TPTP Problem File: SLH0663^1.p

%------------------------------------------------------------------------------
% 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    : FO_Theory_Rewriting/0079_GTT_RRn/prob_00053_002009__18867026_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1898 ( 626 unt; 621 typ;   0 def)
%            Number of atoms       : 3784 (1119 equ;   0 cnn)
%            Maximal formula atoms :   48 (   2 avg)
%            Number of connectives : 12923 ( 325   ~;  65   |; 226   &;10781   @)
%                                         (   0 <=>;1526  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   26 (   7 avg)
%            Number of types       :  122 ( 121 usr)
%            Number of type conns  : 2762 (2762   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  503 ( 500 usr;  22 con; 0-4 aty)
%            Number of variables   : 4669 ( 911   ^;3685   !;  73   ?;4669   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 12:02:56.964
%------------------------------------------------------------------------------
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thf(ty_n_t__Tree____Automata__Oreg_It__Sum____Type__Osum_Itf__a_M_Eo_J_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_J_J,type,
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thf(ty_n_t__Tree____Automata__Oreg_It__Option__Ooption_Itf__a_J_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_Mtf__b_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Option__Ooption_Itf__b_J_J_Mt__Set__Oset_It__Option__Ooption_Itf__b_J_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J_J,type,
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thf(ty_n_t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J,type,
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thf(ty_n_t__Ground____Terms__Ogterm_Itf__b_J,type,
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thf(ty_n_t__FOR____Certificate__Opos____step,type,
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thf(ty_n_t__FOR____Certificate__Oext____step,type,
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thf(ty_n_t__Set__Oset_Itf__b_J,type,
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thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_tf__b,type,
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% Explicit typings (500)
thf(sy_c_Abstract__Rewriting_Omeet_001t__Option__Ooption_Itf__b_J,type,
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thf(sy_c_Abstract__Rewriting_Orestrict_001t__Option__Ooption_Itf__b_J,type,
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thf(sy_c_Abstract__Rewriting_Orestrict_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
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thf(sy_c_Abstract__Rewriting_Orestrict_001t__Set__Oset_It__Option__Ooption_Itf__b_J_J,type,
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thf(sy_c_Abstract__Rewriting_Ostrongly__confluent__on_001t__Option__Ooption_Itf__b_J,type,
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thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Option__Ooption_Itf__b_J_001t__Nat__Onat,type,
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thf(sy_c_Context__Extensions_Ogctxtex__onp_001tf__b,type,
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thf(sy_c_FOR__Certificate_Oext__step_OEParallel,type,
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thf(sy_c_FOR__Certificate_Oext__step_OESingle,type,
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thf(sy_c_FOR__Certificate_Oext__step_OEStrictParallel,type,
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thf(sy_c_FOR__Certificate_Opos__step_OPAny,type,
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thf(sy_c_FOR__Certificate_Opos__step_OPNonRoot,type,
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thf(sy_c_FOR__Certificate_Opos__step_OPRoot,type,
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thf(sy_c_FSet_Ofimage_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_FSet_Ofimage_001t__Option__Ooption_Itf__b_J_001t__Option__Ooption_Itf__b_J,type,
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thf(sy_c_FSet_Ofimage_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
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thf(sy_c_FSet_Ofimage_001tf__b_001tf__b,type,
    fimage_b_b: ( b > b ) > fset_b > fset_b ).

thf(sy_c_FSet_Ofset_Ofset_001t__Nat__Onat,type,
    fset_nat2: fset_nat > set_nat ).

thf(sy_c_FSet_Ofset_Ofset_001t__Option__Ooption_Itf__b_J,type,
    fset_option_b2: fset_option_b > set_option_b ).

thf(sy_c_FSet_Ofset_Ofset_001t__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J,type,
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thf(sy_c_FSet_Ofset_Ofset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_Mt__Nat__Onat_J,type,
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thf(sy_c_FSet_Ofset_Ofset_001t__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J,type,
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thf(sy_c_FSet_Ofset_Ofset_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
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thf(sy_c_FSet_Ofset_Ofset_001tf__b,type,
    fset_b2: fset_b > set_b ).

thf(sy_c_Fun__Def_Oin__rel_001t__Option__Ooption_Itf__b_J_001t__Option__Ooption_Itf__b_J,type,
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thf(sy_c_Fun__Def_Oin__rel_001t__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_001t__Nat__Onat,type,
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thf(sy_c_Fun__Def_Oin__rel_001tf__b_001t__Nat__Onat,type,
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thf(sy_c_Fun__Def_Orp__inv__image_001t__Option__Ooption_Itf__b_J_001t__Option__Ooption_Itf__b_J,type,
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thf(sy_c_Ground__Closure_Ogcomp__rel_001tf__b,type,
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thf(sy_c_Ground__Ctxt_Ofunas__gctxt_001tf__b,type,
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thf(sy_c_Ground__Ctxt_Ogctxt_OGHole_001tf__b,type,
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thf(sy_c_Ground__MCtxt_Ofunas__gmctxt_001tf__b,type,
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thf(sy_c_Ground__MCtxt_Ogmctxt_OGMHole_001tf__b,type,
    ground_GMHole_b: ground_gmctxt_b ).

thf(sy_c_Ground__MCtxt_Onum__gholes_001tf__b,type,
    ground_num_gholes_b: ground_gmctxt_b > nat ).

thf(sy_c_Ground__Terms_O_092_060T_062_092_060_094sub_062G_001tf__b,type,
    ground_T_G_b: set_Pr1307281990691478580_b_nat > set_Ground_gterm_b ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

thf(sy_c_HOL_OThe_001t__Nat__Onat,type,
    the_nat: ( nat > $o ) > nat ).

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thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_Mt__Nat__Onat_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Option__Ooption_Itf__b_J_M_062_It__Option__Ooption_Itf__b_J_M_Eo_J_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_M_062_It__Nat__Onat_M_Eo_J_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001_062_Itf__b_M_062_It__Nat__Onat_M_Eo_J_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__b_J_Mt__Ground____Terms__Ogterm_Itf__b_J_J_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_062_It__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_M_Eo_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_062_Itf__b_M_Eo_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Option__Ooption_Itf__b_J_M_062_It__Option__Ooption_Itf__b_J_M_Eo_J_J,type,
    sup_su6610355334417212162on_b_o: ( option_b > option_b > $o ) > ( option_b > option_b > $o ) > option_b > option_b > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Option__Ooption_Itf__b_J_M_Eo_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__b_J_Mt__Ground____Terms__Ogterm_Itf__b_J_J_M_Eo_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_M_062_It__Nat__Onat_M_Eo_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_M_Eo_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_J_M_Eo_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Product____Type__Oprod_It__Set__Oset_It__Option__Ooption_Itf__b_J_J_Mt__Set__Oset_It__Option__Ooption_Itf__b_J_J_J_M_Eo_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_J_J_M_Eo_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_M_Eo_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Product____Type__Oprod_Itf__b_Mtf__b_J_M_Eo_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__b_M_062_It__Nat__Onat_M_Eo_J_J,type,
    sup_sup_b_nat_o: ( b > nat > $o ) > ( b > nat > $o ) > b > nat > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__b_M_062_It__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_M_Eo_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__b_M_062_Itf__b_M_Eo_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__b_M_Eo_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__FSet__Ofset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__FSet__Ofset_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
    sup_sup_nat: nat > nat > nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Option__Ooption_I_062_It__Option__Ooption_Itf__b_J_M_062_It__Option__Ooption_Itf__b_J_M_Eo_J_J_J,type,
    sup_su5332351281251056968on_b_o: option818561745025029492on_b_o > option818561745025029492on_b_o > option818561745025029492on_b_o ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Option__Ooption_I_062_It__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_M_062_It__Nat__Onat_M_Eo_J_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Option__Ooption_I_062_Itf__b_M_062_It__Nat__Onat_M_Eo_J_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
    sup_sup_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Option__Ooption_Itf__b_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__b_J_Mt__Ground____Terms__Ogterm_Itf__b_J_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Option__Ooption_Itf__b_J_J_Mt__Set__Oset_It__Option__Ooption_Itf__b_J_J_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_J_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__b_J,type,
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thf(sy_c_Lift__Root__Step_Ogpar__rstepD_H_001t__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J,type,
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thf(sy_c_Lift__Root__Step_Ogpar__rstepD_H_001tf__b,type,
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thf(sy_c_Lift__Root__Step_Ogpar__rstepD_Hp_001tf__b,type,
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thf(sy_c_Lift__Root__Step_Olift__root__step_001tf__b,type,
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thf(sy_c_Lift__Root__Step_Olift__root__step__rel_001tf__b,type,
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thf(sy_c_Option_Ocombine__options_001tf__b,type,
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thf(sy_c_Option_Ooption_ONone_001tf__b,type,
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thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Option__Ooption_Itf__b_J_J_M_062_It__Set__Oset_It__Option__Ooption_Itf__b_J_J_M_Eo_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_I_062_It__Option__Ooption_Itf__b_J_M_062_It__Option__Ooption_Itf__b_J_M_Eo_J_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_I_062_Itf__b_M_062_It__Nat__Onat_M_Eo_J_J_J,type,
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thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J_J,type,
    member8772231139900714834_nat_b: produc4418390255170063017_nat_b > set_Pr4934845982659317257_nat_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Nat__Onat_J_J,type,
    member5433527884241383698_b_nat: produc2647518596149342057_b_nat > set_Pr5878735446571772873_b_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_Mt__Nat__Onat_J,type,
    member7289896697870905981_b_nat: produc1396820459975257318_b_nat > set_Pr6944079702279229084_b_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_J,type,
    member6435960643780680464tion_b: produc6650559869974650087tion_b > set_Pr5366877986981747911tion_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_Mtf__b_J,type,
    member5793571475316979880on_b_b: produc1630303296191104639on_b_b > set_Pr7804114732869508703on_b_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mtf__b_J_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mtf__b_J_J,type,
    member4206488804519221648on_b_b: produc426851384182405223on_b_b > set_Pr902348228576698183on_b_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_Mt__Nat__Onat_J_J,type,
    member4316259128586992598_b_nat: produc7723296494992555821_b_nat > set_Pr8202489598513264781_b_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_J,type,
    member4966892972527255302_b_nat: produc613052087796603485_b_nat > set_Pr5241429435540800445_b_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mt__Option__Ooption_Itf__b_J_J_Mt__Product____Type__Oprod_Itf__b_Mt__Option__Ooption_Itf__b_J_J_J,type,
    member4689101312032709776tion_b: produc909463891695893351tion_b > set_Pr4403999794808447559tion_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_J_J,type,
    member6692499340770384144tion_b: produc4937985361087708391tion_b > set_Pr8840786723410068167tion_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
    member8369738419320101648od_b_b: produc5537630352423442151od_b_b > set_Pr3901141605387707591od_b_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Option__Ooption_Itf__b_J_J_Mt__Set__Oset_It__Option__Ooption_Itf__b_J_J_J,type,
    member9052098846246995088tion_b: produc3122286842528565607tion_b > set_Pr4148925336519852615tion_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_J_J,type,
    member8319496676406272784tion_b: produc4298468441892121831tion_b > set_Pr2609077471296144583tion_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J,type,
    member6959632917342813205_b_nat: product_prod_b_nat > set_Pr1307281990691478580_b_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_J,type,
    member8570655873678976888tion_b: produc4407387694553101647tion_b > set_Pr5422417554659304751tion_b > $o ).

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

thf(sy_c_member_001t__Set__Oset_It__Option__Ooption_Itf__b_J_J,type,
    member_set_option_b: set_option_b > set_set_option_b > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Option__Ooption_Itf__b_J_Mt__Option__Ooption_Itf__b_J_J_J,type,
    member3084166351965065584tion_b: set_Pr258254617715012167tion_b > set_se1183635074333576999tion_b > $o ).

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

thf(sy_v_R,type,
    r: set_Pr3134649703833893895term_b ).

thf(sy_v__092_060A_062,type,
    a: tree_r4611138855727183398tion_b ).

thf(sy_v__092_060F_062,type,
    f: fset_P2289935846317411886_b_nat ).

% Relevant facts (1276)
thf(fact_0_assms,axiom,
    rRn_RR2_spec_a_b_b @ a @ r ).

% assms
thf(fact_1_case__prod__conv,axiom,
    ! [F: option_b > option_b > produc7781606619528532327tion_b > $o,A: option_b,B: option_b] :
      ( ( produc1842704630950797122on_b_o @ F @ ( produc3989389360737052631tion_b @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_2_case__prod__conv,axiom,
    ! [F: set_Pr258254617715012167tion_b > set_Pr258254617715012167tion_b > $o,A: set_Pr258254617715012167tion_b,B: set_Pr258254617715012167tion_b] :
      ( ( produc2671211398375001388on_b_o @ F @ ( produc4709765886195557207tion_b @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_3_case__prod__conv,axiom,
    ! [F: set_option_b > set_option_b > $o,A: set_option_b,B: set_option_b] :
      ( ( produc7796829984862323884on_b_o @ F @ ( produc5242204166755002327tion_b @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_4_case__prod__conv,axiom,
    ! [F: b > nat > product_prod_b_nat,A: b,B: nat] :
      ( ( produc282185899741183267_b_nat @ F @ ( product_Pair_b_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_5_case__prod__conv,axiom,
    ! [F: b > nat > $o,A: b,B: nat] :
      ( ( produc795641402153621683_nat_o @ F @ ( product_Pair_b_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_6_case__prod__conv,axiom,
    ! [F: option_b > option_b > produc7781606619528532327tion_b,A: option_b,B: option_b] :
      ( ( produc5931753611847401595tion_b @ F @ ( produc3989389360737052631tion_b @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_7_case__prod__conv,axiom,
    ! [F: produc7781606619528532327tion_b > nat > produc1396820459975257318_b_nat,A: produc7781606619528532327tion_b,B: nat] :
      ( ( produc1170353205628678643_b_nat @ F @ ( produc2093246604507004384_b_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_8_case__prod__conv,axiom,
    ! [F: b > nat > produc1396820459975257318_b_nat,A: b,B: nat] :
      ( ( produc5650155531262253963_b_nat @ F @ ( product_Pair_b_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_9_fset_Omap__ident,axiom,
    ! [T: fset_P2289935846317411886_b_nat] :
      ( ( fimage6892398636147833211_b_nat
        @ ^ [X: product_prod_b_nat] : X
        @ T )
      = T ) ).

% fset.map_ident
thf(fact_10_fset_Omap__ident,axiom,
    ! [T: fset_P4398644116778547350_b_nat] :
      ( ( fimage1673450118646331979_b_nat
        @ ^ [X: produc1396820459975257318_b_nat] : X
        @ T )
      = T ) ).

% fset.map_ident
thf(fact_11_fimage__ident,axiom,
    ! [Y: fset_P2289935846317411886_b_nat] :
      ( ( fimage6892398636147833211_b_nat
        @ ^ [X: product_prod_b_nat] : X
        @ Y )
      = Y ) ).

% fimage_ident
thf(fact_12_fimage__ident,axiom,
    ! [Y: fset_P4398644116778547350_b_nat] :
      ( ( fimage1673450118646331979_b_nat
        @ ^ [X: produc1396820459975257318_b_nat] : X
        @ Y )
      = Y ) ).

% fimage_ident
thf(fact_13_ctxt__closure__automaton,axiom,
    ! [A2: tree_r4611138855727183398tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR2_spec_a_b_b @ A2 @ R )
     => ( rRn_RR5856091920061875867_o_b_b
        @ ( tA_Clo518742726639014115on_b_a
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PAny @ fOR_ext_ESingle @ R ) ) ) ).

% ctxt_closure_automaton
thf(fact_14_ctxt__closure__automaton,axiom,
    ! [A2: tree_r2154205730188421652tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR7260272242312503694es_b_b @ A2 @ R )
     => ( rRn_RR3067063739481463809_o_b_b
        @ ( tA_Clo6019824460905488909states
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PAny @ fOR_ext_ESingle @ R ) ) ) ).

% ctxt_closure_automaton
thf(fact_15_ctxt__closure__automaton,axiom,
    ! [A2: tree_r6993104598208160480tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR5637769166088770792_a_b_b @ A2 @ R )
     => ( rRn_RR5351431268385054037_o_b_b
        @ ( tA_Clo7166006328798297001tion_a
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PAny @ fOR_ext_ESingle @ R ) ) ) ).

% ctxt_closure_automaton
thf(fact_16_ctxt__closure__automaton,axiom,
    ! [A2: tree_r7504422106042656583tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR5856091920061875867_o_b_b @ A2 @ R )
     => ( rRn_RR6022601609001321268_o_b_b
        @ ( tA_Clo1751284457178519578um_a_o
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PAny @ fOR_ext_ESingle @ R ) ) ) ).

% ctxt_closure_automaton
thf(fact_17_ctxt__closure__automaton,axiom,
    ! [A2: tree_r3013368800753843012tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR225467092841841886es_b_b @ A2 @ R )
     => ( rRn_RR3181898823427072817_o_b_b
        @ ( tA_Clo9033772373668725469states
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PAny @ fOR_ext_ESingle @ R ) ) ) ).

% ctxt_closure_automaton
thf(fact_18_ctxt__closure__automaton,axiom,
    ! [A2: tree_r8506140744905984007tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR3067063739481463809_o_b_b @ A2 @ R )
     => ( rRn_RR8110423076565338236_o_b_b
        @ ( tA_Clo2937445813830582466ates_o
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PAny @ fOR_ext_ESingle @ R ) ) ) ).

% ctxt_closure_automaton
thf(fact_19_ctxt__closure__automaton,axiom,
    ! [A2: tree_r5558399869034676087tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR6598829644733244139_o_b_b @ A2 @ R )
     => ( rRn_RR1516175417905451364_o_b_b
        @ ( tA_Clo5123162488512328170um_a_o
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PAny @ fOR_ext_ESingle @ R ) ) ) ).

% ctxt_closure_automaton
thf(fact_20_ctxt__closure__automaton,axiom,
    ! [A2: tree_r5058894190550810253tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR5351431268385054037_o_b_b @ A2 @ R )
     => ( rRn_RR7496799601987039354_o_b_b
        @ ( tA_Clo4031405743562811732on_a_o
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PAny @ fOR_ext_ESingle @ R ) ) ) ).

% ctxt_closure_automaton
thf(fact_21_ctxt__closure__automaton,axiom,
    ! [A2: tree_r4493376028253769364tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR6022601609001321268_o_b_b @ A2 @ R )
     => ( rRn_RR2879006615815283977_o_b_b
        @ ( tA_Clo1813122786506970869_a_o_o
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PAny @ fOR_ext_ESingle @ R ) ) ) ).

% ctxt_closure_automaton
thf(fact_22_ctxt__closure__automaton,axiom,
    ! [A2: tree_r2365054918505434199tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR3181898823427072817_o_b_b @ A2 @ R )
     => ( rRn_RR4902034076794279628_o_b_b
        @ ( tA_Clo3965253636744424050ates_o
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PAny @ fOR_ext_ESingle @ R ) ) ) ).

% ctxt_closure_automaton
thf(fact_23_parallel__closure__automaton,axiom,
    ! [A2: tree_r4611138855727183398tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR2_spec_a_b_b @ A2 @ R )
     => ( rRn_RR5637769166088770792_a_b_b
        @ ( tA_Clo1704561519649872061on_b_a
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PAny @ fOR_ext_EParallel @ R ) ) ) ).

% parallel_closure_automaton
thf(fact_24_parallel__closure__automaton,axiom,
    ! [A2: tree_r2154205730188421652tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR7260272242312503694es_b_b @ A2 @ R )
     => ( rRn_RR225467092841841886es_b_b
        @ ( tA_Clo9097943769102302643states
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PAny @ fOR_ext_EParallel @ R ) ) ) ).

% parallel_closure_automaton
thf(fact_25_parallel__closure__automaton,axiom,
    ! [A2: tree_r6993104598208160480tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR5637769166088770792_a_b_b @ A2 @ R )
     => ( rRn_RR1656734375441257134_a_b_b
        @ ( tA_Clo3733605555717174403tion_a
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PAny @ fOR_ext_EParallel @ R ) ) ) ).

% parallel_closure_automaton
thf(fact_26_parallel__closure__automaton,axiom,
    ! [A2: tree_r7504422106042656583tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR5856091920061875867_o_b_b @ A2 @ R )
     => ( rRn_RR6598829644733244139_o_b_b
        @ ( tA_Clo185760224333003968um_a_o
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PAny @ fOR_ext_EParallel @ R ) ) ) ).

% parallel_closure_automaton
thf(fact_27_parallel__closure__automaton,axiom,
    ! [A2: tree_r3013368800753843012tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR225467092841841886es_b_b @ A2 @ R )
     => ( rRn_RR2097331226319512750es_b_b
        @ ( tA_Clo9010432012892974467states
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PAny @ fOR_ext_EParallel @ R ) ) ) ).

% parallel_closure_automaton
thf(fact_28_parallel__closure__automaton,axiom,
    ! [A2: tree_r8506140744905984007tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR3067063739481463809_o_b_b @ A2 @ R )
     => ( rRn_RR2318748202180745671_o_b_b
        @ ( tA_Clo7265354579862034588ates_o
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PAny @ fOR_ext_EParallel @ R ) ) ) ).

% parallel_closure_automaton
thf(fact_29_parallel__closure__automaton,axiom,
    ! [A2: tree_r5558399869034676087tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR6598829644733244139_o_b_b @ A2 @ R )
     => ( rRn_RR2516351529612540859_o_b_b
        @ ( tA_Clo6209304465024705936um_a_o
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PAny @ fOR_ext_EParallel @ R ) ) ) ).

% parallel_closure_automaton
thf(fact_30_parallel__closure__automaton,axiom,
    ! [A2: tree_r5058894190550810253tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR5351431268385054037_o_b_b @ A2 @ R )
     => ( rRn_RR1470927249984479525_o_b_b
        @ ( tA_Clo5117547720075189498on_a_o
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PAny @ fOR_ext_EParallel @ R ) ) ) ).

% parallel_closure_automaton
thf(fact_31_parallel__closure__automaton,axiom,
    ! [A2: tree_r4493376028253769364tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR6022601609001321268_o_b_b @ A2 @ R )
     => ( rRn_RR3960834450804983802_o_b_b
        @ ( tA_Clo8581958826018956751_a_o_o
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PAny @ fOR_ext_EParallel @ R ) ) ) ).

% parallel_closure_automaton
thf(fact_32_parallel__closure__automaton,axiom,
    ! [A2: tree_r2365054918505434199tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR3181898823427072817_o_b_b @ A2 @ R )
     => ( rRn_RR3813377194981999735_o_b_b
        @ ( tA_Clo970278096962909004ates_o
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PAny @ fOR_ext_EParallel @ R ) ) ) ).

% parallel_closure_automaton
thf(fact_33_option_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( some_nat @ X2 )
        = ( some_nat @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_34_option_Oinject,axiom,
    ! [X2: b,Y2: b] :
      ( ( ( some_b @ X2 )
        = ( some_b @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_35_case__prod__Pair__iden,axiom,
    ! [P: produc7025179454482589068tion_b] :
      ( ( produc5819465800884139511tion_b @ produc7510438065615382334tion_b @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_36_case__prod__Pair__iden,axiom,
    ! [P: product_prod_nat_nat] :
      ( ( produc2626176000494625587at_nat @ product_Pair_nat_nat @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_37_case__prod__Pair__iden,axiom,
    ! [P: product_prod_nat_b] :
      ( ( produc1854641736588490023_nat_b @ product_Pair_nat_b @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_38_case__prod__Pair__iden,axiom,
    ! [P: produc4407387694553101647tion_b] :
      ( ( produc8305796566076059259tion_b @ produc5187340250978832319tion_b @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_39_case__prod__Pair__iden,axiom,
    ! [P: product_prod_b_b] :
      ( ( produc921320349745534075od_b_b @ product_Pair_b_b @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_40_case__prod__Pair__iden,axiom,
    ! [P: product_prod_b_nat] :
      ( ( produc282185899741183267_b_nat @ product_Pair_b_nat @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_41_case__prod__Pair__iden,axiom,
    ! [P: produc1396820459975257318_b_nat] :
      ( ( produc1170353205628678643_b_nat @ produc2093246604507004384_b_nat @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_42_case__prod__Pair__iden,axiom,
    ! [P: produc7781606619528532327tion_b] :
      ( ( produc5931753611847401595tion_b @ produc3989389360737052631tion_b @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_43_prod_Oinject,axiom,
    ! [X1: nat,X2: produc7781606619528532327tion_b,Y1: nat,Y2: produc7781606619528532327tion_b] :
      ( ( ( produc7510438065615382334tion_b @ X1 @ X2 )
        = ( produc7510438065615382334tion_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_44_prod_Oinject,axiom,
    ! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
      ( ( ( product_Pair_nat_nat @ X1 @ X2 )
        = ( product_Pair_nat_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_45_prod_Oinject,axiom,
    ! [X1: nat,X2: b,Y1: nat,Y2: b] :
      ( ( ( product_Pair_nat_b @ X1 @ X2 )
        = ( product_Pair_nat_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_46_prod_Oinject,axiom,
    ! [X1: b,X2: produc7781606619528532327tion_b,Y1: b,Y2: produc7781606619528532327tion_b] :
      ( ( ( produc5187340250978832319tion_b @ X1 @ X2 )
        = ( produc5187340250978832319tion_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_47_prod_Oinject,axiom,
    ! [X1: b,X2: b,Y1: b,Y2: b] :
      ( ( ( product_Pair_b_b @ X1 @ X2 )
        = ( product_Pair_b_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_48_prod_Oinject,axiom,
    ! [X1: b,X2: nat,Y1: b,Y2: nat] :
      ( ( ( product_Pair_b_nat @ X1 @ X2 )
        = ( product_Pair_b_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_49_prod_Oinject,axiom,
    ! [X1: produc7781606619528532327tion_b,X2: nat,Y1: produc7781606619528532327tion_b,Y2: nat] :
      ( ( ( produc2093246604507004384_b_nat @ X1 @ X2 )
        = ( produc2093246604507004384_b_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_50_prod_Oinject,axiom,
    ! [X1: option_b,X2: option_b,Y1: option_b,Y2: option_b] :
      ( ( ( produc3989389360737052631tion_b @ X1 @ X2 )
        = ( produc3989389360737052631tion_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_51_old_Oprod_Oinject,axiom,
    ! [A: nat,B: produc7781606619528532327tion_b,A3: nat,B2: produc7781606619528532327tion_b] :
      ( ( ( produc7510438065615382334tion_b @ A @ B )
        = ( produc7510438065615382334tion_b @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_52_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A3: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_53_old_Oprod_Oinject,axiom,
    ! [A: nat,B: b,A3: nat,B2: b] :
      ( ( ( product_Pair_nat_b @ A @ B )
        = ( product_Pair_nat_b @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_54_old_Oprod_Oinject,axiom,
    ! [A: b,B: produc7781606619528532327tion_b,A3: b,B2: produc7781606619528532327tion_b] :
      ( ( ( produc5187340250978832319tion_b @ A @ B )
        = ( produc5187340250978832319tion_b @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_55_old_Oprod_Oinject,axiom,
    ! [A: b,B: b,A3: b,B2: b] :
      ( ( ( product_Pair_b_b @ A @ B )
        = ( product_Pair_b_b @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_56_old_Oprod_Oinject,axiom,
    ! [A: b,B: nat,A3: b,B2: nat] :
      ( ( ( product_Pair_b_nat @ A @ B )
        = ( product_Pair_b_nat @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_57_old_Oprod_Oinject,axiom,
    ! [A: produc7781606619528532327tion_b,B: nat,A3: produc7781606619528532327tion_b,B2: nat] :
      ( ( ( produc2093246604507004384_b_nat @ A @ B )
        = ( produc2093246604507004384_b_nat @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_58_old_Oprod_Oinject,axiom,
    ! [A: option_b,B: option_b,A3: option_b,B2: option_b] :
      ( ( ( produc3989389360737052631tion_b @ A @ B )
        = ( produc3989389360737052631tion_b @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_59_case__prodE2,axiom,
    ! [Q: ( produc7781606619528532327tion_b > $o ) > $o,P2: option_b > option_b > produc7781606619528532327tion_b > $o,Z: produc7781606619528532327tion_b] :
      ( ( Q @ ( produc1842704630950797122on_b_o @ P2 @ Z ) )
     => ~ ! [X3: option_b,Y3: option_b] :
            ( ( Z
              = ( produc3989389360737052631tion_b @ X3 @ Y3 ) )
           => ~ ( Q @ ( P2 @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_60_case__prodE2,axiom,
    ! [Q: $o > $o,P2: set_Pr258254617715012167tion_b > set_Pr258254617715012167tion_b > $o,Z: produc4298468441892121831tion_b] :
      ( ( Q @ ( produc2671211398375001388on_b_o @ P2 @ Z ) )
     => ~ ! [X3: set_Pr258254617715012167tion_b,Y3: set_Pr258254617715012167tion_b] :
            ( ( Z
              = ( produc4709765886195557207tion_b @ X3 @ Y3 ) )
           => ~ ( Q @ ( P2 @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_61_case__prodE2,axiom,
    ! [Q: $o > $o,P2: set_option_b > set_option_b > $o,Z: produc3122286842528565607tion_b] :
      ( ( Q @ ( produc7796829984862323884on_b_o @ P2 @ Z ) )
     => ~ ! [X3: set_option_b,Y3: set_option_b] :
            ( ( Z
              = ( produc5242204166755002327tion_b @ X3 @ Y3 ) )
           => ~ ( Q @ ( P2 @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_62_case__prodE2,axiom,
    ! [Q: product_prod_b_nat > $o,P2: b > nat > product_prod_b_nat,Z: product_prod_b_nat] :
      ( ( Q @ ( produc282185899741183267_b_nat @ P2 @ Z ) )
     => ~ ! [X3: b,Y3: nat] :
            ( ( Z
              = ( product_Pair_b_nat @ X3 @ Y3 ) )
           => ~ ( Q @ ( P2 @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_63_case__prodE2,axiom,
    ! [Q: $o > $o,P2: b > nat > $o,Z: product_prod_b_nat] :
      ( ( Q @ ( produc795641402153621683_nat_o @ P2 @ Z ) )
     => ~ ! [X3: b,Y3: nat] :
            ( ( Z
              = ( product_Pair_b_nat @ X3 @ Y3 ) )
           => ~ ( Q @ ( P2 @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_64_case__prodE2,axiom,
    ! [Q: produc1396820459975257318_b_nat > $o,P2: b > nat > produc1396820459975257318_b_nat,Z: product_prod_b_nat] :
      ( ( Q @ ( produc5650155531262253963_b_nat @ P2 @ Z ) )
     => ~ ! [X3: b,Y3: nat] :
            ( ( Z
              = ( product_Pair_b_nat @ X3 @ Y3 ) )
           => ~ ( Q @ ( P2 @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_65_case__prodE2,axiom,
    ! [Q: produc7781606619528532327tion_b > $o,P2: option_b > option_b > produc7781606619528532327tion_b,Z: produc7781606619528532327tion_b] :
      ( ( Q @ ( produc5931753611847401595tion_b @ P2 @ Z ) )
     => ~ ! [X3: option_b,Y3: option_b] :
            ( ( Z
              = ( produc3989389360737052631tion_b @ X3 @ Y3 ) )
           => ~ ( Q @ ( P2 @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_66_case__prodE2,axiom,
    ! [Q: produc1396820459975257318_b_nat > $o,P2: produc7781606619528532327tion_b > nat > produc1396820459975257318_b_nat,Z: produc1396820459975257318_b_nat] :
      ( ( Q @ ( produc1170353205628678643_b_nat @ P2 @ Z ) )
     => ~ ! [X3: produc7781606619528532327tion_b,Y3: nat] :
            ( ( Z
              = ( produc2093246604507004384_b_nat @ X3 @ Y3 ) )
           => ~ ( Q @ ( P2 @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_67_case__prod__eta,axiom,
    ! [F: produc7781606619528532327tion_b > produc7781606619528532327tion_b > $o] :
      ( ( produc1842704630950797122on_b_o
        @ ^ [X: option_b,Y4: option_b] : ( F @ ( produc3989389360737052631tion_b @ X @ Y4 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_68_case__prod__eta,axiom,
    ! [F: produc4298468441892121831tion_b > $o] :
      ( ( produc2671211398375001388on_b_o
        @ ^ [X: set_Pr258254617715012167tion_b,Y4: set_Pr258254617715012167tion_b] : ( F @ ( produc4709765886195557207tion_b @ X @ Y4 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_69_case__prod__eta,axiom,
    ! [F: produc3122286842528565607tion_b > $o] :
      ( ( produc7796829984862323884on_b_o
        @ ^ [X: set_option_b,Y4: set_option_b] : ( F @ ( produc5242204166755002327tion_b @ X @ Y4 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_70_case__prod__eta,axiom,
    ! [F: product_prod_b_nat > product_prod_b_nat] :
      ( ( produc282185899741183267_b_nat
        @ ^ [X: b,Y4: nat] : ( F @ ( product_Pair_b_nat @ X @ Y4 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_71_case__prod__eta,axiom,
    ! [F: product_prod_b_nat > $o] :
      ( ( produc795641402153621683_nat_o
        @ ^ [X: b,Y4: nat] : ( F @ ( product_Pair_b_nat @ X @ Y4 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_72_case__prod__eta,axiom,
    ! [F: product_prod_b_nat > produc1396820459975257318_b_nat] :
      ( ( produc5650155531262253963_b_nat
        @ ^ [X: b,Y4: nat] : ( F @ ( product_Pair_b_nat @ X @ Y4 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_73_case__prod__eta,axiom,
    ! [F: produc7781606619528532327tion_b > produc7781606619528532327tion_b] :
      ( ( produc5931753611847401595tion_b
        @ ^ [X: option_b,Y4: option_b] : ( F @ ( produc3989389360737052631tion_b @ X @ Y4 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_74_case__prod__eta,axiom,
    ! [F: produc1396820459975257318_b_nat > produc1396820459975257318_b_nat] :
      ( ( produc1170353205628678643_b_nat
        @ ^ [X: produc7781606619528532327tion_b,Y4: nat] : ( F @ ( produc2093246604507004384_b_nat @ X @ Y4 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_75_cond__case__prod__eta,axiom,
    ! [F: option_b > option_b > produc7781606619528532327tion_b > $o,G: produc7781606619528532327tion_b > produc7781606619528532327tion_b > $o] :
      ( ! [X3: option_b,Y3: option_b] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( produc3989389360737052631tion_b @ X3 @ Y3 ) ) )
     => ( ( produc1842704630950797122on_b_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_76_cond__case__prod__eta,axiom,
    ! [F: set_Pr258254617715012167tion_b > set_Pr258254617715012167tion_b > $o,G: produc4298468441892121831tion_b > $o] :
      ( ! [X3: set_Pr258254617715012167tion_b,Y3: set_Pr258254617715012167tion_b] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( produc4709765886195557207tion_b @ X3 @ Y3 ) ) )
     => ( ( produc2671211398375001388on_b_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_77_cond__case__prod__eta,axiom,
    ! [F: set_option_b > set_option_b > $o,G: produc3122286842528565607tion_b > $o] :
      ( ! [X3: set_option_b,Y3: set_option_b] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( produc5242204166755002327tion_b @ X3 @ Y3 ) ) )
     => ( ( produc7796829984862323884on_b_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_78_cond__case__prod__eta,axiom,
    ! [F: b > nat > product_prod_b_nat,G: product_prod_b_nat > product_prod_b_nat] :
      ( ! [X3: b,Y3: nat] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( product_Pair_b_nat @ X3 @ Y3 ) ) )
     => ( ( produc282185899741183267_b_nat @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_79_cond__case__prod__eta,axiom,
    ! [F: b > nat > $o,G: product_prod_b_nat > $o] :
      ( ! [X3: b,Y3: nat] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( product_Pair_b_nat @ X3 @ Y3 ) ) )
     => ( ( produc795641402153621683_nat_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_80_cond__case__prod__eta,axiom,
    ! [F: b > nat > produc1396820459975257318_b_nat,G: product_prod_b_nat > produc1396820459975257318_b_nat] :
      ( ! [X3: b,Y3: nat] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( product_Pair_b_nat @ X3 @ Y3 ) ) )
     => ( ( produc5650155531262253963_b_nat @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_81_cond__case__prod__eta,axiom,
    ! [F: option_b > option_b > produc7781606619528532327tion_b,G: produc7781606619528532327tion_b > produc7781606619528532327tion_b] :
      ( ! [X3: option_b,Y3: option_b] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( produc3989389360737052631tion_b @ X3 @ Y3 ) ) )
     => ( ( produc5931753611847401595tion_b @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_82_cond__case__prod__eta,axiom,
    ! [F: produc7781606619528532327tion_b > nat > produc1396820459975257318_b_nat,G: produc1396820459975257318_b_nat > produc1396820459975257318_b_nat] :
      ( ! [X3: produc7781606619528532327tion_b,Y3: nat] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( produc2093246604507004384_b_nat @ X3 @ Y3 ) ) )
     => ( ( produc1170353205628678643_b_nat @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_83_case__prodI2,axiom,
    ! [P: produc7025179454482589068tion_b,C: nat > produc7781606619528532327tion_b > $o] :
      ( ! [A4: nat,B3: produc7781606619528532327tion_b] :
          ( ( P
            = ( produc7510438065615382334tion_b @ A4 @ B3 ) )
         => ( C @ A4 @ B3 ) )
     => ( produc6970957433282489197on_b_o @ C @ P ) ) ).

% case_prodI2
thf(fact_84_case__prodI2,axiom,
    ! [P: product_prod_nat_nat,C: nat > nat > $o] :
      ( ! [A4: nat,B3: nat] :
          ( ( P
            = ( product_Pair_nat_nat @ A4 @ B3 ) )
         => ( C @ A4 @ B3 ) )
     => ( produc6081775807080527818_nat_o @ C @ P ) ) ).

% case_prodI2
thf(fact_85_case__prodI2,axiom,
    ! [P: product_prod_nat_b,C: nat > b > $o] :
      ( ! [A4: nat,B3: b] :
          ( ( P
            = ( product_Pair_nat_b @ A4 @ B3 ) )
         => ( C @ A4 @ B3 ) )
     => ( produc9071082986767181781at_b_o @ C @ P ) ) ).

% case_prodI2
thf(fact_86_case__prodI2,axiom,
    ! [P: produc4407387694553101647tion_b,C: b > produc7781606619528532327tion_b > $o] :
      ( ! [A4: b,B3: produc7781606619528532327tion_b] :
          ( ( P
            = ( produc5187340250978832319tion_b @ A4 @ B3 ) )
         => ( C @ A4 @ B3 ) )
     => ( produc7466096807386998468on_b_o @ C @ P ) ) ).

% case_prodI2
thf(fact_87_case__prodI2,axiom,
    ! [P: product_prod_b_b,C: b > b > $o] :
      ( ! [A4: b,B3: b] :
          ( ( P
            = ( product_Pair_b_b @ A4 @ B3 ) )
         => ( C @ A4 @ B3 ) )
     => ( produc4772849988445941548_b_b_o @ C @ P ) ) ).

% case_prodI2
thf(fact_88_case__prodI2,axiom,
    ! [P: produc4298468441892121831tion_b,C: set_Pr258254617715012167tion_b > set_Pr258254617715012167tion_b > $o] :
      ( ! [A4: set_Pr258254617715012167tion_b,B3: set_Pr258254617715012167tion_b] :
          ( ( P
            = ( produc4709765886195557207tion_b @ A4 @ B3 ) )
         => ( C @ A4 @ B3 ) )
     => ( produc2671211398375001388on_b_o @ C @ P ) ) ).

% case_prodI2
thf(fact_89_case__prodI2,axiom,
    ! [P: produc3122286842528565607tion_b,C: set_option_b > set_option_b > $o] :
      ( ! [A4: set_option_b,B3: set_option_b] :
          ( ( P
            = ( produc5242204166755002327tion_b @ A4 @ B3 ) )
         => ( C @ A4 @ B3 ) )
     => ( produc7796829984862323884on_b_o @ C @ P ) ) ).

% case_prodI2
thf(fact_90_case__prodI2,axiom,
    ! [P: produc1396820459975257318_b_nat,C: produc7781606619528532327tion_b > nat > $o] :
      ( ! [A4: produc7781606619528532327tion_b,B3: nat] :
          ( ( P
            = ( produc2093246604507004384_b_nat @ A4 @ B3 ) )
         => ( C @ A4 @ B3 ) )
     => ( produc5232674473337540683_nat_o @ C @ P ) ) ).

% case_prodI2
thf(fact_91_case__prodI2,axiom,
    ! [P: produc7781606619528532327tion_b,C: option_b > option_b > $o] :
      ( ! [A4: option_b,B3: option_b] :
          ( ( P
            = ( produc3989389360737052631tion_b @ A4 @ B3 ) )
         => ( C @ A4 @ B3 ) )
     => ( produc3378454745690301612on_b_o @ C @ P ) ) ).

% case_prodI2
thf(fact_92_case__prodI2,axiom,
    ! [P: product_prod_b_nat,C: b > nat > $o] :
      ( ! [A4: b,B3: nat] :
          ( ( P
            = ( product_Pair_b_nat @ A4 @ B3 ) )
         => ( C @ A4 @ B3 ) )
     => ( produc795641402153621683_nat_o @ C @ P ) ) ).

% case_prodI2
thf(fact_93_case__prodI,axiom,
    ! [F: nat > produc7781606619528532327tion_b > $o,A: nat,B: produc7781606619528532327tion_b] :
      ( ( F @ A @ B )
     => ( produc6970957433282489197on_b_o @ F @ ( produc7510438065615382334tion_b @ A @ B ) ) ) ).

% case_prodI
thf(fact_94_case__prodI,axiom,
    ! [F: nat > nat > $o,A: nat,B: nat] :
      ( ( F @ A @ B )
     => ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_95_case__prodI,axiom,
    ! [F: nat > b > $o,A: nat,B: b] :
      ( ( F @ A @ B )
     => ( produc9071082986767181781at_b_o @ F @ ( product_Pair_nat_b @ A @ B ) ) ) ).

% case_prodI
thf(fact_96_case__prodI,axiom,
    ! [F: b > produc7781606619528532327tion_b > $o,A: b,B: produc7781606619528532327tion_b] :
      ( ( F @ A @ B )
     => ( produc7466096807386998468on_b_o @ F @ ( produc5187340250978832319tion_b @ A @ B ) ) ) ).

