ ( set_Pr8461946692310444268_a_c_d @ ( zip_Pr428572426551296687_a_c_d @ Xs @ Ys ) ) )
     => ( member8846983874477766826_a_c_d @ X2 @ ( set_Pr2012686464001265470_a_c_d @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1237_set__zip__leftD,axiom,
    ! [X2: a,Y: c > d,Xs: list_a,Ys: list_c_d] :
      ( ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ X2 @ Y ) @ ( set_Pr2012686464001265470_a_c_d @ ( zip_a_c_d @ Xs @ Ys ) ) )
     => ( member_a @ X2 @ ( set_a2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1238_set__zip__rightD,axiom,
    ! [X2: product_prod_a_c_d,Y: option6413918287372586467_a_c_d,Xs: list_P2433834481846699939_a_c_d,Ys: list_o3515477207332150515_a_c_d] :
      ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ X2 @ Y ) @ ( set_Pr8461946692310444268_a_c_d @ ( zip_Pr428572426551296687_a_c_d @ Xs @ Ys ) ) )
     => ( member5926479794665358074_a_c_d @ Y @ ( set_op2507751132114234254_a_c_d @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1239_set__zip__rightD,axiom,
    ! [X2: a,Y: c > d,Xs: list_a,Ys: list_c_d] :
      ( ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ X2 @ Y ) @ ( set_Pr2012686464001265470_a_c_d @ ( zip_a_c_d @ Xs @ Ys ) ) )
     => ( member_c_d @ Y @ ( set_c_d2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1240_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 ) )
     => ~ ! [X4: a,Xs2: list_a] :
            ( ( Xs
              = ( cons_a @ X4 @ Xs2 ) )
           => ! [Y4: a,Ys2: list_a] :
                ( ( Ys
                  = ( cons_a @ Y4 @ Ys2 ) )
               => ( ( Xy
                    = ( product_Pair_a_a @ X4 @ Y4 ) )
                 => ( Xys
                   != ( zip_a_a @ Xs2 @ Ys2 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_1241_zip__eq__ConsE,axiom,
    ! [Xs: list_P2433834481846699939_a_c_d,Ys: list_o3515477207332150515_a_c_d,Xy: produc5278197477302038359_a_c_d,Xys: list_P1929962060191741789_a_c_d] :
      ( ( ( zip_Pr428572426551296687_a_c_d @ Xs @ Ys )
        = ( cons_P4841026258484161293_a_c_d @ Xy @ Xys ) )
     => ~ ! [X4: product_prod_a_c_d,Xs2: list_P2433834481846699939_a_c_d] :
            ( ( Xs
              = ( cons_P1689682164644567517_a_c_d @ X4 @ Xs2 ) )
           => ! [Y4: option6413918287372586467_a_c_d,Ys2: list_o3515477207332150515_a_c_d] :
                ( ( Ys
                  = ( cons_o2716606887476812717_a_c_d @ Y4 @ Ys2 ) )
               => ( ( Xy
                    = ( produc8093790510458973071_a_c_d @ X4 @ Y4 ) )
                 => ( Xys
                   != ( zip_Pr428572426551296687_a_c_d @ Xs2 @ Ys2 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_1242_zip__eq__ConsE,axiom,
    ! [Xs: list_a,Ys: list_c_d,Xy: product_prod_a_c_d,Xys: list_P2433834481846699939_a_c_d] :
      ( ( ( zip_a_c_d @ Xs @ Ys )
        = ( cons_P1689682164644567517_a_c_d @ Xy @ Xys ) )
     => ~ ! [X4: a,Xs2: list_a] :
            ( ( Xs
              = ( cons_a @ X4 @ Xs2 ) )
           => ! [Y4: c > d,Ys2: list_c_d] :
                ( ( Ys
                  = ( cons_c_d @ Y4 @ Ys2 ) )
               => ( ( Xy
                    = ( product_Pair_a_c_d @ X4 @ Y4 ) )
                 => ( Xys
                   != ( zip_a_c_d @ Xs2 @ Ys2 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_1243_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_P2433834481846699939_a_c_d,Ys: list_o3515477207332150515_a_c_d,Y: option6413918287372586467_a_c_d] :
      ( ( ( size_s5659457497308758583_a_c_d @ Xs )
        = ( size_s4259128241386891655_a_c_d @ Ys ) )
     => ( ( member5926479794665358074_a_c_d @ Y @ ( set_op2507751132114234254_a_c_d @ Ys ) )
       => ~ ! [X4: product_prod_a_c_d] :
              ~ ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ X4 @ Y ) @ ( set_Pr8461946692310444268_a_c_d @ ( zip_Pr428572426551296687_a_c_d @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1244_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_a,Ys: list_P1929962060191741789_a_c_d,Y: produc5278197477302038359_a_c_d] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_s7553148943745924553_a_c_d @ Ys ) )
     => ( ( member1180172933830803072_a_c_d @ Y @ ( set_Pr8461946692310444268_a_c_d @ Ys ) )
       => ~ ! [X4: a] :
              ~ ( member6653950053022509865_a_c_d @ ( produc909605494754173808_a_c_d @ X4 @ Y ) @ ( set_Pr6088987051271162261_a_c_d @ ( zip_a_3688402131424189072_a_c_d @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1245_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 ) )
       => ~ ! [X4: a] :
              ~ ( member6937434987665551382tion_a @ ( produc1224194096085666781tion_a @ X4 @ Y ) @ ( set_Pr2114800023962131586tion_a @ ( zip_a_option_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1246_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_a,Ys: list_c_d,Y: c > d] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_c_d @ Ys ) )
     => ( ( member_c_d @ Y @ ( set_c_d2 @ Ys ) )
       => ~ ! [X4: a] :
              ~ ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ X4 @ Y ) @ ( set_Pr2012686464001265470_a_c_d @ ( zip_a_c_d @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1247_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 ) )
       => ~ ! [X4: a] :
              ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1248_Cons__lenlex__iff,axiom,
    ! [M: a,Ms: list_a,N: a,Ns: list_a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ M @ Ms ) @ ( cons_a @ N @ Ns ) ) @ ( lenlex_a @ R ) )
      = ( ( ord_less_nat @ ( size_size_list_a @ Ms ) @ ( size_size_list_a @ Ns ) )
        | ( ( ( size_size_list_a @ Ms )
            = ( size_size_list_a @ Ns ) )
          & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ M @ N ) @ R ) )
        | ( ( M = N )
          & ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ms @ Ns ) @ ( lenlex_a @ R ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_1249_lenlex__length,axiom,
    ! [Ms: list_a,Ns: list_a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ms @ Ns ) @ ( lenlex_a @ R ) )
     => ( ord_less_eq_nat @ ( size_size_list_a @ Ms ) @ ( size_size_list_a @ Ns ) ) ) ).

% lenlex_length
thf(fact_1250_lenlex__trans,axiom,
    ! [X2: list_a,Y: list_a,R: set_Product_prod_a_a,Z2: list_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X2 @ Y ) @ ( lenlex_a @ R ) )
     => ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Y @ Z2 ) @ ( lenlex_a @ R ) )
       => ( ( trans_on_a @ top_top_set_a @ R )
         => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X2 @ Z2 ) @ ( lenlex_a @ R ) ) ) ) ) ).

% lenlex_trans
thf(fact_1251_lenlex__trans,axiom,
    ! [X2: list_option_a,Y: list_option_a,R: set_Pr7585778909603769095tion_a,Z2: list_option_a] :
      ( ( member3116372912924030544tion_a @ ( produc6071813090954082327tion_a @ X2 @ Y ) @ ( lenlex_option_a @ R ) )
     => ( ( member3116372912924030544tion_a @ ( produc6071813090954082327tion_a @ Y @ Z2 ) @ ( lenlex_option_a @ R ) )
       => ( ( trans_on_option_a @ top_top_set_option_a @ R )
         => ( member3116372912924030544tion_a @ ( produc6071813090954082327tion_a @ X2 @ Z2 ) @ ( lenlex_option_a @ R ) ) ) ) ) ).

% lenlex_trans
thf(fact_1252_Cons__in__lex,axiom,
    ! [X2: a,Xs: list_a,Y: a,Ys: list_a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X2 @ Xs ) @ ( cons_a @ Y @ Ys ) ) @ ( lex_a @ R ) )
      = ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
          & ( ( size_size_list_a @ Xs )
            = ( size_size_list_a @ Ys ) ) )
        | ( ( X2 = Y )
          & ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lex_a @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_1253_partition__set,axiom,
    ! [P2: a > $o,Xs: list_a,Yes: list_a,No: list_a] :
      ( ( ( partition_a @ P2 @ Xs )
        = ( produc6837034575241423639list_a @ Yes @ No ) )
     => ( ( sup_sup_set_a @ ( set_a2 @ Yes ) @ ( set_a2 @ No ) )
        = ( set_a2 @ Xs ) ) ) ).