% case_prodI
thf(fact_97_case__prodI,axiom,
    ! [F: b > b > $o,A: b,B: b] :
      ( ( F @ A @ B )
     => ( produc4772849988445941548_b_b_o @ F @ ( product_Pair_b_b @ A @ B ) ) ) ).

% case_prodI
thf(fact_98_case__prodI,axiom,
    ! [F: set_Pr258254617715012167tion_b > set_Pr258254617715012167tion_b > $o,A: set_Pr258254617715012167tion_b,B: set_Pr258254617715012167tion_b] :
      ( ( F @ A @ B )
     => ( produc2671211398375001388on_b_o @ F @ ( produc4709765886195557207tion_b @ A @ B ) ) ) ).

% case_prodI
thf(fact_99_case__prodI,axiom,
    ! [F: set_option_b > set_option_b > $o,A: set_option_b,B: set_option_b] :
      ( ( F @ A @ B )
     => ( produc7796829984862323884on_b_o @ F @ ( produc5242204166755002327tion_b @ A @ B ) ) ) ).

% case_prodI
thf(fact_100_case__prodI,axiom,
    ! [F: produc7781606619528532327tion_b > nat > $o,A: produc7781606619528532327tion_b,B: nat] :
      ( ( F @ A @ B )
     => ( produc5232674473337540683_nat_o @ F @ ( produc2093246604507004384_b_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_101_case__prodI,axiom,
    ! [F: option_b > option_b > $o,A: option_b,B: option_b] :
      ( ( F @ A @ B )
     => ( produc3378454745690301612on_b_o @ F @ ( produc3989389360737052631tion_b @ A @ B ) ) ) ).

% case_prodI
thf(fact_102_case__prodI,axiom,
    ! [F: b > nat > $o,A: b,B: nat] :
      ( ( F @ A @ B )
     => ( produc795641402153621683_nat_o @ F @ ( product_Pair_b_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_103_mem__case__prodI2,axiom,
    ! [P: product_prod_b_nat,Z: nat,C: b > nat > set_nat] :
      ( ! [A4: b,B3: nat] :
          ( ( P
            = ( product_Pair_b_nat @ A4 @ B3 ) )
         => ( member_nat @ Z @ ( C @ A4 @ B3 ) ) )
     => ( member_nat @ Z @ ( produc7337630463249427243et_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_104_mem__case__prodI2,axiom,
    ! [P: product_prod_b_nat,Z: b,C: b > nat > set_b] :
      ( ! [A4: b,B3: nat] :
          ( ( P
            = ( product_Pair_b_nat @ A4 @ B3 ) )
         => ( member_b @ Z @ ( C @ A4 @ B3 ) ) )
     => ( member_b @ Z @ ( produc6760141351533629306_set_b @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_105_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_nat,Z: nat,C: nat > nat > set_nat] :
      ( ! [A4: nat,B3: nat] :
          ( ( P
            = ( product_Pair_nat_nat @ A4 @ B3 ) )
         => ( member_nat @ Z @ ( C @ A4 @ B3 ) ) )
     => ( member_nat @ Z @ ( produc6189476227299908564et_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_106_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_nat,Z: b,C: nat > nat > set_b] :
      ( ! [A4: nat,B3: nat] :
          ( ( P
            = ( product_Pair_nat_nat @ A4 @ B3 ) )
         => ( member_b @ Z @ ( C @ A4 @ B3 ) ) )
     => ( member_b @ Z @ ( produc8052394788132812561_set_b @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_107_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_b,Z: nat,C: nat > b > set_nat] :
      ( ! [A4: nat,B3: b] :
          ( ( P
            = ( product_Pair_nat_b @ A4 @ B3 ) )
         => ( member_nat @ Z @ ( C @ A4 @ B3 ) ) )
     => ( member_nat @ Z @ ( produc7186063458415227017et_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_108_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_b,Z: b,C: nat > b > set_b] :
      ( ! [A4: nat,B3: b] :
          ( ( P
            = ( product_Pair_nat_b @ A4 @ B3 ) )
         => ( member_b @ Z @ ( C @ A4 @ B3 ) ) )
     => ( member_b @ Z @ ( produc4985422997113759132_set_b @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_109_mem__case__prodI2,axiom,
    ! [P: product_prod_b_b,Z: nat,C: b > b > set_nat] :
      ( ! [A4: b,B3: b] :
          ( ( P
            = ( product_Pair_b_b @ A4 @ B3 ) )
         => ( member_nat @ Z @ ( C @ A4 @ B3 ) ) )
     => ( member_nat @ Z @ ( produc5893810021816043762et_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_110_mem__case__prodI2,axiom,
    ! [P: product_prod_b_b,Z: b,C: b > b > set_b] :
      ( ! [A4: b,B3: b] :
          ( ( P
            = ( product_Pair_b_b @ A4 @ B3 ) )
         => ( member_b @ Z @ ( C @ A4 @ B3 ) ) )
     => ( member_b @ Z @ ( produc2614443674381817203_set_b @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_111_mem__case__prodI2,axiom,
    ! [P: product_prod_b_nat,Z: option_b,C: b > nat > set_option_b] :
      ( ! [A4: b,B3: nat] :
          ( ( P
            = ( product_Pair_b_nat @ A4 @ B3 ) )
         => ( member_option_b @ Z @ ( C @ A4 @ B3 ) ) )
     => ( member_option_b @ Z @ ( produc5701421212930722368tion_b @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_112_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_nat,Z: option_b,C: nat > nat > set_option_b] :
      ( ! [A4: nat,B3: nat] :
          ( ( P
            = ( product_Pair_nat_nat @ A4 @ B3 ) )
         => ( member_option_b @ Z @ ( C @ A4 @ B3 ) ) )
     => ( member_option_b @ Z @ ( produc819122955021894487tion_b @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_113_mem__case__prodI,axiom,
    ! [Z: nat,C: b > nat > set_nat,A: b,B: nat] :
      ( ( member_nat @ Z @ ( C @ A @ B ) )
     => ( member_nat @ Z @ ( produc7337630463249427243et_nat @ C @ ( product_Pair_b_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_114_mem__case__prodI,axiom,
    ! [Z: b,C: b > nat > set_b,A: b,B: nat] :
      ( ( member_b @ Z @ ( C @ A @ B ) )
     => ( member_b @ Z @ ( produc6760141351533629306_set_b @ C @ ( product_Pair_b_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_115_mem__case__prodI,axiom,
    ! [Z: nat,C: nat > nat > set_nat,A: nat,B: nat] :
      ( ( member_nat @ Z @ ( C @ A @ B ) )
     => ( member_nat @ Z @ ( produc6189476227299908564et_nat @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_116_mem__case__prodI,axiom,
    ! [Z: b,C: nat > nat > set_b,A: nat,B: nat] :
      ( ( member_b @ Z @ ( C @ A @ B ) )
     => ( member_b @ Z @ ( produc8052394788132812561_set_b @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_117_mem__case__prodI,axiom,
    ! [Z: nat,C: nat > b > set_nat,A: nat,B: b] :
      ( ( member_nat @ Z @ ( C @ A @ B ) )
     => ( member_nat @ Z @ ( produc7186063458415227017et_nat @ C @ ( product_Pair_nat_b @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_118_mem__case__prodI,axiom,
    ! [Z: b,C: nat > b > set_b,A: nat,B: b] :
      ( ( member_b @ Z @ ( C @ A @ B ) )
     => ( member_b @ Z @ ( produc4985422997113759132_set_b @ C @ ( product_Pair_nat_b @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_119_mem__case__prodI,axiom,
    ! [Z: nat,C: b > b > set_nat,A: b,B: b] :
      ( ( member_nat @ Z @ ( C @ A @ B ) )
     => ( member_nat @ Z @ ( produc5893810021816043762et_nat @ C @ ( product_Pair_b_b @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_120_mem__case__prodI,axiom,
    ! [Z: b,C: b > b > set_b,A: b,B: b] :
      ( ( member_b @ Z @ ( C @ A @ B ) )
     => ( member_b @ Z @ ( produc2614443674381817203_set_b @ C @ ( product_Pair_b_b @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_121_mem__case__prodI,axiom,
    ! [Z: option_b,C: b > nat > set_option_b,A: b,B: nat] :
      ( ( member_option_b @ Z @ ( C @ A @ B ) )
     => ( member_option_b @ Z @ ( produc5701421212930722368tion_b @ C @ ( product_Pair_b_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_122_mem__case__prodI,axiom,
    ! [Z: option_b,C: nat > nat > set_option_b,A: nat,B: nat] :
      ( ( member_option_b @ Z @ ( C @ A @ B ) )
     => ( member_option_b @ Z @ ( produc819122955021894487tion_b @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_123_case__prodI2_H,axiom,
    ! [P: produc7781606619528532327tion_b,C: option_b > option_b > produc7781606619528532327tion_b > $o,X4: produc7781606619528532327tion_b] :
      ( ! [A4: option_b,B3: option_b] :
          ( ( ( produc3989389360737052631tion_b @ A4 @ B3 )
            = P )
         => ( C @ A4 @ B3 @ X4 ) )
     => ( produc1842704630950797122on_b_o @ C @ P @ X4 ) ) ).

% case_prodI2'
thf(fact_124_mem__case__prodE,axiom,
    ! [Z: nat,C: b > nat > set_nat,P: product_prod_b_nat] :
      ( ( member_nat @ Z @ ( produc7337630463249427243et_nat @ C @ P ) )
     => ~ ! [X3: b,Y3: nat] :
            ( ( P
              = ( product_Pair_b_nat @ X3 @ Y3 ) )
           => ~ ( member_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_125_mem__case__prodE,axiom,
    ! [Z: b,C: b > nat > set_b,P: product_prod_b_nat] :
      ( ( member_b @ Z @ ( produc6760141351533629306_set_b @ C @ P ) )
     => ~ ! [X3: b,Y3: nat] :
            ( ( P
              = ( product_Pair_b_nat @ X3 @ Y3 ) )
           => ~ ( member_b @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_126_mem__case__prodE,axiom,
    ! [Z: nat,C: nat > nat > set_nat,P: product_prod_nat_nat] :
      ( ( member_nat @ Z @ ( produc6189476227299908564et_nat @ C @ P ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_127_mem__case__prodE,axiom,
    ! [Z: b,C: nat > nat > set_b,P: product_prod_nat_nat] :
      ( ( member_b @ Z @ ( produc8052394788132812561_set_b @ C @ P ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member_b @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_128_mem__case__prodE,axiom,
    ! [Z: nat,C: nat > b > set_nat,P: product_prod_nat_b] :
      ( ( member_nat @ Z @ ( produc7186063458415227017et_nat @ C @ P ) )
     => ~ ! [X3: nat,Y3: b] :
            ( ( P
              = ( product_Pair_nat_b @ X3 @ Y3 ) )
           => ~ ( member_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_129_mem__case__prodE,axiom,
    ! [Z: b,C: nat > b > set_b,P: product_prod_nat_b] :
      ( ( member_b @ Z @ ( produc4985422997113759132_set_b @ C @ P ) )
     => ~ ! [X3: nat,Y3: b] :
            ( ( P
              = ( product_Pair_nat_b @ X3 @ Y3 ) )
           => ~ ( member_b @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_130_mem__case__prodE,axiom,
    ! [Z: nat,C: b > b > set_nat,P: product_prod_b_b] :
      ( ( member_nat @ Z @ ( produc5893810021816043762et_nat @ C @ P ) )
     => ~ ! [X3: b,Y3: b] :
            ( ( P
              = ( product_Pair_b_b @ X3 @ Y3 ) )
           => ~ ( member_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_131_mem__case__prodE,axiom,
    ! [Z: b,C: b > b > set_b,P: product_prod_b_b] :
      ( ( member_b @ Z @ ( produc2614443674381817203_set_b @ C @ P ) )
     => ~ ! [X3: b,Y3: b] :
            ( ( P
              = ( product_Pair_b_b @ X3 @ Y3 ) )
           => ~ ( member_b @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_132_mem__case__prodE,axiom,
    ! [Z: option_b,C: b > nat > set_option_b,P: product_prod_b_nat] :
      ( ( member_option_b @ Z @ ( produc5701421212930722368tion_b @ C @ P ) )
     => ~ ! [X3: b,Y3: nat] :
            ( ( P
              = ( product_Pair_b_nat @ X3 @ Y3 ) )
           => ~ ( member_option_b @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_133_mem__case__prodE,axiom,
    ! [Z: option_b,C: nat > nat > set_option_b,P: product_prod_nat_nat] :
      ( ( member_option_b @ Z @ ( produc819122955021894487tion_b @ C @ P ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member_option_b @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_134_case__prodE,axiom,
    ! [C: nat > produc7781606619528532327tion_b > $o,P: produc7025179454482589068tion_b] :
      ( ( produc6970957433282489197on_b_o @ C @ P )
     => ~ ! [X3: nat,Y3: produc7781606619528532327tion_b] :
            ( ( P
              = ( produc7510438065615382334tion_b @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_135_case__prodE,axiom,
    ! [C: nat > nat > $o,P: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o @ C @ P )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_136_case__prodE,axiom,
    ! [C: nat > b > $o,P: product_prod_nat_b] :
      ( ( produc9071082986767181781at_b_o @ C @ P )
     => ~ ! [X3: nat,Y3: b] :
            ( ( P
              = ( product_Pair_nat_b @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_137_case__prodE,axiom,
    ! [C: b > produc7781606619528532327tion_b > $o,P: produc4407387694553101647tion_b] :
      ( ( produc7466096807386998468on_b_o @ C @ P )
     => ~ ! [X3: b,Y3: produc7781606619528532327tion_b] :
            ( ( P
              = ( produc5187340250978832319tion_b @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_138_case__prodE,axiom,
    ! [C: b > b > $o,P: product_prod_b_b] :
      ( ( produc4772849988445941548_b_b_o @ C @ P )
     => ~ ! [X3: b,Y3: b] :
            ( ( P
              = ( product_Pair_b_b @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_139_case__prodE,axiom,
    ! [C: set_Pr258254617715012167tion_b > set_Pr258254617715012167tion_b > $o,P: produc4298468441892121831tion_b] :
      ( ( produc2671211398375001388on_b_o @ C @ P )
     => ~ ! [X3: set_Pr258254617715012167tion_b,Y3: set_Pr258254617715012167tion_b] :
            ( ( P
              = ( produc4709765886195557207tion_b @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_140_case__prodE,axiom,
    ! [C: set_option_b > set_option_b > $o,P: produc3122286842528565607tion_b] :
      ( ( produc7796829984862323884on_b_o @ C @ P )
     => ~ ! [X3: set_option_b,Y3: set_option_b] :
            ( ( P
              = ( produc5242204166755002327tion_b @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_141_case__prodE,axiom,
    ! [C: produc7781606619528532327tion_b > nat > $o,P: produc1396820459975257318_b_nat] :
      ( ( produc5232674473337540683_nat_o @ C @ P )
     => ~ ! [X3: produc7781606619528532327tion_b,Y3: nat] :
            ( ( P
              = ( produc2093246604507004384_b_nat @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_142_case__prodE,axiom,
    ! [C: option_b > option_b > $o,P: produc7781606619528532327tion_b] :
      ( ( produc3378454745690301612on_b_o @ C @ P )
     => ~ ! [X3: option_b,Y3: option_b] :
            ( ( P
              = ( produc3989389360737052631tion_b @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_143_case__prodE,axiom,
    ! [C: b > nat > $o,P: product_prod_b_nat] :
      ( ( produc795641402153621683_nat_o @ C @ P )
     => ~ ! [X3: b,Y3: nat] :
            ( ( P
              = ( product_Pair_b_nat @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_144_case__prodD,axiom,
    ! [F: nat > produc7781606619528532327tion_b > $o,A: nat,B: produc7781606619528532327tion_b] :
      ( ( produc6970957433282489197on_b_o @ F @ ( produc7510438065615382334tion_b @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_145_case__prodD,axiom,
    ! [F: nat > nat > $o,A: nat,B: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_146_case__prodD,axiom,
    ! [F: nat > b > $o,A: nat,B: b] :
      ( ( produc9071082986767181781at_b_o @ F @ ( product_Pair_nat_b @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_147_case__prodD,axiom,
    ! [F: b > produc7781606619528532327tion_b > $o,A: b,B: produc7781606619528532327tion_b] :
      ( ( produc7466096807386998468on_b_o @ F @ ( produc5187340250978832319tion_b @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_148_case__prodD,axiom,
    ! [F: b > b > $o,A: b,B: b] :
      ( ( produc4772849988445941548_b_b_o @ F @ ( product_Pair_b_b @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_149_case__prodD,axiom,
    ! [F: set_Pr258254617715012167tion_b > set_Pr258254617715012167tion_b > $o,A: set_Pr258254617715012167tion_b,B: set_Pr258254617715012167tion_b] :
      ( ( produc2671211398375001388on_b_o @ F @ ( produc4709765886195557207tion_b @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_150_case__prodD,axiom,
    ! [F: set_option_b > set_option_b > $o,A: set_option_b,B: set_option_b] :
      ( ( produc7796829984862323884on_b_o @ F @ ( produc5242204166755002327tion_b @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_151_case__prodD,axiom,
    ! [F: produc7781606619528532327tion_b > nat > $o,A: produc7781606619528532327tion_b,B: nat] :
      ( ( produc5232674473337540683_nat_o @ F @ ( produc2093246604507004384_b_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_152_case__prodD,axiom,
    ! [F: option_b > option_b > $o,A: option_b,B: option_b] :
      ( ( produc3378454745690301612on_b_o @ F @ ( produc3989389360737052631tion_b @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_153_case__prodD,axiom,
    ! [F: b > nat > $o,A: b,B: nat] :
      ( ( produc795641402153621683_nat_o @ F @ ( product_Pair_b_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_154_case__prodE_H,axiom,
    ! [C: option_b > option_b > produc7781606619528532327tion_b > $o,P: produc7781606619528532327tion_b,Z: produc7781606619528532327tion_b] :
      ( ( produc1842704630950797122on_b_o @ C @ P @ Z )
     => ~ ! [X3: option_b,Y3: option_b] :
            ( ( P
              = ( produc3989389360737052631tion_b @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 @ Z ) ) ) ).

% case_prodE'
thf(fact_155_case__prodD_H,axiom,
    ! [R: option_b > option_b > produc7781606619528532327tion_b > $o,A: option_b,B: option_b,C: produc7781606619528532327tion_b] :
      ( ( produc1842704630950797122on_b_o @ R @ ( produc3989389360737052631tion_b @ A @ B ) @ C )
     => ( R @ A @ B @ C ) ) ).

% case_prodD'
thf(fact_156_prod__induct3,axiom,
    ! [P2: produc7025179454482589068tion_b > $o,X4: produc7025179454482589068tion_b] :
      ( ! [A4: nat,B3: option_b,C2: option_b] : ( P2 @ ( produc7510438065615382334tion_b @ A4 @ ( produc3989389360737052631tion_b @ B3 @ C2 ) ) )
     => ( P2 @ X4 ) ) ).

% prod_induct3
thf(fact_157_prod__induct3,axiom,
    ! [P2: produc4407387694553101647tion_b > $o,X4: produc4407387694553101647tion_b] :
      ( ! [A4: b,B3: option_b,C2: option_b] : ( P2 @ ( produc5187340250978832319tion_b @ A4 @ ( produc3989389360737052631tion_b @ B3 @ C2 ) ) )
     => ( P2 @ X4 ) ) ).

% prod_induct3
thf(fact_158_prod__cases3,axiom,
    ! [Y5: produc7025179454482589068tion_b] :
      ~ ! [A4: nat,B3: option_b,C2: option_b] :
          ( Y5
         != ( produc7510438065615382334tion_b @ A4 @ ( produc3989389360737052631tion_b @ B3 @ C2 ) ) ) ).

% prod_cases3
thf(fact_159_prod__cases3,axiom,
    ! [Y5: produc4407387694553101647tion_b] :
      ~ ! [A4: b,B3: option_b,C2: option_b] :
          ( Y5
         != ( produc5187340250978832319tion_b @ A4 @ ( produc3989389360737052631tion_b @ B3 @ C2 ) ) ) ).

% prod_cases3
thf(fact_160_Pair__inject,axiom,
    ! [A: nat,B: produc7781606619528532327tion_b,A3: nat,B2: produc7781606619528532327tion_b] :
      ( ( ( produc7510438065615382334tion_b @ A @ B )
        = ( produc7510438065615382334tion_b @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_161_Pair__inject,axiom,
    ! [A: nat,B: nat,A3: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_162_Pair__inject,axiom,
    ! [A: nat,B: b,A3: nat,B2: b] :
      ( ( ( product_Pair_nat_b @ A @ B )
        = ( product_Pair_nat_b @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_163_Pair__inject,axiom,
    ! [A: b,B: produc7781606619528532327tion_b,A3: b,B2: produc7781606619528532327tion_b] :
      ( ( ( produc5187340250978832319tion_b @ A @ B )
        = ( produc5187340250978832319tion_b @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_164_Pair__inject,axiom,
    ! [A: b,B: b,A3: b,B2: b] :
      ( ( ( product_Pair_b_b @ A @ B )
        = ( product_Pair_b_b @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_165_Pair__inject,axiom,
    ! [A: produc7781606619528532327tion_b,B: nat,A3: produc7781606619528532327tion_b,B2: nat] :
      ( ( ( produc2093246604507004384_b_nat @ A @ B )
        = ( produc2093246604507004384_b_nat @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_166_Pair__inject,axiom,
    ! [A: option_b,B: option_b,A3: option_b,B2: option_b] :
      ( ( ( produc3989389360737052631tion_b @ A @ B )
        = ( produc3989389360737052631tion_b @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_167_Pair__inject,axiom,
    ! [A: b,B: nat,A3: b,B2: nat] :
      ( ( ( product_Pair_b_nat @ A @ B )
        = ( product_Pair_b_nat @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_168_prod__cases,axiom,
    ! [P2: produc7025179454482589068tion_b > $o,P: produc7025179454482589068tion_b] :
      ( ! [A4: nat,B3: produc7781606619528532327tion_b] : ( P2 @ ( produc7510438065615382334tion_b @ A4 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_169_prod__cases,axiom,
    ! [P2: product_prod_nat_nat > $o,P: product_prod_nat_nat] :
      ( ! [A4: nat,B3: nat] : ( P2 @ ( product_Pair_nat_nat @ A4 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_170_prod__cases,axiom,
    ! [P2: product_prod_nat_b > $o,P: product_prod_nat_b] :
      ( ! [A4: nat,B3: b] : ( P2 @ ( product_Pair_nat_b @ A4 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_171_prod__cases,axiom,
    ! [P2: produc4407387694553101647tion_b > $o,P: produc4407387694553101647tion_b] :
      ( ! [A4: b,B3: produc7781606619528532327tion_b] : ( P2 @ ( produc5187340250978832319tion_b @ A4 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_172_prod__cases,axiom,
    ! [P2: product_prod_b_b > $o,P: product_prod_b_b] :
      ( ! [A4: b,B3: b] : ( P2 @ ( product_Pair_b_b @ A4 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_173_prod__cases,axiom,
    ! [P2: produc1396820459975257318_b_nat > $o,P: produc1396820459975257318_b_nat] :
      ( ! [A4: produc7781606619528532327tion_b,B3: nat] : ( P2 @ ( produc2093246604507004384_b_nat @ A4 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_174_prod__cases,axiom,
    ! [P2: produc7781606619528532327tion_b > $o,P: produc7781606619528532327tion_b] :
      ( ! [A4: option_b,B3: option_b] : ( P2 @ ( produc3989389360737052631tion_b @ A4 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_175_prod__cases,axiom,
    ! [P2: product_prod_b_nat > $o,P: product_prod_b_nat] :
      ( ! [A4: b,B3: nat] : ( P2 @ ( product_Pair_b_nat @ A4 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_176_surj__pair,axiom,
    ! [P: produc7025179454482589068tion_b] :
    ? [X3: nat,Y3: produc7781606619528532327tion_b] :
      ( P
      = ( produc7510438065615382334tion_b @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_177_surj__pair,axiom,
    ! [P: product_prod_nat_nat] :
    ? [X3: nat,Y3: nat] :
      ( P
      = ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_178_surj__pair,axiom,
    ! [P: product_prod_nat_b] :
    ? [X3: nat,Y3: b] :
      ( P
      = ( product_Pair_nat_b @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_179_surj__pair,axiom,
    ! [P: produc4407387694553101647tion_b] :
    ? [X3: b,Y3: produc7781606619528532327tion_b] :
      ( P
      = ( produc5187340250978832319tion_b @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_180_surj__pair,axiom,
    ! [P: product_prod_b_b] :
    ? [X3: b,Y3: b] :
      ( P
      = ( product_Pair_b_b @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_181_surj__pair,axiom,
    ! [P: produc1396820459975257318_b_nat] :
    ? [X3: produc7781606619528532327tion_b,Y3: nat] :
      ( P
      = ( produc2093246604507004384_b_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_182_surj__pair,axiom,
    ! [P: produc7781606619528532327tion_b] :
    ? [X3: option_b,Y3: option_b] :
      ( P
      = ( produc3989389360737052631tion_b @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_183_surj__pair,axiom,
    ! [P: product_prod_b_nat] :
    ? [X3: b,Y3: nat] :
      ( P
      = ( product_Pair_b_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_184_old_Oprod_Oexhaust,axiom,
    ! [Y5: produc7025179454482589068tion_b] :
      ~ ! [A4: nat,B3: produc7781606619528532327tion_b] :
          ( Y5
         != ( produc7510438065615382334tion_b @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_185_old_Oprod_Oexhaust,axiom,
    ! [Y5: product_prod_nat_nat] :
      ~ ! [A4: nat,B3: nat] :
          ( Y5
         != ( product_Pair_nat_nat @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_186_old_Oprod_Oexhaust,axiom,
    ! [Y5: product_prod_nat_b] :
      ~ ! [A4: nat,B3: b] :
          ( Y5
         != ( product_Pair_nat_b @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_187_old_Oprod_Oexhaust,axiom,
    ! [Y5: produc4407387694553101647tion_b] :
      ~ ! [A4: b,B3: produc7781606619528532327tion_b] :
          ( Y5
         != ( produc5187340250978832319tion_b @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_188_old_Oprod_Oexhaust,axiom,
    ! [Y5: product_prod_b_b] :
      ~ ! [A4: b,B3: b] :
          ( Y5
         != ( product_Pair_b_b @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_189_old_Oprod_Oexhaust,axiom,
    ! [Y5: produc1396820459975257318_b_nat] :
      ~ ! [A4: produc7781606619528532327tion_b,B3: nat] :
          ( Y5
         != ( produc2093246604507004384_b_nat @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_190_old_Oprod_Oexhaust,axiom,
    ! [Y5: produc7781606619528532327tion_b] :
      ~ ! [A4: option_b,B3: option_b] :
          ( Y5
         != ( produc3989389360737052631tion_b @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_191_old_Oprod_Oexhaust,axiom,
    ! [Y5: product_prod_b_nat] :
      ~ ! [A4: b,B3: nat] :
          ( Y5
         != ( product_Pair_b_nat @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_192_fset__cong,axiom,
    ! [X4: fset_P2289935846317411886_b_nat,Y5: fset_P2289935846317411886_b_nat] :
      ( ( ( fset_P7858563055952297295_b_nat @ X4 )
        = ( fset_P7858563055952297295_b_nat @ Y5 ) )
      = ( X4 = Y5 ) ) ).

% fset_cong
thf(fact_193_case__prod__app,axiom,
    ( produc1842704630950797122on_b_o
    = ( ^ [F3: option_b > option_b > produc7781606619528532327tion_b > $o,X: produc7781606619528532327tion_b,Y4: produc7781606619528532327tion_b] :
          ( produc3378454745690301612on_b_o
          @ ^ [L: option_b,R2: option_b] : ( F3 @ L @ R2 @ Y4 )
          @ X ) ) ) ).

% case_prod_app
thf(fact_194_prod_Ocase__distrib,axiom,
    ! [H: produc7781606619528532327tion_b > produc7781606619528532327tion_b,F: option_b > option_b > produc7781606619528532327tion_b,Prod: produc7781606619528532327tion_b] :
      ( ( H @ ( produc5931753611847401595tion_b @ F @ Prod ) )
      = ( produc5931753611847401595tion_b
        @ ^ [X12: option_b,X22: option_b] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_195_prod_Ocase__distrib,axiom,
    ! [H: produc1396820459975257318_b_nat > produc1396820459975257318_b_nat,F: b > nat > produc1396820459975257318_b_nat,Prod: product_prod_b_nat] :
      ( ( H @ ( produc5650155531262253963_b_nat @ F @ Prod ) )
      = ( produc5650155531262253963_b_nat
        @ ^ [X12: b,X22: nat] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_196_prod_Ocase__distrib,axiom,
    ! [H: produc1396820459975257318_b_nat > produc1396820459975257318_b_nat,F: produc7781606619528532327tion_b > nat > produc1396820459975257318_b_nat,Prod: produc1396820459975257318_b_nat] :
      ( ( H @ ( produc1170353205628678643_b_nat @ F @ Prod ) )
      = ( produc1170353205628678643_b_nat
        @ ^ [X12: produc7781606619528532327tion_b,X22: nat] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_197_prod_Ocase__distrib,axiom,
    ! [H: $o > $o,F: b > nat > $o,Prod: product_prod_b_nat] :
      ( ( H @ ( produc795641402153621683_nat_o @ F @ Prod ) )
      = ( produc795641402153621683_nat_o
        @ ^ [X12: b,X22: nat] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_198_prod_Ocase__distrib,axiom,
    ! [H: product_prod_b_nat > $o,F: b > nat > product_prod_b_nat,Prod: product_prod_b_nat] :
      ( ( H @ ( produc282185899741183267_b_nat @ F @ Prod ) )
      = ( produc795641402153621683_nat_o
        @ ^ [X12: b,X22: nat] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_199_prod_Ocase__distrib,axiom,
    ! [H: $o > product_prod_b_nat,F: b > nat > $o,Prod: product_prod_b_nat] :
      ( ( H @ ( produc795641402153621683_nat_o @ F @ Prod ) )
      = ( produc282185899741183267_b_nat
        @ ^ [X12: b,X22: nat] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_200_prod_Ocase__distrib,axiom,
    ! [H: $o > $o,F: set_option_b > set_option_b > $o,Prod: produc3122286842528565607tion_b] :
      ( ( H @ ( produc7796829984862323884on_b_o @ F @ Prod ) )
      = ( produc7796829984862323884on_b_o
        @ ^ [X12: set_option_b,X22: set_option_b] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_201_prod_Ocase__distrib,axiom,
    ! [H: product_prod_b_nat > product_prod_b_nat,F: b > nat > product_prod_b_nat,Prod: product_prod_b_nat] :
      ( ( H @ ( produc282185899741183267_b_nat @ F @ Prod ) )
      = ( produc282185899741183267_b_nat
        @ ^ [X12: b,X22: nat] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_202_prod_Ocase__distrib,axiom,
    ! [H: produc1396820459975257318_b_nat > $o,F: b > nat > produc1396820459975257318_b_nat,Prod: product_prod_b_nat] :
      ( ( H @ ( produc5650155531262253963_b_nat @ F @ Prod ) )
      = ( produc795641402153621683_nat_o
        @ ^ [X12: b,X22: nat] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_203_prod_Ocase__distrib,axiom,
    ! [H: $o > produc1396820459975257318_b_nat,F: b > nat > $o,Prod: product_prod_b_nat] :
      ( ( H @ ( produc795641402153621683_nat_o @ F @ Prod ) )
      = ( produc5650155531262253963_b_nat
        @ ^ [X12: b,X22: nat] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_204_fimage__fimage,axiom,
    ! [F: produc1396820459975257318_b_nat > product_prod_b_nat,G: product_prod_b_nat > produc1396820459975257318_b_nat,A5: fset_P2289935846317411886_b_nat] :
      ( ( fimage7252752143858342883_b_nat @ F @ ( fimage4266613729464776675_b_nat @ G @ A5 ) )
      = ( fimage6892398636147833211_b_nat
        @ ^ [X: product_prod_b_nat] : ( F @ ( G @ X ) )
        @ A5 ) ) ).

% fimage_fimage
thf(fact_205_fimage__fimage,axiom,
    ! [F: product_prod_b_nat > produc1396820459975257318_b_nat,G: produc1396820459975257318_b_nat > product_prod_b_nat,A5: fset_P4398644116778547350_b_nat] :
      ( ( fimage4266613729464776675_b_nat @ F @ ( fimage7252752143858342883_b_nat @ G @ A5 ) )
      = ( fimage1673450118646331979_b_nat
        @ ^ [X: produc1396820459975257318_b_nat] : ( F @ ( G @ X ) )
        @ A5 ) ) ).

% fimage_fimage
thf(fact_206_fimage__fimage,axiom,
    ! [F: product_prod_b_nat > product_prod_b_nat,G: product_prod_b_nat > product_prod_b_nat,A5: fset_P2289935846317411886_b_nat] :
      ( ( fimage6892398636147833211_b_nat @ F @ ( fimage6892398636147833211_b_nat @ G @ A5 ) )
      = ( fimage6892398636147833211_b_nat
        @ ^ [X: product_prod_b_nat] : ( F @ ( G @ X ) )
        @ A5 ) ) ).

% fimage_fimage
thf(fact_207_fimage__fimage,axiom,
    ! [F: produc1396820459975257318_b_nat > produc1396820459975257318_b_nat,G: produc1396820459975257318_b_nat > produc1396820459975257318_b_nat,A5: fset_P4398644116778547350_b_nat] :
      ( ( fimage1673450118646331979_b_nat @ F @ ( fimage1673450118646331979_b_nat @ G @ A5 ) )
      = ( fimage1673450118646331979_b_nat
        @ ^ [X: produc1396820459975257318_b_nat] : ( F @ ( G @ X ) )
        @ A5 ) ) ).

% fimage_fimage
thf(fact_208_fimage__fimage,axiom,
    ! [F: produc1396820459975257318_b_nat > produc1396820459975257318_b_nat,G: product_prod_b_nat > produc1396820459975257318_b_nat,A5: fset_P2289935846317411886_b_nat] :
      ( ( fimage1673450118646331979_b_nat @ F @ ( fimage4266613729464776675_b_nat @ G @ A5 ) )
      = ( fimage4266613729464776675_b_nat
        @ ^ [X: product_prod_b_nat] : ( F @ ( G @ X ) )
        @ A5 ) ) ).

% fimage_fimage
thf(fact_209_fimage__fimage,axiom,
    ! [F: product_prod_b_nat > produc1396820459975257318_b_nat,G: product_prod_b_nat > product_prod_b_nat,A5: fset_P2289935846317411886_b_nat] :
      ( ( fimage4266613729464776675_b_nat @ F @ ( fimage6892398636147833211_b_nat @ G @ A5 ) )
      = ( fimage4266613729464776675_b_nat
        @ ^ [X: product_prod_b_nat] : ( F @ ( G @ X ) )
        @ A5 ) ) ).

% fimage_fimage
thf(fact_210_old_Oprod_Ocase,axiom,
    ! [F: option_b > option_b > produc7781606619528532327tion_b > $o,X1: option_b,X2: option_b] :
      ( ( produc1842704630950797122on_b_o @ F @ ( produc3989389360737052631tion_b @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_211_old_Oprod_Ocase,axiom,
    ! [F: set_Pr258254617715012167tion_b > set_Pr258254617715012167tion_b > $o,X1: set_Pr258254617715012167tion_b,X2: set_Pr258254617715012167tion_b] :
      ( ( produc2671211398375001388on_b_o @ F @ ( produc4709765886195557207tion_b @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_212_old_Oprod_Ocase,axiom,
    ! [F: set_option_b > set_option_b > $o,X1: set_option_b,X2: set_option_b] :
      ( ( produc7796829984862323884on_b_o @ F @ ( produc5242204166755002327tion_b @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_213_old_Oprod_Ocase,axiom,
    ! [F: b > nat > product_prod_b_nat,X1: b,X2: nat] :
      ( ( produc282185899741183267_b_nat @ F @ ( product_Pair_b_nat @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_214_old_Oprod_Ocase,axiom,
    ! [F: b > nat > $o,X1: b,X2: nat] :
      ( ( produc795641402153621683_nat_o @ F @ ( product_Pair_b_nat @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_215_old_Oprod_Ocase,axiom,
    ! [F: b > nat > produc1396820459975257318_b_nat,X1: b,X2: nat] :
      ( ( produc5650155531262253963_b_nat @ F @ ( product_Pair_b_nat @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_216_old_Oprod_Ocase,axiom,
    ! [F: option_b > option_b > produc7781606619528532327tion_b,X1: option_b,X2: option_b] :
      ( ( produc5931753611847401595tion_b @ F @ ( produc3989389360737052631tion_b @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_217_old_Oprod_Ocase,axiom,
    ! [F: produc7781606619528532327tion_b > nat > produc1396820459975257318_b_nat,X1: produc7781606619528532327tion_b,X2: nat] :
      ( ( produc1170353205628678643_b_nat @ F @ ( produc2093246604507004384_b_nat @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_218_fset_Omap__ident__strong,axiom,
    ! [T: fset_P8578720700757663565od_b_b,F: product_prod_b_b > product_prod_b_b] :
      ( ! [Z2: product_prod_b_b] :
          ( ( member7862447936710763792od_b_b @ Z2 @ ( fset_P8805262604002016854od_b_b @ T ) )
         => ( ( F @ Z2 )
            = Z2 ) )
     => ( ( fimage4331940131092207933od_b_b @ F @ T )
        = T ) ) ).

% fset.map_ident_strong
thf(fact_219_fset_Omap__ident__strong,axiom,
    ! [T: fset_option_b,F: option_b > option_b] :
      ( ! [Z2: option_b] :
          ( ( member_option_b @ Z2 @ ( fset_option_b2 @ T ) )
         => ( ( F @ Z2 )
            = Z2 ) )
     => ( ( fimage6747786463419672509tion_b @ F @ T )
        = T ) ) ).

% fset.map_ident_strong
thf(fact_220_fset_Omap__ident__strong,axiom,
    ! [T: fset_nat,F: nat > nat] :
      ( ! [Z2: nat] :
          ( ( member_nat @ Z2 @ ( fset_nat2 @ T ) )
         => ( ( F @ Z2 )
            = Z2 ) )
     => ( ( fimage_nat_nat @ F @ T )
        = T ) ) ).

% fset.map_ident_strong
thf(fact_221_fset_Omap__ident__strong,axiom,
    ! [T: fset_b,F: b > b] :
      ( ! [Z2: b] :
          ( ( member_b @ Z2 @ ( fset_b2 @ T ) )
         => ( ( F @ Z2 )
            = Z2 ) )
     => ( ( fimage_b_b @ F @ T )
        = T ) ) ).

% fset.map_ident_strong
thf(fact_222_fset_Omap__ident__strong,axiom,
    ! [T: fset_P4398644116778547350_b_nat,F: produc1396820459975257318_b_nat > produc1396820459975257318_b_nat] :
      ( ! [Z2: produc1396820459975257318_b_nat] :
          ( ( member7289896697870905981_b_nat @ Z2 @ ( fset_P628457320923261879_b_nat @ T ) )
         => ( ( F @ Z2 )
            = Z2 ) )
     => ( ( fimage1673450118646331979_b_nat @ F @ T )
        = T ) ) ).

% fset.map_ident_strong
thf(fact_223_fset_Omap__ident__strong,axiom,
    ! [T: fset_P2289935846317411886_b_nat,F: product_prod_b_nat > product_prod_b_nat] :
      ( ! [Z2: product_prod_b_nat] :
          ( ( member6959632917342813205_b_nat @ Z2 @ ( fset_P7858563055952297295_b_nat @ T ) )
         => ( ( F @ Z2 )
            = Z2 ) )
     => ( ( fimage6892398636147833211_b_nat @ F @ T )
        = T ) ) ).

% fset.map_ident_strong
thf(fact_224_fset_Oinj__map__strong,axiom,
    ! [X4: fset_P4398644116778547350_b_nat,Xa: fset_P4398644116778547350_b_nat,F: produc1396820459975257318_b_nat > produc1396820459975257318_b_nat,Fa: produc1396820459975257318_b_nat > produc1396820459975257318_b_nat] :
      ( ! [Z2: produc1396820459975257318_b_nat,Za: produc1396820459975257318_b_nat] :
          ( ( member7289896697870905981_b_nat @ Z2 @ ( fset_P628457320923261879_b_nat @ X4 ) )
         => ( ( member7289896697870905981_b_nat @ Za @ ( fset_P628457320923261879_b_nat @ Xa ) )
           => ( ( ( F @ Z2 )
                = ( Fa @ Za ) )
             => ( Z2 = Za ) ) ) )
     => ( ( ( fimage1673450118646331979_b_nat @ F @ X4 )
          = ( fimage1673450118646331979_b_nat @ Fa @ Xa ) )
       => ( X4 = Xa ) ) ) ).

% fset.inj_map_strong
thf(fact_225_fset_Oinj__map__strong,axiom,
    ! [X4: fset_P2289935846317411886_b_nat,Xa: fset_P2289935846317411886_b_nat,F: product_prod_b_nat > product_prod_b_nat,Fa: product_prod_b_nat > product_prod_b_nat] :
      ( ! [Z2: product_prod_b_nat,Za: product_prod_b_nat] :
          ( ( member6959632917342813205_b_nat @ Z2 @ ( fset_P7858563055952297295_b_nat @ X4 ) )
         => ( ( member6959632917342813205_b_nat @ Za @ ( fset_P7858563055952297295_b_nat @ Xa ) )
           => ( ( ( F @ Z2 )
                = ( Fa @ Za ) )
             => ( Z2 = Za ) ) ) )
     => ( ( ( fimage6892398636147833211_b_nat @ F @ X4 )
          = ( fimage6892398636147833211_b_nat @ Fa @ Xa ) )
       => ( X4 = Xa ) ) ) ).