% partition_set
thf(fact_1254_partition__P,axiom,
    ! [P2: a > $o,Xs: list_a,Yes: list_a,No: list_a] :
      ( ( ( partition_a @ P2 @ Xs )
        = ( produc6837034575241423639list_a @ Yes @ No ) )
     => ( ! [X8: a] :
            ( ( member_a @ X8 @ ( set_a2 @ Yes ) )
           => ( P2 @ X8 ) )
        & ! [X8: a] :
            ( ( member_a @ X8 @ ( set_a2 @ No ) )
           => ~ ( P2 @ X8 ) ) ) ) ).

% partition_P
thf(fact_1255_lex__take__index,axiom,
    ! [Xs: list_a,Ys: list_a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lex_a @ R ) )
     => ~ ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
           => ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ys ) )
             => ( ( ( take_a @ I2 @ Xs )
                  = ( take_a @ I2 @ Ys ) )
               => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ I2 ) @ ( nth_a @ Ys @ I2 ) ) @ R ) ) ) ) ) ).

% lex_take_index
thf(fact_1256_ran__map__of__zip,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( distinct_a @ Xs )
       => ( ( ran_a_a @ ( map_of_a_a @ ( zip_a_a @ Xs @ Ys ) ) )
          = ( set_a2 @ Ys ) ) ) ) ).

% ran_map_of_zip
thf(fact_1257_nth__zip,axiom,
    ! [I: nat,Xs: list_P2433834481846699939_a_c_d,Ys: list_o3515477207332150515_a_c_d] :
      ( ( ord_less_nat @ I @ ( size_s5659457497308758583_a_c_d @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s4259128241386891655_a_c_d @ Ys ) )
       => ( ( nth_Pr7302098727961027038_a_c_d @ ( zip_Pr428572426551296687_a_c_d @ Xs @ Ys ) @ I )
          = ( produc8093790510458973071_a_c_d @ ( nth_Pr8468873534830408652_a_c_d @ Xs @ I ) @ ( nth_op5023121364616438556_a_c_d @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1258_nth__zip,axiom,
    ! [I: nat,Xs: list_a,Ys: list_c_d] :
      ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_c_d @ Ys ) )
       => ( ( nth_Pr8468873534830408652_a_c_d @ ( zip_a_c_d @ Xs @ Ys ) @ I )
          = ( product_Pair_a_c_d @ ( nth_a @ Xs @ I ) @ ( nth_c_d @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1259_nth__zip,axiom,
    ! [I: nat,Xs: list_a,Ys: list_a] :
      ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_a @ Ys ) )
       => ( ( nth_Product_prod_a_a @ ( zip_a_a @ Xs @ Ys ) @ I )
          = ( product_Pair_a_a @ ( nth_a @ Xs @ I ) @ ( nth_a @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1260_map__of__zip__nth,axiom,
    ! [Xs: list_a,Ys: list_P2433834481846699939_a_c_d,I: nat] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_s5659457497308758583_a_c_d @ Ys ) )
     => ( ( distinct_a @ Xs )
       => ( ( ord_less_nat @ I @ ( size_s5659457497308758583_a_c_d @ Ys ) )
         => ( ( map_of7052448288080188526_a_c_d @ ( zip_a_1300947307190184090_a_c_d @ Xs @ Ys ) @ ( nth_a @ Xs @ I ) )
            = ( some_P1084500821511757806_a_c_d @ ( nth_Pr8468873534830408652_a_c_d @ Ys @ I ) ) ) ) ) ) ).

% map_of_zip_nth
thf(fact_1261_map__of__zip__nth,axiom,
    ! [Xs: list_a,Ys: list_a,I: nat] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( distinct_a @ Xs )
       => ( ( ord_less_nat @ I @ ( size_size_list_a @ Ys ) )
         => ( ( map_of_a_a @ ( zip_a_a @ Xs @ Ys ) @ ( nth_a @ Xs @ I ) )
            = ( some_a @ ( nth_a @ Ys @ I ) ) ) ) ) ) ).

% map_of_zip_nth
thf(fact_1262_map__of__zip__inject,axiom,
    ! [Ys: list_a,Xs: list_a,Zs: list_a] :
      ( ( ( size_size_list_a @ Ys )
        = ( size_size_list_a @ Xs ) )
     => ( ( ( size_size_list_a @ Zs )
          = ( size_size_list_a @ Xs ) )
       => ( ( distinct_a @ Xs )
         => ( ( ( map_of_a_a @ ( zip_a_a @ Xs @ Ys ) )
              = ( map_of_a_a @ ( zip_a_a @ Xs @ Zs ) ) )
           => ( Ys = Zs ) ) ) ) ) ).