% fset.inj_map_strong
thf(fact_226_fset_Oinj__map__strong,axiom,
    ! [X4: fset_P2289935846317411886_b_nat,Xa: fset_P2289935846317411886_b_nat,F: product_prod_b_nat > produc1396820459975257318_b_nat,Fa: product_prod_b_nat > produc1396820459975257318_b_nat] :
      ( ! [Z2: product_prod_b_nat,Za: product_prod_b_nat] :
          ( ( member6959632917342813205_b_nat @ Z2 @ ( fset_P7858563055952297295_b_nat @ X4 ) )
         => ( ( member6959632917342813205_b_nat @ Za @ ( fset_P7858563055952297295_b_nat @ Xa ) )
           => ( ( ( F @ Z2 )
                = ( Fa @ Za ) )
             => ( Z2 = Za ) ) ) )
     => ( ( ( fimage4266613729464776675_b_nat @ F @ X4 )
          = ( fimage4266613729464776675_b_nat @ Fa @ Xa ) )
       => ( X4 = Xa ) ) ) ).

% fset.inj_map_strong
thf(fact_227_mem__Collect__eq,axiom,
    ! [A: product_prod_b_nat,P2: product_prod_b_nat > $o] :
      ( ( member6959632917342813205_b_nat @ A @ ( collec5699578864178162003_b_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_228_mem__Collect__eq,axiom,
    ! [A: product_prod_b_b,P2: product_prod_b_b > $o] :
      ( ( member7862447936710763792od_b_b @ A @ ( collec548942219715005266od_b_b @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_229_mem__Collect__eq,axiom,
    ! [A: option_b,P2: option_b > $o] :
      ( ( member_option_b @ A @ ( collect_option_b @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_230_mem__Collect__eq,axiom,
    ! [A: nat,P2: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_231_mem__Collect__eq,axiom,
    ! [A: b,P2: b > $o] :
      ( ( member_b @ A @ ( collect_b @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_232_mem__Collect__eq,axiom,
    ! [A: produc4514605509320855335term_b,P2: produc4514605509320855335term_b > $o] :
      ( ( member7340517243570908752term_b @ A @ ( collec478437889140176530term_b @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_233_mem__Collect__eq,axiom,
    ! [A: produc6650559869974650087tion_b,P2: produc6650559869974650087tion_b > $o] :
      ( ( member6435960643780680464tion_b @ A @ ( collec447916338801455442tion_b @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_234_mem__Collect__eq,axiom,
    ! [A: produc7781606619528532327tion_b,P2: produc7781606619528532327tion_b > $o] :
      ( ( member547026996284216464tion_b @ A @ ( collec8407377912107443410tion_b @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_235_mem__Collect__eq,axiom,
    ! [A: produc4298468441892121831tion_b,P2: produc4298468441892121831tion_b > $o] :
      ( ( member8319496676406272784tion_b @ A @ ( collec1918043184897894738tion_b @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_236_mem__Collect__eq,axiom,
    ! [A: produc3122286842528565607tion_b,P2: produc3122286842528565607tion_b > $o] :
      ( ( member9052098846246995088tion_b @ A @ ( collec6622971782277219538tion_b @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_237_Collect__mem__eq,axiom,
    ! [A5: set_Pr1307281990691478580_b_nat] :
      ( ( collec5699578864178162003_b_nat
        @ ^ [X: product_prod_b_nat] : ( member6959632917342813205_b_nat @ X @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_238_Collect__mem__eq,axiom,
    ! [A5: set_Product_prod_b_b] :
      ( ( collec548942219715005266od_b_b
        @ ^ [X: product_prod_b_b] : ( member7862447936710763792od_b_b @ X @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_239_Collect__mem__eq,axiom,
    ! [A5: set_option_b] :
      ( ( collect_option_b
        @ ^ [X: option_b] : ( member_option_b @ X @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_240_Collect__mem__eq,axiom,
    ! [A5: set_nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( member_nat @ X @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_241_Collect__mem__eq,axiom,
    ! [A5: set_b] :
      ( ( collect_b
        @ ^ [X: b] : ( member_b @ X @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_242_Collect__mem__eq,axiom,
    ! [A5: set_Pr3134649703833893895term_b] :
      ( ( collec478437889140176530term_b
        @ ^ [X: produc4514605509320855335term_b] : ( member7340517243570908752term_b @ X @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_243_Collect__mem__eq,axiom,
    ! [A5: set_Pr5366877986981747911tion_b] :
      ( ( collec447916338801455442tion_b
        @ ^ [X: produc6650559869974650087tion_b] : ( member6435960643780680464tion_b @ X @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_244_Collect__mem__eq,axiom,
    ! [A5: set_Pr258254617715012167tion_b] :
      ( ( collec8407377912107443410tion_b
        @ ^ [X: produc7781606619528532327tion_b] : ( member547026996284216464tion_b @ X @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_245_Collect__mem__eq,axiom,
    ! [A5: set_Pr2609077471296144583tion_b] :
      ( ( collec1918043184897894738tion_b
        @ ^ [X: produc4298468441892121831tion_b] : ( member8319496676406272784tion_b @ X @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_246_Collect__mem__eq,axiom,
    ! [A5: set_Pr4148925336519852615tion_b] :
      ( ( collec6622971782277219538tion_b
        @ ^ [X: produc3122286842528565607tion_b] : ( member9052098846246995088tion_b @ X @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_247_Collect__cong,axiom,
    ! [P2: produc4514605509320855335term_b > $o,Q: produc4514605509320855335term_b > $o] :
      ( ! [X3: produc4514605509320855335term_b] :
          ( ( P2 @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec478437889140176530term_b @ P2 )
        = ( collec478437889140176530term_b @ Q ) ) ) ).

% Collect_cong
thf(fact_248_Collect__cong,axiom,
    ! [P2: produc6650559869974650087tion_b > $o,Q: produc6650559869974650087tion_b > $o] :
      ( ! [X3: produc6650559869974650087tion_b] :
          ( ( P2 @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec447916338801455442tion_b @ P2 )
        = ( collec447916338801455442tion_b @ Q ) ) ) ).

% Collect_cong
thf(fact_249_Collect__cong,axiom,
    ! [P2: produc7781606619528532327tion_b > $o,Q: produc7781606619528532327tion_b > $o] :
      ( ! [X3: produc7781606619528532327tion_b] :
          ( ( P2 @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec8407377912107443410tion_b @ P2 )
        = ( collec8407377912107443410tion_b @ Q ) ) ) ).

% Collect_cong
thf(fact_250_Collect__cong,axiom,
    ! [P2: produc4298468441892121831tion_b > $o,Q: produc4298468441892121831tion_b > $o] :
      ( ! [X3: produc4298468441892121831tion_b] :
          ( ( P2 @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec1918043184897894738tion_b @ P2 )
        = ( collec1918043184897894738tion_b @ Q ) ) ) ).

% Collect_cong
thf(fact_251_Collect__cong,axiom,
    ! [P2: produc3122286842528565607tion_b > $o,Q: produc3122286842528565607tion_b > $o] :
      ( ! [X3: produc3122286842528565607tion_b] :
          ( ( P2 @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec6622971782277219538tion_b @ P2 )
        = ( collec6622971782277219538tion_b @ Q ) ) ) ).

% Collect_cong
thf(fact_252_fset_Omap__cong0,axiom,
    ! [X4: fset_P4398644116778547350_b_nat,F: produc1396820459975257318_b_nat > produc1396820459975257318_b_nat,G: produc1396820459975257318_b_nat > produc1396820459975257318_b_nat] :
      ( ! [Z2: produc1396820459975257318_b_nat] :
          ( ( member7289896697870905981_b_nat @ Z2 @ ( fset_P628457320923261879_b_nat @ X4 ) )
         => ( ( F @ Z2 )
            = ( G @ Z2 ) ) )
     => ( ( fimage1673450118646331979_b_nat @ F @ X4 )
        = ( fimage1673450118646331979_b_nat @ G @ X4 ) ) ) ).

% fset.map_cong0
thf(fact_253_fset_Omap__cong0,axiom,
    ! [X4: fset_P2289935846317411886_b_nat,F: product_prod_b_nat > product_prod_b_nat,G: product_prod_b_nat > product_prod_b_nat] :
      ( ! [Z2: product_prod_b_nat] :
          ( ( member6959632917342813205_b_nat @ Z2 @ ( fset_P7858563055952297295_b_nat @ X4 ) )
         => ( ( F @ Z2 )
            = ( G @ Z2 ) ) )
     => ( ( fimage6892398636147833211_b_nat @ F @ X4 )
        = ( fimage6892398636147833211_b_nat @ G @ X4 ) ) ) ).

% fset.map_cong0
thf(fact_254_fset_Omap__cong0,axiom,
    ! [X4: fset_P2289935846317411886_b_nat,F: product_prod_b_nat > produc1396820459975257318_b_nat,G: product_prod_b_nat > produc1396820459975257318_b_nat] :
      ( ! [Z2: product_prod_b_nat] :
          ( ( member6959632917342813205_b_nat @ Z2 @ ( fset_P7858563055952297295_b_nat @ X4 ) )
         => ( ( F @ Z2 )
            = ( G @ Z2 ) ) )
     => ( ( fimage4266613729464776675_b_nat @ F @ X4 )
        = ( fimage4266613729464776675_b_nat @ G @ X4 ) ) ) ).

% fset.map_cong0
thf(fact_255_fset_Omap__cong,axiom,
    ! [X4: fset_P4398644116778547350_b_nat,Ya: fset_P4398644116778547350_b_nat,F: produc1396820459975257318_b_nat > produc1396820459975257318_b_nat,G: produc1396820459975257318_b_nat > produc1396820459975257318_b_nat] :
      ( ( X4 = Ya )
     => ( ! [Z2: produc1396820459975257318_b_nat] :
            ( ( member7289896697870905981_b_nat @ Z2 @ ( fset_P628457320923261879_b_nat @ Ya ) )
           => ( ( F @ Z2 )
              = ( G @ Z2 ) ) )
       => ( ( fimage1673450118646331979_b_nat @ F @ X4 )
          = ( fimage1673450118646331979_b_nat @ G @ Ya ) ) ) ) ).

% fset.map_cong
thf(fact_256_fset_Omap__cong,axiom,
    ! [X4: fset_P2289935846317411886_b_nat,Ya: fset_P2289935846317411886_b_nat,F: product_prod_b_nat > product_prod_b_nat,G: product_prod_b_nat > product_prod_b_nat] :
      ( ( X4 = Ya )
     => ( ! [Z2: product_prod_b_nat] :
            ( ( member6959632917342813205_b_nat @ Z2 @ ( fset_P7858563055952297295_b_nat @ Ya ) )
           => ( ( F @ Z2 )
              = ( G @ Z2 ) ) )
       => ( ( fimage6892398636147833211_b_nat @ F @ X4 )
          = ( fimage6892398636147833211_b_nat @ G @ Ya ) ) ) ) ).

% fset.map_cong
thf(fact_257_fset_Omap__cong,axiom,
    ! [X4: fset_P2289935846317411886_b_nat,Ya: fset_P2289935846317411886_b_nat,F: product_prod_b_nat > produc1396820459975257318_b_nat,G: product_prod_b_nat > produc1396820459975257318_b_nat] :
      ( ( X4 = Ya )
     => ( ! [Z2: product_prod_b_nat] :
            ( ( member6959632917342813205_b_nat @ Z2 @ ( fset_P7858563055952297295_b_nat @ Ya ) )
           => ( ( F @ Z2 )
              = ( G @ Z2 ) ) )
       => ( ( fimage4266613729464776675_b_nat @ F @ X4 )
          = ( fimage4266613729464776675_b_nat @ G @ Ya ) ) ) ) ).

% fset.map_cong
thf(fact_258_ext__step_Oexhaust,axiom,
    ! [Y5: fOR_ext_step] :
      ( ( Y5 != fOR_ext_ESingle )
     => ( ( Y5 != fOR_ext_EParallel )
       => ( Y5 = fOR_ex5070955383060788274rallel ) ) ) ).

% ext_step.exhaust
thf(fact_259_reflcl__automaton,axiom,
    ! [A2: tree_r4611138855727183398tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR2_spec_a_b_b @ A2 @ R )
     => ( rRn_RR5637769166088770792_a_b_b
        @ ( tA_Clo8488081378127098019on_b_a
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PRoot @ fOR_ext_EParallel @ R ) ) ) ).

% reflcl_automaton
thf(fact_260_reflcl__automaton,axiom,
    ! [A2: tree_r6993104598208160480tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR5637769166088770792_a_b_b @ A2 @ R )
     => ( rRn_RR1656734375441257134_a_b_b
        @ ( tA_Clo6554009024366082921tion_a
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PRoot @ fOR_ext_EParallel @ R ) ) ) ).

% reflcl_automaton
thf(fact_261_reflcl__automaton,axiom,
    ! [A2: tree_r2154205730188421652tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR7260272242312503694es_b_b @ A2 @ R )
     => ( rRn_RR225467092841841886es_b_b
        @ ( tA_Clo416856889220764237states
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PRoot @ fOR_ext_EParallel @ R ) ) ) ).

% reflcl_automaton
thf(fact_262_reflcl__automaton,axiom,
    ! [A2: tree_r7504422106042656583tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR5856091920061875867_o_b_b @ A2 @ R )
     => ( rRn_RR6598829644733244139_o_b_b
        @ ( tA_Clo1721423454482933850um_a_o
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PRoot @ fOR_ext_EParallel @ R ) ) ) ).

% reflcl_automaton
thf(fact_263_reflcl__automaton,axiom,
    ! [A2: tree_r3013368800753843012tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR225467092841841886es_b_b @ A2 @ R )
     => ( rRn_RR2097331226319512750es_b_b
        @ ( tA_Clo2362788092263830301states
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PRoot @ fOR_ext_EParallel @ R ) ) ) ).

% reflcl_automaton
thf(fact_264_reflcl__automaton,axiom,
    ! [A2: tree_r8506140744905984007tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR3067063739481463809_o_b_b @ A2 @ R )
     => ( rRn_RR2318748202180745671_o_b_b
        @ ( tA_Clo6735527661874405506ates_o
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PRoot @ fOR_ext_EParallel @ R ) ) ) ).

% reflcl_automaton
thf(fact_265_reflcl__automaton,axiom,
    ! [A2: tree_r5558399869034676087tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR6598829644733244139_o_b_b @ A2 @ R )
     => ( rRn_RR2516351529612540859_o_b_b
        @ ( tA_Clo4459576460116894250um_a_o
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PRoot @ fOR_ext_EParallel @ R ) ) ) ).

% reflcl_automaton
thf(fact_266_reflcl__automaton,axiom,
    ! [A2: tree_r5058894190550810253tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR5351431268385054037_o_b_b @ A2 @ R )
     => ( rRn_RR1470927249984479525_o_b_b
        @ ( tA_Clo3367819715167377812on_a_o
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PRoot @ fOR_ext_EParallel @ R ) ) ) ).

% reflcl_automaton
thf(fact_267_reflcl__automaton,axiom,
    ! [A2: tree_r4493376028253769364tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR6022601609001321268_o_b_b @ A2 @ R )
     => ( rRn_RR3960834450804983802_o_b_b
        @ ( tA_Clo5397621994544338613_a_o_o
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PRoot @ fOR_ext_EParallel @ R ) ) ) ).

% reflcl_automaton
thf(fact_268_reflcl__automaton,axiom,
    ! [A2: tree_r2365054918505434199tion_b,R: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat] :
      ( ( rRn_RR3181898823427072817_o_b_b @ A2 @ R )
     => ( rRn_RR3813377194981999735_o_b_b
        @ ( tA_Clo1026746659789498930ates_o
          @ ( fimage4266613729464776675_b_nat
            @ ( produc5650155531262253963_b_nat
              @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
            @ F2 )
          @ A2 )
        @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ F2 ) @ fOR_pos_PRoot @ fOR_ext_EParallel @ R ) ) ) ).

% reflcl_automaton
thf(fact_269_ext__step_Odistinct_I3_J,axiom,
    fOR_ext_ESingle != fOR_ex5070955383060788274rallel ).

% ext_step.distinct(3)
thf(fact_270_ext__step_Odistinct_I5_J,axiom,
    fOR_ext_EParallel != fOR_ex5070955383060788274rallel ).

% ext_step.distinct(5)
thf(fact_271_ext__step_Odistinct_I1_J,axiom,
    fOR_ext_ESingle != fOR_ext_EParallel ).

% ext_step.distinct(1)
thf(fact_272_split__cong,axiom,
    ! [Q2: produc7781606619528532327tion_b,F: option_b > option_b > produc7781606619528532327tion_b > $o,G: option_b > option_b > produc7781606619528532327tion_b > $o,P: produc7781606619528532327tion_b] :
      ( ! [X3: option_b,Y3: option_b] :
          ( ( ( produc3989389360737052631tion_b @ X3 @ Y3 )
            = Q2 )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P = Q2 )
       => ( ( produc1842704630950797122on_b_o @ F @ P )
          = ( produc1842704630950797122on_b_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_273_split__cong,axiom,
    ! [Q2: produc4298468441892121831tion_b,F: set_Pr258254617715012167tion_b > set_Pr258254617715012167tion_b > $o,G: set_Pr258254617715012167tion_b > set_Pr258254617715012167tion_b > $o,P: produc4298468441892121831tion_b] :
      ( ! [X3: set_Pr258254617715012167tion_b,Y3: set_Pr258254617715012167tion_b] :
          ( ( ( produc4709765886195557207tion_b @ X3 @ Y3 )
            = Q2 )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P = Q2 )
       => ( ( produc2671211398375001388on_b_o @ F @ P )
          = ( produc2671211398375001388on_b_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_274_split__cong,axiom,
    ! [Q2: produc3122286842528565607tion_b,F: set_option_b > set_option_b > $o,G: set_option_b > set_option_b > $o,P: produc3122286842528565607tion_b] :
      ( ! [X3: set_option_b,Y3: set_option_b] :
          ( ( ( produc5242204166755002327tion_b @ X3 @ Y3 )
            = Q2 )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P = Q2 )
       => ( ( produc7796829984862323884on_b_o @ F @ P )
          = ( produc7796829984862323884on_b_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_275_split__cong,axiom,
    ! [Q2: product_prod_b_nat,F: b > nat > product_prod_b_nat,G: b > nat > product_prod_b_nat,P: product_prod_b_nat] :
      ( ! [X3: b,Y3: nat] :
          ( ( ( product_Pair_b_nat @ X3 @ Y3 )
            = Q2 )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P = Q2 )
       => ( ( produc282185899741183267_b_nat @ F @ P )
          = ( produc282185899741183267_b_nat @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_276_split__cong,axiom,
    ! [Q2: product_prod_b_nat,F: b > nat > $o,G: b > nat > $o,P: product_prod_b_nat] :
      ( ! [X3: b,Y3: nat] :
          ( ( ( product_Pair_b_nat @ X3 @ Y3 )
            = Q2 )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P = Q2 )
       => ( ( produc795641402153621683_nat_o @ F @ P )
          = ( produc795641402153621683_nat_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_277_split__cong,axiom,
    ! [Q2: product_prod_b_nat,F: b > nat > produc1396820459975257318_b_nat,G: b > nat > produc1396820459975257318_b_nat,P: product_prod_b_nat] :
      ( ! [X3: b,Y3: nat] :
          ( ( ( product_Pair_b_nat @ X3 @ Y3 )
            = Q2 )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P = Q2 )
       => ( ( produc5650155531262253963_b_nat @ F @ P )
          = ( produc5650155531262253963_b_nat @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_278_split__cong,axiom,
    ! [Q2: produc7781606619528532327tion_b,F: option_b > option_b > produc7781606619528532327tion_b,G: option_b > option_b > produc7781606619528532327tion_b,P: produc7781606619528532327tion_b] :
      ( ! [X3: option_b,Y3: option_b] :
          ( ( ( produc3989389360737052631tion_b @ X3 @ Y3 )
            = Q2 )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P = Q2 )
       => ( ( produc5931753611847401595tion_b @ F @ P )
          = ( produc5931753611847401595tion_b @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_279_split__cong,axiom,
    ! [Q2: produc1396820459975257318_b_nat,F: produc7781606619528532327tion_b > nat > produc1396820459975257318_b_nat,G: produc7781606619528532327tion_b > nat > produc1396820459975257318_b_nat,P: produc1396820459975257318_b_nat] :
      ( ! [X3: produc7781606619528532327tion_b,Y3: nat] :
          ( ( ( produc2093246604507004384_b_nat @ X3 @ Y3 )
            = Q2 )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P = Q2 )
       => ( ( produc1170353205628678643_b_nat @ F @ P )
          = ( produc1170353205628678643_b_nat @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_280_internal__case__prod__def,axiom,
    produc2468970574363758872on_b_o = produc1842704630950797122on_b_o ).

% internal_case_prod_def
thf(fact_281_internal__case__prod__def,axiom,
    produc1096406401732013782on_b_o = produc2671211398375001388on_b_o ).

% internal_case_prod_def
thf(fact_282_internal__case__prod__def,axiom,
    produc4678270042593847382on_b_o = produc7796829984862323884on_b_o ).

% internal_case_prod_def
thf(fact_283_internal__case__prod__def,axiom,
    produc3217404686050991309_b_nat = produc282185899741183267_b_nat ).

% internal_case_prod_def
thf(fact_284_internal__case__prod__def,axiom,
    produc1917031114693025929_nat_o = produc795641402153621683_nat_o ).

% internal_case_prod_def
thf(fact_285_internal__case__prod__def,axiom,
    produc4597801630894830389_b_nat = produc5650155531262253963_b_nat ).

% internal_case_prod_def
thf(fact_286_internal__case__prod__def,axiom,
    produc7948586220478986021tion_b = produc5931753611847401595tion_b ).

% internal_case_prod_def
thf(fact_287_internal__case__prod__def,axiom,
    produc6415579403031724445_b_nat = produc1170353205628678643_b_nat ).

% internal_case_prod_def
thf(fact_288_root__strictparallel__automaton,axiom,
    ! [A2: tree_r4611138855727183398tion_b,R: set_Pr3134649703833893895term_b,F2: set_Pr1307281990691478580_b_nat] :
      ( ( rRn_RR2_spec_a_b_b @ A2 @ R )
     => ( rRn_RR2_spec_a_b_b @ A2 @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PRoot @ fOR_ex5070955383060788274rallel @ R ) ) ) ).

% root_strictparallel_automaton
thf(fact_289_root__strictparallel__automaton,axiom,
    ! [A2: tree_r6993104598208160480tion_b,R: set_Pr3134649703833893895term_b,F2: set_Pr1307281990691478580_b_nat] :
      ( ( rRn_RR5637769166088770792_a_b_b @ A2 @ R )
     => ( rRn_RR5637769166088770792_a_b_b @ A2 @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PRoot @ fOR_ex5070955383060788274rallel @ R ) ) ) ).

% root_strictparallel_automaton
thf(fact_290_root__strictparallel__automaton,axiom,
    ! [A2: tree_r2154205730188421652tion_b,R: set_Pr3134649703833893895term_b,F2: set_Pr1307281990691478580_b_nat] :
      ( ( rRn_RR7260272242312503694es_b_b @ A2 @ R )
     => ( rRn_RR7260272242312503694es_b_b @ A2 @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PRoot @ fOR_ex5070955383060788274rallel @ R ) ) ) ).

% root_strictparallel_automaton
thf(fact_291_root__strictparallel__automaton,axiom,
    ! [A2: tree_r7504422106042656583tion_b,R: set_Pr3134649703833893895term_b,F2: set_Pr1307281990691478580_b_nat] :
      ( ( rRn_RR5856091920061875867_o_b_b @ A2 @ R )
     => ( rRn_RR5856091920061875867_o_b_b @ A2 @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PRoot @ fOR_ex5070955383060788274rallel @ R ) ) ) ).

% root_strictparallel_automaton
thf(fact_292_root__strictparallel__automaton,axiom,
    ! [A2: tree_r3013368800753843012tion_b,R: set_Pr3134649703833893895term_b,F2: set_Pr1307281990691478580_b_nat] :
      ( ( rRn_RR225467092841841886es_b_b @ A2 @ R )
     => ( rRn_RR225467092841841886es_b_b @ A2 @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PRoot @ fOR_ex5070955383060788274rallel @ R ) ) ) ).

% root_strictparallel_automaton
thf(fact_293_root__strictparallel__automaton,axiom,
    ! [A2: tree_r8506140744905984007tion_b,R: set_Pr3134649703833893895term_b,F2: set_Pr1307281990691478580_b_nat] :
      ( ( rRn_RR3067063739481463809_o_b_b @ A2 @ R )
     => ( rRn_RR3067063739481463809_o_b_b @ A2 @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PRoot @ fOR_ex5070955383060788274rallel @ R ) ) ) ).

% root_strictparallel_automaton
thf(fact_294_root__strictparallel__automaton,axiom,
    ! [A2: tree_r5558399869034676087tion_b,R: set_Pr3134649703833893895term_b,F2: set_Pr1307281990691478580_b_nat] :
      ( ( rRn_RR6598829644733244139_o_b_b @ A2 @ R )
     => ( rRn_RR6598829644733244139_o_b_b @ A2 @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PRoot @ fOR_ex5070955383060788274rallel @ R ) ) ) ).

% root_strictparallel_automaton
thf(fact_295_root__strictparallel__automaton,axiom,
    ! [A2: tree_r5058894190550810253tion_b,R: set_Pr3134649703833893895term_b,F2: set_Pr1307281990691478580_b_nat] :
      ( ( rRn_RR5351431268385054037_o_b_b @ A2 @ R )
     => ( rRn_RR5351431268385054037_o_b_b @ A2 @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PRoot @ fOR_ex5070955383060788274rallel @ R ) ) ) ).

% root_strictparallel_automaton
thf(fact_296_root__strictparallel__automaton,axiom,
    ! [A2: tree_r4493376028253769364tion_b,R: set_Pr3134649703833893895term_b,F2: set_Pr1307281990691478580_b_nat] :
      ( ( rRn_RR6022601609001321268_o_b_b @ A2 @ R )
     => ( rRn_RR6022601609001321268_o_b_b @ A2 @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PRoot @ fOR_ex5070955383060788274rallel @ R ) ) ) ).

% root_strictparallel_automaton
thf(fact_297_root__strictparallel__automaton,axiom,
    ! [A2: tree_r2365054918505434199tion_b,R: set_Pr3134649703833893895term_b,F2: set_Pr1307281990691478580_b_nat] :
      ( ( rRn_RR3181898823427072817_o_b_b @ A2 @ R )
     => ( rRn_RR3181898823427072817_o_b_b @ A2 @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PRoot @ fOR_ex5070955383060788274rallel @ R ) ) ) ).

% root_strictparallel_automaton
thf(fact_298_root__single__automaton,axiom,
    ! [A2: tree_r4611138855727183398tion_b,R: set_Pr3134649703833893895term_b,F2: set_Pr1307281990691478580_b_nat] :
      ( ( rRn_RR2_spec_a_b_b @ A2 @ R )
     => ( rRn_RR2_spec_a_b_b @ A2 @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PRoot @ fOR_ext_ESingle @ R ) ) ) ).

% root_single_automaton
thf(fact_299_root__single__automaton,axiom,
    ! [A2: tree_r6993104598208160480tion_b,R: set_Pr3134649703833893895term_b,F2: set_Pr1307281990691478580_b_nat] :
      ( ( rRn_RR5637769166088770792_a_b_b @ A2 @ R )
     => ( rRn_RR5637769166088770792_a_b_b @ A2 @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PRoot @ fOR_ext_ESingle @ R ) ) ) ).

% root_single_automaton
thf(fact_300_root__single__automaton,axiom,
    ! [A2: tree_r2154205730188421652tion_b,R: set_Pr3134649703833893895term_b,F2: set_Pr1307281990691478580_b_nat] :
      ( ( rRn_RR7260272242312503694es_b_b @ A2 @ R )
     => ( rRn_RR7260272242312503694es_b_b @ A2 @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PRoot @ fOR_ext_ESingle @ R ) ) ) ).

% root_single_automaton
thf(fact_301_root__single__automaton,axiom,
    ! [A2: tree_r7504422106042656583tion_b,R: set_Pr3134649703833893895term_b,F2: set_Pr1307281990691478580_b_nat] :
      ( ( rRn_RR5856091920061875867_o_b_b @ A2 @ R )
     => ( rRn_RR5856091920061875867_o_b_b @ A2 @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PRoot @ fOR_ext_ESingle @ R ) ) ) ).

% root_single_automaton
thf(fact_302_root__single__automaton,axiom,
    ! [A2: tree_r3013368800753843012tion_b,R: set_Pr3134649703833893895term_b,F2: set_Pr1307281990691478580_b_nat] :
      ( ( rRn_RR225467092841841886es_b_b @ A2 @ R )
     => ( rRn_RR225467092841841886es_b_b @ A2 @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PRoot @ fOR_ext_ESingle @ R ) ) ) ).

% root_single_automaton
thf(fact_303_root__single__automaton,axiom,
    ! [A2: tree_r8506140744905984007tion_b,R: set_Pr3134649703833893895term_b,F2: set_Pr1307281990691478580_b_nat] :
      ( ( rRn_RR3067063739481463809_o_b_b @ A2 @ R )
     => ( rRn_RR3067063739481463809_o_b_b @ A2 @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PRoot @ fOR_ext_ESingle @ R ) ) ) ).

% root_single_automaton
thf(fact_304_root__single__automaton,axiom,
    ! [A2: tree_r5558399869034676087tion_b,R: set_Pr3134649703833893895term_b,F2: set_Pr1307281990691478580_b_nat] :
      ( ( rRn_RR6598829644733244139_o_b_b @ A2 @ R )
     => ( rRn_RR6598829644733244139_o_b_b @ A2 @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PRoot @ fOR_ext_ESingle @ R ) ) ) ).

% root_single_automaton
thf(fact_305_root__single__automaton,axiom,
    ! [A2: tree_r5058894190550810253tion_b,R: set_Pr3134649703833893895term_b,F2: set_Pr1307281990691478580_b_nat] :
      ( ( rRn_RR5351431268385054037_o_b_b @ A2 @ R )
     => ( rRn_RR5351431268385054037_o_b_b @ A2 @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PRoot @ fOR_ext_ESingle @ R ) ) ) ).

% root_single_automaton
thf(fact_306_root__single__automaton,axiom,
    ! [A2: tree_r4493376028253769364tion_b,R: set_Pr3134649703833893895term_b,F2: set_Pr1307281990691478580_b_nat] :
      ( ( rRn_RR6022601609001321268_o_b_b @ A2 @ R )
     => ( rRn_RR6022601609001321268_o_b_b @ A2 @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PRoot @ fOR_ext_ESingle @ R ) ) ) ).

% root_single_automaton
thf(fact_307_root__single__automaton,axiom,
    ! [A2: tree_r2365054918505434199tion_b,R: set_Pr3134649703833893895term_b,F2: set_Pr1307281990691478580_b_nat] :
      ( ( rRn_RR3181898823427072817_o_b_b @ A2 @ R )
     => ( rRn_RR3181898823427072817_o_b_b @ A2 @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PRoot @ fOR_ext_ESingle @ R ) ) ) ).

% root_single_automaton
thf(fact_308_pred__equals__eq2,axiom,
    ! [R: set_Pr4178509052509721666tion_b,S: set_Pr4178509052509721666tion_b] :
      ( ( ( ^ [X: nat,Y4: produc7781606619528532327tion_b] : ( member3694883655523461923tion_b @ ( produc7510438065615382334tion_b @ X @ Y4 ) @ R ) )
        = ( ^ [X: nat,Y4: produc7781606619528532327tion_b] : ( member3694883655523461923tion_b @ ( produc7510438065615382334tion_b @ X @ Y4 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_309_pred__equals__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ R ) )
        = ( ^ [X: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_310_pred__equals__eq2,axiom,
    ! [R: set_Pr4264375888882495962_nat_b,S: set_Pr4264375888882495962_nat_b] :
      ( ( ( ^ [X: nat,Y4: b] : ( member8962352056413324475_nat_b @ ( product_Pair_nat_b @ X @ Y4 ) @ R ) )
        = ( ^ [X: nat,Y4: b] : ( member8962352056413324475_nat_b @ ( product_Pair_nat_b @ X @ Y4 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_311_pred__equals__eq2,axiom,
    ! [R: set_Pr5422417554659304751tion_b,S: set_Pr5422417554659304751tion_b] :
      ( ( ( ^ [X: b,Y4: produc7781606619528532327tion_b] : ( member8570655873678976888tion_b @ ( produc5187340250978832319tion_b @ X @ Y4 ) @ R ) )
        = ( ^ [X: b,Y4: produc7781606619528532327tion_b] : ( member8570655873678976888tion_b @ ( produc5187340250978832319tion_b @ X @ Y4 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_312_pred__equals__eq2,axiom,
    ! [R: set_Product_prod_b_b,S: set_Product_prod_b_b] :
      ( ( ( ^ [X: b,Y4: b] : ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y4 ) @ R ) )
        = ( ^ [X: b,Y4: b] : ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y4 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_313_pred__equals__eq2,axiom,
    ! [R: set_Pr6944079702279229084_b_nat,S: set_Pr6944079702279229084_b_nat] :
      ( ( ( ^ [X: produc7781606619528532327tion_b,Y4: nat] : ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ X @ Y4 ) @ R ) )
        = ( ^ [X: produc7781606619528532327tion_b,Y4: nat] : ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ X @ Y4 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_314_pred__equals__eq2,axiom,
    ! [R: set_Pr258254617715012167tion_b,S: set_Pr258254617715012167tion_b] :
      ( ( ( ^ [X: option_b,Y4: option_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ Y4 ) @ R ) )
        = ( ^ [X: option_b,Y4: option_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ Y4 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_315_pred__equals__eq2,axiom,
    ! [R: set_Pr1307281990691478580_b_nat,S: set_Pr1307281990691478580_b_nat] :
      ( ( ( ^ [X: b,Y4: nat] : ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ X @ Y4 ) @ R ) )
        = ( ^ [X: b,Y4: nat] : ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ X @ Y4 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_316_split__part,axiom,
    ! [P2: $o,Q: set_Pr258254617715012167tion_b > set_Pr258254617715012167tion_b > $o] :
      ( ( produc2671211398375001388on_b_o
        @ ^ [A6: set_Pr258254617715012167tion_b,B4: set_Pr258254617715012167tion_b] :
            ( P2
            & ( Q @ A6 @ B4 ) ) )
      = ( ^ [Ab: produc4298468441892121831tion_b] :
            ( P2
            & ( produc2671211398375001388on_b_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_317_split__part,axiom,
    ! [P2: $o,Q: set_option_b > set_option_b > $o] :
      ( ( produc7796829984862323884on_b_o
        @ ^ [A6: set_option_b,B4: set_option_b] :
            ( P2
            & ( Q @ A6 @ B4 ) ) )
      = ( ^ [Ab: produc3122286842528565607tion_b] :
            ( P2
            & ( produc7796829984862323884on_b_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_318_split__part,axiom,
    ! [P2: $o,Q: b > nat > $o] :
      ( ( produc795641402153621683_nat_o
        @ ^ [A6: b,B4: nat] :
            ( P2
            & ( Q @ A6 @ B4 ) ) )
      = ( ^ [Ab: product_prod_b_nat] :
            ( P2
            & ( produc795641402153621683_nat_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_319_prod_Odisc__eq__case,axiom,
    ! [Prod: produc4298468441892121831tion_b] :
      ( produc2671211398375001388on_b_o
      @ ^ [Uu: set_Pr258254617715012167tion_b,Uv: set_Pr258254617715012167tion_b] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_320_prod_Odisc__eq__case,axiom,
    ! [Prod: produc3122286842528565607tion_b] :
      ( produc7796829984862323884on_b_o
      @ ^ [Uu: set_option_b,Uv: set_option_b] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_321_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_b_nat] :
      ( produc795641402153621683_nat_o
      @ ^ [Uu: b,Uv: nat] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_322_pos__step_Odistinct_I3_J,axiom,
    fOR_pos_PRoot != fOR_pos_PAny ).

% pos_step.distinct(3)
thf(fact_323_internal__case__prod__conv,axiom,
    ! [C: produc7781606619528532327tion_b > nat > produc1396820459975257318_b_nat,A: produc7781606619528532327tion_b,B: nat] :
      ( ( produc6415579403031724445_b_nat @ C @ ( produc2093246604507004384_b_nat @ A @ B ) )
      = ( C @ A @ B ) ) ).

% internal_case_prod_conv
thf(fact_324_internal__case__prod__conv,axiom,
    ! [C: option_b > option_b > produc7781606619528532327tion_b,A: option_b,B: option_b] :
      ( ( produc7948586220478986021tion_b @ C @ ( produc3989389360737052631tion_b @ A @ B ) )
      = ( C @ A @ B ) ) ).

% internal_case_prod_conv
thf(fact_325_internal__case__prod__conv,axiom,
    ! [C: b > nat > produc1396820459975257318_b_nat,A: b,B: nat] :
      ( ( produc4597801630894830389_b_nat @ C @ ( product_Pair_b_nat @ A @ B ) )
      = ( C @ A @ B ) ) ).

% internal_case_prod_conv
thf(fact_326_lift__root__step_Osimps_I8_J,axiom,
    ! [F2: set_Pr1307281990691478580_b_nat,R3: set_Pr3134649703833893895term_b] :
      ( ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PRoot @ fOR_ex5070955383060788274rallel @ R3 )
      = R3 ) ).

% lift_root_step.simps(8)
thf(fact_327_lift__root__step__PRoot__EStrictParallel,axiom,
    ! [F2: set_Pr1307281990691478580_b_nat,R3: set_Pr3134649703833893895term_b] :
      ( ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PRoot @ fOR_ex5070955383060788274rallel @ R3 )
      = R3 ) ).

% lift_root_step_PRoot_EStrictParallel
thf(fact_328_lift__root__step_Osimps_I7_J,axiom,
    ! [F2: set_Pr1307281990691478580_b_nat,R3: set_Pr3134649703833893895term_b] :
      ( ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PRoot @ fOR_ext_ESingle @ R3 )
      = R3 ) ).

% lift_root_step.simps(7)
thf(fact_329_lift__root__step__PRoot__ESingle,axiom,
    ! [F2: set_Pr1307281990691478580_b_nat,R3: set_Pr3134649703833893895term_b] :
      ( ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PRoot @ fOR_ext_ESingle @ R3 )
      = R3 ) ).

% lift_root_step_PRoot_ESingle
thf(fact_330_inv__image__def,axiom,
    ( inv_im4790136988661274508term_b
    = ( ^ [R2: set_Pr1261947904930325089at_nat,F3: ground_gterm_b > nat] :
          ( collec478437889140176530term_b
          @ ( produc1717101583779059052rm_b_o
            @ ^ [X: ground_gterm_b,Y4: ground_gterm_b] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ X ) @ ( F3 @ Y4 ) ) @ R2 ) ) ) ) ) ).

% inv_image_def
thf(fact_331_inv__image__def,axiom,
    ( inv_im4670702844647688481tion_b
    = ( ^ [R2: set_Pr1261947904930325089at_nat,F3: option_b > nat] :
          ( collec8407377912107443410tion_b
          @ ( produc3378454745690301612on_b_o
            @ ^ [X: option_b,Y4: option_b] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ X ) @ ( F3 @ Y4 ) ) @ R2 ) ) ) ) ) ).

% inv_image_def
thf(fact_332_inv__image__def,axiom,
    ( inv_im2307231402264022467term_b
    = ( ^ [R2: set_Product_prod_b_b,F3: ground_gterm_b > b] :
          ( collec478437889140176530term_b
          @ ( produc1717101583779059052rm_b_o
            @ ^ [X: ground_gterm_b,Y4: ground_gterm_b] : ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( F3 @ X ) @ ( F3 @ Y4 ) ) @ R2 ) ) ) ) ) ).

% inv_image_def
thf(fact_333_inv__image__def,axiom,
    ( inv_image_b_option_b
    = ( ^ [R2: set_Product_prod_b_b,F3: option_b > b] :
          ( collec8407377912107443410tion_b
          @ ( produc3378454745690301612on_b_o
            @ ^ [X: option_b,Y4: option_b] : ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( F3 @ X ) @ ( F3 @ Y4 ) ) @ R2 ) ) ) ) ) ).

% inv_image_def
thf(fact_334_inv__image__def,axiom,
    ( inv_im6470721272166791677term_b
    = ( ^ [R2: set_Pr258254617715012167tion_b,F3: ground_gterm_b > option_b] :
          ( collec478437889140176530term_b
          @ ( produc1717101583779059052rm_b_o
            @ ^ [X: ground_gterm_b,Y4: ground_gterm_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ ( F3 @ X ) @ ( F3 @ Y4 ) ) @ R2 ) ) ) ) ) ).

% inv_image_def
thf(fact_335_inv__image__def,axiom,
    ( inv_im5784823153951788434tion_b
    = ( ^ [R2: set_Pr258254617715012167tion_b,F3: option_b > option_b] :
          ( collec8407377912107443410tion_b
          @ ( produc3378454745690301612on_b_o
            @ ^ [X: option_b,Y4: option_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ ( F3 @ X ) @ ( F3 @ Y4 ) ) @ R2 ) ) ) ) ) ).

% inv_image_def
thf(fact_336_inv__image__def,axiom,
    ( inv_im2082697534779480961tion_b
    = ( ^ [R2: set_Pr1261947904930325089at_nat,F3: set_option_b > nat] :
          ( collec6622971782277219538tion_b
          @ ( produc7796829984862323884on_b_o
            @ ^ [X: set_option_b,Y4: set_option_b] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ X ) @ ( F3 @ Y4 ) ) @ R2 ) ) ) ) ) ).

% inv_image_def
thf(fact_337_inv__image__def,axiom,
    ( inv_im2848783409229996984tion_b
    = ( ^ [R2: set_Product_prod_b_b,F3: set_option_b > b] :
          ( collec6622971782277219538tion_b
          @ ( produc7796829984862323884on_b_o
            @ ^ [X: set_option_b,Y4: set_option_b] : ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( F3 @ X ) @ ( F3 @ Y4 ) ) @ R2 ) ) ) ) ) ).

% inv_image_def
thf(fact_338_inv__image__def,axiom,
    ( inv_im5183776136373017586tion_b
    = ( ^ [R2: set_Pr258254617715012167tion_b,F3: set_option_b > option_b] :
          ( collec6622971782277219538tion_b
          @ ( produc7796829984862323884on_b_o
            @ ^ [X: set_option_b,Y4: set_option_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ ( F3 @ X ) @ ( F3 @ Y4 ) ) @ R2 ) ) ) ) ) ).

% inv_image_def
thf(fact_339_inv__image__def,axiom,
    ( inv_im7179320181675035587tion_b
    = ( ^ [R2: set_Pr1261947904930325089at_nat,F3: produc7781606619528532327tion_b > nat] :
          ( collec447916338801455442tion_b
          @ ( produc8054777514177824044on_b_o
            @ ^ [X: produc7781606619528532327tion_b,Y4: produc7781606619528532327tion_b] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ X ) @ ( F3 @ Y4 ) ) @ R2 ) ) ) ) ) ).

% inv_image_def
thf(fact_340_restrict__def,axiom,
    ( abstra6440977867845469302_b_nat
    = ( ^ [R2: set_Pr5241429435540800445_b_nat,S2: set_Pr1307281990691478580_b_nat] :
          ( collec6052715720417950280_b_nat
          @ ( produc3203605564640457326_nat_o
            @ ^ [X: product_prod_b_nat,Y4: product_prod_b_nat] :
                ( ( member6959632917342813205_b_nat @ X @ S2 )
                & ( member6959632917342813205_b_nat @ Y4 @ S2 )
                & ( member4966892972527255302_b_nat @ ( produc2793586353817733269_b_nat @ X @ Y4 ) @ R2 ) ) ) ) ) ) ).

% restrict_def
thf(fact_341_restrict__def,axiom,
    ( abstra2172771641121501935od_b_b
    = ( ^ [R2: set_Pr3901141605387707591od_b_b,S2: set_Product_prod_b_b] :
          ( collec2049399640006090066od_b_b
          @ ( produc7890132440258884908_b_b_o
            @ ^ [X: product_prod_b_b,Y4: product_prod_b_b] :
                ( ( member7862447936710763792od_b_b @ X @ S2 )
                & ( member7862447936710763792od_b_b @ Y4 @ S2 )
                & ( member8369738419320101648od_b_b @ ( produc7702952596710024023od_b_b @ X @ Y4 ) @ R2 ) ) ) ) ) ) ).

% restrict_def
thf(fact_342_restrict__def,axiom,
    ( abstra354823189701009208term_b
    = ( ^ [R2: set_Pr3134649703833893895term_b,S2: set_Ground_gterm_b] :
          ( collec478437889140176530term_b
          @ ( produc1717101583779059052rm_b_o
            @ ^ [X: ground_gterm_b,Y4: ground_gterm_b] :
                ( ( member5261315027475796697term_b @ X @ S2 )
                & ( member5261315027475796697term_b @ Y4 @ S2 )
                & ( member7340517243570908752term_b @ ( produc1113259529098229527term_b @ X @ Y4 ) @ R2 ) ) ) ) ) ) ).

% restrict_def
thf(fact_343_restrict__def,axiom,
    ( abstra2148236097575398767tion_b
    = ( ^ [R2: set_Pr5366877986981747911tion_b,S2: set_Pr258254617715012167tion_b] :
          ( collec447916338801455442tion_b
          @ ( produc8054777514177824044on_b_o
            @ ^ [X: produc7781606619528532327tion_b,Y4: produc7781606619528532327tion_b] :
                ( ( member547026996284216464tion_b @ X @ S2 )
                & ( member547026996284216464tion_b @ Y4 @ S2 )
                & ( member6435960643780680464tion_b @ ( produc7954421581189286743tion_b @ X @ Y4 ) @ R2 ) ) ) ) ) ) ).

% restrict_def
thf(fact_344_restrict__def,axiom,
    ( abstra1219450721576609224ct_nat
    = ( ^ [R2: set_Pr1261947904930325089at_nat,S2: set_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X: nat,Y4: nat] :
                ( ( member_nat @ X @ S2 )
                & ( member_nat @ Y4 @ S2 )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ R2 ) ) ) ) ) ) ).

% restrict_def
thf(fact_345_restrict__def,axiom,
    ( abstract_restrict_b
    = ( ^ [R2: set_Product_prod_b_b,S2: set_b] :
          ( collec548942219715005266od_b_b
          @ ( produc4772849988445941548_b_b_o
            @ ^ [X: b,Y4: b] :
                ( ( member_b @ X @ S2 )
                & ( member_b @ Y4 @ S2 )
                & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y4 ) @ R2 ) ) ) ) ) ) ).

% restrict_def
thf(fact_346_restrict__def,axiom,
    ( abstra5044752225732226639tion_b
    = ( ^ [R2: set_Pr2609077471296144583tion_b,S2: set_se1183635074333576999tion_b] :
          ( collec1918043184897894738tion_b
          @ ( produc2671211398375001388on_b_o
            @ ^ [X: set_Pr258254617715012167tion_b,Y4: set_Pr258254617715012167tion_b] :
                ( ( member3084166351965065584tion_b @ X @ S2 )
                & ( member3084166351965065584tion_b @ Y4 @ S2 )
                & ( member8319496676406272784tion_b @ ( produc4709765886195557207tion_b @ X @ Y4 ) @ R2 ) ) ) ) ) ) ).

% restrict_def
thf(fact_347_restrict__def,axiom,
    ( abstra7455183742696442669tion_b
    = ( ^ [R2: set_Pr4148925336519852615tion_b,S2: set_set_option_b] :
          ( collec6622971782277219538tion_b
          @ ( produc7796829984862323884on_b_o
            @ ^ [X: set_option_b,Y4: set_option_b] :
                ( ( member_set_option_b @ X @ S2 )
                & ( member_set_option_b @ Y4 @ S2 )
                & ( member9052098846246995088tion_b @ ( produc5242204166755002327tion_b @ X @ Y4 ) @ R2 ) ) ) ) ) ) ).

% restrict_def
thf(fact_348_restrict__def,axiom,
    ( abstra3747389286633826509tion_b
    = ( ^ [R2: set_Pr258254617715012167tion_b,S2: set_option_b] :
          ( collec8407377912107443410tion_b
          @ ( produc3378454745690301612on_b_o
            @ ^ [X: option_b,Y4: option_b] :
                ( ( member_option_b @ X @ S2 )
                & ( member_option_b @ Y4 @ S2 )
                & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ Y4 ) @ R2 ) ) ) ) ) ) ).

% restrict_def
thf(fact_349_same__fst__def,axiom,
    ( same_f1374672915113801675_b_nat
    = ( ^ [P3: produc7781606619528532327tion_b > $o,R4: produc7781606619528532327tion_b > set_Pr1261947904930325089at_nat] :
          ( collec2051687263103543064_b_nat
          @ ( produc4199811631454031262_nat_o
            @ ( produc1254906330614685212_nat_o
              @ ^ [X5: produc7781606619528532327tion_b,Y6: nat] :
                  ( produc5232674473337540683_nat_o
                  @ ^ [X: produc7781606619528532327tion_b,Y4: nat] :
                      ( ( X5 = X )
                      & ( P3 @ X )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y6 @ Y4 ) @ ( R4 @ X ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_350_same__fst__def,axiom,
    ( same_f2319420875927667244tion_b
    = ( ^ [P3: set_Pr258254617715012167tion_b > $o,R4: set_Pr258254617715012167tion_b > set_Pr2609077471296144583tion_b] :
          ( collec8403462119798873426tion_b
          @ ( produc4170300588352091436on_b_o
            @ ( produc4337731742241352770on_b_o
              @ ^ [X5: set_Pr258254617715012167tion_b,Y6: set_Pr258254617715012167tion_b] :
                  ( produc2671211398375001388on_b_o
                  @ ^ [X: set_Pr258254617715012167tion_b,Y4: set_Pr258254617715012167tion_b] :
                      ( ( X5 = X )
                      & ( P3 @ X )
                      & ( member8319496676406272784tion_b @ ( produc4709765886195557207tion_b @ Y6 @ Y4 ) @ ( R4 @ X ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_351_same__fst__def,axiom,
    ( same_f1674158784740640684tion_b
    = ( ^ [P3: set_option_b > $o,R4: set_option_b > set_Pr4148925336519852615tion_b] :
          ( collec1476746889905017170tion_b
          @ ( produc3294476002735424812on_b_o
            @ ( produc2735982765467381570on_b_o
              @ ^ [X5: set_option_b,Y6: set_option_b] :
                  ( produc7796829984862323884on_b_o
                  @ ^ [X: set_option_b,Y4: set_option_b] :
                      ( ( X5 = X )
                      & ( P3 @ X )
                      & ( member9052098846246995088tion_b @ ( produc5242204166755002327tion_b @ Y6 @ Y4 ) @ ( R4 @ X ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_352_same__fst__def,axiom,
    ( same_fst_b_nat
    = ( ^ [P3: b > $o,R4: b > set_Pr1261947904930325089at_nat] :
          ( collec6052715720417950280_b_nat
          @ ( produc3203605564640457326_nat_o
            @ ( produc4535354591796149996_nat_o
              @ ^ [X5: b,Y6: nat] :
                  ( produc795641402153621683_nat_o
                  @ ^ [X: b,Y4: nat] :
                      ( ( X5 = X )
                      & ( P3 @ X )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y6 @ Y4 ) @ ( R4 @ X ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_353_same__fst__def,axiom,
    ( same_f9095071695510155180tion_b
    = ( ^ [P3: option_b > $o,R4: option_b > set_Pr258254617715012167tion_b] :
          ( collec447916338801455442tion_b
          @ ( produc8054777514177824044on_b_o
            @ ( produc1842704630950797122on_b_o
              @ ^ [X5: option_b,Y6: option_b] :
                  ( produc3378454745690301612on_b_o
                  @ ^ [X: option_b,Y4: option_b] :
                      ( ( X5 = X )
                      & ( P3 @ X )
                      & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y6 @ Y4 ) @ ( R4 @ X ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_354_lift__root__step__Single__to__Parallel,axiom,
    ! [F2: set_Pr1307281990691478580_b_nat,R: set_Pr3134649703833893895term_b] : ( ord_le2408605988743369639term_b @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PAny @ fOR_ext_ESingle @ R ) @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PAny @ fOR_ext_EParallel @ R ) ) ).

% lift_root_step_Single_to_Parallel
thf(fact_355_in__inv__image,axiom,
    ! [X4: option_b,Y5: option_b,R5: set_Pr1261947904930325089at_nat,F: option_b > nat] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ ( inv_im4670702844647688481tion_b @ R5 @ F ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) @ R5 ) ) ).

% in_inv_image
thf(fact_356_in__inv__image,axiom,
    ! [X4: option_b,Y5: option_b,R5: set_Product_prod_b_b,F: option_b > b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ ( inv_image_b_option_b @ R5 @ F ) )
      = ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( F @ X4 ) @ ( F @ Y5 ) ) @ R5 ) ) ).

% in_inv_image
thf(fact_357_in__inv__image,axiom,
    ! [X4: nat,Y5: nat,R5: set_Pr258254617715012167tion_b,F: nat > option_b] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ ( inv_im7642071220286529091_b_nat @ R5 @ F ) )
      = ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ ( F @ X4 ) @ ( F @ Y5 ) ) @ R5 ) ) ).

% in_inv_image
thf(fact_358_in__inv__image,axiom,
    ! [X4: nat,Y5: nat,R5: set_Pr1261947904930325089at_nat,F: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ ( inv_image_nat_nat @ R5 @ F ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) @ R5 ) ) ).

% in_inv_image
thf(fact_359_in__inv__image,axiom,
    ! [X4: nat,Y5: nat,R5: set_Product_prod_b_b,F: nat > b] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ ( inv_image_b_nat @ R5 @ F ) )
      = ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( F @ X4 ) @ ( F @ Y5 ) ) @ R5 ) ) ).

% in_inv_image
thf(fact_360_in__inv__image,axiom,
    ! [X4: b,Y5: b,R5: set_Pr258254617715012167tion_b,F: b > option_b] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X4 @ Y5 ) @ ( inv_image_option_b_b @ R5 @ F ) )
      = ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ ( F @ X4 ) @ ( F @ Y5 ) ) @ R5 ) ) ).

% in_inv_image
thf(fact_361_in__inv__image,axiom,
    ! [X4: b,Y5: b,R5: set_Pr1261947904930325089at_nat,F: b > nat] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X4 @ Y5 ) @ ( inv_image_nat_b @ R5 @ F ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) @ R5 ) ) ).

% in_inv_image
thf(fact_362_in__inv__image,axiom,
    ! [X4: b,Y5: b,R5: set_Product_prod_b_b,F: b > b] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X4 @ Y5 ) @ ( inv_image_b_b @ R5 @ F ) )
      = ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( F @ X4 ) @ ( F @ Y5 ) ) @ R5 ) ) ).

% in_inv_image
thf(fact_363_in__inv__image,axiom,
    ! [X4: option_b,Y5: option_b,R5: set_Pr258254617715012167tion_b,F: option_b > option_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ ( inv_im5784823153951788434tion_b @ R5 @ F ) )
      = ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ ( F @ X4 ) @ ( F @ Y5 ) ) @ R5 ) ) ).

% in_inv_image
thf(fact_364_same__fstI,axiom,
    ! [P2: nat > $o,X4: nat,Y7: produc7781606619528532327tion_b,Y5: produc7781606619528532327tion_b,R: nat > set_Pr5366877986981747911tion_b] :
      ( ( P2 @ X4 )
     => ( ( member6435960643780680464tion_b @ ( produc7954421581189286743tion_b @ Y7 @ Y5 ) @ ( R @ X4 ) )
       => ( member3972112514548880418tion_b @ ( produc6629285643181589681tion_b @ ( produc7510438065615382334tion_b @ X4 @ Y7 ) @ ( produc7510438065615382334tion_b @ X4 @ Y5 ) ) @ ( same_f6791864376222179625tion_b @ P2 @ R ) ) ) ) ).

% same_fstI
thf(fact_365_same__fstI,axiom,
    ! [P2: b > $o,X4: b,Y7: produc7781606619528532327tion_b,Y5: produc7781606619528532327tion_b,R: b > set_Pr5366877986981747911tion_b] :
      ( ( P2 @ X4 )
     => ( ( member6435960643780680464tion_b @ ( produc7954421581189286743tion_b @ Y7 @ Y5 ) @ ( R @ X4 ) )
       => ( member6692499340770384144tion_b @ ( produc1683552709489456471tion_b @ ( produc5187340250978832319tion_b @ X4 @ Y7 ) @ ( produc5187340250978832319tion_b @ X4 @ Y5 ) ) @ ( same_f3701940369486346900tion_b @ P2 @ R ) ) ) ) ).

% same_fstI
thf(fact_366_same__fstI,axiom,
    ! [P2: nat > $o,X4: nat,Y7: nat,Y5: nat,R: nat > set_Pr1261947904930325089at_nat] :
      ( ( P2 @ X4 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y7 @ Y5 ) @ ( R @ X4 ) )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X4 @ Y7 ) @ ( product_Pair_nat_nat @ X4 @ Y5 ) ) @ ( same_fst_nat_nat @ P2 @ R ) ) ) ) ).

% same_fstI
thf(fact_367_same__fstI,axiom,
    ! [P2: nat > $o,X4: nat,Y7: b,Y5: b,R: nat > set_Product_prod_b_b] :
      ( ( P2 @ X4 )
     => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y7 @ Y5 ) @ ( R @ X4 ) )
       => ( member8772231139900714834_nat_b @ ( produc5644549762070686689_nat_b @ ( product_Pair_nat_b @ X4 @ Y7 ) @ ( product_Pair_nat_b @ X4 @ Y5 ) ) @ ( same_fst_nat_b @ P2 @ R ) ) ) ) ).

% same_fstI
thf(fact_368_same__fstI,axiom,
    ! [P2: b > $o,X4: b,Y7: b,Y5: b,R: b > set_Product_prod_b_b] :
      ( ( P2 @ X4 )
     => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y7 @ Y5 ) @ ( R @ X4 ) )
       => ( member8369738419320101648od_b_b @ ( produc7702952596710024023od_b_b @ ( product_Pair_b_b @ X4 @ Y7 ) @ ( product_Pair_b_b @ X4 @ Y5 ) ) @ ( same_fst_b_b @ P2 @ R ) ) ) ) ).

% same_fstI
thf(fact_369_same__fstI,axiom,
    ! [P2: produc7781606619528532327tion_b > $o,X4: produc7781606619528532327tion_b,Y7: nat,Y5: nat,R: produc7781606619528532327tion_b > set_Pr1261947904930325089at_nat] :
      ( ( P2 @ X4 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y7 @ Y5 ) @ ( R @ X4 ) )
       => ( member4316259128586992598_b_nat @ ( produc7802874649797638501_b_nat @ ( produc2093246604507004384_b_nat @ X4 @ Y7 ) @ ( produc2093246604507004384_b_nat @ X4 @ Y5 ) ) @ ( same_f1374672915113801675_b_nat @ P2 @ R ) ) ) ) ).

% same_fstI
thf(fact_370_same__fstI,axiom,
    ! [P2: b > $o,X4: b,Y7: nat,Y5: nat,R: b > set_Pr1261947904930325089at_nat] :
      ( ( P2 @ X4 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y7 @ Y5 ) @ ( R @ X4 ) )
       => ( member4966892972527255302_b_nat @ ( produc2793586353817733269_b_nat @ ( product_Pair_b_nat @ X4 @ Y7 ) @ ( product_Pair_b_nat @ X4 @ Y5 ) ) @ ( same_fst_b_nat @ P2 @ R ) ) ) ) ).

% same_fstI
thf(fact_371_same__fstI,axiom,
    ! [P2: option_b > $o,X4: option_b,Y7: option_b,Y5: option_b,R: option_b > set_Pr258254617715012167tion_b] :
      ( ( P2 @ X4 )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y7 @ Y5 ) @ ( R @ X4 ) )
       => ( member6435960643780680464tion_b @ ( produc7954421581189286743tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y7 ) @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) ) @ ( same_f9095071695510155180tion_b @ P2 @ R ) ) ) ) ).

% same_fstI
thf(fact_372_lift__root__step__incr,axiom,
    ! [R3: set_Pr3134649703833893895term_b,S3: set_Pr3134649703833893895term_b,F2: set_Pr1307281990691478580_b_nat,W: fOR_pos_step,X6: fOR_ext_step] :
      ( ( ord_le2408605988743369639term_b @ R3 @ S3 )
     => ( ord_le2408605988743369639term_b @ ( lift_R8854095231239425056step_b @ F2 @ W @ X6 @ R3 ) @ ( lift_R8854095231239425056step_b @ F2 @ W @ X6 @ S3 ) ) ) ).

% lift_root_step_incr
thf(fact_373_subrelI,axiom,
    ! [R5: set_Pr4178509052509721666tion_b,S4: set_Pr4178509052509721666tion_b] :
      ( ! [X3: nat,Y3: produc7781606619528532327tion_b] :
          ( ( member3694883655523461923tion_b @ ( produc7510438065615382334tion_b @ X3 @ Y3 ) @ R5 )
         => ( member3694883655523461923tion_b @ ( produc7510438065615382334tion_b @ X3 @ Y3 ) @ S4 ) )
     => ( ord_le1167626946193011874tion_b @ R5 @ S4 ) ) ).

% subrelI
thf(fact_374_subrelI,axiom,
    ! [R5: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R5 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ S4 ) )
     => ( ord_le3146513528884898305at_nat @ R5 @ S4 ) ) ).

% subrelI
thf(fact_375_subrelI,axiom,
    ! [R5: set_Pr4264375888882495962_nat_b,S4: set_Pr4264375888882495962_nat_b] :
      ( ! [X3: nat,Y3: b] :
          ( ( member8962352056413324475_nat_b @ ( product_Pair_nat_b @ X3 @ Y3 ) @ R5 )
         => ( member8962352056413324475_nat_b @ ( product_Pair_nat_b @ X3 @ Y3 ) @ S4 ) )
     => ( ord_le7995947752535495226_nat_b @ R5 @ S4 ) ) ).

% subrelI
thf(fact_376_subrelI,axiom,
    ! [R5: set_Pr5422417554659304751tion_b,S4: set_Pr5422417554659304751tion_b] :
      ( ! [X3: b,Y3: produc7781606619528532327tion_b] :
          ( ( member8570655873678976888tion_b @ ( produc5187340250978832319tion_b @ X3 @ Y3 ) @ R5 )
         => ( member8570655873678976888tion_b @ ( produc5187340250978832319tion_b @ X3 @ Y3 ) @ S4 ) )
     => ( ord_le7044419192447376591tion_b @ R5 @ S4 ) ) ).

% subrelI
thf(fact_377_subrelI,axiom,
    ! [R5: set_Product_prod_b_b,S4: set_Product_prod_b_b] :
      ( ! [X3: b,Y3: b] :
          ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X3 @ Y3 ) @ R5 )
         => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X3 @ Y3 ) @ S4 ) )
     => ( ord_le182087997850975847od_b_b @ R5 @ S4 ) ) ).

% subrelI
thf(fact_378_subrelI,axiom,
    ! [R5: set_Pr6944079702279229084_b_nat,S4: set_Pr6944079702279229084_b_nat] :
      ( ! [X3: produc7781606619528532327tion_b,Y3: nat] :
          ( ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ X3 @ Y3 ) @ R5 )
         => ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ X3 @ Y3 ) @ S4 ) )
     => ( ord_le3933197595962519292_b_nat @ R5 @ S4 ) ) ).

% subrelI
thf(fact_379_subrelI,axiom,
    ! [R5: set_Pr258254617715012167tion_b,S4: set_Pr258254617715012167tion_b] :
      ( ! [X3: option_b,Y3: option_b] :
          ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X3 @ Y3 ) @ R5 )
         => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X3 @ Y3 ) @ S4 ) )
     => ( ord_le6367397903258896871tion_b @ R5 @ S4 ) ) ).

% subrelI
thf(fact_380_subrelI,axiom,
    ! [R5: set_Pr1307281990691478580_b_nat,S4: set_Pr1307281990691478580_b_nat] :
      ( ! [X3: b,Y3: nat] :
          ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ X3 @ Y3 ) @ R5 )
         => ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ X3 @ Y3 ) @ S4 ) )
     => ( ord_le5038853854344477844_b_nat @ R5 @ S4 ) ) ).

% subrelI
thf(fact_381_relax__pos__lift__root__step,axiom,
    ! [F2: set_Pr1307281990691478580_b_nat,W: fOR_pos_step,X6: fOR_ext_step,R: set_Pr3134649703833893895term_b] : ( ord_le2408605988743369639term_b @ ( lift_R8854095231239425056step_b @ F2 @ W @ X6 @ R ) @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PAny @ X6 @ R ) ) ).

% relax_pos_lift_root_step
thf(fact_382_relax__ext__lift__root__step,axiom,
    ! [F2: set_Pr1307281990691478580_b_nat,W: fOR_pos_step,X6: fOR_ext_step,R: set_Pr3134649703833893895term_b] : ( ord_le2408605988743369639term_b @ ( lift_R8854095231239425056step_b @ F2 @ W @ X6 @ R ) @ ( lift_R8854095231239425056step_b @ F2 @ W @ fOR_ext_EParallel @ R ) ) ).

% relax_ext_lift_root_step
thf(fact_383_pos__step_Odistinct_I5_J,axiom,
    fOR_pos_PNonRoot != fOR_pos_PAny ).

% pos_step.distinct(5)
thf(fact_384_pos__step_Odistinct_I1_J,axiom,
    fOR_pos_PRoot != fOR_pos_PNonRoot ).

% pos_step.distinct(1)
thf(fact_385_pos__step_Oexhaust,axiom,
    ! [Y5: fOR_pos_step] :
      ( ( Y5 != fOR_pos_PRoot )
     => ( ( Y5 != fOR_pos_PNonRoot )
       => ( Y5 = fOR_pos_PAny ) ) ) ).

% pos_step.exhaust
thf(fact_386_less__eq__option__Some,axiom,
    ! [X4: set_Pr258254617715012167tion_b,Y5: set_Pr258254617715012167tion_b] :
      ( ( ord_le832834698364216237tion_b @ ( some_s4708006284228339500tion_b @ X4 ) @ ( some_s4708006284228339500tion_b @ Y5 ) )
      = ( ord_le6367397903258896871tion_b @ X4 @ Y5 ) ) ).

% less_eq_option_Some
thf(fact_387_less__eq__option__Some,axiom,
    ! [X4: set_Pr1307281990691478580_b_nat,Y5: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le7941364901598255844_b_nat @ ( some_s7477018756347572495_b_nat @ X4 ) @ ( some_s7477018756347572495_b_nat @ Y5 ) )
      = ( ord_le5038853854344477844_b_nat @ X4 @ Y5 ) ) ).

% less_eq_option_Some
thf(fact_388_less__eq__option__Some,axiom,
    ! [X4: produc7781606619528532327tion_b > nat > $o,Y5: produc7781606619528532327tion_b > nat > $o] :
      ( ( ord_le1159229661344459373_nat_o @ ( some_P4610421320244625688_nat_o @ X4 ) @ ( some_P4610421320244625688_nat_o @ Y5 ) )
      = ( ord_le6556145504227509917_nat_o @ X4 @ Y5 ) ) ).

% less_eq_option_Some
thf(fact_389_less__eq__option__Some,axiom,
    ! [X4: option_b > option_b > $o,Y5: option_b > option_b > $o] :
      ( ( ord_le325562215255571732on_b_o @ ( some_o5225106107430977555on_b_o @ X4 ) @ ( some_o5225106107430977555on_b_o @ Y5 ) )
      = ( ord_le3674354805995996366on_b_o @ X4 @ Y5 ) ) ).

% less_eq_option_Some
thf(fact_390_less__eq__option__Some,axiom,
    ! [X4: b > nat > $o,Y5: b > nat > $o] :
      ( ( ord_le662391368169153237_nat_o @ ( some_b_nat_o @ X4 ) @ ( some_b_nat_o @ Y5 ) )
      = ( ord_less_eq_b_nat_o @ X4 @ Y5 ) ) ).

% less_eq_option_Some
thf(fact_391_less__eq__option__Some,axiom,
    ! [X4: nat,Y5: nat] :
      ( ( ord_le5914376470875661696on_nat @ ( some_nat @ X4 ) @ ( some_nat @ Y5 ) )
      = ( ord_less_eq_nat @ X4 @ Y5 ) ) ).

% less_eq_option_Some
thf(fact_392_subsetI,axiom,
    ! [A5: set_Product_prod_b_b,B5: set_Product_prod_b_b] :
      ( ! [X3: product_prod_b_b] :
          ( ( member7862447936710763792od_b_b @ X3 @ A5 )
         => ( member7862447936710763792od_b_b @ X3 @ B5 ) )
     => ( ord_le182087997850975847od_b_b @ A5 @ B5 ) ) ).

% subsetI
thf(fact_393_subsetI,axiom,
    ! [A5: set_option_b,B5: set_option_b] :
      ( ! [X3: option_b] :
          ( ( member_option_b @ X3 @ A5 )
         => ( member_option_b @ X3 @ B5 ) )
     => ( ord_le2026170893118325445tion_b @ A5 @ B5 ) ) ).

% subsetI
thf(fact_394_subsetI,axiom,
    ! [A5: set_nat,B5: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A5 )
         => ( member_nat @ X3 @ B5 ) )
     => ( ord_less_eq_set_nat @ A5 @ B5 ) ) ).

% subsetI
thf(fact_395_subsetI,axiom,
    ! [A5: set_b,B5: set_b] :
      ( ! [X3: b] :
          ( ( member_b @ X3 @ A5 )
         => ( member_b @ X3 @ B5 ) )
     => ( ord_less_eq_set_b @ A5 @ B5 ) ) ).

% subsetI
thf(fact_396_subsetI,axiom,
    ! [A5: set_Pr258254617715012167tion_b,B5: set_Pr258254617715012167tion_b] :
      ( ! [X3: produc7781606619528532327tion_b] :
          ( ( member547026996284216464tion_b @ X3 @ A5 )
         => ( member547026996284216464tion_b @ X3 @ B5 ) )
     => ( ord_le6367397903258896871tion_b @ A5 @ B5 ) ) ).

% subsetI
thf(fact_397_subsetI,axiom,
    ! [A5: set_Pr1307281990691478580_b_nat,B5: set_Pr1307281990691478580_b_nat] :
      ( ! [X3: product_prod_b_nat] :
          ( ( member6959632917342813205_b_nat @ X3 @ A5 )
         => ( member6959632917342813205_b_nat @ X3 @ B5 ) )
     => ( ord_le5038853854344477844_b_nat @ A5 @ B5 ) ) ).

% subsetI
thf(fact_398_subset__antisym,axiom,
    ! [A5: set_Pr258254617715012167tion_b,B5: set_Pr258254617715012167tion_b] :
      ( ( ord_le6367397903258896871tion_b @ A5 @ B5 )
     => ( ( ord_le6367397903258896871tion_b @ B5 @ A5 )
       => ( A5 = B5 ) ) ) ).

% subset_antisym
thf(fact_399_subset__antisym,axiom,
    ! [A5: set_Pr1307281990691478580_b_nat,B5: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le5038853854344477844_b_nat @ A5 @ B5 )
     => ( ( ord_le5038853854344477844_b_nat @ B5 @ A5 )
       => ( A5 = B5 ) ) ) ).

% subset_antisym
thf(fact_400_rp__inv__image__def,axiom,
    ( fun_rp2348643092286788506tion_b
    = ( produc6119899154636713037tion_b
      @ ^ [R4: set_Pr258254617715012167tion_b,S2: set_Pr258254617715012167tion_b,F3: option_b > option_b] : ( produc4709765886195557207tion_b @ ( inv_im5784823153951788434tion_b @ R4 @ F3 ) @ ( inv_im5784823153951788434tion_b @ S2 @ F3 ) ) ) ) ).

% rp_inv_image_def
thf(fact_401_order__refl,axiom,
    ! [X4: set_Pr258254617715012167tion_b] : ( ord_le6367397903258896871tion_b @ X4 @ X4 ) ).

% order_refl
thf(fact_402_order__refl,axiom,
    ! [X4: set_Pr1307281990691478580_b_nat] : ( ord_le5038853854344477844_b_nat @ X4 @ X4 ) ).

% order_refl
thf(fact_403_order__refl,axiom,
    ! [X4: produc7781606619528532327tion_b > nat > $o] : ( ord_le6556145504227509917_nat_o @ X4 @ X4 ) ).

% order_refl
thf(fact_404_order__refl,axiom,
    ! [X4: option_b > option_b > $o] : ( ord_le3674354805995996366on_b_o @ X4 @ X4 ) ).

% order_refl
thf(fact_405_order__refl,axiom,
    ! [X4: b > nat > $o] : ( ord_less_eq_b_nat_o @ X4 @ X4 ) ).

% order_refl
thf(fact_406_order__refl,axiom,
    ! [X4: nat] : ( ord_less_eq_nat @ X4 @ X4 ) ).

% order_refl
thf(fact_407_dual__order_Orefl,axiom,
    ! [A: set_Pr258254617715012167tion_b] : ( ord_le6367397903258896871tion_b @ A @ A ) ).

% dual_order.refl
thf(fact_408_dual__order_Orefl,axiom,
    ! [A: set_Pr1307281990691478580_b_nat] : ( ord_le5038853854344477844_b_nat @ A @ A ) ).

% dual_order.refl
thf(fact_409_dual__order_Orefl,axiom,
    ! [A: produc7781606619528532327tion_b > nat > $o] : ( ord_le6556145504227509917_nat_o @ A @ A ) ).

% dual_order.refl
thf(fact_410_dual__order_Orefl,axiom,
    ! [A: option_b > option_b > $o] : ( ord_le3674354805995996366on_b_o @ A @ A ) ).

% dual_order.refl
thf(fact_411_dual__order_Orefl,axiom,
    ! [A: b > nat > $o] : ( ord_less_eq_b_nat_o @ A @ A ) ).

% dual_order.refl
thf(fact_412_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_413_lift__root__Any__EStrict__eq,axiom,
    ! [F2: set_Pr1307281990691478580_b_nat,R: set_Pr3134649703833893895term_b] :
      ( ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PAny @ fOR_ex5070955383060788274rallel @ R )
      = ( sup_su1163039302778497499term_b @ R @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PNonRoot @ fOR_ex5070955383060788274rallel @ R ) ) ) ).

% lift_root_Any_EStrict_eq
thf(fact_414_lex__prod__def,axiom,
    ( lex_pr810045972678751215tion_b
    = ( ^ [Ra: set_Pr258254617715012167tion_b,Rb: set_Pr258254617715012167tion_b] :
          ( collec447916338801455442tion_b
          @ ( produc8054777514177824044on_b_o
            @ ( produc1842704630950797122on_b_o
              @ ^ [A6: option_b,B4: option_b] :
                  ( produc3378454745690301612on_b_o
                  @ ^ [A7: option_b,B6: option_b] :
                      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A6 @ A7 ) @ Ra )
                      | ( ( A6 = A7 )
                        & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_415_lex__prod__def,axiom,
    ( lex_prod_nat_nat
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr1261947904930325089at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ( produc8739625826339149834_nat_o
              @ ^ [A6: nat,B4: nat] :
                  ( produc6081775807080527818_nat_o
                  @ ^ [A7: nat,B6: nat] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ A7 ) @ Ra )
                      | ( ( A6 = A7 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_416_lex__prod__def,axiom,
    ( lex_prod_nat_b
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Product_prod_b_b] :
          ( collec634681850936634004_nat_b
          @ ( produc8972256381162492962at_b_o
            @ ( produc593422043868379880at_b_o
              @ ^ [A6: nat,B4: b] :
                  ( produc9071082986767181781at_b_o
                  @ ^ [A7: nat,B6: b] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ A7 ) @ Ra )
                      | ( ( A6 = A7 )
                        & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_417_lex__prod__def,axiom,
    ( lex_prod_b_b
    = ( ^ [Ra: set_Product_prod_b_b,Rb: set_Product_prod_b_b] :
          ( collec2049399640006090066od_b_b
          @ ( produc7890132440258884908_b_b_o
            @ ( produc5105606039670327362_b_b_o
              @ ^ [A6: b,B4: b] :
                  ( produc4772849988445941548_b_b_o
                  @ ^ [A7: b,B6: b] :
                      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A6 @ A7 ) @ Ra )
                      | ( ( A6 = A7 )
                        & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_418_lex__prod__def,axiom,
    ( lex_prod_b_nat
    = ( ^ [Ra: set_Product_prod_b_b,Rb: set_Pr1261947904930325089at_nat] :
          ( collec6052715720417950280_b_nat
          @ ( produc3203605564640457326_nat_o
            @ ( produc4535354591796149996_nat_o
              @ ^ [A6: b,B4: nat] :
                  ( produc795641402153621683_nat_o
                  @ ^ [A7: b,B6: nat] :
                      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A6 @ A7 ) @ Ra )
                      | ( ( A6 = A7 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_419_lex__prod__def,axiom,
    ( lex_pr7803610914482903846_b_nat
    = ( ^ [Ra: set_Pr258254617715012167tion_b,Rb: set_Pr1261947904930325089at_nat] :
          ( collec3708845310148990548_b_nat
          @ ( produc2149759016953554530_nat_o
            @ ( produc7072454349396750048_nat_o
              @ ^ [A6: option_b,B4: nat] :
                  ( produc1228302092434924397_nat_o
                  @ ^ [A7: option_b,B6: nat] :
                      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A6 @ A7 ) @ Ra )
                      | ( ( A6 = A7 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_420_lex__prod__def,axiom,
    ( lex_prod_option_b_b
    = ( ^ [Ra: set_Pr258254617715012167tion_b,Rb: set_Product_prod_b_b] :
          ( collec4248788493952665042on_b_b
          @ ( produc1427124677488583084_b_b_o
            @ ( produc1812080453570669762_b_b_o
              @ ^ [A6: option_b,B4: b] :
                  ( produc4210729135258984178_b_b_o
                  @ ^ [A7: option_b,B6: b] :
                      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A6 @ A7 ) @ Ra )
                      | ( ( A6 = A7 )
                        & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_421_lex__prod__def,axiom,
    ( lex_pr4832242538844063236tion_b
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr258254617715012167tion_b] :
          ( collec9128185009117097632tion_b
          @ ( produc1951005263183667222on_b_o
            @ ( produc395474374274420188on_b_o
              @ ^ [A6: nat,B4: option_b] :
                  ( produc1894228119609269519on_b_o
                  @ ^ [A7: nat,B6: option_b] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ A7 ) @ Ra )
                      | ( ( A6 = A7 )
                        & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_422_lex__prod__def,axiom,
    ( lex_prod_b_option_b
    = ( ^ [Ra: set_Product_prod_b_b,Rb: set_Pr258254617715012167tion_b] :
          ( collec4731401001466153170tion_b
          @ ( produc460095736079718572on_b_o
            @ ( produc1150986030574074562on_b_o
              @ ^ [A6: b,B4: option_b] :
                  ( produc6673172643028850150on_b_o
                  @ ^ [A7: b,B6: option_b] :
                      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A6 @ A7 ) @ Ra )
                      | ( ( A6 = A7 )
                        & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_423_lex__prod__def,axiom,
    ( lex_pr314275984586175432_b_nat
    = ( ^ [Ra: set_Pr5366877986981747911tion_b,Rb: set_Pr1261947904930325089at_nat] :
          ( collec2051687263103543064_b_nat
          @ ( produc4199811631454031262_nat_o
            @ ( produc1254906330614685212_nat_o
              @ ^ [A6: produc7781606619528532327tion_b,B4: nat] :
                  ( produc5232674473337540683_nat_o
                  @ ^ [A7: produc7781606619528532327tion_b,B6: nat] :
                      ( ( member6435960643780680464tion_b @ ( produc7954421581189286743tion_b @ A6 @ A7 ) @ Ra )
                      | ( ( A6 = A7 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_424_lift__root__Any__Single__eq,axiom,
    ! [F2: set_Pr1307281990691478580_b_nat,R: set_Pr3134649703833893895term_b] :
      ( ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PAny @ fOR_ext_ESingle @ R )
      = ( sup_su1163039302778497499term_b @ R @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PNonRoot @ fOR_ext_ESingle @ R ) ) ) ).

% lift_root_Any_Single_eq
thf(fact_425_Collect__subset,axiom,
    ! [A5: set_Product_prod_b_b,P2: product_prod_b_b > $o] :
      ( ord_le182087997850975847od_b_b
      @ ( collec548942219715005266od_b_b
        @ ^ [X: product_prod_b_b] :
            ( ( member7862447936710763792od_b_b @ X @ A5 )
            & ( P2 @ X ) ) )
      @ A5 ) ).

% Collect_subset
thf(fact_426_Collect__subset,axiom,
    ! [A5: set_option_b,P2: option_b > $o] :
      ( ord_le2026170893118325445tion_b
      @ ( collect_option_b
        @ ^ [X: option_b] :
            ( ( member_option_b @ X @ A5 )
            & ( P2 @ X ) ) )
      @ A5 ) ).

% Collect_subset
thf(fact_427_Collect__subset,axiom,
    ! [A5: set_nat,P2: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ A5 )
            & ( P2 @ X ) ) )
      @ A5 ) ).

% Collect_subset
thf(fact_428_Collect__subset,axiom,
    ! [A5: set_b,P2: b > $o] :
      ( ord_less_eq_set_b
      @ ( collect_b
        @ ^ [X: b] :
            ( ( member_b @ X @ A5 )
            & ( P2 @ X ) ) )
      @ A5 ) ).

% Collect_subset
thf(fact_429_Collect__subset,axiom,
    ! [A5: set_Pr3134649703833893895term_b,P2: produc4514605509320855335term_b > $o] :
      ( ord_le2408605988743369639term_b
      @ ( collec478437889140176530term_b
        @ ^ [X: produc4514605509320855335term_b] :
            ( ( member7340517243570908752term_b @ X @ A5 )
            & ( P2 @ X ) ) )
      @ A5 ) ).

% Collect_subset
thf(fact_430_Collect__subset,axiom,
    ! [A5: set_Pr5366877986981747911tion_b,P2: produc6650559869974650087tion_b > $o] :
      ( ord_le5090656177574270055tion_b
      @ ( collec447916338801455442tion_b
        @ ^ [X: produc6650559869974650087tion_b] :
            ( ( member6435960643780680464tion_b @ X @ A5 )
            & ( P2 @ X ) ) )
      @ A5 ) ).

% Collect_subset
thf(fact_431_Collect__subset,axiom,
    ! [A5: set_Pr2609077471296144583tion_b,P2: produc4298468441892121831tion_b > $o] :
      ( ord_le3057591301425134695tion_b
      @ ( collec1918043184897894738tion_b
        @ ^ [X: produc4298468441892121831tion_b] :
            ( ( member8319496676406272784tion_b @ X @ A5 )
            & ( P2 @ X ) ) )
      @ A5 ) ).

% Collect_subset
thf(fact_432_Collect__subset,axiom,
    ! [A5: set_Pr4148925336519852615tion_b,P2: produc3122286842528565607tion_b > $o] :
      ( ord_le4306526642958997991tion_b
      @ ( collec6622971782277219538tion_b
        @ ^ [X: produc3122286842528565607tion_b] :
            ( ( member9052098846246995088tion_b @ X @ A5 )
            & ( P2 @ X ) ) )
      @ A5 ) ).

% Collect_subset
thf(fact_433_Collect__subset,axiom,
    ! [A5: set_Pr258254617715012167tion_b,P2: produc7781606619528532327tion_b > $o] :
      ( ord_le6367397903258896871tion_b
      @ ( collec8407377912107443410tion_b
        @ ^ [X: produc7781606619528532327tion_b] :
            ( ( member547026996284216464tion_b @ X @ A5 )
            & ( P2 @ X ) ) )
      @ A5 ) ).

% Collect_subset
thf(fact_434_Collect__subset,axiom,
    ! [A5: set_Pr1307281990691478580_b_nat,P2: product_prod_b_nat > $o] :
      ( ord_le5038853854344477844_b_nat
      @ ( collec5699578864178162003_b_nat
        @ ^ [X: product_prod_b_nat] :
            ( ( member6959632917342813205_b_nat @ X @ A5 )
            & ( P2 @ X ) ) )
      @ A5 ) ).

% Collect_subset
thf(fact_435_Un__iff,axiom,
    ! [C: product_prod_b_b,A5: set_Product_prod_b_b,B5: set_Product_prod_b_b] :
      ( ( member7862447936710763792od_b_b @ C @ ( sup_su2483643821041016987od_b_b @ A5 @ B5 ) )
      = ( ( member7862447936710763792od_b_b @ C @ A5 )
        | ( member7862447936710763792od_b_b @ C @ B5 ) ) ) ).