% map_of_zip_inject
thf(fact_1263_map__of__is__SomeI,axiom,
    ! [Xys: list_P2433834481846699939_a_c_d,X2: a,Y: c > d] :
      ( ( distinct_a @ ( map_Pr199199480951861093_c_d_a @ product_fst_a_c_d @ Xys ) )
     => ( ( member8846983874477766826_a_c_d @ ( product_Pair_a_c_d @ X2 @ Y ) @ ( set_Pr2012686464001265470_a_c_d @ Xys ) )
       => ( ( map_of_a_c_d @ Xys @ X2 )
          = ( some_c_d @ Y ) ) ) ) ).

% map_of_is_SomeI
thf(fact_1264_find__Some__iff2,axiom,
    ! [X2: a,P2: a > $o,Xs: list_a] :
      ( ( ( some_a @ X2 )
        = ( find_a @ P2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
            & ( P2 @ ( nth_a @ Xs @ I3 ) )
            & ( X2
              = ( nth_a @ Xs @ I3 ) )
            & ! [J2: nat] :
                ( ( ord_less_nat @ J2 @ I3 )
               => ~ ( P2 @ ( nth_a @ Xs @ J2 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_1265_find__Some__iff,axiom,
    ! [P2: a > $o,Xs: list_a,X2: a] :
      ( ( ( find_a @ P2 @ Xs )
        = ( some_a @ X2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
            & ( P2 @ ( nth_a @ Xs @ I3 ) )
            & ( X2
              = ( nth_a @ Xs @ I3 ) )
            & ! [J2: nat] :
                ( ( ord_less_nat @ J2 @ I3 )
               => ~ ( P2 @ ( nth_a @ Xs @ J2 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_1266_find_Osimps_I2_J,axiom,
    ! [P2: a > $o,X2: a,Xs: list_a] :
      ( ( ( P2 @ X2 )
       => ( ( find_a @ P2 @ ( cons_a @ X2 @ Xs ) )
          = ( some_a @ X2 ) ) )
      & ( ~ ( P2 @ X2 )
       => ( ( find_a @ P2 @ ( cons_a @ X2 @ Xs ) )
          = ( find_a @ P2 @ Xs ) ) ) ) ).

% find.simps(2)
thf(fact_1267_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_1268_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_1269_find_Osimps_I1_J,axiom,
    ! [Uu: a > $o] :
      ( ( find_a @ Uu @ nil_a )
      = none_a ) ).

% find.simps(1)
thf(fact_1270_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)
thf(fact_1271_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_1272_Option_Othese__def,axiom,
    ( these_a
    = ( ^ [A8: set_option_a] :
          ( image_option_a_a2 @ the_a
          @ ( collect_option_a
            @ ^ [X: option_a] :
                ( ( member_option_a @ X @ A8 )
                & ( X != none_a ) ) ) ) ) ) ).

% Option.these_def
thf(fact_1273_Some__image__these__eq,axiom,
    ! [A6: set_option_a] :
      ( ( image_a_option_a2 @ some_a @ ( these_a @ A6 ) )
      = ( collect_option_a
        @ ^ [X: option_a] :
            ( ( member_option_a @ X @ A6 )
            & ( X != none_a ) ) ) ) ).

% Some_image_these_eq
thf(fact_1274_case__optionE,axiom,
    ! [P2: $o,Q2: a > $o,X2: option_a] :
      ( ( case_option_o_a @ P2 @ Q2 @ X2 )
     => ( ( ( X2 = none_a )
         => ~ P2 )
       => ~ ! [Y4: a] :
              ( ( X2
                = ( some_a @ Y4 ) )
             => ~ ( Q2 @ Y4 ) ) ) ) ).

% case_optionE

% Helper facts (5)
thf(help_If_2_1_If_001tf__d_T,axiom,
    ! [X2: d,Y: d] :
      ( ( if_d @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001tf__d_T,axiom,
    ! [X2: d,Y: d] :
      ( ( if_d @ $true @ X2 @ Y )
      = X2 ) ).

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,
    ! [X2: option_a,Y: option_a] :
      ( ( if_option_a @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_Itf__a_J_T,axiom,
    ! [X2: option_a,Y: option_a] :
      ( ( if_option_a @ $true @ X2 @ Y )
      = X2 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( a22
    = ( mult @ p @ aaa ) ) ).

%------------------------------------------------------------------------------