% Un_iff
thf(fact_436_Un__iff,axiom,
    ! [C: option_b,A5: set_option_b,B5: set_option_b] :
      ( ( member_option_b @ C @ ( sup_sup_set_option_b @ A5 @ B5 ) )
      = ( ( member_option_b @ C @ A5 )
        | ( member_option_b @ C @ B5 ) ) ) ).

% Un_iff
thf(fact_437_Un__iff,axiom,
    ! [C: nat,A5: set_nat,B5: set_nat] :
      ( ( member_nat @ C @ ( sup_sup_set_nat @ A5 @ B5 ) )
      = ( ( member_nat @ C @ A5 )
        | ( member_nat @ C @ B5 ) ) ) ).

% Un_iff
thf(fact_438_Un__iff,axiom,
    ! [C: b,A5: set_b,B5: set_b] :
      ( ( member_b @ C @ ( sup_sup_set_b @ A5 @ B5 ) )
      = ( ( member_b @ C @ A5 )
        | ( member_b @ C @ B5 ) ) ) ).

% Un_iff
thf(fact_439_Un__iff,axiom,
    ! [C: product_prod_b_nat,A5: set_Pr1307281990691478580_b_nat,B5: set_Pr1307281990691478580_b_nat] :
      ( ( member6959632917342813205_b_nat @ C @ ( sup_su6056130500584125792_b_nat @ A5 @ B5 ) )
      = ( ( member6959632917342813205_b_nat @ C @ A5 )
        | ( member6959632917342813205_b_nat @ C @ B5 ) ) ) ).

% Un_iff
thf(fact_440_UnCI,axiom,
    ! [C: product_prod_b_b,B5: set_Product_prod_b_b,A5: set_Product_prod_b_b] :
      ( ( ~ ( member7862447936710763792od_b_b @ C @ B5 )
       => ( member7862447936710763792od_b_b @ C @ A5 ) )
     => ( member7862447936710763792od_b_b @ C @ ( sup_su2483643821041016987od_b_b @ A5 @ B5 ) ) ) ).

% UnCI
thf(fact_441_UnCI,axiom,
    ! [C: option_b,B5: set_option_b,A5: set_option_b] :
      ( ( ~ ( member_option_b @ C @ B5 )
       => ( member_option_b @ C @ A5 ) )
     => ( member_option_b @ C @ ( sup_sup_set_option_b @ A5 @ B5 ) ) ) ).

% UnCI
thf(fact_442_UnCI,axiom,
    ! [C: nat,B5: set_nat,A5: set_nat] :
      ( ( ~ ( member_nat @ C @ B5 )
       => ( member_nat @ C @ A5 ) )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A5 @ B5 ) ) ) ).

% UnCI
thf(fact_443_UnCI,axiom,
    ! [C: b,B5: set_b,A5: set_b] :
      ( ( ~ ( member_b @ C @ B5 )
       => ( member_b @ C @ A5 ) )
     => ( member_b @ C @ ( sup_sup_set_b @ A5 @ B5 ) ) ) ).

% UnCI
thf(fact_444_UnCI,axiom,
    ! [C: product_prod_b_nat,B5: set_Pr1307281990691478580_b_nat,A5: set_Pr1307281990691478580_b_nat] :
      ( ( ~ ( member6959632917342813205_b_nat @ C @ B5 )
       => ( member6959632917342813205_b_nat @ C @ A5 ) )
     => ( member6959632917342813205_b_nat @ C @ ( sup_su6056130500584125792_b_nat @ A5 @ B5 ) ) ) ).

% UnCI
thf(fact_445_Un__subset__iff,axiom,
    ! [A5: set_Pr258254617715012167tion_b,B5: set_Pr258254617715012167tion_b,C3: set_Pr258254617715012167tion_b] :
      ( ( ord_le6367397903258896871tion_b @ ( sup_su3110286242275913755tion_b @ A5 @ B5 ) @ C3 )
      = ( ( ord_le6367397903258896871tion_b @ A5 @ C3 )
        & ( ord_le6367397903258896871tion_b @ B5 @ C3 ) ) ) ).

% Un_subset_iff
thf(fact_446_Un__subset__iff,axiom,
    ! [A5: set_Pr1307281990691478580_b_nat,B5: set_Pr1307281990691478580_b_nat,C3: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le5038853854344477844_b_nat @ ( sup_su6056130500584125792_b_nat @ A5 @ B5 ) @ C3 )
      = ( ( ord_le5038853854344477844_b_nat @ A5 @ C3 )
        & ( ord_le5038853854344477844_b_nat @ B5 @ C3 ) ) ) ).

% Un_subset_iff
thf(fact_447_sup__Some,axiom,
    ! [X4: set_Pr1307281990691478580_b_nat,Y5: set_Pr1307281990691478580_b_nat] :
      ( ( sup_su8012392926425379248_b_nat @ ( some_s7477018756347572495_b_nat @ X4 ) @ ( some_s7477018756347572495_b_nat @ Y5 ) )
      = ( some_s7477018756347572495_b_nat @ ( sup_su6056130500584125792_b_nat @ X4 @ Y5 ) ) ) ).

% sup_Some
thf(fact_448_sup__Some,axiom,
    ! [X4: nat,Y5: nat] :
      ( ( sup_sup_option_nat @ ( some_nat @ X4 ) @ ( some_nat @ Y5 ) )
      = ( some_nat @ ( sup_sup_nat @ X4 @ Y5 ) ) ) ).

% sup_Some
thf(fact_449_sup__Some,axiom,
    ! [X4: produc7781606619528532327tion_b > nat > $o,Y5: produc7781606619528532327tion_b > nat > $o] :
      ( ( sup_su7858367213232150841_nat_o @ ( some_P4610421320244625688_nat_o @ X4 ) @ ( some_P4610421320244625688_nat_o @ Y5 ) )
      = ( some_P4610421320244625688_nat_o @ ( sup_su6759296390479959913_nat_o @ X4 @ Y5 ) ) ) ).

% sup_Some
thf(fact_450_sup__Some,axiom,
    ! [X4: option_b > option_b > $o,Y5: option_b > option_b > $o] :
      ( ( sup_su5332351281251056968on_b_o @ ( some_o5225106107430977555on_b_o @ X4 ) @ ( some_o5225106107430977555on_b_o @ Y5 ) )
      = ( some_o5225106107430977555on_b_o @ ( sup_su6610355334417212162on_b_o @ X4 @ Y5 ) ) ) ).

% sup_Some
thf(fact_451_sup__Some,axiom,
    ! [X4: b > nat > $o,Y5: b > nat > $o] :
      ( ( sup_su2869153249045692833_nat_o @ ( some_b_nat_o @ X4 ) @ ( some_b_nat_o @ Y5 ) )
      = ( some_b_nat_o @ ( sup_sup_b_nat_o @ X4 @ Y5 ) ) ) ).

% sup_Some
thf(fact_452_in__lex__prod,axiom,
    ! [A: b,B: nat,A3: b,B2: nat,R5: set_Product_prod_b_b,S4: set_Pr1261947904930325089at_nat] :
      ( ( member4966892972527255302_b_nat @ ( produc2793586353817733269_b_nat @ ( product_Pair_b_nat @ A @ B ) @ ( product_Pair_b_nat @ A3 @ B2 ) ) @ ( lex_prod_b_nat @ R5 @ S4 ) )
      = ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ A3 ) @ R5 )
        | ( ( A = A3 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B2 ) @ S4 ) ) ) ) ).

% in_lex_prod
thf(fact_453_in__lex__prod,axiom,
    ! [A: option_b,B: option_b,A3: option_b,B2: option_b,R5: set_Pr258254617715012167tion_b,S4: set_Pr258254617715012167tion_b] :
      ( ( member6435960643780680464tion_b @ ( produc7954421581189286743tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ ( produc3989389360737052631tion_b @ A3 @ B2 ) ) @ ( lex_pr810045972678751215tion_b @ R5 @ S4 ) )
      = ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ A3 ) @ R5 )
        | ( ( A = A3 )
          & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ B @ B2 ) @ S4 ) ) ) ) ).

% in_lex_prod
thf(fact_454_in__lex__prod,axiom,
    ! [A: produc7781606619528532327tion_b,B: nat,A3: produc7781606619528532327tion_b,B2: nat,R5: set_Pr5366877986981747911tion_b,S4: set_Pr1261947904930325089at_nat] :
      ( ( member4316259128586992598_b_nat @ ( produc7802874649797638501_b_nat @ ( produc2093246604507004384_b_nat @ A @ B ) @ ( produc2093246604507004384_b_nat @ A3 @ B2 ) ) @ ( lex_pr314275984586175432_b_nat @ R5 @ S4 ) )
      = ( ( member6435960643780680464tion_b @ ( produc7954421581189286743tion_b @ A @ A3 ) @ R5 )
        | ( ( A = A3 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B2 ) @ S4 ) ) ) ) ).

% in_lex_prod
thf(fact_455_in__lex__prod,axiom,
    ! [A: nat,B: nat,A3: nat,B2: nat,R5: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ A3 @ B2 ) ) @ ( lex_prod_nat_nat @ R5 @ S4 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A3 ) @ R5 )
        | ( ( A = A3 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B2 ) @ S4 ) ) ) ) ).

% in_lex_prod
thf(fact_456_in__lex__prod,axiom,
    ! [A: nat,B: b,A3: nat,B2: b,R5: set_Pr1261947904930325089at_nat,S4: set_Product_prod_b_b] :
      ( ( member8772231139900714834_nat_b @ ( produc5644549762070686689_nat_b @ ( product_Pair_nat_b @ A @ B ) @ ( product_Pair_nat_b @ A3 @ B2 ) ) @ ( lex_prod_nat_b @ R5 @ S4 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A3 ) @ R5 )
        | ( ( A = A3 )
          & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B @ B2 ) @ S4 ) ) ) ) ).

% in_lex_prod
thf(fact_457_in__lex__prod,axiom,
    ! [A: b,B: b,A3: b,B2: b,R5: set_Product_prod_b_b,S4: set_Product_prod_b_b] :
      ( ( member8369738419320101648od_b_b @ ( produc7702952596710024023od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( product_Pair_b_b @ A3 @ B2 ) ) @ ( lex_prod_b_b @ R5 @ S4 ) )
      = ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ A3 ) @ R5 )
        | ( ( A = A3 )
          & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B @ B2 ) @ S4 ) ) ) ) ).

% in_lex_prod
thf(fact_458_in__lex__prod,axiom,
    ! [A: option_b,B: nat,A3: option_b,B2: nat,R5: set_Pr258254617715012167tion_b,S4: set_Pr1261947904930325089at_nat] :
      ( ( member5433527884241383698_b_nat @ ( produc8955844040950097313_b_nat @ ( produc9173639105284866366_b_nat @ A @ B ) @ ( produc9173639105284866366_b_nat @ A3 @ B2 ) ) @ ( lex_pr7803610914482903846_b_nat @ R5 @ S4 ) )
      = ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ A3 ) @ R5 )
        | ( ( A = A3 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B2 ) @ S4 ) ) ) ) ).

% in_lex_prod
thf(fact_459_in__lex__prod,axiom,
    ! [A: option_b,B: b,A3: option_b,B2: b,R5: set_Pr258254617715012167tion_b,S4: set_Product_prod_b_b] :
      ( ( member4206488804519221648on_b_b @ ( produc6567937323074884311on_b_b @ ( produc2811058977019475217on_b_b @ A @ B ) @ ( produc2811058977019475217on_b_b @ A3 @ B2 ) ) @ ( lex_prod_option_b_b @ R5 @ S4 ) )
      = ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ A3 ) @ R5 )
        | ( ( A = A3 )
          & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B @ B2 ) @ S4 ) ) ) ) ).

% in_lex_prod
thf(fact_460_in__lex__prod,axiom,
    ! [A: nat,B: option_b,A3: nat,B2: option_b,R5: set_Pr1261947904930325089at_nat,S4: set_Pr258254617715012167tion_b] :
      ( ( member1629495546354714974tion_b @ ( produc7554641163325534445tion_b @ ( produc6202270729646025756tion_b @ A @ B ) @ ( produc6202270729646025756tion_b @ A3 @ B2 ) ) @ ( lex_pr4832242538844063236tion_b @ R5 @ S4 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A3 ) @ R5 )
        | ( ( A = A3 )
          & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ B @ B2 ) @ S4 ) ) ) ) ).

% in_lex_prod
thf(fact_461_in__lex__prod,axiom,
    ! [A: b,B: option_b,A3: b,B2: option_b,R5: set_Product_prod_b_b,S4: set_Pr258254617715012167tion_b] :
      ( ( member4689101312032709776tion_b @ ( produc268218253536058327tion_b @ ( produc7361809895253676829tion_b @ A @ B ) @ ( produc7361809895253676829tion_b @ A3 @ B2 ) ) @ ( lex_prod_b_option_b @ R5 @ S4 ) )
      = ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ A3 ) @ R5 )
        | ( ( A = A3 )
          & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ B @ B2 ) @ S4 ) ) ) ) ).

% in_lex_prod
thf(fact_462_Un__left__commute,axiom,
    ! [A5: set_Pr1307281990691478580_b_nat,B5: set_Pr1307281990691478580_b_nat,C3: set_Pr1307281990691478580_b_nat] :
      ( ( sup_su6056130500584125792_b_nat @ A5 @ ( sup_su6056130500584125792_b_nat @ B5 @ C3 ) )
      = ( sup_su6056130500584125792_b_nat @ B5 @ ( sup_su6056130500584125792_b_nat @ A5 @ C3 ) ) ) ).

% Un_left_commute
thf(fact_463_Collect__disj__eq,axiom,
    ! [P2: produc4514605509320855335term_b > $o,Q: produc4514605509320855335term_b > $o] :
      ( ( collec478437889140176530term_b
        @ ^ [X: produc4514605509320855335term_b] :
            ( ( P2 @ X )
            | ( Q @ X ) ) )
      = ( sup_su1163039302778497499term_b @ ( collec478437889140176530term_b @ P2 ) @ ( collec478437889140176530term_b @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_464_Collect__disj__eq,axiom,
    ! [P2: produc6650559869974650087tion_b > $o,Q: produc6650559869974650087tion_b > $o] :
      ( ( collec447916338801455442tion_b
        @ ^ [X: produc6650559869974650087tion_b] :
            ( ( P2 @ X )
            | ( Q @ X ) ) )
      = ( sup_su2925575103159965339tion_b @ ( collec447916338801455442tion_b @ P2 ) @ ( collec447916338801455442tion_b @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_465_Collect__disj__eq,axiom,
    ! [P2: produc7781606619528532327tion_b > $o,Q: produc7781606619528532327tion_b > $o] :
      ( ( collec8407377912107443410tion_b
        @ ^ [X: produc7781606619528532327tion_b] :
            ( ( P2 @ X )
            | ( Q @ X ) ) )
      = ( sup_su3110286242275913755tion_b @ ( collec8407377912107443410tion_b @ P2 ) @ ( collec8407377912107443410tion_b @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_466_Collect__disj__eq,axiom,
    ! [P2: produc4298468441892121831tion_b > $o,Q: produc4298468441892121831tion_b > $o] :
      ( ( collec1918043184897894738tion_b
        @ ^ [X: produc4298468441892121831tion_b] :
            ( ( P2 @ X )
            | ( Q @ X ) ) )
      = ( sup_su3944740340205657755tion_b @ ( collec1918043184897894738tion_b @ P2 ) @ ( collec1918043184897894738tion_b @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_467_Collect__disj__eq,axiom,
    ! [P2: produc3122286842528565607tion_b > $o,Q: produc3122286842528565607tion_b > $o] :
      ( ( collec6622971782277219538tion_b
        @ ^ [X: produc3122286842528565607tion_b] :
            ( ( P2 @ X )
            | ( Q @ X ) ) )
      = ( sup_su8287916483737133083tion_b @ ( collec6622971782277219538tion_b @ P2 ) @ ( collec6622971782277219538tion_b @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_468_Collect__disj__eq,axiom,
    ! [P2: product_prod_b_nat > $o,Q: product_prod_b_nat > $o] :
      ( ( collec5699578864178162003_b_nat
        @ ^ [X: product_prod_b_nat] :
            ( ( P2 @ X )
            | ( Q @ X ) ) )
      = ( sup_su6056130500584125792_b_nat @ ( collec5699578864178162003_b_nat @ P2 ) @ ( collec5699578864178162003_b_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_469_Un__left__absorb,axiom,
    ! [A5: set_Pr1307281990691478580_b_nat,B5: set_Pr1307281990691478580_b_nat] :
      ( ( sup_su6056130500584125792_b_nat @ A5 @ ( sup_su6056130500584125792_b_nat @ A5 @ B5 ) )
      = ( sup_su6056130500584125792_b_nat @ A5 @ B5 ) ) ).

% Un_left_absorb
thf(fact_470_Un__commute,axiom,
    ( sup_su6056130500584125792_b_nat
    = ( ^ [A8: set_Pr1307281990691478580_b_nat,B7: set_Pr1307281990691478580_b_nat] : ( sup_su6056130500584125792_b_nat @ B7 @ A8 ) ) ) ).

% Un_commute
thf(fact_471_Un__absorb,axiom,
    ! [A5: set_Pr1307281990691478580_b_nat] :
      ( ( sup_su6056130500584125792_b_nat @ A5 @ A5 )
      = A5 ) ).

% Un_absorb
thf(fact_472_Un__assoc,axiom,
    ! [A5: set_Pr1307281990691478580_b_nat,B5: set_Pr1307281990691478580_b_nat,C3: set_Pr1307281990691478580_b_nat] :
      ( ( sup_su6056130500584125792_b_nat @ ( sup_su6056130500584125792_b_nat @ A5 @ B5 ) @ C3 )
      = ( sup_su6056130500584125792_b_nat @ A5 @ ( sup_su6056130500584125792_b_nat @ B5 @ C3 ) ) ) ).

% Un_assoc
thf(fact_473_ball__Un,axiom,
    ! [A5: set_Pr1307281990691478580_b_nat,B5: set_Pr1307281990691478580_b_nat,P2: product_prod_b_nat > $o] :
      ( ( ! [X: product_prod_b_nat] :
            ( ( member6959632917342813205_b_nat @ X @ ( sup_su6056130500584125792_b_nat @ A5 @ B5 ) )
           => ( P2 @ X ) ) )
      = ( ! [X: product_prod_b_nat] :
            ( ( member6959632917342813205_b_nat @ X @ A5 )
           => ( P2 @ X ) )
        & ! [X: product_prod_b_nat] :
            ( ( member6959632917342813205_b_nat @ X @ B5 )
           => ( P2 @ X ) ) ) ) ).

% ball_Un
thf(fact_474_bex__Un,axiom,
    ! [A5: set_Pr1307281990691478580_b_nat,B5: set_Pr1307281990691478580_b_nat,P2: product_prod_b_nat > $o] :
      ( ( ? [X: product_prod_b_nat] :
            ( ( member6959632917342813205_b_nat @ X @ ( sup_su6056130500584125792_b_nat @ A5 @ B5 ) )
            & ( P2 @ X ) ) )
      = ( ? [X: product_prod_b_nat] :
            ( ( member6959632917342813205_b_nat @ X @ A5 )
            & ( P2 @ X ) )
        | ? [X: product_prod_b_nat] :
            ( ( member6959632917342813205_b_nat @ X @ B5 )
            & ( P2 @ X ) ) ) ) ).

% bex_Un
thf(fact_475_Un__def,axiom,
    ( sup_su2483643821041016987od_b_b
    = ( ^ [A8: set_Product_prod_b_b,B7: set_Product_prod_b_b] :
          ( collec548942219715005266od_b_b
          @ ^ [X: product_prod_b_b] :
              ( ( member7862447936710763792od_b_b @ X @ A8 )
              | ( member7862447936710763792od_b_b @ X @ B7 ) ) ) ) ) ).

% Un_def
thf(fact_476_Un__def,axiom,
    ( sup_sup_set_option_b
    = ( ^ [A8: set_option_b,B7: set_option_b] :
          ( collect_option_b
          @ ^ [X: option_b] :
              ( ( member_option_b @ X @ A8 )
              | ( member_option_b @ X @ B7 ) ) ) ) ) ).

% Un_def
thf(fact_477_Un__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A8: set_nat,B7: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A8 )
              | ( member_nat @ X @ B7 ) ) ) ) ) ).

% Un_def
thf(fact_478_Un__def,axiom,
    ( sup_sup_set_b
    = ( ^ [A8: set_b,B7: set_b] :
          ( collect_b
          @ ^ [X: b] :
              ( ( member_b @ X @ A8 )
              | ( member_b @ X @ B7 ) ) ) ) ) ).

% Un_def
thf(fact_479_Un__def,axiom,
    ( sup_su1163039302778497499term_b
    = ( ^ [A8: set_Pr3134649703833893895term_b,B7: set_Pr3134649703833893895term_b] :
          ( collec478437889140176530term_b
          @ ^ [X: produc4514605509320855335term_b] :
              ( ( member7340517243570908752term_b @ X @ A8 )
              | ( member7340517243570908752term_b @ X @ B7 ) ) ) ) ) ).

% Un_def
thf(fact_480_Un__def,axiom,
    ( sup_su2925575103159965339tion_b
    = ( ^ [A8: set_Pr5366877986981747911tion_b,B7: set_Pr5366877986981747911tion_b] :
          ( collec447916338801455442tion_b
          @ ^ [X: produc6650559869974650087tion_b] :
              ( ( member6435960643780680464tion_b @ X @ A8 )
              | ( member6435960643780680464tion_b @ X @ B7 ) ) ) ) ) ).

% Un_def
thf(fact_481_Un__def,axiom,
    ( sup_su3110286242275913755tion_b
    = ( ^ [A8: set_Pr258254617715012167tion_b,B7: set_Pr258254617715012167tion_b] :
          ( collec8407377912107443410tion_b
          @ ^ [X: produc7781606619528532327tion_b] :
              ( ( member547026996284216464tion_b @ X @ A8 )
              | ( member547026996284216464tion_b @ X @ B7 ) ) ) ) ) ).

% Un_def
thf(fact_482_Un__def,axiom,
    ( sup_su3944740340205657755tion_b
    = ( ^ [A8: set_Pr2609077471296144583tion_b,B7: set_Pr2609077471296144583tion_b] :
          ( collec1918043184897894738tion_b
          @ ^ [X: produc4298468441892121831tion_b] :
              ( ( member8319496676406272784tion_b @ X @ A8 )
              | ( member8319496676406272784tion_b @ X @ B7 ) ) ) ) ) ).

% Un_def
thf(fact_483_Un__def,axiom,
    ( sup_su8287916483737133083tion_b
    = ( ^ [A8: set_Pr4148925336519852615tion_b,B7: set_Pr4148925336519852615tion_b] :
          ( collec6622971782277219538tion_b
          @ ^ [X: produc3122286842528565607tion_b] :
              ( ( member9052098846246995088tion_b @ X @ A8 )
              | ( member9052098846246995088tion_b @ X @ B7 ) ) ) ) ) ).

% Un_def
thf(fact_484_Un__def,axiom,
    ( sup_su6056130500584125792_b_nat
    = ( ^ [A8: set_Pr1307281990691478580_b_nat,B7: set_Pr1307281990691478580_b_nat] :
          ( collec5699578864178162003_b_nat
          @ ^ [X: product_prod_b_nat] :
              ( ( member6959632917342813205_b_nat @ X @ A8 )
              | ( member6959632917342813205_b_nat @ X @ B7 ) ) ) ) ) ).

% Un_def
thf(fact_485_UnI2,axiom,
    ! [C: product_prod_b_b,B5: set_Product_prod_b_b,A5: set_Product_prod_b_b] :
      ( ( member7862447936710763792od_b_b @ C @ B5 )
     => ( member7862447936710763792od_b_b @ C @ ( sup_su2483643821041016987od_b_b @ A5 @ B5 ) ) ) ).

% UnI2
thf(fact_486_UnI2,axiom,
    ! [C: option_b,B5: set_option_b,A5: set_option_b] :
      ( ( member_option_b @ C @ B5 )
     => ( member_option_b @ C @ ( sup_sup_set_option_b @ A5 @ B5 ) ) ) ).

% UnI2
thf(fact_487_UnI2,axiom,
    ! [C: nat,B5: set_nat,A5: set_nat] :
      ( ( member_nat @ C @ B5 )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A5 @ B5 ) ) ) ).

% UnI2
thf(fact_488_UnI2,axiom,
    ! [C: b,B5: set_b,A5: set_b] :
      ( ( member_b @ C @ B5 )
     => ( member_b @ C @ ( sup_sup_set_b @ A5 @ B5 ) ) ) ).

% UnI2
thf(fact_489_UnI2,axiom,
    ! [C: product_prod_b_nat,B5: set_Pr1307281990691478580_b_nat,A5: set_Pr1307281990691478580_b_nat] :
      ( ( member6959632917342813205_b_nat @ C @ B5 )
     => ( member6959632917342813205_b_nat @ C @ ( sup_su6056130500584125792_b_nat @ A5 @ B5 ) ) ) ).

% UnI2
thf(fact_490_UnI1,axiom,
    ! [C: product_prod_b_b,A5: set_Product_prod_b_b,B5: set_Product_prod_b_b] :
      ( ( member7862447936710763792od_b_b @ C @ A5 )
     => ( member7862447936710763792od_b_b @ C @ ( sup_su2483643821041016987od_b_b @ A5 @ B5 ) ) ) ).

% UnI1
thf(fact_491_UnI1,axiom,
    ! [C: option_b,A5: set_option_b,B5: set_option_b] :
      ( ( member_option_b @ C @ A5 )
     => ( member_option_b @ C @ ( sup_sup_set_option_b @ A5 @ B5 ) ) ) ).

% UnI1
thf(fact_492_UnI1,axiom,
    ! [C: nat,A5: set_nat,B5: set_nat] :
      ( ( member_nat @ C @ A5 )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A5 @ B5 ) ) ) ).

% UnI1
thf(fact_493_UnI1,axiom,
    ! [C: b,A5: set_b,B5: set_b] :
      ( ( member_b @ C @ A5 )
     => ( member_b @ C @ ( sup_sup_set_b @ A5 @ B5 ) ) ) ).

% UnI1
thf(fact_494_UnI1,axiom,
    ! [C: product_prod_b_nat,A5: set_Pr1307281990691478580_b_nat,B5: set_Pr1307281990691478580_b_nat] :
      ( ( member6959632917342813205_b_nat @ C @ A5 )
     => ( member6959632917342813205_b_nat @ C @ ( sup_su6056130500584125792_b_nat @ A5 @ B5 ) ) ) ).

% UnI1
thf(fact_495_UnE,axiom,
    ! [C: product_prod_b_b,A5: set_Product_prod_b_b,B5: set_Product_prod_b_b] :
      ( ( member7862447936710763792od_b_b @ C @ ( sup_su2483643821041016987od_b_b @ A5 @ B5 ) )
     => ( ~ ( member7862447936710763792od_b_b @ C @ A5 )
       => ( member7862447936710763792od_b_b @ C @ B5 ) ) ) ).

% UnE
thf(fact_496_UnE,axiom,
    ! [C: option_b,A5: set_option_b,B5: set_option_b] :
      ( ( member_option_b @ C @ ( sup_sup_set_option_b @ A5 @ B5 ) )
     => ( ~ ( member_option_b @ C @ A5 )
       => ( member_option_b @ C @ B5 ) ) ) ).

% UnE
thf(fact_497_UnE,axiom,
    ! [C: nat,A5: set_nat,B5: set_nat] :
      ( ( member_nat @ C @ ( sup_sup_set_nat @ A5 @ B5 ) )
     => ( ~ ( member_nat @ C @ A5 )
       => ( member_nat @ C @ B5 ) ) ) ).

% UnE
thf(fact_498_UnE,axiom,
    ! [C: b,A5: set_b,B5: set_b] :
      ( ( member_b @ C @ ( sup_sup_set_b @ A5 @ B5 ) )
     => ( ~ ( member_b @ C @ A5 )
       => ( member_b @ C @ B5 ) ) ) ).

% UnE
thf(fact_499_UnE,axiom,
    ! [C: product_prod_b_nat,A5: set_Pr1307281990691478580_b_nat,B5: set_Pr1307281990691478580_b_nat] :
      ( ( member6959632917342813205_b_nat @ C @ ( sup_su6056130500584125792_b_nat @ A5 @ B5 ) )
     => ( ~ ( member6959632917342813205_b_nat @ C @ A5 )
       => ( member6959632917342813205_b_nat @ C @ B5 ) ) ) ).

% UnE
thf(fact_500_subset__Un__eq,axiom,
    ( ord_le6367397903258896871tion_b
    = ( ^ [A8: set_Pr258254617715012167tion_b,B7: set_Pr258254617715012167tion_b] :
          ( ( sup_su3110286242275913755tion_b @ A8 @ B7 )
          = B7 ) ) ) ).

% subset_Un_eq
thf(fact_501_subset__Un__eq,axiom,
    ( ord_le5038853854344477844_b_nat
    = ( ^ [A8: set_Pr1307281990691478580_b_nat,B7: set_Pr1307281990691478580_b_nat] :
          ( ( sup_su6056130500584125792_b_nat @ A8 @ B7 )
          = B7 ) ) ) ).

% subset_Un_eq
thf(fact_502_subset__UnE,axiom,
    ! [C3: set_Pr258254617715012167tion_b,A5: set_Pr258254617715012167tion_b,B5: set_Pr258254617715012167tion_b] :
      ( ( ord_le6367397903258896871tion_b @ C3 @ ( sup_su3110286242275913755tion_b @ A5 @ B5 ) )
     => ~ ! [A9: set_Pr258254617715012167tion_b] :
            ( ( ord_le6367397903258896871tion_b @ A9 @ A5 )
           => ! [B8: set_Pr258254617715012167tion_b] :
                ( ( ord_le6367397903258896871tion_b @ B8 @ B5 )
               => ( C3
                 != ( sup_su3110286242275913755tion_b @ A9 @ B8 ) ) ) ) ) ).

% subset_UnE
thf(fact_503_subset__UnE,axiom,
    ! [C3: set_Pr1307281990691478580_b_nat,A5: set_Pr1307281990691478580_b_nat,B5: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le5038853854344477844_b_nat @ C3 @ ( sup_su6056130500584125792_b_nat @ A5 @ B5 ) )
     => ~ ! [A9: set_Pr1307281990691478580_b_nat] :
            ( ( ord_le5038853854344477844_b_nat @ A9 @ A5 )
           => ! [B8: set_Pr1307281990691478580_b_nat] :
                ( ( ord_le5038853854344477844_b_nat @ B8 @ B5 )
               => ( C3
                 != ( sup_su6056130500584125792_b_nat @ A9 @ B8 ) ) ) ) ) ).

% subset_UnE
thf(fact_504_Un__absorb2,axiom,
    ! [B5: set_Pr258254617715012167tion_b,A5: set_Pr258254617715012167tion_b] :
      ( ( ord_le6367397903258896871tion_b @ B5 @ A5 )
     => ( ( sup_su3110286242275913755tion_b @ A5 @ B5 )
        = A5 ) ) ).

% Un_absorb2
thf(fact_505_Un__absorb2,axiom,
    ! [B5: set_Pr1307281990691478580_b_nat,A5: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le5038853854344477844_b_nat @ B5 @ A5 )
     => ( ( sup_su6056130500584125792_b_nat @ A5 @ B5 )
        = A5 ) ) ).

% Un_absorb2
thf(fact_506_Un__absorb1,axiom,
    ! [A5: set_Pr258254617715012167tion_b,B5: set_Pr258254617715012167tion_b] :
      ( ( ord_le6367397903258896871tion_b @ A5 @ B5 )
     => ( ( sup_su3110286242275913755tion_b @ A5 @ B5 )
        = B5 ) ) ).

% Un_absorb1
thf(fact_507_Un__absorb1,axiom,
    ! [A5: set_Pr1307281990691478580_b_nat,B5: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le5038853854344477844_b_nat @ A5 @ B5 )
     => ( ( sup_su6056130500584125792_b_nat @ A5 @ B5 )
        = B5 ) ) ).

% Un_absorb1
thf(fact_508_Un__upper2,axiom,
    ! [B5: set_Pr258254617715012167tion_b,A5: set_Pr258254617715012167tion_b] : ( ord_le6367397903258896871tion_b @ B5 @ ( sup_su3110286242275913755tion_b @ A5 @ B5 ) ) ).

% Un_upper2
thf(fact_509_Un__upper2,axiom,
    ! [B5: set_Pr1307281990691478580_b_nat,A5: set_Pr1307281990691478580_b_nat] : ( ord_le5038853854344477844_b_nat @ B5 @ ( sup_su6056130500584125792_b_nat @ A5 @ B5 ) ) ).

% Un_upper2
thf(fact_510_Un__upper1,axiom,
    ! [A5: set_Pr258254617715012167tion_b,B5: set_Pr258254617715012167tion_b] : ( ord_le6367397903258896871tion_b @ A5 @ ( sup_su3110286242275913755tion_b @ A5 @ B5 ) ) ).

% Un_upper1
thf(fact_511_Un__upper1,axiom,
    ! [A5: set_Pr1307281990691478580_b_nat,B5: set_Pr1307281990691478580_b_nat] : ( ord_le5038853854344477844_b_nat @ A5 @ ( sup_su6056130500584125792_b_nat @ A5 @ B5 ) ) ).

% Un_upper1
thf(fact_512_Un__least,axiom,
    ! [A5: set_Pr258254617715012167tion_b,C3: set_Pr258254617715012167tion_b,B5: set_Pr258254617715012167tion_b] :
      ( ( ord_le6367397903258896871tion_b @ A5 @ C3 )
     => ( ( ord_le6367397903258896871tion_b @ B5 @ C3 )
       => ( ord_le6367397903258896871tion_b @ ( sup_su3110286242275913755tion_b @ A5 @ B5 ) @ C3 ) ) ) ).

% Un_least
thf(fact_513_Un__least,axiom,
    ! [A5: set_Pr1307281990691478580_b_nat,C3: set_Pr1307281990691478580_b_nat,B5: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le5038853854344477844_b_nat @ A5 @ C3 )
     => ( ( ord_le5038853854344477844_b_nat @ B5 @ C3 )
       => ( ord_le5038853854344477844_b_nat @ ( sup_su6056130500584125792_b_nat @ A5 @ B5 ) @ C3 ) ) ) ).

% Un_least
thf(fact_514_Un__mono,axiom,
    ! [A5: set_Pr258254617715012167tion_b,C3: set_Pr258254617715012167tion_b,B5: set_Pr258254617715012167tion_b,D: set_Pr258254617715012167tion_b] :
      ( ( ord_le6367397903258896871tion_b @ A5 @ C3 )
     => ( ( ord_le6367397903258896871tion_b @ B5 @ D )
       => ( ord_le6367397903258896871tion_b @ ( sup_su3110286242275913755tion_b @ A5 @ B5 ) @ ( sup_su3110286242275913755tion_b @ C3 @ D ) ) ) ) ).

% Un_mono
thf(fact_515_Un__mono,axiom,
    ! [A5: set_Pr1307281990691478580_b_nat,C3: set_Pr1307281990691478580_b_nat,B5: set_Pr1307281990691478580_b_nat,D: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le5038853854344477844_b_nat @ A5 @ C3 )
     => ( ( ord_le5038853854344477844_b_nat @ B5 @ D )
       => ( ord_le5038853854344477844_b_nat @ ( sup_su6056130500584125792_b_nat @ A5 @ B5 ) @ ( sup_su6056130500584125792_b_nat @ C3 @ D ) ) ) ) ).

% Un_mono
thf(fact_516_less__eq__set__def,axiom,
    ( ord_le182087997850975847od_b_b
    = ( ^ [A8: set_Product_prod_b_b,B7: set_Product_prod_b_b] :
          ( ord_le39139162152160566_b_b_o
          @ ^ [X: product_prod_b_b] : ( member7862447936710763792od_b_b @ X @ A8 )
          @ ^ [X: product_prod_b_b] : ( member7862447936710763792od_b_b @ X @ B7 ) ) ) ) ).

% less_eq_set_def
thf(fact_517_less__eq__set__def,axiom,
    ( ord_le2026170893118325445tion_b
    = ( ^ [A8: set_option_b,B7: set_option_b] :
          ( ord_le6102088707553712216on_b_o
          @ ^ [X: option_b] : ( member_option_b @ X @ A8 )
          @ ^ [X: option_b] : ( member_option_b @ X @ B7 ) ) ) ) ).

% less_eq_set_def
thf(fact_518_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A8: set_nat,B7: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X: nat] : ( member_nat @ X @ A8 )
          @ ^ [X: nat] : ( member_nat @ X @ B7 ) ) ) ) ).

% less_eq_set_def
thf(fact_519_less__eq__set__def,axiom,
    ( ord_less_eq_set_b
    = ( ^ [A8: set_b,B7: set_b] :
          ( ord_less_eq_b_o
          @ ^ [X: b] : ( member_b @ X @ A8 )
          @ ^ [X: b] : ( member_b @ X @ B7 ) ) ) ) ).

% less_eq_set_def
thf(fact_520_less__eq__set__def,axiom,
    ( ord_le6367397903258896871tion_b
    = ( ^ [A8: set_Pr258254617715012167tion_b,B7: set_Pr258254617715012167tion_b] :
          ( ord_le3144871161510034870on_b_o
          @ ^ [X: produc7781606619528532327tion_b] : ( member547026996284216464tion_b @ X @ A8 )
          @ ^ [X: produc7781606619528532327tion_b] : ( member547026996284216464tion_b @ X @ B7 ) ) ) ) ).

% less_eq_set_def
thf(fact_521_less__eq__set__def,axiom,
    ( ord_le5038853854344477844_b_nat
    = ( ^ [A8: set_Pr1307281990691478580_b_nat,B7: set_Pr1307281990691478580_b_nat] :
          ( ord_le6065810571798950129_nat_o
          @ ^ [X: product_prod_b_nat] : ( member6959632917342813205_b_nat @ X @ A8 )
          @ ^ [X: product_prod_b_nat] : ( member6959632917342813205_b_nat @ X @ B7 ) ) ) ) ).

% less_eq_set_def
thf(fact_522_fimage__mono,axiom,
    ! [A5: fset_P2289935846317411886_b_nat,B5: fset_P2289935846317411886_b_nat,F: product_prod_b_nat > product_prod_b_nat] :
      ( ( ord_le6837747134398578318_b_nat @ A5 @ B5 )
     => ( ord_le6837747134398578318_b_nat @ ( fimage6892398636147833211_b_nat @ F @ A5 ) @ ( fimage6892398636147833211_b_nat @ F @ B5 ) ) ) ).

% fimage_mono
thf(fact_523_fimage__mono,axiom,
    ! [A5: fset_P4398644116778547350_b_nat,B5: fset_P4398644116778547350_b_nat,F: produc1396820459975257318_b_nat > produc1396820459975257318_b_nat] :
      ( ( ord_le1476955799578339574_b_nat @ A5 @ B5 )
     => ( ord_le1476955799578339574_b_nat @ ( fimage1673450118646331979_b_nat @ F @ A5 ) @ ( fimage1673450118646331979_b_nat @ F @ B5 ) ) ) ).

% fimage_mono
thf(fact_524_fimage__mono,axiom,
    ! [A5: fset_P2289935846317411886_b_nat,B5: fset_P2289935846317411886_b_nat,F: product_prod_b_nat > produc1396820459975257318_b_nat] :
      ( ( ord_le6837747134398578318_b_nat @ A5 @ B5 )
     => ( ord_le1476955799578339574_b_nat @ ( fimage4266613729464776675_b_nat @ F @ A5 ) @ ( fimage4266613729464776675_b_nat @ F @ B5 ) ) ) ).

% fimage_mono
thf(fact_525_subset__fimage__iff,axiom,
    ! [B5: fset_P2289935846317411886_b_nat,F: product_prod_b_nat > product_prod_b_nat,A5: fset_P2289935846317411886_b_nat] :
      ( ( ord_le6837747134398578318_b_nat @ B5 @ ( fimage6892398636147833211_b_nat @ F @ A5 ) )
      = ( ? [AA: fset_P2289935846317411886_b_nat] :
            ( ( ord_le6837747134398578318_b_nat @ AA @ A5 )
            & ( B5
              = ( fimage6892398636147833211_b_nat @ F @ AA ) ) ) ) ) ).

% subset_fimage_iff
thf(fact_526_subset__fimage__iff,axiom,
    ! [B5: fset_P4398644116778547350_b_nat,F: produc1396820459975257318_b_nat > produc1396820459975257318_b_nat,A5: fset_P4398644116778547350_b_nat] :
      ( ( ord_le1476955799578339574_b_nat @ B5 @ ( fimage1673450118646331979_b_nat @ F @ A5 ) )
      = ( ? [AA: fset_P4398644116778547350_b_nat] :
            ( ( ord_le1476955799578339574_b_nat @ AA @ A5 )
            & ( B5
              = ( fimage1673450118646331979_b_nat @ F @ AA ) ) ) ) ) ).

% subset_fimage_iff
thf(fact_527_subset__fimage__iff,axiom,
    ! [B5: fset_P4398644116778547350_b_nat,F: product_prod_b_nat > produc1396820459975257318_b_nat,A5: fset_P2289935846317411886_b_nat] :
      ( ( ord_le1476955799578339574_b_nat @ B5 @ ( fimage4266613729464776675_b_nat @ F @ A5 ) )
      = ( ? [AA: fset_P2289935846317411886_b_nat] :
            ( ( ord_le6837747134398578318_b_nat @ AA @ A5 )
            & ( B5
              = ( fimage4266613729464776675_b_nat @ F @ AA ) ) ) ) ) ).

% subset_fimage_iff
thf(fact_528_pred__subset__eq,axiom,
    ! [R: set_Product_prod_b_b,S: set_Product_prod_b_b] :
      ( ( ord_le39139162152160566_b_b_o
        @ ^ [X: product_prod_b_b] : ( member7862447936710763792od_b_b @ X @ R )
        @ ^ [X: product_prod_b_b] : ( member7862447936710763792od_b_b @ X @ S ) )
      = ( ord_le182087997850975847od_b_b @ R @ S ) ) ).

% pred_subset_eq
thf(fact_529_pred__subset__eq,axiom,
    ! [R: set_option_b,S: set_option_b] :
      ( ( ord_le6102088707553712216on_b_o
        @ ^ [X: option_b] : ( member_option_b @ X @ R )
        @ ^ [X: option_b] : ( member_option_b @ X @ S ) )
      = ( ord_le2026170893118325445tion_b @ R @ S ) ) ).

% pred_subset_eq
thf(fact_530_pred__subset__eq,axiom,
    ! [R: set_nat,S: set_nat] :
      ( ( ord_less_eq_nat_o
        @ ^ [X: nat] : ( member_nat @ X @ R )
        @ ^ [X: nat] : ( member_nat @ X @ S ) )
      = ( ord_less_eq_set_nat @ R @ S ) ) ).

% pred_subset_eq
thf(fact_531_pred__subset__eq,axiom,
    ! [R: set_b,S: set_b] :
      ( ( ord_less_eq_b_o
        @ ^ [X: b] : ( member_b @ X @ R )
        @ ^ [X: b] : ( member_b @ X @ S ) )
      = ( ord_less_eq_set_b @ R @ S ) ) ).

% pred_subset_eq
thf(fact_532_pred__subset__eq,axiom,
    ! [R: set_Pr258254617715012167tion_b,S: set_Pr258254617715012167tion_b] :
      ( ( ord_le3144871161510034870on_b_o
        @ ^ [X: produc7781606619528532327tion_b] : ( member547026996284216464tion_b @ X @ R )
        @ ^ [X: produc7781606619528532327tion_b] : ( member547026996284216464tion_b @ X @ S ) )
      = ( ord_le6367397903258896871tion_b @ R @ S ) ) ).

% pred_subset_eq
thf(fact_533_pred__subset__eq,axiom,
    ! [R: set_Pr1307281990691478580_b_nat,S: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le6065810571798950129_nat_o
        @ ^ [X: product_prod_b_nat] : ( member6959632917342813205_b_nat @ X @ R )
        @ ^ [X: product_prod_b_nat] : ( member6959632917342813205_b_nat @ X @ S ) )
      = ( ord_le5038853854344477844_b_nat @ R @ S ) ) ).

% pred_subset_eq
thf(fact_534_Collect__case__prod__mono,axiom,
    ! [A5: ground_gterm_b > ground_gterm_b > $o,B5: ground_gterm_b > ground_gterm_b > $o] :
      ( ( ord_le4898483182178144398rm_b_o @ A5 @ B5 )
     => ( ord_le2408605988743369639term_b @ ( collec478437889140176530term_b @ ( produc1717101583779059052rm_b_o @ A5 ) ) @ ( collec478437889140176530term_b @ ( produc1717101583779059052rm_b_o @ B5 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_535_Collect__case__prod__mono,axiom,
    ! [A5: produc7781606619528532327tion_b > produc7781606619528532327tion_b > $o,B5: produc7781606619528532327tion_b > produc7781606619528532327tion_b > $o] :
      ( ( ord_le5770911127500734158on_b_o @ A5 @ B5 )
     => ( ord_le5090656177574270055tion_b @ ( collec447916338801455442tion_b @ ( produc8054777514177824044on_b_o @ A5 ) ) @ ( collec447916338801455442tion_b @ ( produc8054777514177824044on_b_o @ B5 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_536_Collect__case__prod__mono,axiom,
    ! [A5: produc7781606619528532327tion_b > nat > $o,B5: produc7781606619528532327tion_b > nat > $o] :
      ( ( ord_le6556145504227509917_nat_o @ A5 @ B5 )
     => ( ord_le3933197595962519292_b_nat @ ( collec4074094950137202619_b_nat @ ( produc5232674473337540683_nat_o @ A5 ) ) @ ( collec4074094950137202619_b_nat @ ( produc5232674473337540683_nat_o @ B5 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_537_Collect__case__prod__mono,axiom,
    ! [A5: option_b > option_b > $o,B5: option_b > option_b > $o] :
      ( ( ord_le3674354805995996366on_b_o @ A5 @ B5 )
     => ( ord_le6367397903258896871tion_b @ ( collec8407377912107443410tion_b @ ( produc3378454745690301612on_b_o @ A5 ) ) @ ( collec8407377912107443410tion_b @ ( produc3378454745690301612on_b_o @ B5 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_538_Collect__case__prod__mono,axiom,
    ! [A5: set_Pr258254617715012167tion_b > set_Pr258254617715012167tion_b > $o,B5: set_Pr258254617715012167tion_b > set_Pr258254617715012167tion_b > $o] :
      ( ( ord_le7070035895940199118on_b_o @ A5 @ B5 )
     => ( ord_le3057591301425134695tion_b @ ( collec1918043184897894738tion_b @ ( produc2671211398375001388on_b_o @ A5 ) ) @ ( collec1918043184897894738tion_b @ ( produc2671211398375001388on_b_o @ B5 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_539_Collect__case__prod__mono,axiom,
    ! [A5: set_option_b > set_option_b > $o,B5: set_option_b > set_option_b > $o] :
      ( ( ord_le6060424294837784782on_b_o @ A5 @ B5 )
     => ( ord_le4306526642958997991tion_b @ ( collec6622971782277219538tion_b @ ( produc7796829984862323884on_b_o @ A5 ) ) @ ( collec6622971782277219538tion_b @ ( produc7796829984862323884on_b_o @ B5 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_540_Collect__case__prod__mono,axiom,
    ! [A5: b > nat > $o,B5: b > nat > $o] :
      ( ( ord_less_eq_b_nat_o @ A5 @ B5 )
     => ( ord_le5038853854344477844_b_nat @ ( collec5699578864178162003_b_nat @ ( produc795641402153621683_nat_o @ A5 ) ) @ ( collec5699578864178162003_b_nat @ ( produc795641402153621683_nat_o @ B5 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_541_less__eq__fset_Orep__eq,axiom,
    ( ord_le8085106467073953901tion_b
    = ( ^ [X: fset_P9170875836021835981tion_b,Xa2: fset_P9170875836021835981tion_b] : ( ord_le6367397903258896871tion_b @ ( fset_P85497291108707286tion_b @ X ) @ ( fset_P85497291108707286tion_b @ Xa2 ) ) ) ) ).

% less_eq_fset.rep_eq
thf(fact_542_less__eq__fset_Orep__eq,axiom,
    ( ord_le6837747134398578318_b_nat
    = ( ^ [X: fset_P2289935846317411886_b_nat,Xa2: fset_P2289935846317411886_b_nat] : ( ord_le5038853854344477844_b_nat @ ( fset_P7858563055952297295_b_nat @ X ) @ ( fset_P7858563055952297295_b_nat @ Xa2 ) ) ) ) ).

% less_eq_fset.rep_eq
thf(fact_543_pred__subset__eq2,axiom,
    ! [R: set_Pr4178509052509721666tion_b,S: set_Pr4178509052509721666tion_b] :
      ( ( ord_le5222810953289671205on_b_o
        @ ^ [X: nat,Y4: produc7781606619528532327tion_b] : ( member3694883655523461923tion_b @ ( produc7510438065615382334tion_b @ X @ Y4 ) @ R )
        @ ^ [X: nat,Y4: produc7781606619528532327tion_b] : ( member3694883655523461923tion_b @ ( produc7510438065615382334tion_b @ X @ Y4 ) @ S ) )
      = ( ord_le1167626946193011874tion_b @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_544_pred__subset__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [X: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ R )
        @ ^ [X: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ S ) )
      = ( ord_le3146513528884898305at_nat @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_545_pred__subset__eq2,axiom,
    ! [R: set_Pr4264375888882495962_nat_b,S: set_Pr4264375888882495962_nat_b] :
      ( ( ord_less_eq_nat_b_o
        @ ^ [X: nat,Y4: b] : ( member8962352056413324475_nat_b @ ( product_Pair_nat_b @ X @ Y4 ) @ R )
        @ ^ [X: nat,Y4: b] : ( member8962352056413324475_nat_b @ ( product_Pair_nat_b @ X @ Y4 ) @ S ) )
      = ( ord_le7995947752535495226_nat_b @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_546_pred__subset__eq2,axiom,
    ! [R: set_Pr5422417554659304751tion_b,S: set_Pr5422417554659304751tion_b] :
      ( ( ord_le1601408845547135078on_b_o
        @ ^ [X: b,Y4: produc7781606619528532327tion_b] : ( member8570655873678976888tion_b @ ( produc5187340250978832319tion_b @ X @ Y4 ) @ R )
        @ ^ [X: b,Y4: produc7781606619528532327tion_b] : ( member8570655873678976888tion_b @ ( produc5187340250978832319tion_b @ X @ Y4 ) @ S ) )
      = ( ord_le7044419192447376591tion_b @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_547_pred__subset__eq2,axiom,
    ! [R: set_Product_prod_b_b,S: set_Product_prod_b_b] :
      ( ( ord_less_eq_b_b_o
        @ ^ [X: b,Y4: b] : ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y4 ) @ R )
        @ ^ [X: b,Y4: b] : ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y4 ) @ S ) )
      = ( ord_le182087997850975847od_b_b @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_548_pred__subset__eq2,axiom,
    ! [R: set_Pr6944079702279229084_b_nat,S: set_Pr6944079702279229084_b_nat] :
      ( ( ord_le6556145504227509917_nat_o
        @ ^ [X: produc7781606619528532327tion_b,Y4: nat] : ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ X @ Y4 ) @ R )
        @ ^ [X: produc7781606619528532327tion_b,Y4: nat] : ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ X @ Y4 ) @ S ) )
      = ( ord_le3933197595962519292_b_nat @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_549_pred__subset__eq2,axiom,
    ! [R: set_Pr258254617715012167tion_b,S: set_Pr258254617715012167tion_b] :
      ( ( ord_le3674354805995996366on_b_o
        @ ^ [X: option_b,Y4: option_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ Y4 ) @ R )
        @ ^ [X: option_b,Y4: option_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ Y4 ) @ S ) )
      = ( ord_le6367397903258896871tion_b @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_550_pred__subset__eq2,axiom,
    ! [R: set_Pr1307281990691478580_b_nat,S: set_Pr1307281990691478580_b_nat] :
      ( ( ord_less_eq_b_nat_o
        @ ^ [X: b,Y4: nat] : ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ X @ Y4 ) @ R )
        @ ^ [X: b,Y4: nat] : ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ X @ Y4 ) @ S ) )
      = ( ord_le5038853854344477844_b_nat @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_551_lift__root__step__mono,axiom,
    ! [F2: set_Pr1307281990691478580_b_nat,G2: set_Pr1307281990691478580_b_nat,W: fOR_pos_step,X6: fOR_ext_step,R3: set_Pr3134649703833893895term_b] :
      ( ( ord_le5038853854344477844_b_nat @ F2 @ G2 )
     => ( ord_le2408605988743369639term_b @ ( lift_R8854095231239425056step_b @ F2 @ W @ X6 @ R3 ) @ ( lift_R8854095231239425056step_b @ G2 @ W @ X6 @ R3 ) ) ) ).

% lift_root_step_mono
thf(fact_552_order__antisym__conv,axiom,
    ! [Y5: set_Pr258254617715012167tion_b,X4: set_Pr258254617715012167tion_b] :
      ( ( ord_le6367397903258896871tion_b @ Y5 @ X4 )
     => ( ( ord_le6367397903258896871tion_b @ X4 @ Y5 )
        = ( X4 = Y5 ) ) ) ).

% order_antisym_conv
thf(fact_553_order__antisym__conv,axiom,
    ! [Y5: set_Pr1307281990691478580_b_nat,X4: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le5038853854344477844_b_nat @ Y5 @ X4 )
     => ( ( ord_le5038853854344477844_b_nat @ X4 @ Y5 )
        = ( X4 = Y5 ) ) ) ).

% order_antisym_conv
thf(fact_554_order__antisym__conv,axiom,
    ! [Y5: produc7781606619528532327tion_b > nat > $o,X4: produc7781606619528532327tion_b > nat > $o] :
      ( ( ord_le6556145504227509917_nat_o @ Y5 @ X4 )
     => ( ( ord_le6556145504227509917_nat_o @ X4 @ Y5 )
        = ( X4 = Y5 ) ) ) ).

% order_antisym_conv
thf(fact_555_order__antisym__conv,axiom,
    ! [Y5: option_b > option_b > $o,X4: option_b > option_b > $o] :
      ( ( ord_le3674354805995996366on_b_o @ Y5 @ X4 )
     => ( ( ord_le3674354805995996366on_b_o @ X4 @ Y5 )
        = ( X4 = Y5 ) ) ) ).

% order_antisym_conv
thf(fact_556_order__antisym__conv,axiom,
    ! [Y5: b > nat > $o,X4: b > nat > $o] :
      ( ( ord_less_eq_b_nat_o @ Y5 @ X4 )
     => ( ( ord_less_eq_b_nat_o @ X4 @ Y5 )
        = ( X4 = Y5 ) ) ) ).

% order_antisym_conv
thf(fact_557_order__antisym__conv,axiom,
    ! [Y5: nat,X4: nat] :
      ( ( ord_less_eq_nat @ Y5 @ X4 )
     => ( ( ord_less_eq_nat @ X4 @ Y5 )
        = ( X4 = Y5 ) ) ) ).

% order_antisym_conv
thf(fact_558_linorder__le__cases,axiom,
    ! [X4: nat,Y5: nat] :
      ( ~ ( ord_less_eq_nat @ X4 @ Y5 )
     => ( ord_less_eq_nat @ Y5 @ X4 ) ) ).

% linorder_le_cases
thf(fact_559_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_560_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > set_Pr1307281990691478580_b_nat,C: set_Pr1307281990691478580_b_nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_le5038853854344477844_b_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le5038853854344477844_b_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_561_ord__le__eq__subst,axiom,
    ! [A: set_Pr1307281990691478580_b_nat,B: set_Pr1307281990691478580_b_nat,F: set_Pr1307281990691478580_b_nat > nat,C: nat] :
      ( ( ord_le5038853854344477844_b_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: set_Pr1307281990691478580_b_nat,Y3: set_Pr1307281990691478580_b_nat] :
              ( ( ord_le5038853854344477844_b_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_562_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > b > nat > $o,C: b > nat > $o] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_b_nat_o @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_b_nat_o @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_563_ord__le__eq__subst,axiom,
    ! [A: b > nat > $o,B: b > nat > $o,F: ( b > nat > $o ) > nat,C: nat] :
      ( ( ord_less_eq_b_nat_o @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: b > nat > $o,Y3: b > nat > $o] :
              ( ( ord_less_eq_b_nat_o @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_564_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > set_Pr258254617715012167tion_b,C: set_Pr258254617715012167tion_b] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_le6367397903258896871tion_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le6367397903258896871tion_b @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_565_ord__le__eq__subst,axiom,
    ! [A: set_Pr258254617715012167tion_b,B: set_Pr258254617715012167tion_b,F: set_Pr258254617715012167tion_b > nat,C: nat] :
      ( ( ord_le6367397903258896871tion_b @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: set_Pr258254617715012167tion_b,Y3: set_Pr258254617715012167tion_b] :
              ( ( ord_le6367397903258896871tion_b @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_566_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > option_b > option_b > $o,C: option_b > option_b > $o] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_le3674354805995996366on_b_o @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le3674354805995996366on_b_o @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_567_ord__le__eq__subst,axiom,
    ! [A: set_Pr1307281990691478580_b_nat,B: set_Pr1307281990691478580_b_nat,F: set_Pr1307281990691478580_b_nat > set_Pr1307281990691478580_b_nat,C: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le5038853854344477844_b_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: set_Pr1307281990691478580_b_nat,Y3: set_Pr1307281990691478580_b_nat] :
              ( ( ord_le5038853854344477844_b_nat @ X3 @ Y3 )
             => ( ord_le5038853854344477844_b_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le5038853854344477844_b_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_568_ord__le__eq__subst,axiom,
    ! [A: option_b > option_b > $o,B: option_b > option_b > $o,F: ( option_b > option_b > $o ) > nat,C: nat] :
      ( ( ord_le3674354805995996366on_b_o @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: option_b > option_b > $o,Y3: option_b > option_b > $o] :
              ( ( ord_le3674354805995996366on_b_o @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_569_ord__eq__le__subst,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_570_ord__eq__le__subst,axiom,
    ! [A: set_Pr1307281990691478580_b_nat,F: nat > set_Pr1307281990691478580_b_nat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_le5038853854344477844_b_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le5038853854344477844_b_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_571_ord__eq__le__subst,axiom,
    ! [A: nat,F: set_Pr1307281990691478580_b_nat > nat,B: set_Pr1307281990691478580_b_nat,C: set_Pr1307281990691478580_b_nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_le5038853854344477844_b_nat @ B @ C )
       => ( ! [X3: set_Pr1307281990691478580_b_nat,Y3: set_Pr1307281990691478580_b_nat] :
              ( ( ord_le5038853854344477844_b_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_572_ord__eq__le__subst,axiom,
    ! [A: b > nat > $o,F: nat > b > nat > $o,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_b_nat_o @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_b_nat_o @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_573_ord__eq__le__subst,axiom,
    ! [A: nat,F: ( b > nat > $o ) > nat,B: b > nat > $o,C: b > nat > $o] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_b_nat_o @ B @ C )
       => ( ! [X3: b > nat > $o,Y3: b > nat > $o] :
              ( ( ord_less_eq_b_nat_o @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_574_ord__eq__le__subst,axiom,
    ! [A: set_Pr258254617715012167tion_b,F: nat > set_Pr258254617715012167tion_b,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_le6367397903258896871tion_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le6367397903258896871tion_b @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_575_ord__eq__le__subst,axiom,
    ! [A: nat,F: set_Pr258254617715012167tion_b > nat,B: set_Pr258254617715012167tion_b,C: set_Pr258254617715012167tion_b] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_le6367397903258896871tion_b @ B @ C )
       => ( ! [X3: set_Pr258254617715012167tion_b,Y3: set_Pr258254617715012167tion_b] :
              ( ( ord_le6367397903258896871tion_b @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_576_ord__eq__le__subst,axiom,
    ! [A: option_b > option_b > $o,F: nat > option_b > option_b > $o,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_le3674354805995996366on_b_o @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le3674354805995996366on_b_o @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_577_ord__eq__le__subst,axiom,
    ! [A: set_Pr1307281990691478580_b_nat,F: set_Pr1307281990691478580_b_nat > set_Pr1307281990691478580_b_nat,B: set_Pr1307281990691478580_b_nat,C: set_Pr1307281990691478580_b_nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_le5038853854344477844_b_nat @ B @ C )
       => ( ! [X3: set_Pr1307281990691478580_b_nat,Y3: set_Pr1307281990691478580_b_nat] :
              ( ( ord_le5038853854344477844_b_nat @ X3 @ Y3 )
             => ( ord_le5038853854344477844_b_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le5038853854344477844_b_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_578_ord__eq__le__subst,axiom,
    ! [A: nat,F: ( option_b > option_b > $o ) > nat,B: option_b > option_b > $o,C: option_b > option_b > $o] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_le3674354805995996366on_b_o @ B @ C )
       => ( ! [X3: option_b > option_b > $o,Y3: option_b > option_b > $o] :
              ( ( ord_le3674354805995996366on_b_o @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_579_linorder__linear,axiom,
    ! [X4: nat,Y5: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y5 )
      | ( ord_less_eq_nat @ Y5 @ X4 ) ) ).

% linorder_linear
thf(fact_580_order__eq__refl,axiom,
    ! [X4: set_Pr258254617715012167tion_b,Y5: set_Pr258254617715012167tion_b] :
      ( ( X4 = Y5 )
     => ( ord_le6367397903258896871tion_b @ X4 @ Y5 ) ) ).

% order_eq_refl
thf(fact_581_order__eq__refl,axiom,
    ! [X4: set_Pr1307281990691478580_b_nat,Y5: set_Pr1307281990691478580_b_nat] :
      ( ( X4 = Y5 )
     => ( ord_le5038853854344477844_b_nat @ X4 @ Y5 ) ) ).

% order_eq_refl
thf(fact_582_order__eq__refl,axiom,
    ! [X4: produc7781606619528532327tion_b > nat > $o,Y5: produc7781606619528532327tion_b > nat > $o] :
      ( ( X4 = Y5 )
     => ( ord_le6556145504227509917_nat_o @ X4 @ Y5 ) ) ).

% order_eq_refl
thf(fact_583_order__eq__refl,axiom,
    ! [X4: option_b > option_b > $o,Y5: option_b > option_b > $o] :
      ( ( X4 = Y5 )
     => ( ord_le3674354805995996366on_b_o @ X4 @ Y5 ) ) ).

% order_eq_refl
thf(fact_584_order__eq__refl,axiom,
    ! [X4: b > nat > $o,Y5: b > nat > $o] :
      ( ( X4 = Y5 )
     => ( ord_less_eq_b_nat_o @ X4 @ Y5 ) ) ).

% order_eq_refl
thf(fact_585_order__eq__refl,axiom,
    ! [X4: nat,Y5: nat] :
      ( ( X4 = Y5 )
     => ( ord_less_eq_nat @ X4 @ Y5 ) ) ).

% order_eq_refl
thf(fact_586_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_587_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > set_Pr1307281990691478580_b_nat,C: set_Pr1307281990691478580_b_nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_le5038853854344477844_b_nat @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_le5038853854344477844_b_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le5038853854344477844_b_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_588_order__subst2,axiom,
    ! [A: set_Pr1307281990691478580_b_nat,B: set_Pr1307281990691478580_b_nat,F: set_Pr1307281990691478580_b_nat > nat,C: nat] :
      ( ( ord_le5038853854344477844_b_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X3: set_Pr1307281990691478580_b_nat,Y3: set_Pr1307281990691478580_b_nat] :
              ( ( ord_le5038853854344477844_b_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_589_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > b > nat > $o,C: b > nat > $o] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_b_nat_o @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_b_nat_o @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_b_nat_o @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_590_order__subst2,axiom,
    ! [A: b > nat > $o,B: b > nat > $o,F: ( b > nat > $o ) > nat,C: nat] :
      ( ( ord_less_eq_b_nat_o @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X3: b > nat > $o,Y3: b > nat > $o] :
              ( ( ord_less_eq_b_nat_o @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_591_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > set_Pr258254617715012167tion_b,C: set_Pr258254617715012167tion_b] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_le6367397903258896871tion_b @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_le6367397903258896871tion_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le6367397903258896871tion_b @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_592_order__subst2,axiom,
    ! [A: set_Pr258254617715012167tion_b,B: set_Pr258254617715012167tion_b,F: set_Pr258254617715012167tion_b > nat,C: nat] :
      ( ( ord_le6367397903258896871tion_b @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X3: set_Pr258254617715012167tion_b,Y3: set_Pr258254617715012167tion_b] :
              ( ( ord_le6367397903258896871tion_b @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_593_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > option_b > option_b > $o,C: option_b > option_b > $o] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_le3674354805995996366on_b_o @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_le3674354805995996366on_b_o @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le3674354805995996366on_b_o @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_594_order__subst2,axiom,
    ! [A: set_Pr1307281990691478580_b_nat,B: set_Pr1307281990691478580_b_nat,F: set_Pr1307281990691478580_b_nat > set_Pr1307281990691478580_b_nat,C: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le5038853854344477844_b_nat @ A @ B )
     => ( ( ord_le5038853854344477844_b_nat @ ( F @ B ) @ C )
       => ( ! [X3: set_Pr1307281990691478580_b_nat,Y3: set_Pr1307281990691478580_b_nat] :
              ( ( ord_le5038853854344477844_b_nat @ X3 @ Y3 )
             => ( ord_le5038853854344477844_b_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le5038853854344477844_b_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_595_order__subst2,axiom,
    ! [A: option_b > option_b > $o,B: option_b > option_b > $o,F: ( option_b > option_b > $o ) > nat,C: nat] :
      ( ( ord_le3674354805995996366on_b_o @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X3: option_b > option_b > $o,Y3: option_b > option_b > $o] :
              ( ( ord_le3674354805995996366on_b_o @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_596_order__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_597_order__subst1,axiom,
    ! [A: nat,F: set_Pr1307281990691478580_b_nat > nat,B: set_Pr1307281990691478580_b_nat,C: set_Pr1307281990691478580_b_nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_le5038853854344477844_b_nat @ B @ C )
       => ( ! [X3: set_Pr1307281990691478580_b_nat,Y3: set_Pr1307281990691478580_b_nat] :
              ( ( ord_le5038853854344477844_b_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_598_order__subst1,axiom,
    ! [A: set_Pr1307281990691478580_b_nat,F: nat > set_Pr1307281990691478580_b_nat,B: nat,C: nat] :
      ( ( ord_le5038853854344477844_b_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_le5038853854344477844_b_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le5038853854344477844_b_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_599_order__subst1,axiom,
    ! [A: nat,F: ( b > nat > $o ) > nat,B: b > nat > $o,C: b > nat > $o] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_b_nat_o @ B @ C )
       => ( ! [X3: b > nat > $o,Y3: b > nat > $o] :
              ( ( ord_less_eq_b_nat_o @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_600_order__subst1,axiom,
    ! [A: b > nat > $o,F: nat > b > nat > $o,B: nat,C: nat] :
      ( ( ord_less_eq_b_nat_o @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_b_nat_o @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_b_nat_o @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_601_order__subst1,axiom,
    ! [A: nat,F: set_Pr258254617715012167tion_b > nat,B: set_Pr258254617715012167tion_b,C: set_Pr258254617715012167tion_b] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_le6367397903258896871tion_b @ B @ C )
       => ( ! [X3: set_Pr258254617715012167tion_b,Y3: set_Pr258254617715012167tion_b] :
              ( ( ord_le6367397903258896871tion_b @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_602_order__subst1,axiom,
    ! [A: set_Pr258254617715012167tion_b,F: nat > set_Pr258254617715012167tion_b,B: nat,C: nat] :
      ( ( ord_le6367397903258896871tion_b @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_le6367397903258896871tion_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le6367397903258896871tion_b @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_603_order__subst1,axiom,
    ! [A: nat,F: ( option_b > option_b > $o ) > nat,B: option_b > option_b > $o,C: option_b > option_b > $o] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_le3674354805995996366on_b_o @ B @ C )
       => ( ! [X3: option_b > option_b > $o,Y3: option_b > option_b > $o] :
              ( ( ord_le3674354805995996366on_b_o @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_604_order__subst1,axiom,
    ! [A: set_Pr1307281990691478580_b_nat,F: set_Pr1307281990691478580_b_nat > set_Pr1307281990691478580_b_nat,B: set_Pr1307281990691478580_b_nat,C: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le5038853854344477844_b_nat @ A @ ( F @ B ) )
     => ( ( ord_le5038853854344477844_b_nat @ B @ C )
       => ( ! [X3: set_Pr1307281990691478580_b_nat,Y3: set_Pr1307281990691478580_b_nat] :
              ( ( ord_le5038853854344477844_b_nat @ X3 @ Y3 )
             => ( ord_le5038853854344477844_b_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le5038853854344477844_b_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_605_order__subst1,axiom,
    ! [A: option_b > option_b > $o,F: nat > option_b > option_b > $o,B: nat,C: nat] :
      ( ( ord_le3674354805995996366on_b_o @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_le3674354805995996366on_b_o @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le3674354805995996366on_b_o @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_606_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y8: set_Pr258254617715012167tion_b,Z3: set_Pr258254617715012167tion_b] : ( Y8 = Z3 ) )
    = ( ^ [A6: set_Pr258254617715012167tion_b,B4: set_Pr258254617715012167tion_b] :
          ( ( ord_le6367397903258896871tion_b @ A6 @ B4 )
          & ( ord_le6367397903258896871tion_b @ B4 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_607_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y8: set_Pr1307281990691478580_b_nat,Z3: set_Pr1307281990691478580_b_nat] : ( Y8 = Z3 ) )
    = ( ^ [A6: set_Pr1307281990691478580_b_nat,B4: set_Pr1307281990691478580_b_nat] :
          ( ( ord_le5038853854344477844_b_nat @ A6 @ B4 )
          & ( ord_le5038853854344477844_b_nat @ B4 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_608_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y8: produc7781606619528532327tion_b > nat > $o,Z3: produc7781606619528532327tion_b > nat > $o] : ( Y8 = Z3 ) )
    = ( ^ [A6: produc7781606619528532327tion_b > nat > $o,B4: produc7781606619528532327tion_b > nat > $o] :
          ( ( ord_le6556145504227509917_nat_o @ A6 @ B4 )
          & ( ord_le6556145504227509917_nat_o @ B4 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_609_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y8: option_b > option_b > $o,Z3: option_b > option_b > $o] : ( Y8 = Z3 ) )
    = ( ^ [A6: option_b > option_b > $o,B4: option_b > option_b > $o] :
          ( ( ord_le3674354805995996366on_b_o @ A6 @ B4 )
          & ( ord_le3674354805995996366on_b_o @ B4 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_610_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y8: b > nat > $o,Z3: b > nat > $o] : ( Y8 = Z3 ) )
    = ( ^ [A6: b > nat > $o,B4: b > nat > $o] :
          ( ( ord_less_eq_b_nat_o @ A6 @ B4 )
          & ( ord_less_eq_b_nat_o @ B4 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_611_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y8: nat,Z3: nat] : ( Y8 = Z3 ) )
    = ( ^ [A6: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A6 @ B4 )
          & ( ord_less_eq_nat @ B4 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_612_le__fun__def,axiom,
    ( ord_le6556145504227509917_nat_o
    = ( ^ [F3: produc7781606619528532327tion_b > nat > $o,G3: produc7781606619528532327tion_b > nat > $o] :
        ! [X: produc7781606619528532327tion_b] : ( ord_less_eq_nat_o @ ( F3 @ X ) @ ( G3 @ X ) ) ) ) ).

% le_fun_def
thf(fact_613_le__fun__def,axiom,
    ( ord_le3674354805995996366on_b_o
    = ( ^ [F3: option_b > option_b > $o,G3: option_b > option_b > $o] :
        ! [X: option_b] : ( ord_le6102088707553712216on_b_o @ ( F3 @ X ) @ ( G3 @ X ) ) ) ) ).

% le_fun_def
thf(fact_614_le__fun__def,axiom,
    ( ord_less_eq_b_nat_o
    = ( ^ [F3: b > nat > $o,G3: b > nat > $o] :
        ! [X: b] : ( ord_less_eq_nat_o @ ( F3 @ X ) @ ( G3 @ X ) ) ) ) ).

% le_fun_def
thf(fact_615_le__funI,axiom,
    ! [F: produc7781606619528532327tion_b > nat > $o,G: produc7781606619528532327tion_b > nat > $o] :
      ( ! [X3: produc7781606619528532327tion_b] : ( ord_less_eq_nat_o @ ( F @ X3 ) @ ( G @ X3 ) )
     => ( ord_le6556145504227509917_nat_o @ F @ G ) ) ).

% le_funI
thf(fact_616_le__funI,axiom,
    ! [F: option_b > option_b > $o,G: option_b > option_b > $o] :
      ( ! [X3: option_b] : ( ord_le6102088707553712216on_b_o @ ( F @ X3 ) @ ( G @ X3 ) )
     => ( ord_le3674354805995996366on_b_o @ F @ G ) ) ).

% le_funI
thf(fact_617_le__funI,axiom,
    ! [F: b > nat > $o,G: b > nat > $o] :
      ( ! [X3: b] : ( ord_less_eq_nat_o @ ( F @ X3 ) @ ( G @ X3 ) )
     => ( ord_less_eq_b_nat_o @ F @ G ) ) ).

% le_funI
thf(fact_618_le__funE,axiom,
    ! [F: produc7781606619528532327tion_b > nat > $o,G: produc7781606619528532327tion_b > nat > $o,X4: produc7781606619528532327tion_b] :
      ( ( ord_le6556145504227509917_nat_o @ F @ G )
     => ( ord_less_eq_nat_o @ ( F @ X4 ) @ ( G @ X4 ) ) ) ).

% le_funE
thf(fact_619_le__funE,axiom,
    ! [F: option_b > option_b > $o,G: option_b > option_b > $o,X4: option_b] :
      ( ( ord_le3674354805995996366on_b_o @ F @ G )
     => ( ord_le6102088707553712216on_b_o @ ( F @ X4 ) @ ( G @ X4 ) ) ) ).

% le_funE
thf(fact_620_le__funE,axiom,
    ! [F: b > nat > $o,G: b > nat > $o,X4: b] :
      ( ( ord_less_eq_b_nat_o @ F @ G )
     => ( ord_less_eq_nat_o @ ( F @ X4 ) @ ( G @ X4 ) ) ) ).

% le_funE
thf(fact_621_le__funD,axiom,
    ! [F: produc7781606619528532327tion_b > nat > $o,G: produc7781606619528532327tion_b > nat > $o,X4: produc7781606619528532327tion_b] :
      ( ( ord_le6556145504227509917_nat_o @ F @ G )
     => ( ord_less_eq_nat_o @ ( F @ X4 ) @ ( G @ X4 ) ) ) ).

% le_funD
thf(fact_622_le__funD,axiom,
    ! [F: option_b > option_b > $o,G: option_b > option_b > $o,X4: option_b] :
      ( ( ord_le3674354805995996366on_b_o @ F @ G )
     => ( ord_le6102088707553712216on_b_o @ ( F @ X4 ) @ ( G @ X4 ) ) ) ).

% le_funD
thf(fact_623_le__funD,axiom,
    ! [F: b > nat > $o,G: b > nat > $o,X4: b] :
      ( ( ord_less_eq_b_nat_o @ F @ G )
     => ( ord_less_eq_nat_o @ ( F @ X4 ) @ ( G @ X4 ) ) ) ).

% le_funD
thf(fact_624_antisym,axiom,
    ! [A: set_Pr258254617715012167tion_b,B: set_Pr258254617715012167tion_b] :
      ( ( ord_le6367397903258896871tion_b @ A @ B )
     => ( ( ord_le6367397903258896871tion_b @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_625_antisym,axiom,
    ! [A: set_Pr1307281990691478580_b_nat,B: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le5038853854344477844_b_nat @ A @ B )
     => ( ( ord_le5038853854344477844_b_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_626_antisym,axiom,
    ! [A: produc7781606619528532327tion_b > nat > $o,B: produc7781606619528532327tion_b > nat > $o] :
      ( ( ord_le6556145504227509917_nat_o @ A @ B )
     => ( ( ord_le6556145504227509917_nat_o @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_627_antisym,axiom,
    ! [A: option_b > option_b > $o,B: option_b > option_b > $o] :
      ( ( ord_le3674354805995996366on_b_o @ A @ B )
     => ( ( ord_le3674354805995996366on_b_o @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_628_antisym,axiom,
    ! [A: b > nat > $o,B: b > nat > $o] :
      ( ( ord_less_eq_b_nat_o @ A @ B )
     => ( ( ord_less_eq_b_nat_o @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_629_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_630_dual__order_Otrans,axiom,
    ! [B: set_Pr258254617715012167tion_b,A: set_Pr258254617715012167tion_b,C: set_Pr258254617715012167tion_b] :
      ( ( ord_le6367397903258896871tion_b @ B @ A )
     => ( ( ord_le6367397903258896871tion_b @ C @ B )
       => ( ord_le6367397903258896871tion_b @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_631_dual__order_Otrans,axiom,
    ! [B: set_Pr1307281990691478580_b_nat,A: set_Pr1307281990691478580_b_nat,C: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le5038853854344477844_b_nat @ B @ A )
     => ( ( ord_le5038853854344477844_b_nat @ C @ B )
       => ( ord_le5038853854344477844_b_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_632_dual__order_Otrans,axiom,
    ! [B: produc7781606619528532327tion_b > nat > $o,A: produc7781606619528532327tion_b > nat > $o,C: produc7781606619528532327tion_b > nat > $o] :
      ( ( ord_le6556145504227509917_nat_o @ B @ A )
     => ( ( ord_le6556145504227509917_nat_o @ C @ B )
       => ( ord_le6556145504227509917_nat_o @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_633_dual__order_Otrans,axiom,
    ! [B: option_b > option_b > $o,A: option_b > option_b > $o,C: option_b > option_b > $o] :
      ( ( ord_le3674354805995996366on_b_o @ B @ A )
     => ( ( ord_le3674354805995996366on_b_o @ C @ B )
       => ( ord_le3674354805995996366on_b_o @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_634_dual__order_Otrans,axiom,
    ! [B: b > nat > $o,A: b > nat > $o,C: b > nat > $o] :
      ( ( ord_less_eq_b_nat_o @ B @ A )
     => ( ( ord_less_eq_b_nat_o @ C @ B )
       => ( ord_less_eq_b_nat_o @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_635_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_636_dual__order_Oantisym,axiom,
    ! [B: set_Pr258254617715012167tion_b,A: set_Pr258254617715012167tion_b] :
      ( ( ord_le6367397903258896871tion_b @ B @ A )
     => ( ( ord_le6367397903258896871tion_b @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_637_dual__order_Oantisym,axiom,
    ! [B: set_Pr1307281990691478580_b_nat,A: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le5038853854344477844_b_nat @ B @ A )
     => ( ( ord_le5038853854344477844_b_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_638_dual__order_Oantisym,axiom,
    ! [B: produc7781606619528532327tion_b > nat > $o,A: produc7781606619528532327tion_b > nat > $o] :
      ( ( ord_le6556145504227509917_nat_o @ B @ A )
     => ( ( ord_le6556145504227509917_nat_o @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_639_dual__order_Oantisym,axiom,
    ! [B: option_b > option_b > $o,A: option_b > option_b > $o] :
      ( ( ord_le3674354805995996366on_b_o @ B @ A )
     => ( ( ord_le3674354805995996366on_b_o @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_640_dual__order_Oantisym,axiom,
    ! [B: b > nat > $o,A: b > nat > $o] :
      ( ( ord_less_eq_b_nat_o @ B @ A )
     => ( ( ord_less_eq_b_nat_o @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_641_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_642_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y8: set_Pr258254617715012167tion_b,Z3: set_Pr258254617715012167tion_b] : ( Y8 = Z3 ) )
    = ( ^ [A6: set_Pr258254617715012167tion_b,B4: set_Pr258254617715012167tion_b] :
          ( ( ord_le6367397903258896871tion_b @ B4 @ A6 )
          & ( ord_le6367397903258896871tion_b @ A6 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_643_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y8: set_Pr1307281990691478580_b_nat,Z3: set_Pr1307281990691478580_b_nat] : ( Y8 = Z3 ) )
    = ( ^ [A6: set_Pr1307281990691478580_b_nat,B4: set_Pr1307281990691478580_b_nat] :
          ( ( ord_le5038853854344477844_b_nat @ B4 @ A6 )
          & ( ord_le5038853854344477844_b_nat @ A6 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_644_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y8: produc7781606619528532327tion_b > nat > $o,Z3: produc7781606619528532327tion_b > nat > $o] : ( Y8 = Z3 ) )
    = ( ^ [A6: produc7781606619528532327tion_b > nat > $o,B4: produc7781606619528532327tion_b > nat > $o] :
          ( ( ord_le6556145504227509917_nat_o @ B4 @ A6 )
          & ( ord_le6556145504227509917_nat_o @ A6 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_645_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y8: option_b > option_b > $o,Z3: option_b > option_b > $o] : ( Y8 = Z3 ) )
    = ( ^ [A6: option_b > option_b > $o,B4: option_b > option_b > $o] :
          ( ( ord_le3674354805995996366on_b_o @ B4 @ A6 )
          & ( ord_le3674354805995996366on_b_o @ A6 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_646_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y8: b > nat > $o,Z3: b > nat > $o] : ( Y8 = Z3 ) )
    = ( ^ [A6: b > nat > $o,B4: b > nat > $o] :
          ( ( ord_less_eq_b_nat_o @ B4 @ A6 )
          & ( ord_less_eq_b_nat_o @ A6 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_647_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y8: nat,Z3: nat] : ( Y8 = Z3 ) )
    = ( ^ [A6: nat,B4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A6 )
          & ( ord_less_eq_nat @ A6 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_648_linorder__wlog,axiom,
    ! [P2: nat > nat > $o,A: nat,B: nat] :
      ( ! [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A4 @ B3 )
         => ( P2 @ A4 @ B3 ) )
     => ( ! [A4: nat,B3: nat] :
            ( ( P2 @ B3 @ A4 )
           => ( P2 @ A4 @ B3 ) )
       => ( P2 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_649_order__trans,axiom,
    ! [X4: set_Pr258254617715012167tion_b,Y5: set_Pr258254617715012167tion_b,Z: set_Pr258254617715012167tion_b] :
      ( ( ord_le6367397903258896871tion_b @ X4 @ Y5 )
     => ( ( ord_le6367397903258896871tion_b @ Y5 @ Z )
       => ( ord_le6367397903258896871tion_b @ X4 @ Z ) ) ) ).

% order_trans
thf(fact_650_order__trans,axiom,
    ! [X4: set_Pr1307281990691478580_b_nat,Y5: set_Pr1307281990691478580_b_nat,Z: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le5038853854344477844_b_nat @ X4 @ Y5 )
     => ( ( ord_le5038853854344477844_b_nat @ Y5 @ Z )
       => ( ord_le5038853854344477844_b_nat @ X4 @ Z ) ) ) ).

% order_trans
thf(fact_651_order__trans,axiom,
    ! [X4: produc7781606619528532327tion_b > nat > $o,Y5: produc7781606619528532327tion_b > nat > $o,Z: produc7781606619528532327tion_b > nat > $o] :
      ( ( ord_le6556145504227509917_nat_o @ X4 @ Y5 )
     => ( ( ord_le6556145504227509917_nat_o @ Y5 @ Z )
       => ( ord_le6556145504227509917_nat_o @ X4 @ Z ) ) ) ).

% order_trans
thf(fact_652_order__trans,axiom,
    ! [X4: option_b > option_b > $o,Y5: option_b > option_b > $o,Z: option_b > option_b > $o] :
      ( ( ord_le3674354805995996366on_b_o @ X4 @ Y5 )
     => ( ( ord_le3674354805995996366on_b_o @ Y5 @ Z )
       => ( ord_le3674354805995996366on_b_o @ X4 @ Z ) ) ) ).

% order_trans
thf(fact_653_order__trans,axiom,
    ! [X4: b > nat > $o,Y5: b > nat > $o,Z: b > nat > $o] :
      ( ( ord_less_eq_b_nat_o @ X4 @ Y5 )
     => ( ( ord_less_eq_b_nat_o @ Y5 @ Z )
       => ( ord_less_eq_b_nat_o @ X4 @ Z ) ) ) ).

% order_trans
thf(fact_654_order__trans,axiom,
    ! [X4: nat,Y5: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y5 )
     => ( ( ord_less_eq_nat @ Y5 @ Z )
       => ( ord_less_eq_nat @ X4 @ Z ) ) ) ).

% order_trans
thf(fact_655_order_Otrans,axiom,
    ! [A: set_Pr258254617715012167tion_b,B: set_Pr258254617715012167tion_b,C: set_Pr258254617715012167tion_b] :
      ( ( ord_le6367397903258896871tion_b @ A @ B )
     => ( ( ord_le6367397903258896871tion_b @ B @ C )
       => ( ord_le6367397903258896871tion_b @ A @ C ) ) ) ).

% order.trans
thf(fact_656_order_Otrans,axiom,
    ! [A: set_Pr1307281990691478580_b_nat,B: set_Pr1307281990691478580_b_nat,C: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le5038853854344477844_b_nat @ A @ B )
     => ( ( ord_le5038853854344477844_b_nat @ B @ C )
       => ( ord_le5038853854344477844_b_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_657_order_Otrans,axiom,
    ! [A: produc7781606619528532327tion_b > nat > $o,B: produc7781606619528532327tion_b > nat > $o,C: produc7781606619528532327tion_b > nat > $o] :
      ( ( ord_le6556145504227509917_nat_o @ A @ B )
     => ( ( ord_le6556145504227509917_nat_o @ B @ C )
       => ( ord_le6556145504227509917_nat_o @ A @ C ) ) ) ).

% order.trans
thf(fact_658_order_Otrans,axiom,
    ! [A: option_b > option_b > $o,B: option_b > option_b > $o,C: option_b > option_b > $o] :
      ( ( ord_le3674354805995996366on_b_o @ A @ B )
     => ( ( ord_le3674354805995996366on_b_o @ B @ C )
       => ( ord_le3674354805995996366on_b_o @ A @ C ) ) ) ).

% order.trans
thf(fact_659_order_Otrans,axiom,
    ! [A: b > nat > $o,B: b > nat > $o,C: b > nat > $o] :
      ( ( ord_less_eq_b_nat_o @ A @ B )
     => ( ( ord_less_eq_b_nat_o @ B @ C )
       => ( ord_less_eq_b_nat_o @ A @ C ) ) ) ).

% order.trans
thf(fact_660_order_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_661_order__antisym,axiom,
    ! [X4: set_Pr258254617715012167tion_b,Y5: set_Pr258254617715012167tion_b] :
      ( ( ord_le6367397903258896871tion_b @ X4 @ Y5 )
     => ( ( ord_le6367397903258896871tion_b @ Y5 @ X4 )
       => ( X4 = Y5 ) ) ) ).

% order_antisym
thf(fact_662_order__antisym,axiom,
    ! [X4: set_Pr1307281990691478580_b_nat,Y5: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le5038853854344477844_b_nat @ X4 @ Y5 )
     => ( ( ord_le5038853854344477844_b_nat @ Y5 @ X4 )
       => ( X4 = Y5 ) ) ) ).

% order_antisym
thf(fact_663_order__antisym,axiom,
    ! [X4: produc7781606619528532327tion_b > nat > $o,Y5: produc7781606619528532327tion_b > nat > $o] :
      ( ( ord_le6556145504227509917_nat_o @ X4 @ Y5 )
     => ( ( ord_le6556145504227509917_nat_o @ Y5 @ X4 )
       => ( X4 = Y5 ) ) ) ).

% order_antisym
thf(fact_664_order__antisym,axiom,
    ! [X4: option_b > option_b > $o,Y5: option_b > option_b > $o] :
      ( ( ord_le3674354805995996366on_b_o @ X4 @ Y5 )
     => ( ( ord_le3674354805995996366on_b_o @ Y5 @ X4 )
       => ( X4 = Y5 ) ) ) ).

% order_antisym
thf(fact_665_order__antisym,axiom,
    ! [X4: b > nat > $o,Y5: b > nat > $o] :
      ( ( ord_less_eq_b_nat_o @ X4 @ Y5 )
     => ( ( ord_less_eq_b_nat_o @ Y5 @ X4 )
       => ( X4 = Y5 ) ) ) ).

% order_antisym
thf(fact_666_order__antisym,axiom,
    ! [X4: nat,Y5: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y5 )
     => ( ( ord_less_eq_nat @ Y5 @ X4 )
       => ( X4 = Y5 ) ) ) ).

% order_antisym
thf(fact_667_ord__le__eq__trans,axiom,
    ! [A: set_Pr258254617715012167tion_b,B: set_Pr258254617715012167tion_b,C: set_Pr258254617715012167tion_b] :
      ( ( ord_le6367397903258896871tion_b @ A @ B )
     => ( ( B = C )
       => ( ord_le6367397903258896871tion_b @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_668_ord__le__eq__trans,axiom,
    ! [A: set_Pr1307281990691478580_b_nat,B: set_Pr1307281990691478580_b_nat,C: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le5038853854344477844_b_nat @ A @ B )
     => ( ( B = C )
       => ( ord_le5038853854344477844_b_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_669_ord__le__eq__trans,axiom,
    ! [A: produc7781606619528532327tion_b > nat > $o,B: produc7781606619528532327tion_b > nat > $o,C: produc7781606619528532327tion_b > nat > $o] :
      ( ( ord_le6556145504227509917_nat_o @ A @ B )
     => ( ( B = C )
       => ( ord_le6556145504227509917_nat_o @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_670_ord__le__eq__trans,axiom,
    ! [A: option_b > option_b > $o,B: option_b > option_b > $o,C: option_b > option_b > $o] :
      ( ( ord_le3674354805995996366on_b_o @ A @ B )
     => ( ( B = C )
       => ( ord_le3674354805995996366on_b_o @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_671_ord__le__eq__trans,axiom,
    ! [A: b > nat > $o,B: b > nat > $o,C: b > nat > $o] :
      ( ( ord_less_eq_b_nat_o @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_b_nat_o @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_672_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_673_ord__eq__le__trans,axiom,
    ! [A: set_Pr258254617715012167tion_b,B: set_Pr258254617715012167tion_b,C: set_Pr258254617715012167tion_b] :
      ( ( A = B )
     => ( ( ord_le6367397903258896871tion_b @ B @ C )
       => ( ord_le6367397903258896871tion_b @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_674_ord__eq__le__trans,axiom,
    ! [A: set_Pr1307281990691478580_b_nat,B: set_Pr1307281990691478580_b_nat,C: set_Pr1307281990691478580_b_nat] :
      ( ( A = B )
     => ( ( ord_le5038853854344477844_b_nat @ B @ C )
       => ( ord_le5038853854344477844_b_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_675_ord__eq__le__trans,axiom,
    ! [A: produc7781606619528532327tion_b > nat > $o,B: produc7781606619528532327tion_b > nat > $o,C: produc7781606619528532327tion_b > nat > $o] :
      ( ( A = B )
     => ( ( ord_le6556145504227509917_nat_o @ B @ C )
       => ( ord_le6556145504227509917_nat_o @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_676_ord__eq__le__trans,axiom,
    ! [A: option_b > option_b > $o,B: option_b > option_b > $o,C: option_b > option_b > $o] :
      ( ( A = B )
     => ( ( ord_le3674354805995996366on_b_o @ B @ C )
       => ( ord_le3674354805995996366on_b_o @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_677_ord__eq__le__trans,axiom,
    ! [A: b > nat > $o,B: b > nat > $o,C: b > nat > $o] :
      ( ( A = B )
     => ( ( ord_less_eq_b_nat_o @ B @ C )
       => ( ord_less_eq_b_nat_o @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_678_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_679_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y8: set_Pr258254617715012167tion_b,Z3: set_Pr258254617715012167tion_b] : ( Y8 = Z3 ) )
    = ( ^ [X: set_Pr258254617715012167tion_b,Y4: set_Pr258254617715012167tion_b] :
          ( ( ord_le6367397903258896871tion_b @ X @ Y4 )
          & ( ord_le6367397903258896871tion_b @ Y4 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_680_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y8: set_Pr1307281990691478580_b_nat,Z3: set_Pr1307281990691478580_b_nat] : ( Y8 = Z3 ) )
    = ( ^ [X: set_Pr1307281990691478580_b_nat,Y4: set_Pr1307281990691478580_b_nat] :
          ( ( ord_le5038853854344477844_b_nat @ X @ Y4 )
          & ( ord_le5038853854344477844_b_nat @ Y4 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_681_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y8: produc7781606619528532327tion_b > nat > $o,Z3: produc7781606619528532327tion_b > nat > $o] : ( Y8 = Z3 ) )
    = ( ^ [X: produc7781606619528532327tion_b > nat > $o,Y4: produc7781606619528532327tion_b > nat > $o] :
          ( ( ord_le6556145504227509917_nat_o @ X @ Y4 )
          & ( ord_le6556145504227509917_nat_o @ Y4 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_682_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y8: option_b > option_b > $o,Z3: option_b > option_b > $o] : ( Y8 = Z3 ) )
    = ( ^ [X: option_b > option_b > $o,Y4: option_b > option_b > $o] :
          ( ( ord_le3674354805995996366on_b_o @ X @ Y4 )
          & ( ord_le3674354805995996366on_b_o @ Y4 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_683_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y8: b > nat > $o,Z3: b > nat > $o] : ( Y8 = Z3 ) )
    = ( ^ [X: b > nat > $o,Y4: b > nat > $o] :
          ( ( ord_less_eq_b_nat_o @ X @ Y4 )
          & ( ord_less_eq_b_nat_o @ Y4 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_684_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y8: nat,Z3: nat] : ( Y8 = Z3 ) )
    = ( ^ [X: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X @ Y4 )
          & ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_685_le__cases3,axiom,
    ! [X4: nat,Y5: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X4 @ Y5 )
       => ~ ( ord_less_eq_nat @ Y5 @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y5 @ X4 )
         => ~ ( ord_less_eq_nat @ X4 @ Z ) )
       => ( ( ( ord_less_eq_nat @ X4 @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y5 ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y5 )
             => ~ ( ord_less_eq_nat @ Y5 @ X4 ) )
           => ( ( ( ord_less_eq_nat @ Y5 @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X4 ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X4 )
                 => ~ ( ord_less_eq_nat @ X4 @ Y5 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_686_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B ) )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_687_Collect__mono__iff,axiom,
    ! [P2: produc4514605509320855335term_b > $o,Q: produc4514605509320855335term_b > $o] :
      ( ( ord_le2408605988743369639term_b @ ( collec478437889140176530term_b @ P2 ) @ ( collec478437889140176530term_b @ Q ) )
      = ( ! [X: produc4514605509320855335term_b] :
            ( ( P2 @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_688_Collect__mono__iff,axiom,
    ! [P2: produc6650559869974650087tion_b > $o,Q: produc6650559869974650087tion_b > $o] :
      ( ( ord_le5090656177574270055tion_b @ ( collec447916338801455442tion_b @ P2 ) @ ( collec447916338801455442tion_b @ Q ) )
      = ( ! [X: produc6650559869974650087tion_b] :
            ( ( P2 @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_689_Collect__mono__iff,axiom,
    ! [P2: produc4298468441892121831tion_b > $o,Q: produc4298468441892121831tion_b > $o] :
      ( ( ord_le3057591301425134695tion_b @ ( collec1918043184897894738tion_b @ P2 ) @ ( collec1918043184897894738tion_b @ Q ) )
      = ( ! [X: produc4298468441892121831tion_b] :
            ( ( P2 @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_690_Collect__mono__iff,axiom,
    ! [P2: produc3122286842528565607tion_b > $o,Q: produc3122286842528565607tion_b > $o] :
      ( ( ord_le4306526642958997991tion_b @ ( collec6622971782277219538tion_b @ P2 ) @ ( collec6622971782277219538tion_b @ Q ) )
      = ( ! [X: produc3122286842528565607tion_b] :
            ( ( P2 @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_691_Collect__mono__iff,axiom,
    ! [P2: produc7781606619528532327tion_b > $o,Q: produc7781606619528532327tion_b > $o] :
      ( ( ord_le6367397903258896871tion_b @ ( collec8407377912107443410tion_b @ P2 ) @ ( collec8407377912107443410tion_b @ Q ) )
      = ( ! [X: produc7781606619528532327tion_b] :
            ( ( P2 @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_692_Collect__mono__iff,axiom,
    ! [P2: product_prod_b_nat > $o,Q: product_prod_b_nat > $o] :
      ( ( ord_le5038853854344477844_b_nat @ ( collec5699578864178162003_b_nat @ P2 ) @ ( collec5699578864178162003_b_nat @ Q ) )
      = ( ! [X: product_prod_b_nat] :
            ( ( P2 @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_693_set__eq__subset,axiom,
    ( ( ^ [Y8: set_Pr258254617715012167tion_b,Z3: set_Pr258254617715012167tion_b] : ( Y8 = Z3 ) )
    = ( ^ [A8: set_Pr258254617715012167tion_b,B7: set_Pr258254617715012167tion_b] :
          ( ( ord_le6367397903258896871tion_b @ A8 @ B7 )
          & ( ord_le6367397903258896871tion_b @ B7 @ A8 ) ) ) ) ).

% set_eq_subset
thf(fact_694_set__eq__subset,axiom,
    ( ( ^ [Y8: set_Pr1307281990691478580_b_nat,Z3: set_Pr1307281990691478580_b_nat] : ( Y8 = Z3 ) )
    = ( ^ [A8: set_Pr1307281990691478580_b_nat,B7: set_Pr1307281990691478580_b_nat] :
          ( ( ord_le5038853854344477844_b_nat @ A8 @ B7 )
          & ( ord_le5038853854344477844_b_nat @ B7 @ A8 ) ) ) ) ).

% set_eq_subset
thf(fact_695_subset__trans,axiom,
    ! [A5: set_Pr258254617715012167tion_b,B5: set_Pr258254617715012167tion_b,C3: set_Pr258254617715012167tion_b] :
      ( ( ord_le6367397903258896871tion_b @ A5 @ B5 )
     => ( ( ord_le6367397903258896871tion_b @ B5 @ C3 )
       => ( ord_le6367397903258896871tion_b @ A5 @ C3 ) ) ) ).

% subset_trans
thf(fact_696_subset__trans,axiom,
    ! [A5: set_Pr1307281990691478580_b_nat,B5: set_Pr1307281990691478580_b_nat,C3: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le5038853854344477844_b_nat @ A5 @ B5 )
     => ( ( ord_le5038853854344477844_b_nat @ B5 @ C3 )
       => ( ord_le5038853854344477844_b_nat @ A5 @ C3 ) ) ) ).

% subset_trans
thf(fact_697_Collect__mono,axiom,
    ! [P2: produc4514605509320855335term_b > $o,Q: produc4514605509320855335term_b > $o] :
      ( ! [X3: produc4514605509320855335term_b] :
          ( ( P2 @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le2408605988743369639term_b @ ( collec478437889140176530term_b @ P2 ) @ ( collec478437889140176530term_b @ Q ) ) ) ).

% Collect_mono
thf(fact_698_Collect__mono,axiom,
    ! [P2: produc6650559869974650087tion_b > $o,Q: produc6650559869974650087tion_b > $o] :
      ( ! [X3: produc6650559869974650087tion_b] :
          ( ( P2 @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le5090656177574270055tion_b @ ( collec447916338801455442tion_b @ P2 ) @ ( collec447916338801455442tion_b @ Q ) ) ) ).

% Collect_mono
thf(fact_699_Collect__mono,axiom,
    ! [P2: produc4298468441892121831tion_b > $o,Q: produc4298468441892121831tion_b > $o] :
      ( ! [X3: produc4298468441892121831tion_b] :
          ( ( P2 @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le3057591301425134695tion_b @ ( collec1918043184897894738tion_b @ P2 ) @ ( collec1918043184897894738tion_b @ Q ) ) ) ).

% Collect_mono
thf(fact_700_Collect__mono,axiom,
    ! [P2: produc3122286842528565607tion_b > $o,Q: produc3122286842528565607tion_b > $o] :
      ( ! [X3: produc3122286842528565607tion_b] :
          ( ( P2 @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le4306526642958997991tion_b @ ( collec6622971782277219538tion_b @ P2 ) @ ( collec6622971782277219538tion_b @ Q ) ) ) ).

% Collect_mono
thf(fact_701_Collect__mono,axiom,
    ! [P2: produc7781606619528532327tion_b > $o,Q: produc7781606619528532327tion_b > $o] :
      ( ! [X3: produc7781606619528532327tion_b] :
          ( ( P2 @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le6367397903258896871tion_b @ ( collec8407377912107443410tion_b @ P2 ) @ ( collec8407377912107443410tion_b @ Q ) ) ) ).

% Collect_mono
thf(fact_702_Collect__mono,axiom,
    ! [P2: product_prod_b_nat > $o,Q: product_prod_b_nat > $o] :
      ( ! [X3: product_prod_b_nat] :
          ( ( P2 @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le5038853854344477844_b_nat @ ( collec5699578864178162003_b_nat @ P2 ) @ ( collec5699578864178162003_b_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_703_subset__refl,axiom,
    ! [A5: set_Pr258254617715012167tion_b] : ( ord_le6367397903258896871tion_b @ A5 @ A5 ) ).

% subset_refl
thf(fact_704_subset__refl,axiom,
    ! [A5: set_Pr1307281990691478580_b_nat] : ( ord_le5038853854344477844_b_nat @ A5 @ A5 ) ).

% subset_refl
thf(fact_705_subset__iff,axiom,
    ( ord_le182087997850975847od_b_b
    = ( ^ [A8: set_Product_prod_b_b,B7: set_Product_prod_b_b] :
        ! [T2: product_prod_b_b] :
          ( ( member7862447936710763792od_b_b @ T2 @ A8 )
         => ( member7862447936710763792od_b_b @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_706_subset__iff,axiom,
    ( ord_le2026170893118325445tion_b
    = ( ^ [A8: set_option_b,B7: set_option_b] :
        ! [T2: option_b] :
          ( ( member_option_b @ T2 @ A8 )
         => ( member_option_b @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_707_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A8: set_nat,B7: set_nat] :
        ! [T2: nat] :
          ( ( member_nat @ T2 @ A8 )
         => ( member_nat @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_708_subset__iff,axiom,
    ( ord_less_eq_set_b
    = ( ^ [A8: set_b,B7: set_b] :
        ! [T2: b] :
          ( ( member_b @ T2 @ A8 )
         => ( member_b @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_709_subset__iff,axiom,
    ( ord_le6367397903258896871tion_b
    = ( ^ [A8: set_Pr258254617715012167tion_b,B7: set_Pr258254617715012167tion_b] :
        ! [T2: produc7781606619528532327tion_b] :
          ( ( member547026996284216464tion_b @ T2 @ A8 )
         => ( member547026996284216464tion_b @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_710_subset__iff,axiom,
    ( ord_le5038853854344477844_b_nat
    = ( ^ [A8: set_Pr1307281990691478580_b_nat,B7: set_Pr1307281990691478580_b_nat] :
        ! [T2: product_prod_b_nat] :
          ( ( member6959632917342813205_b_nat @ T2 @ A8 )
         => ( member6959632917342813205_b_nat @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_711_equalityD2,axiom,
    ! [A5: set_Pr258254617715012167tion_b,B5: set_Pr258254617715012167tion_b] :
      ( ( A5 = B5 )
     => ( ord_le6367397903258896871tion_b @ B5 @ A5 ) ) ).

% equalityD2
thf(fact_712_equalityD2,axiom,
    ! [A5: set_Pr1307281990691478580_b_nat,B5: set_Pr1307281990691478580_b_nat] :
      ( ( A5 = B5 )
     => ( ord_le5038853854344477844_b_nat @ B5 @ A5 ) ) ).

% equalityD2
thf(fact_713_equalityD1,axiom,
    ! [A5: set_Pr258254617715012167tion_b,B5: set_Pr258254617715012167tion_b] :
      ( ( A5 = B5 )
     => ( ord_le6367397903258896871tion_b @ A5 @ B5 ) ) ).

% equalityD1
thf(fact_714_equalityD1,axiom,
    ! [A5: set_Pr1307281990691478580_b_nat,B5: set_Pr1307281990691478580_b_nat] :
      ( ( A5 = B5 )
     => ( ord_le5038853854344477844_b_nat @ A5 @ B5 ) ) ).

% equalityD1
thf(fact_715_subset__eq,axiom,
    ( ord_le182087997850975847od_b_b
    = ( ^ [A8: set_Product_prod_b_b,B7: set_Product_prod_b_b] :
        ! [X: product_prod_b_b] :
          ( ( member7862447936710763792od_b_b @ X @ A8 )
         => ( member7862447936710763792od_b_b @ X @ B7 ) ) ) ) ).

% subset_eq
thf(fact_716_subset__eq,axiom,
    ( ord_le2026170893118325445tion_b
    = ( ^ [A8: set_option_b,B7: set_option_b] :
        ! [X: option_b] :
          ( ( member_option_b @ X @ A8 )
         => ( member_option_b @ X @ B7 ) ) ) ) ).

% subset_eq
thf(fact_717_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A8: set_nat,B7: set_nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ A8 )
         => ( member_nat @ X @ B7 ) ) ) ) ).

% subset_eq
thf(fact_718_subset__eq,axiom,
    ( ord_less_eq_set_b
    = ( ^ [A8: set_b,B7: set_b] :
        ! [X: b] :
          ( ( member_b @ X @ A8 )
         => ( member_b @ X @ B7 ) ) ) ) ).

% subset_eq
thf(fact_719_subset__eq,axiom,
    ( ord_le6367397903258896871tion_b
    = ( ^ [A8: set_Pr258254617715012167tion_b,B7: set_Pr258254617715012167tion_b] :
        ! [X: produc7781606619528532327tion_b] :
          ( ( member547026996284216464tion_b @ X @ A8 )
         => ( member547026996284216464tion_b @ X @ B7 ) ) ) ) ).

% subset_eq
thf(fact_720_subset__eq,axiom,
    ( ord_le5038853854344477844_b_nat
    = ( ^ [A8: set_Pr1307281990691478580_b_nat,B7: set_Pr1307281990691478580_b_nat] :
        ! [X: product_prod_b_nat] :
          ( ( member6959632917342813205_b_nat @ X @ A8 )
         => ( member6959632917342813205_b_nat @ X @ B7 ) ) ) ) ).

% subset_eq
thf(fact_721_equalityE,axiom,
    ! [A5: set_Pr258254617715012167tion_b,B5: set_Pr258254617715012167tion_b] :
      ( ( A5 = B5 )
     => ~ ( ( ord_le6367397903258896871tion_b @ A5 @ B5 )
         => ~ ( ord_le6367397903258896871tion_b @ B5 @ A5 ) ) ) ).

% equalityE
thf(fact_722_equalityE,axiom,
    ! [A5: set_Pr1307281990691478580_b_nat,B5: set_Pr1307281990691478580_b_nat] :
      ( ( A5 = B5 )
     => ~ ( ( ord_le5038853854344477844_b_nat @ A5 @ B5 )
         => ~ ( ord_le5038853854344477844_b_nat @ B5 @ A5 ) ) ) ).

% equalityE
thf(fact_723_subsetD,axiom,
    ! [A5: set_Product_prod_b_b,B5: set_Product_prod_b_b,C: product_prod_b_b] :
      ( ( ord_le182087997850975847od_b_b @ A5 @ B5 )
     => ( ( member7862447936710763792od_b_b @ C @ A5 )
       => ( member7862447936710763792od_b_b @ C @ B5 ) ) ) ).

% subsetD
thf(fact_724_subsetD,axiom,
    ! [A5: set_option_b,B5: set_option_b,C: option_b] :
      ( ( ord_le2026170893118325445tion_b @ A5 @ B5 )
     => ( ( member_option_b @ C @ A5 )
       => ( member_option_b @ C @ B5 ) ) ) ).

% subsetD
thf(fact_725_subsetD,axiom,
    ! [A5: set_nat,B5: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A5 @ B5 )
     => ( ( member_nat @ C @ A5 )
       => ( member_nat @ C @ B5 ) ) ) ).

% subsetD
thf(fact_726_subsetD,axiom,
    ! [A5: set_b,B5: set_b,C: b] :
      ( ( ord_less_eq_set_b @ A5 @ B5 )
     => ( ( member_b @ C @ A5 )
       => ( member_b @ C @ B5 ) ) ) ).

% subsetD
thf(fact_727_subsetD,axiom,
    ! [A5: set_Pr258254617715012167tion_b,B5: set_Pr258254617715012167tion_b,C: produc7781606619528532327tion_b] :
      ( ( ord_le6367397903258896871tion_b @ A5 @ B5 )
     => ( ( member547026996284216464tion_b @ C @ A5 )
       => ( member547026996284216464tion_b @ C @ B5 ) ) ) ).

% subsetD
thf(fact_728_subsetD,axiom,
    ! [A5: set_Pr1307281990691478580_b_nat,B5: set_Pr1307281990691478580_b_nat,C: product_prod_b_nat] :
      ( ( ord_le5038853854344477844_b_nat @ A5 @ B5 )
     => ( ( member6959632917342813205_b_nat @ C @ A5 )
       => ( member6959632917342813205_b_nat @ C @ B5 ) ) ) ).

% subsetD
thf(fact_729_in__mono,axiom,
    ! [A5: set_Product_prod_b_b,B5: set_Product_prod_b_b,X4: product_prod_b_b] :
      ( ( ord_le182087997850975847od_b_b @ A5 @ B5 )
     => ( ( member7862447936710763792od_b_b @ X4 @ A5 )
       => ( member7862447936710763792od_b_b @ X4 @ B5 ) ) ) ).

% in_mono
thf(fact_730_in__mono,axiom,
    ! [A5: set_option_b,B5: set_option_b,X4: option_b] :
      ( ( ord_le2026170893118325445tion_b @ A5 @ B5 )
     => ( ( member_option_b @ X4 @ A5 )
       => ( member_option_b @ X4 @ B5 ) ) ) ).

% in_mono
thf(fact_731_in__mono,axiom,
    ! [A5: set_nat,B5: set_nat,X4: nat] :
      ( ( ord_less_eq_set_nat @ A5 @ B5 )
     => ( ( member_nat @ X4 @ A5 )
       => ( member_nat @ X4 @ B5 ) ) ) ).

% in_mono
thf(fact_732_in__mono,axiom,
    ! [A5: set_b,B5: set_b,X4: b] :
      ( ( ord_less_eq_set_b @ A5 @ B5 )
     => ( ( member_b @ X4 @ A5 )
       => ( member_b @ X4 @ B5 ) ) ) ).

% in_mono
thf(fact_733_in__mono,axiom,
    ! [A5: set_Pr258254617715012167tion_b,B5: set_Pr258254617715012167tion_b,X4: produc7781606619528532327tion_b] :
      ( ( ord_le6367397903258896871tion_b @ A5 @ B5 )
     => ( ( member547026996284216464tion_b @ X4 @ A5 )
       => ( member547026996284216464tion_b @ X4 @ B5 ) ) ) ).

% in_mono
thf(fact_734_in__mono,axiom,
    ! [A5: set_Pr1307281990691478580_b_nat,B5: set_Pr1307281990691478580_b_nat,X4: product_prod_b_nat] :
      ( ( ord_le5038853854344477844_b_nat @ A5 @ B5 )
     => ( ( member6959632917342813205_b_nat @ X4 @ A5 )
       => ( member6959632917342813205_b_nat @ X4 @ B5 ) ) ) ).

% in_mono
thf(fact_735_sup_Obounded__iff,axiom,
    ! [B: set_Pr258254617715012167tion_b,C: set_Pr258254617715012167tion_b,A: set_Pr258254617715012167tion_b] :
      ( ( ord_le6367397903258896871tion_b @ ( sup_su3110286242275913755tion_b @ B @ C ) @ A )
      = ( ( ord_le6367397903258896871tion_b @ B @ A )
        & ( ord_le6367397903258896871tion_b @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_736_sup_Obounded__iff,axiom,
    ! [B: set_Pr1307281990691478580_b_nat,C: set_Pr1307281990691478580_b_nat,A: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le5038853854344477844_b_nat @ ( sup_su6056130500584125792_b_nat @ B @ C ) @ A )
      = ( ( ord_le5038853854344477844_b_nat @ B @ A )
        & ( ord_le5038853854344477844_b_nat @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_737_sup_Obounded__iff,axiom,
    ! [B: produc7781606619528532327tion_b > nat > $o,C: produc7781606619528532327tion_b > nat > $o,A: produc7781606619528532327tion_b > nat > $o] :
      ( ( ord_le6556145504227509917_nat_o @ ( sup_su6759296390479959913_nat_o @ B @ C ) @ A )
      = ( ( ord_le6556145504227509917_nat_o @ B @ A )
        & ( ord_le6556145504227509917_nat_o @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_738_sup_Obounded__iff,axiom,
    ! [B: option_b > option_b > $o,C: option_b > option_b > $o,A: option_b > option_b > $o] :
      ( ( ord_le3674354805995996366on_b_o @ ( sup_su6610355334417212162on_b_o @ B @ C ) @ A )
      = ( ( ord_le3674354805995996366on_b_o @ B @ A )
        & ( ord_le3674354805995996366on_b_o @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_739_sup_Obounded__iff,axiom,
    ! [B: b > nat > $o,C: b > nat > $o,A: b > nat > $o] :
      ( ( ord_less_eq_b_nat_o @ ( sup_sup_b_nat_o @ B @ C ) @ A )
      = ( ( ord_less_eq_b_nat_o @ B @ A )
        & ( ord_less_eq_b_nat_o @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_740_sup_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( ord_less_eq_nat @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_741_le__sup__iff,axiom,
    ! [X4: set_Pr258254617715012167tion_b,Y5: set_Pr258254617715012167tion_b,Z: set_Pr258254617715012167tion_b] :
      ( ( ord_le6367397903258896871tion_b @ ( sup_su3110286242275913755tion_b @ X4 @ Y5 ) @ Z )
      = ( ( ord_le6367397903258896871tion_b @ X4 @ Z )
        & ( ord_le6367397903258896871tion_b @ Y5 @ Z ) ) ) ).

% le_sup_iff
thf(fact_742_le__sup__iff,axiom,
    ! [X4: set_Pr1307281990691478580_b_nat,Y5: set_Pr1307281990691478580_b_nat,Z: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le5038853854344477844_b_nat @ ( sup_su6056130500584125792_b_nat @ X4 @ Y5 ) @ Z )
      = ( ( ord_le5038853854344477844_b_nat @ X4 @ Z )
        & ( ord_le5038853854344477844_b_nat @ Y5 @ Z ) ) ) ).

% le_sup_iff
thf(fact_743_le__sup__iff,axiom,
    ! [X4: produc7781606619528532327tion_b > nat > $o,Y5: produc7781606619528532327tion_b > nat > $o,Z: produc7781606619528532327tion_b > nat > $o] :
      ( ( ord_le6556145504227509917_nat_o @ ( sup_su6759296390479959913_nat_o @ X4 @ Y5 ) @ Z )
      = ( ( ord_le6556145504227509917_nat_o @ X4 @ Z )
        & ( ord_le6556145504227509917_nat_o @ Y5 @ Z ) ) ) ).

% le_sup_iff
thf(fact_744_le__sup__iff,axiom,
    ! [X4: option_b > option_b > $o,Y5: option_b > option_b > $o,Z: option_b > option_b > $o] :
      ( ( ord_le3674354805995996366on_b_o @ ( sup_su6610355334417212162on_b_o @ X4 @ Y5 ) @ Z )
      = ( ( ord_le3674354805995996366on_b_o @ X4 @ Z )
        & ( ord_le3674354805995996366on_b_o @ Y5 @ Z ) ) ) ).

% le_sup_iff
thf(fact_745_le__sup__iff,axiom,
    ! [X4: b > nat > $o,Y5: b > nat > $o,Z: b > nat > $o] :
      ( ( ord_less_eq_b_nat_o @ ( sup_sup_b_nat_o @ X4 @ Y5 ) @ Z )
      = ( ( ord_less_eq_b_nat_o @ X4 @ Z )
        & ( ord_less_eq_b_nat_o @ Y5 @ Z ) ) ) ).

% le_sup_iff
thf(fact_746_le__sup__iff,axiom,
    ! [X4: nat,Y5: nat,Z: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ X4 @ Y5 ) @ Z )
      = ( ( ord_less_eq_nat @ X4 @ Z )
        & ( ord_less_eq_nat @ Y5 @ Z ) ) ) ).

% le_sup_iff
thf(fact_747_sup__apply,axiom,
    ( sup_su6759296390479959913_nat_o
    = ( ^ [F3: produc7781606619528532327tion_b > nat > $o,G3: produc7781606619528532327tion_b > nat > $o,X: produc7781606619528532327tion_b] : ( sup_sup_nat_o @ ( F3 @ X ) @ ( G3 @ X ) ) ) ) ).

% sup_apply
thf(fact_748_sup__apply,axiom,
    ( sup_su6610355334417212162on_b_o
    = ( ^ [F3: option_b > option_b > $o,G3: option_b > option_b > $o,X: option_b] : ( sup_sup_option_b_o @ ( F3 @ X ) @ ( G3 @ X ) ) ) ) ).

% sup_apply
thf(fact_749_sup__apply,axiom,
    ( sup_sup_b_nat_o
    = ( ^ [F3: b > nat > $o,G3: b > nat > $o,X: b] : ( sup_sup_nat_o @ ( F3 @ X ) @ ( G3 @ X ) ) ) ) ).

% sup_apply
thf(fact_750_sup_Oright__idem,axiom,
    ! [A: set_Pr1307281990691478580_b_nat,B: set_Pr1307281990691478580_b_nat] :
      ( ( sup_su6056130500584125792_b_nat @ ( sup_su6056130500584125792_b_nat @ A @ B ) @ B )
      = ( sup_su6056130500584125792_b_nat @ A @ B ) ) ).

% sup.right_idem
thf(fact_751_sup_Oright__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( sup_sup_nat @ ( sup_sup_nat @ A @ B ) @ B )
      = ( sup_sup_nat @ A @ B ) ) ).

% sup.right_idem
thf(fact_752_sup_Oright__idem,axiom,
    ! [A: produc7781606619528532327tion_b > nat > $o,B: produc7781606619528532327tion_b > nat > $o] :
      ( ( sup_su6759296390479959913_nat_o @ ( sup_su6759296390479959913_nat_o @ A @ B ) @ B )
      = ( sup_su6759296390479959913_nat_o @ A @ B ) ) ).

% sup.right_idem
thf(fact_753_sup_Oright__idem,axiom,
    ! [A: option_b > option_b > $o,B: option_b > option_b > $o] :
      ( ( sup_su6610355334417212162on_b_o @ ( sup_su6610355334417212162on_b_o @ A @ B ) @ B )
      = ( sup_su6610355334417212162on_b_o @ A @ B ) ) ).

% sup.right_idem
thf(fact_754_sup_Oright__idem,axiom,
    ! [A: b > nat > $o,B: b > nat > $o] :
      ( ( sup_sup_b_nat_o @ ( sup_sup_b_nat_o @ A @ B ) @ B )
      = ( sup_sup_b_nat_o @ A @ B ) ) ).

% sup.right_idem
thf(fact_755_sup__left__idem,axiom,
    ! [X4: set_Pr1307281990691478580_b_nat,Y5: set_Pr1307281990691478580_b_nat] :
      ( ( sup_su6056130500584125792_b_nat @ X4 @ ( sup_su6056130500584125792_b_nat @ X4 @ Y5 ) )
      = ( sup_su6056130500584125792_b_nat @ X4 @ Y5 ) ) ).

% sup_left_idem
thf(fact_756_sup__left__idem,axiom,
    ! [X4: nat,Y5: nat] :
      ( ( sup_sup_nat @ X4 @ ( sup_sup_nat @ X4 @ Y5 ) )
      = ( sup_sup_nat @ X4 @ Y5 ) ) ).

% sup_left_idem
thf(fact_757_sup__left__idem,axiom,
    ! [X4: produc7781606619528532327tion_b > nat > $o,Y5: produc7781606619528532327tion_b > nat > $o] :
      ( ( sup_su6759296390479959913_nat_o @ X4 @ ( sup_su6759296390479959913_nat_o @ X4 @ Y5 ) )
      = ( sup_su6759296390479959913_nat_o @ X4 @ Y5 ) ) ).

% sup_left_idem
thf(fact_758_sup__left__idem,axiom,
    ! [X4: option_b > option_b > $o,Y5: option_b > option_b > $o] :
      ( ( sup_su6610355334417212162on_b_o @ X4 @ ( sup_su6610355334417212162on_b_o @ X4 @ Y5 ) )
      = ( sup_su6610355334417212162on_b_o @ X4 @ Y5 ) ) ).

% sup_left_idem
thf(fact_759_sup__left__idem,axiom,
    ! [X4: b > nat > $o,Y5: b > nat > $o] :
      ( ( sup_sup_b_nat_o @ X4 @ ( sup_sup_b_nat_o @ X4 @ Y5 ) )
      = ( sup_sup_b_nat_o @ X4 @ Y5 ) ) ).

% sup_left_idem
thf(fact_760_sup_Oleft__idem,axiom,
    ! [A: set_Pr1307281990691478580_b_nat,B: set_Pr1307281990691478580_b_nat] :
      ( ( sup_su6056130500584125792_b_nat @ A @ ( sup_su6056130500584125792_b_nat @ A @ B ) )
      = ( sup_su6056130500584125792_b_nat @ A @ B ) ) ).

% sup.left_idem
thf(fact_761_sup_Oleft__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( sup_sup_nat @ A @ ( sup_sup_nat @ A @ B ) )
      = ( sup_sup_nat @ A @ B ) ) ).

% sup.left_idem
thf(fact_762_sup_Oleft__idem,axiom,
    ! [A: produc7781606619528532327tion_b > nat > $o,B: produc7781606619528532327tion_b > nat > $o] :
      ( ( sup_su6759296390479959913_nat_o @ A @ ( sup_su6759296390479959913_nat_o @ A @ B ) )
      = ( sup_su6759296390479959913_nat_o @ A @ B ) ) ).

% sup.left_idem
thf(fact_763_sup_Oleft__idem,axiom,
    ! [A: option_b > option_b > $o,B: option_b > option_b > $o] :
      ( ( sup_su6610355334417212162on_b_o @ A @ ( sup_su6610355334417212162on_b_o @ A @ B ) )
      = ( sup_su6610355334417212162on_b_o @ A @ B ) ) ).

% sup.left_idem
thf(fact_764_sup_Oleft__idem,axiom,
    ! [A: b > nat > $o,B: b > nat > $o] :
      ( ( sup_sup_b_nat_o @ A @ ( sup_sup_b_nat_o @ A @ B ) )
      = ( sup_sup_b_nat_o @ A @ B ) ) ).

% sup.left_idem
thf(fact_765_sup__idem,axiom,
    ! [X4: set_Pr1307281990691478580_b_nat] :
      ( ( sup_su6056130500584125792_b_nat @ X4 @ X4 )
      = X4 ) ).

% sup_idem
thf(fact_766_sup__idem,axiom,
    ! [X4: nat] :
      ( ( sup_sup_nat @ X4 @ X4 )
      = X4 ) ).

% sup_idem
thf(fact_767_sup__idem,axiom,
    ! [X4: produc7781606619528532327tion_b > nat > $o] :
      ( ( sup_su6759296390479959913_nat_o @ X4 @ X4 )
      = X4 ) ).

% sup_idem
thf(fact_768_sup__idem,axiom,
    ! [X4: option_b > option_b > $o] :
      ( ( sup_su6610355334417212162on_b_o @ X4 @ X4 )
      = X4 ) ).

% sup_idem
thf(fact_769_sup__idem,axiom,
    ! [X4: b > nat > $o] :
      ( ( sup_sup_b_nat_o @ X4 @ X4 )
      = X4 ) ).

% sup_idem
thf(fact_770_sup_Oidem,axiom,
    ! [A: set_Pr1307281990691478580_b_nat] :
      ( ( sup_su6056130500584125792_b_nat @ A @ A )
      = A ) ).

% sup.idem
thf(fact_771_sup_Oidem,axiom,
    ! [A: nat] :
      ( ( sup_sup_nat @ A @ A )
      = A ) ).

% sup.idem
thf(fact_772_sup_Oidem,axiom,
    ! [A: produc7781606619528532327tion_b > nat > $o] :
      ( ( sup_su6759296390479959913_nat_o @ A @ A )
      = A ) ).

% sup.idem
thf(fact_773_sup_Oidem,axiom,
    ! [A: option_b > option_b > $o] :
      ( ( sup_su6610355334417212162on_b_o @ A @ A )
      = A ) ).

% sup.idem
thf(fact_774_sup_Oidem,axiom,
    ! [A: b > nat > $o] :
      ( ( sup_sup_b_nat_o @ A @ A )
      = A ) ).

% sup.idem
thf(fact_775_sup_OcoboundedI2,axiom,
    ! [C: set_Pr258254617715012167tion_b,B: set_Pr258254617715012167tion_b,A: set_Pr258254617715012167tion_b] :
      ( ( ord_le6367397903258896871tion_b @ C @ B )
     => ( ord_le6367397903258896871tion_b @ C @ ( sup_su3110286242275913755tion_b @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_776_sup_OcoboundedI2,axiom,
    ! [C: set_Pr1307281990691478580_b_nat,B: set_Pr1307281990691478580_b_nat,A: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le5038853854344477844_b_nat @ C @ B )
     => ( ord_le5038853854344477844_b_nat @ C @ ( sup_su6056130500584125792_b_nat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_777_sup_OcoboundedI2,axiom,
    ! [C: produc7781606619528532327tion_b > nat > $o,B: produc7781606619528532327tion_b > nat > $o,A: produc7781606619528532327tion_b > nat > $o] :
      ( ( ord_le6556145504227509917_nat_o @ C @ B )
     => ( ord_le6556145504227509917_nat_o @ C @ ( sup_su6759296390479959913_nat_o @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_778_sup_OcoboundedI2,axiom,
    ! [C: option_b > option_b > $o,B: option_b > option_b > $o,A: option_b > option_b > $o] :
      ( ( ord_le3674354805995996366on_b_o @ C @ B )
     => ( ord_le3674354805995996366on_b_o @ C @ ( sup_su6610355334417212162on_b_o @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_779_sup_OcoboundedI2,axiom,
    ! [C: b > nat > $o,B: b > nat > $o,A: b > nat > $o] :
      ( ( ord_less_eq_b_nat_o @ C @ B )
     => ( ord_less_eq_b_nat_o @ C @ ( sup_sup_b_nat_o @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_780_sup_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ B )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_781_sup_OcoboundedI1,axiom,
    ! [C: set_Pr258254617715012167tion_b,A: set_Pr258254617715012167tion_b,B: set_Pr258254617715012167tion_b] :
      ( ( ord_le6367397903258896871tion_b @ C @ A )
     => ( ord_le6367397903258896871tion_b @ C @ ( sup_su3110286242275913755tion_b @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_782_sup_OcoboundedI1,axiom,
    ! [C: set_Pr1307281990691478580_b_nat,A: set_Pr1307281990691478580_b_nat,B: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le5038853854344477844_b_nat @ C @ A )
     => ( ord_le5038853854344477844_b_nat @ C @ ( sup_su6056130500584125792_b_nat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_783_sup_OcoboundedI1,axiom,
    ! [C: produc7781606619528532327tion_b > nat > $o,A: produc7781606619528532327tion_b > nat > $o,B: produc7781606619528532327tion_b > nat > $o] :
      ( ( ord_le6556145504227509917_nat_o @ C @ A )
     => ( ord_le6556145504227509917_nat_o @ C @ ( sup_su6759296390479959913_nat_o @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_784_sup_OcoboundedI1,axiom,
    ! [C: option_b > option_b > $o,A: option_b > option_b > $o,B: option_b > option_b > $o] :
      ( ( ord_le3674354805995996366on_b_o @ C @ A )
     => ( ord_le3674354805995996366on_b_o @ C @ ( sup_su6610355334417212162on_b_o @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_785_sup_OcoboundedI1,axiom,
    ! [C: b > nat > $o,A: b > nat > $o,B: b > nat > $o] :
      ( ( ord_less_eq_b_nat_o @ C @ A )
     => ( ord_less_eq_b_nat_o @ C @ ( sup_sup_b_nat_o @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_786_sup_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_787_predicate2I,axiom,
    ! [P2: produc7781606619528532327tion_b > nat > $o,Q: produc7781606619528532327tion_b > nat > $o] :
      ( ! [X3: produc7781606619528532327tion_b,Y3: nat] :
          ( ( P2 @ X3 @ Y3 )
         => ( Q @ X3 @ Y3 ) )
     => ( ord_le6556145504227509917_nat_o @ P2 @ Q ) ) ).

% predicate2I
thf(fact_788_predicate2I,axiom,
    ! [P2: option_b > option_b > $o,Q: option_b > option_b > $o] :
      ( ! [X3: option_b,Y3: option_b] :
          ( ( P2 @ X3 @ Y3 )
         => ( Q @ X3 @ Y3 ) )
     => ( ord_le3674354805995996366on_b_o @ P2 @ Q ) ) ).

% predicate2I
thf(fact_789_predicate2I,axiom,
    ! [P2: b > nat > $o,Q: b > nat > $o] :
      ( ! [X3: b,Y3: nat] :
          ( ( P2 @ X3 @ Y3 )
         => ( Q @ X3 @ Y3 ) )
     => ( ord_less_eq_b_nat_o @ P2 @ Q ) ) ).

% predicate2I
thf(fact_790_union__fset,axiom,
    ! [X4: fset_P2289935846317411886_b_nat,Xa: fset_P2289935846317411886_b_nat] :
      ( ( fset_P7858563055952297295_b_nat @ ( sup_su4437341427378704730_b_nat @ X4 @ Xa ) )
      = ( sup_su6056130500584125792_b_nat @ ( fset_P7858563055952297295_b_nat @ X4 ) @ ( fset_P7858563055952297295_b_nat @ Xa ) ) ) ).

% union_fset
thf(fact_791_predicate2D,axiom,
    ! [P2: produc7781606619528532327tion_b > nat > $o,Q: produc7781606619528532327tion_b > nat > $o,X4: produc7781606619528532327tion_b,Y5: nat] :
      ( ( ord_le6556145504227509917_nat_o @ P2 @ Q )
     => ( ( P2 @ X4 @ Y5 )
       => ( Q @ X4 @ Y5 ) ) ) ).

% predicate2D
thf(fact_792_predicate2D,axiom,
    ! [P2: option_b > option_b > $o,Q: option_b > option_b > $o,X4: option_b,Y5: option_b] :
      ( ( ord_le3674354805995996366on_b_o @ P2 @ Q )
     => ( ( P2 @ X4 @ Y5 )
       => ( Q @ X4 @ Y5 ) ) ) ).

% predicate2D
thf(fact_793_predicate2D,axiom,
    ! [P2: b > nat > $o,Q: b > nat > $o,X4: b,Y5: nat] :
      ( ( ord_less_eq_b_nat_o @ P2 @ Q )
     => ( ( P2 @ X4 @ Y5 )
       => ( Q @ X4 @ Y5 ) ) ) ).

% predicate2D
thf(fact_794_rev__predicate2D,axiom,
    ! [P2: produc7781606619528532327tion_b > nat > $o,X4: produc7781606619528532327tion_b,Y5: nat,Q: produc7781606619528532327tion_b > nat > $o] :
      ( ( P2 @ X4 @ Y5 )
     => ( ( ord_le6556145504227509917_nat_o @ P2 @ Q )
       => ( Q @ X4 @ Y5 ) ) ) ).

% rev_predicate2D
thf(fact_795_rev__predicate2D,axiom,
    ! [P2: option_b > option_b > $o,X4: option_b,Y5: option_b,Q: option_b > option_b > $o] :
      ( ( P2 @ X4 @ Y5 )
     => ( ( ord_le3674354805995996366on_b_o @ P2 @ Q )
       => ( Q @ X4 @ Y5 ) ) ) ).

% rev_predicate2D
thf(fact_796_rev__predicate2D,axiom,
    ! [P2: b > nat > $o,X4: b,Y5: nat,Q: b > nat > $o] :
      ( ( P2 @ X4 @ Y5 )
     => ( ( ord_less_eq_b_nat_o @ P2 @ Q )
       => ( Q @ X4 @ Y5 ) ) ) ).

% rev_predicate2D
thf(fact_797_predicate2D__conj,axiom,
    ! [P2: produc7781606619528532327tion_b > nat > $o,Q: produc7781606619528532327tion_b > nat > $o,R: $o,X4: produc7781606619528532327tion_b,Y5: nat] :
      ( ( ( ord_le6556145504227509917_nat_o @ P2 @ Q )
        & R )
     => ( R
        & ( ( P2 @ X4 @ Y5 )
         => ( Q @ X4 @ Y5 ) ) ) ) ).

% predicate2D_conj
thf(fact_798_predicate2D__conj,axiom,
    ! [P2: option_b > option_b > $o,Q: option_b > option_b > $o,R: $o,X4: option_b,Y5: option_b] :
      ( ( ( ord_le3674354805995996366on_b_o @ P2 @ Q )
        & R )
     => ( R
        & ( ( P2 @ X4 @ Y5 )
         => ( Q @ X4 @ Y5 ) ) ) ) ).

% predicate2D_conj
thf(fact_799_predicate2D__conj,axiom,
    ! [P2: b > nat > $o,Q: b > nat > $o,R: $o,X4: b,Y5: nat] :
      ( ( ( ord_less_eq_b_nat_o @ P2 @ Q )
        & R )
     => ( R
        & ( ( P2 @ X4 @ Y5 )
         => ( Q @ X4 @ Y5 ) ) ) ) ).

% predicate2D_conj
thf(fact_800_eq__subset,axiom,
    ! [P2: option_b > option_b > $o] :
      ( ord_le3674354805995996366on_b_o
      @ ^ [Y8: option_b,Z3: option_b] : ( Y8 = Z3 )
      @ ^ [A6: option_b,B4: option_b] :
          ( ( P2 @ A6 @ B4 )
          | ( A6 = B4 ) ) ) ).

% eq_subset
thf(fact_801_fimage__funion,axiom,
    ! [F: product_prod_b_nat > product_prod_b_nat,A5: fset_P2289935846317411886_b_nat,B5: fset_P2289935846317411886_b_nat] :
      ( ( fimage6892398636147833211_b_nat @ F @ ( sup_su4437341427378704730_b_nat @ A5 @ B5 ) )
      = ( sup_su4437341427378704730_b_nat @ ( fimage6892398636147833211_b_nat @ F @ A5 ) @ ( fimage6892398636147833211_b_nat @ F @ B5 ) ) ) ).

% fimage_funion
thf(fact_802_fimage__funion,axiom,
    ! [F: produc1396820459975257318_b_nat > produc1396820459975257318_b_nat,A5: fset_P4398644116778547350_b_nat,B5: fset_P4398644116778547350_b_nat] :
      ( ( fimage1673450118646331979_b_nat @ F @ ( sup_su6170363988329532354_b_nat @ A5 @ B5 ) )
      = ( sup_su6170363988329532354_b_nat @ ( fimage1673450118646331979_b_nat @ F @ A5 ) @ ( fimage1673450118646331979_b_nat @ F @ B5 ) ) ) ).

% fimage_funion
thf(fact_803_fimage__funion,axiom,
    ! [F: product_prod_b_nat > produc1396820459975257318_b_nat,A5: fset_P2289935846317411886_b_nat,B5: fset_P2289935846317411886_b_nat] :
      ( ( fimage4266613729464776675_b_nat @ F @ ( sup_su4437341427378704730_b_nat @ A5 @ B5 ) )
      = ( sup_su6170363988329532354_b_nat @ ( fimage4266613729464776675_b_nat @ F @ A5 ) @ ( fimage4266613729464776675_b_nat @ F @ B5 ) ) ) ).

% fimage_funion
thf(fact_804_sup__Un__eq,axiom,
    ! [R: set_Product_prod_b_b,S: set_Product_prod_b_b] :
      ( ( sup_su88143223436887042_b_b_o
        @ ^ [X: product_prod_b_b] : ( member7862447936710763792od_b_b @ X @ R )
        @ ^ [X: product_prod_b_b] : ( member7862447936710763792od_b_b @ X @ S ) )
      = ( ^ [X: product_prod_b_b] : ( member7862447936710763792od_b_b @ X @ ( sup_su2483643821041016987od_b_b @ R @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_805_sup__Un__eq,axiom,
    ! [R: set_option_b,S: set_option_b] :
      ( ( sup_sup_option_b_o
        @ ^ [X: option_b] : ( member_option_b @ X @ R )
        @ ^ [X: option_b] : ( member_option_b @ X @ S ) )
      = ( ^ [X: option_b] : ( member_option_b @ X @ ( sup_sup_set_option_b @ R @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_806_sup__Un__eq,axiom,
    ! [R: set_nat,S: set_nat] :
      ( ( sup_sup_nat_o
        @ ^ [X: nat] : ( member_nat @ X @ R )
        @ ^ [X: nat] : ( member_nat @ X @ S ) )
      = ( ^ [X: nat] : ( member_nat @ X @ ( sup_sup_set_nat @ R @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_807_sup__Un__eq,axiom,
    ! [R: set_b,S: set_b] :
      ( ( sup_sup_b_o
        @ ^ [X: b] : ( member_b @ X @ R )
        @ ^ [X: b] : ( member_b @ X @ S ) )
      = ( ^ [X: b] : ( member_b @ X @ ( sup_sup_set_b @ R @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_808_sup__Un__eq,axiom,
    ! [R: set_Pr1307281990691478580_b_nat,S: set_Pr1307281990691478580_b_nat] :
      ( ( sup_su346563364047904549_nat_o
        @ ^ [X: product_prod_b_nat] : ( member6959632917342813205_b_nat @ X @ R )
        @ ^ [X: product_prod_b_nat] : ( member6959632917342813205_b_nat @ X @ S ) )
      = ( ^ [X: product_prod_b_nat] : ( member6959632917342813205_b_nat @ X @ ( sup_su6056130500584125792_b_nat @ R @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_809_sup__set__def,axiom,
    ( sup_su2483643821041016987od_b_b
    = ( ^ [A8: set_Product_prod_b_b,B7: set_Product_prod_b_b] :
          ( collec548942219715005266od_b_b
          @ ( sup_su88143223436887042_b_b_o
            @ ^ [X: product_prod_b_b] : ( member7862447936710763792od_b_b @ X @ A8 )
            @ ^ [X: product_prod_b_b] : ( member7862447936710763792od_b_b @ X @ B7 ) ) ) ) ) ).

% sup_set_def
thf(fact_810_sup__set__def,axiom,
    ( sup_sup_set_option_b
    = ( ^ [A8: set_option_b,B7: set_option_b] :
          ( collect_option_b
          @ ( sup_sup_option_b_o
            @ ^ [X: option_b] : ( member_option_b @ X @ A8 )
            @ ^ [X: option_b] : ( member_option_b @ X @ B7 ) ) ) ) ) ).

% sup_set_def
thf(fact_811_sup__set__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A8: set_nat,B7: set_nat] :
          ( collect_nat
          @ ( sup_sup_nat_o
            @ ^ [X: nat] : ( member_nat @ X @ A8 )
            @ ^ [X: nat] : ( member_nat @ X @ B7 ) ) ) ) ) ).

% sup_set_def
thf(fact_812_sup__set__def,axiom,
    ( sup_sup_set_b
    = ( ^ [A8: set_b,B7: set_b] :
          ( collect_b
          @ ( sup_sup_b_o
            @ ^ [X: b] : ( member_b @ X @ A8 )
            @ ^ [X: b] : ( member_b @ X @ B7 ) ) ) ) ) ).

% sup_set_def
thf(fact_813_sup__set__def,axiom,
    ( sup_su1163039302778497499term_b
    = ( ^ [A8: set_Pr3134649703833893895term_b,B7: set_Pr3134649703833893895term_b] :
          ( collec478437889140176530term_b
          @ ( sup_su4689331601617090242rm_b_o
            @ ^ [X: produc4514605509320855335term_b] : ( member7340517243570908752term_b @ X @ A8 )
            @ ^ [X: produc4514605509320855335term_b] : ( member7340517243570908752term_b @ X @ B7 ) ) ) ) ) ).

% sup_set_def
thf(fact_814_sup__set__def,axiom,
    ( sup_su2925575103159965339tion_b
    = ( ^ [A8: set_Pr5366877986981747911tion_b,B7: set_Pr5366877986981747911tion_b] :
          ( collec447916338801455442tion_b
          @ ( sup_su8546542134409276930on_b_o
            @ ^ [X: produc6650559869974650087tion_b] : ( member6435960643780680464tion_b @ X @ A8 )
            @ ^ [X: produc6650559869974650087tion_b] : ( member6435960643780680464tion_b @ X @ B7 ) ) ) ) ) ).

% sup_set_def
thf(fact_815_sup__set__def,axiom,
    ( sup_su3110286242275913755tion_b
    = ( ^ [A8: set_Pr258254617715012167tion_b,B7: set_Pr258254617715012167tion_b] :
          ( collec8407377912107443410tion_b
          @ ( sup_su1564436729930002562on_b_o
            @ ^ [X: produc7781606619528532327tion_b] : ( member547026996284216464tion_b @ X @ A8 )
            @ ^ [X: produc7781606619528532327tion_b] : ( member547026996284216464tion_b @ X @ B7 ) ) ) ) ) ).

% sup_set_def
thf(fact_816_sup__set__def,axiom,
    ( sup_su3944740340205657755tion_b
    = ( ^ [A8: set_Pr2609077471296144583tion_b,B7: set_Pr2609077471296144583tion_b] :
          ( collec1918043184897894738tion_b
          @ ( sup_su3443213708705613314on_b_o
            @ ^ [X: produc4298468441892121831tion_b] : ( member8319496676406272784tion_b @ X @ A8 )
            @ ^ [X: produc4298468441892121831tion_b] : ( member8319496676406272784tion_b @ X @ B7 ) ) ) ) ) ).

% sup_set_def
thf(fact_817_sup__set__def,axiom,
    ( sup_su8287916483737133083tion_b
    = ( ^ [A8: set_Pr4148925336519852615tion_b,B7: set_Pr4148925336519852615tion_b] :
          ( collec6622971782277219538tion_b
          @ ( sup_su6183354604956295298on_b_o
            @ ^ [X: produc3122286842528565607tion_b] : ( member9052098846246995088tion_b @ X @ A8 )
            @ ^ [X: produc3122286842528565607tion_b] : ( member9052098846246995088tion_b @ X @ B7 ) ) ) ) ) ).

% sup_set_def
thf(fact_818_sup__set__def,axiom,
    ( sup_su6056130500584125792_b_nat
    = ( ^ [A8: set_Pr1307281990691478580_b_nat,B7: set_Pr1307281990691478580_b_nat] :
          ( collec5699578864178162003_b_nat
          @ ( sup_su346563364047904549_nat_o
            @ ^ [X: product_prod_b_nat] : ( member6959632917342813205_b_nat @ X @ A8 )
            @ ^ [X: product_prod_b_nat] : ( member6959632917342813205_b_nat @ X @ B7 ) ) ) ) ) ).

% sup_set_def
thf(fact_819_sup__Un__eq2,axiom,
    ! [R: set_Pr4178509052509721666tion_b,S: set_Pr4178509052509721666tion_b] :
      ( ( sup_su5425961839542121201on_b_o
        @ ^ [X: nat,Y4: produc7781606619528532327tion_b] : ( member3694883655523461923tion_b @ ( produc7510438065615382334tion_b @ X @ Y4 ) @ R )
        @ ^ [X: nat,Y4: produc7781606619528532327tion_b] : ( member3694883655523461923tion_b @ ( produc7510438065615382334tion_b @ X @ Y4 ) @ S ) )
      = ( ^ [X: nat,Y4: produc7781606619528532327tion_b] : ( member3694883655523461923tion_b @ ( produc7510438065615382334tion_b @ X @ Y4 ) @ ( sup_su7392266752968523118tion_b @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_820_sup__Un__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( sup_sup_nat_nat_o
        @ ^ [X: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ R )
        @ ^ [X: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ S ) )
      = ( ^ [X: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ ( sup_su6327502436637775413at_nat @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_821_sup__Un__eq2,axiom,
    ! [R: set_Pr4264375888882495962_nat_b,S: set_Pr4264375888882495962_nat_b] :
      ( ( sup_sup_nat_b_o
        @ ^ [X: nat,Y4: b] : ( member8962352056413324475_nat_b @ ( product_Pair_nat_b @ X @ Y4 ) @ R )
        @ ^ [X: nat,Y4: b] : ( member8962352056413324475_nat_b @ ( product_Pair_nat_b @ X @ Y4 ) @ S ) )
      = ( ^ [X: nat,Y4: b] : ( member8962352056413324475_nat_b @ ( product_Pair_nat_b @ X @ Y4 ) @ ( sup_su9013224398775143174_nat_b @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_822_sup__Un__eq2,axiom,
    ! [R: set_Pr5422417554659304751tion_b,S: set_Pr5422417554659304751tion_b] :
      ( ( sup_su5268273525949000346on_b_o
        @ ^ [X: b,Y4: produc7781606619528532327tion_b] : ( member8570655873678976888tion_b @ ( produc5187340250978832319tion_b @ X @ Y4 ) @ R )
        @ ^ [X: b,Y4: produc7781606619528532327tion_b] : ( member8570655873678976888tion_b @ ( produc5187340250978832319tion_b @ X @ Y4 ) @ S ) )
      = ( ^ [X: b,Y4: produc7781606619528532327tion_b] : ( member8570655873678976888tion_b @ ( produc5187340250978832319tion_b @ X @ Y4 ) @ ( sup_su3446553804406812419tion_b @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_823_sup__Un__eq2,axiom,
    ! [R: set_Product_prod_b_b,S: set_Product_prod_b_b] :
      ( ( sup_sup_b_b_o
        @ ^ [X: b,Y4: b] : ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y4 ) @ R )
        @ ^ [X: b,Y4: b] : ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y4 ) @ S ) )
      = ( ^ [X: b,Y4: b] : ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y4 ) @ ( sup_su2483643821041016987od_b_b @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_824_sup__Un__eq2,axiom,
    ! [R: set_Pr6944079702279229084_b_nat,S: set_Pr6944079702279229084_b_nat] :
      ( ( sup_su6759296390479959913_nat_o
        @ ^ [X: produc7781606619528532327tion_b,Y4: nat] : ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ X @ Y4 ) @ R )
        @ ^ [X: produc7781606619528532327tion_b,Y4: nat] : ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ X @ Y4 ) @ S ) )
      = ( ^ [X: produc7781606619528532327tion_b,Y4: nat] : ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ X @ Y4 ) @ ( sup_su934465365883254728_b_nat @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_825_sup__Un__eq2,axiom,
    ! [R: set_Pr258254617715012167tion_b,S: set_Pr258254617715012167tion_b] :
      ( ( sup_su6610355334417212162on_b_o
        @ ^ [X: option_b,Y4: option_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ Y4 ) @ R )
        @ ^ [X: option_b,Y4: option_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ Y4 ) @ S ) )
      = ( ^ [X: option_b,Y4: option_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ Y4 ) @ ( sup_su3110286242275913755tion_b @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_826_sup__Un__eq2,axiom,
    ! [R: set_Pr1307281990691478580_b_nat,S: set_Pr1307281990691478580_b_nat] :
      ( ( sup_sup_b_nat_o
        @ ^ [X: b,Y4: nat] : ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ X @ Y4 ) @ R )
        @ ^ [X: b,Y4: nat] : ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ X @ Y4 ) @ S ) )
      = ( ^ [X: b,Y4: nat] : ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ X @ Y4 ) @ ( sup_su6056130500584125792_b_nat @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_827_inf__sup__aci_I8_J,axiom,
    ! [X4: set_Pr1307281990691478580_b_nat,Y5: set_Pr1307281990691478580_b_nat] :
      ( ( sup_su6056130500584125792_b_nat @ X4 @ ( sup_su6056130500584125792_b_nat @ X4 @ Y5 ) )
      = ( sup_su6056130500584125792_b_nat @ X4 @ Y5 ) ) ).

% inf_sup_aci(8)
thf(fact_828_inf__sup__aci_I8_J,axiom,
    ! [X4: nat,Y5: nat] :
      ( ( sup_sup_nat @ X4 @ ( sup_sup_nat @ X4 @ Y5 ) )
      = ( sup_sup_nat @ X4 @ Y5 ) ) ).

% inf_sup_aci(8)
thf(fact_829_inf__sup__aci_I8_J,axiom,
    ! [X4: produc7781606619528532327tion_b > nat > $o,Y5: produc7781606619528532327tion_b > nat > $o] :
      ( ( sup_su6759296390479959913_nat_o @ X4 @ ( sup_su6759296390479959913_nat_o @ X4 @ Y5 ) )
      = ( sup_su6759296390479959913_nat_o @ X4 @ Y5 ) ) ).

% inf_sup_aci(8)
thf(fact_830_inf__sup__aci_I8_J,axiom,
    ! [X4: option_b > option_b > $o,Y5: option_b > option_b > $o] :
      ( ( sup_su6610355334417212162on_b_o @ X4 @ ( sup_su6610355334417212162on_b_o @ X4 @ Y5 ) )
      = ( sup_su6610355334417212162on_b_o @ X4 @ Y5 ) ) ).

% inf_sup_aci(8)
thf(fact_831_inf__sup__aci_I8_J,axiom,
    ! [X4: b > nat > $o,Y5: b > nat > $o] :
      ( ( sup_sup_b_nat_o @ X4 @ ( sup_sup_b_nat_o @ X4 @ Y5 ) )
      = ( sup_sup_b_nat_o @ X4 @ Y5 ) ) ).

% inf_sup_aci(8)
thf(fact_832_inf__sup__aci_I7_J,axiom,
    ! [X4: set_Pr1307281990691478580_b_nat,Y5: set_Pr1307281990691478580_b_nat,Z: set_Pr1307281990691478580_b_nat] :
      ( ( sup_su6056130500584125792_b_nat @ X4 @ ( sup_su6056130500584125792_b_nat @ Y5 @ Z ) )
      = ( sup_su6056130500584125792_b_nat @ Y5 @ ( sup_su6056130500584125792_b_nat @ X4 @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_833_inf__sup__aci_I7_J,axiom,
    ! [X4: nat,Y5: nat,Z: nat] :
      ( ( sup_sup_nat @ X4 @ ( sup_sup_nat @ Y5 @ Z ) )
      = ( sup_sup_nat @ Y5 @ ( sup_sup_nat @ X4 @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_834_inf__sup__aci_I7_J,axiom,
    ! [X4: produc7781606619528532327tion_b > nat > $o,Y5: produc7781606619528532327tion_b > nat > $o,Z: produc7781606619528532327tion_b > nat > $o] :
      ( ( sup_su6759296390479959913_nat_o @ X4 @ ( sup_su6759296390479959913_nat_o @ Y5 @ Z ) )
      = ( sup_su6759296390479959913_nat_o @ Y5 @ ( sup_su6759296390479959913_nat_o @ X4 @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_835_inf__sup__aci_I7_J,axiom,
    ! [X4: option_b > option_b > $o,Y5: option_b > option_b > $o,Z: option_b > option_b > $o] :
      ( ( sup_su6610355334417212162on_b_o @ X4 @ ( sup_su6610355334417212162on_b_o @ Y5 @ Z ) )
      = ( sup_su6610355334417212162on_b_o @ Y5 @ ( sup_su6610355334417212162on_b_o @ X4 @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_836_inf__sup__aci_I7_J,axiom,
    ! [X4: b > nat > $o,Y5: b > nat > $o,Z: b > nat > $o] :
      ( ( sup_sup_b_nat_o @ X4 @ ( sup_sup_b_nat_o @ Y5 @ Z ) )
      = ( sup_sup_b_nat_o @ Y5 @ ( sup_sup_b_nat_o @ X4 @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_837_inf__sup__aci_I6_J,axiom,
    ! [X4: set_Pr1307281990691478580_b_nat,Y5: set_Pr1307281990691478580_b_nat,Z: set_Pr1307281990691478580_b_nat] :
      ( ( sup_su6056130500584125792_b_nat @ ( sup_su6056130500584125792_b_nat @ X4 @ Y5 ) @ Z )
      = ( sup_su6056130500584125792_b_nat @ X4 @ ( sup_su6056130500584125792_b_nat @ Y5 @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_838_inf__sup__aci_I6_J,axiom,
    ! [X4: nat,Y5: nat,Z: nat] :
      ( ( sup_sup_nat @ ( sup_sup_nat @ X4 @ Y5 ) @ Z )
      = ( sup_sup_nat @ X4 @ ( sup_sup_nat @ Y5 @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_839_inf__sup__aci_I6_J,axiom,
    ! [X4: produc7781606619528532327tion_b > nat > $o,Y5: produc7781606619528532327tion_b > nat > $o,Z: produc7781606619528532327tion_b > nat > $o] :
      ( ( sup_su6759296390479959913_nat_o @ ( sup_su6759296390479959913_nat_o @ X4 @ Y5 ) @ Z )
      = ( sup_su6759296390479959913_nat_o @ X4 @ ( sup_su6759296390479959913_nat_o @ Y5 @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_840_inf__sup__aci_I6_J,axiom,
    ! [X4: option_b > option_b > $o,Y5: option_b > option_b > $o,Z: option_b > option_b > $o] :
      ( ( sup_su6610355334417212162on_b_o @ ( sup_su6610355334417212162on_b_o @ X4 @ Y5 ) @ Z )
      = ( sup_su6610355334417212162on_b_o @ X4 @ ( sup_su6610355334417212162on_b_o @ Y5 @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_841_inf__sup__aci_I6_J,axiom,
    ! [X4: b > nat > $o,Y5: b > nat > $o,Z: b > nat > $o] :
      ( ( sup_sup_b_nat_o @ ( sup_sup_b_nat_o @ X4 @ Y5 ) @ Z )
      = ( sup_sup_b_nat_o @ X4 @ ( sup_sup_b_nat_o @ Y5 @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_842_inf__sup__aci_I5_J,axiom,
    ( sup_su6056130500584125792_b_nat
    = ( ^ [X: set_Pr1307281990691478580_b_nat,Y4: set_Pr1307281990691478580_b_nat] : ( sup_su6056130500584125792_b_nat @ Y4 @ X ) ) ) ).

% inf_sup_aci(5)
thf(fact_843_inf__sup__aci_I5_J,axiom,
    ( sup_sup_nat
    = ( ^ [X: nat,Y4: nat] : ( sup_sup_nat @ Y4 @ X ) ) ) ).

% inf_sup_aci(5)
thf(fact_844_inf__sup__aci_I5_J,axiom,
    ( sup_su6759296390479959913_nat_o
    = ( ^ [X: produc7781606619528532327tion_b > nat > $o,Y4: produc7781606619528532327tion_b > nat > $o] : ( sup_su6759296390479959913_nat_o @ Y4 @ X ) ) ) ).

% inf_sup_aci(5)
thf(fact_845_inf__sup__aci_I5_J,axiom,
    ( sup_su6610355334417212162on_b_o
    = ( ^ [X: option_b > option_b > $o,Y4: option_b > option_b > $o] : ( sup_su6610355334417212162on_b_o @ Y4 @ X ) ) ) ).

% inf_sup_aci(5)
thf(fact_846_inf__sup__aci_I5_J,axiom,
    ( sup_sup_b_nat_o
    = ( ^ [X: b > nat > $o,Y4: b > nat > $o] : ( sup_sup_b_nat_o @ Y4 @ X ) ) ) ).

% inf_sup_aci(5)
thf(fact_847_sup_Oassoc,axiom,
    ! [A: set_Pr1307281990691478580_b_nat,B: set_Pr1307281990691478580_b_nat,C: set_Pr1307281990691478580_b_nat] :
      ( ( sup_su6056130500584125792_b_nat @ ( sup_su6056130500584125792_b_nat @ A @ B ) @ C )
      = ( sup_su6056130500584125792_b_nat @ A @ ( sup_su6056130500584125792_b_nat @ B @ C ) ) ) ).

% sup.assoc
thf(fact_848_sup_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( sup_sup_nat @ ( sup_sup_nat @ A @ B ) @ C )
      = ( sup_sup_nat @ A @ ( sup_sup_nat @ B @ C ) ) ) ).

% sup.assoc
thf(fact_849_sup_Oassoc,axiom,
    ! [A: produc7781606619528532327tion_b > nat > $o,B: produc7781606619528532327tion_b > nat > $o,C: produc7781606619528532327tion_b > nat > $o] :
      ( ( sup_su6759296390479959913_nat_o @ ( sup_su6759296390479959913_nat_o @ A @ B ) @ C )
      = ( sup_su6759296390479959913_nat_o @ A @ ( sup_su6759296390479959913_nat_o @ B @ C ) ) ) ).

% sup.assoc
thf(fact_850_sup_Oassoc,axiom,
    ! [A: option_b > option_b > $o,B: option_b > option_b > $o,C: option_b > option_b > $o] :
      ( ( sup_su6610355334417212162on_b_o @ ( sup_su6610355334417212162on_b_o @ A @ B ) @ C )
      = ( sup_su6610355334417212162on_b_o @ A @ ( sup_su6610355334417212162on_b_o @ B @ C ) ) ) ).

% sup.assoc
thf(fact_851_sup_Oassoc,axiom,
    ! [A: b > nat > $o,B: b > nat > $o,C: b > nat > $o] :
      ( ( sup_sup_b_nat_o @ ( sup_sup_b_nat_o @ A @ B ) @ C )
      = ( sup_sup_b_nat_o @ A @ ( sup_sup_b_nat_o @ B @ C ) ) ) ).

% sup.assoc
thf(fact_852_sup__assoc,axiom,
    ! [X4: set_Pr1307281990691478580_b_nat,Y5: set_Pr1307281990691478580_b_nat,Z: set_Pr1307281990691478580_b_nat] :
      ( ( sup_su6056130500584125792_b_nat @ ( sup_su6056130500584125792_b_nat @ X4 @ Y5 ) @ Z )
      = ( sup_su6056130500584125792_b_nat @ X4 @ ( sup_su6056130500584125792_b_nat @ Y5 @ Z ) ) ) ).

% sup_assoc
thf(fact_853_sup__assoc,axiom,
    ! [X4: nat,Y5: nat,Z: nat] :
      ( ( sup_sup_nat @ ( sup_sup_nat @ X4 @ Y5 ) @ Z )
      = ( sup_sup_nat @ X4 @ ( sup_sup_nat @ Y5 @ Z ) ) ) ).

% sup_assoc
thf(fact_854_sup__assoc,axiom,
    ! [X4: produc7781606619528532327tion_b > nat > $o,Y5: produc7781606619528532327tion_b > nat > $o,Z: produc7781606619528532327tion_b > nat > $o] :
      ( ( sup_su6759296390479959913_nat_o @ ( sup_su6759296390479959913_nat_o @ X4 @ Y5 ) @ Z )
      = ( sup_su6759296390479959913_nat_o @ X4 @ ( sup_su6759296390479959913_nat_o @ Y5 @ Z ) ) ) ).

% sup_assoc
thf(fact_855_sup__assoc,axiom,
    ! [X4: option_b > option_b > $o,Y5: option_b > option_b > $o,Z: option_b > option_b > $o] :
      ( ( sup_su6610355334417212162on_b_o @ ( sup_su6610355334417212162on_b_o @ X4 @ Y5 ) @ Z )
      = ( sup_su6610355334417212162on_b_o @ X4 @ ( sup_su6610355334417212162on_b_o @ Y5 @ Z ) ) ) ).

% sup_assoc
thf(fact_856_sup__assoc,axiom,
    ! [X4: b > nat > $o,Y5: b > nat > $o,Z: b > nat > $o] :
      ( ( sup_sup_b_nat_o @ ( sup_sup_b_nat_o @ X4 @ Y5 ) @ Z )
      = ( sup_sup_b_nat_o @ X4 @ ( sup_sup_b_nat_o @ Y5 @ Z ) ) ) ).

% sup_assoc
thf(fact_857_sup_Ocommute,axiom,
    ( sup_su6759296390479959913_nat_o
    = ( ^ [A6: produc7781606619528532327tion_b > nat > $o,B4: produc7781606619528532327tion_b > nat > $o] : ( sup_su6759296390479959913_nat_o @ B4 @ A6 ) ) ) ).

% sup.commute
thf(fact_858_sup_Ocommute,axiom,
    ( sup_su6610355334417212162on_b_o
    = ( ^ [A6: option_b > option_b > $o,B4: option_b > option_b > $o] : ( sup_su6610355334417212162on_b_o @ B4 @ A6 ) ) ) ).

% sup.commute
thf(fact_859_sup_Ocommute,axiom,
    ( sup_sup_b_nat_o
    = ( ^ [A6: b > nat > $o,B4: b > nat > $o] : ( sup_sup_b_nat_o @ B4 @ A6 ) ) ) ).

% sup.commute
thf(fact_860_inf__sup__ord_I4_J,axiom,
    ! [Y5: nat,X4: nat] : ( ord_less_eq_nat @ Y5 @ ( sup_sup_nat @ X4 @ Y5 ) ) ).

% inf_sup_ord(4)
thf(fact_861_inf__sup__ord_I3_J,axiom,
    ! [X4: nat,Y5: nat] : ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ X4 @ Y5 ) ) ).

% inf_sup_ord(3)
thf(fact_862_le__supE,axiom,
    ! [A: nat,B: nat,X4: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X4 )
     => ~ ( ( ord_less_eq_nat @ A @ X4 )
         => ~ ( ord_less_eq_nat @ B @ X4 ) ) ) ).

% le_supE
thf(fact_863_le__supI,axiom,
    ! [A: nat,X4: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ X4 )
     => ( ( ord_less_eq_nat @ B @ X4 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X4 ) ) ) ).

% le_supI
thf(fact_864_sup__ge1,axiom,
    ! [X4: nat,Y5: nat] : ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ X4 @ Y5 ) ) ).

% sup_ge1
thf(fact_865_sup__ge2,axiom,
    ! [Y5: nat,X4: nat] : ( ord_less_eq_nat @ Y5 @ ( sup_sup_nat @ X4 @ Y5 ) ) ).

% sup_ge2
thf(fact_866_le__supI1,axiom,
    ! [X4: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X4 @ A )
     => ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ A @ B ) ) ) ).

% le_supI1
thf(fact_867_le__supI2,axiom,
    ! [X4: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ X4 @ B )
     => ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ A @ B ) ) ) ).

% le_supI2
thf(fact_868_sup_Omono,axiom,
    ! [C: nat,A: nat,D2: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ( ord_less_eq_nat @ D2 @ B )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_869_sup__mono,axiom,
    ! [A: nat,C: nat,B: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ D2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_870_sup__least,axiom,
    ! [Y5: nat,X4: nat,Z: nat] :
      ( ( ord_less_eq_nat @ Y5 @ X4 )
     => ( ( ord_less_eq_nat @ Z @ X4 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ Y5 @ Z ) @ X4 ) ) ) ).

% sup_least
thf(fact_871_le__iff__sup,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y4: nat] :
          ( ( sup_sup_nat @ X @ Y4 )
          = Y4 ) ) ) ).

% le_iff_sup
thf(fact_872_sup_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( sup_sup_nat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_873_sup_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( sup_sup_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% sup.orderI
thf(fact_874_sup__unique,axiom,
    ! [F: nat > nat > nat,X4: nat,Y5: nat] :
      ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y3 ) )
     => ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ ( F @ X3 @ Y3 ) )
       => ( ! [X3: nat,Y3: nat,Z2: nat] :
              ( ( ord_less_eq_nat @ Y3 @ X3 )
             => ( ( ord_less_eq_nat @ Z2 @ X3 )
               => ( ord_less_eq_nat @ ( F @ Y3 @ Z2 ) @ X3 ) ) )
         => ( ( sup_sup_nat @ X4 @ Y5 )
            = ( F @ X4 @ Y5 ) ) ) ) ) ).

% sup_unique
thf(fact_875_sup_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( sup_sup_nat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_876_sup_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( sup_sup_nat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_877_sup__absorb1,axiom,
    ! [Y5: nat,X4: nat] :
      ( ( ord_less_eq_nat @ Y5 @ X4 )
     => ( ( sup_sup_nat @ X4 @ Y5 )
        = X4 ) ) ).

% sup_absorb1
thf(fact_878_sup__absorb2,axiom,
    ! [X4: nat,Y5: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y5 )
     => ( ( sup_sup_nat @ X4 @ Y5 )
        = Y5 ) ) ).

% sup_absorb2
thf(fact_879_sup_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_nat @ B @ A )
         => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_880_sup_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_881_sup_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A6: nat] :
          ( A6
          = ( sup_sup_nat @ A6 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_882_sup_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_883_sup_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_884_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A6: nat] :
          ( ( sup_sup_nat @ A6 @ B4 )
          = A6 ) ) ) ).

% sup.absorb_iff1
thf(fact_885_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A6: nat,B4: nat] :
          ( ( sup_sup_nat @ A6 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_886_relChain__def,axiom,
    ( bNF_Ca5450492268418663793_b_nat
    = ( ^ [R2: set_Pr258254617715012167tion_b,As: option_b > nat] :
        ! [I: option_b,J: option_b] :
          ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ I @ J ) @ R2 )
         => ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_887_ssubst__Pair__rhs,axiom,
    ! [R5: produc7781606619528532327tion_b,S4: nat,R: set_Pr6944079702279229084_b_nat,S5: nat] :
      ( ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ R5 @ S4 ) @ R )
     => ( ( S5 = S4 )
       => ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ R5 @ S5 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_888_ssubst__Pair__rhs,axiom,
    ! [R5: option_b,S4: option_b,R: set_Pr258254617715012167tion_b,S5: option_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ R5 @ S4 ) @ R )
     => ( ( S5 = S4 )
       => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ R5 @ S5 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_889_ssubst__Pair__rhs,axiom,
    ! [R5: b,S4: nat,R: set_Pr1307281990691478580_b_nat,S5: nat] :
      ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ R5 @ S4 ) @ R )
     => ( ( S5 = S4 )
       => ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ R5 @ S5 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_890_less__eq__option__def,axiom,
    ( ord_le5914376470875661696on_nat
    = ( ^ [X: option_nat,Y4: option_nat] :
          ( case_option_o_nat @ $true
          @ ^ [Z4: nat] : ( case_option_o_nat @ $false @ ( ord_less_eq_nat @ Z4 ) @ Y4 )
          @ X ) ) ) ).

% less_eq_option_def
thf(fact_891_combine__options__def,axiom,
    ( combine_options_b
    = ( ^ [F3: b > b > b,X: option_b,Y4: option_b] :
          ( case_o2513330393899446277on_b_b @ Y4
          @ ^ [Z4: b] :
              ( case_o2513330393899446277on_b_b @ ( some_b @ Z4 )
              @ ^ [Aa: b] : ( some_b @ ( F3 @ Z4 @ Aa ) )
              @ Y4 )
          @ X ) ) ) ).

% combine_options_def
thf(fact_892_reflcl__set__eq,axiom,
    ! [R5: set_Pr258254617715012167tion_b] :
      ( ( sup_su6610355334417212162on_b_o
        @ ^ [X: option_b,Y4: option_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ Y4 ) @ R5 )
        @ ^ [Y8: option_b,Z3: option_b] : ( Y8 = Z3 ) )
      = ( ^ [X: option_b,Y4: option_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ Y4 ) @ ( sup_su3110286242275913755tion_b @ R5 @ id_option_b ) ) ) ) ).

% reflcl_set_eq
thf(fact_893_IdI,axiom,
    ! [A: option_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ A ) @ id_option_b ) ).

% IdI
thf(fact_894_pair__in__Id__conv,axiom,
    ! [A: option_b,B: option_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ id_option_b )
      = ( A = B ) ) ).

% pair_in_Id_conv
thf(fact_895_combine__options__simps_I3_J,axiom,
    ! [F: b > b > b,A: b,B: b] :
      ( ( combine_options_b @ F @ ( some_b @ A ) @ ( some_b @ B ) )
      = ( some_b @ ( F @ A @ B ) ) ) ).

% combine_options_simps(3)
thf(fact_896_IdD,axiom,
    ! [A: option_b,B: option_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ id_option_b )
     => ( A = B ) ) ).

% IdD
thf(fact_897_IdE,axiom,
    ! [P: produc7781606619528532327tion_b] :
      ( ( member547026996284216464tion_b @ P @ id_option_b )
     => ~ ! [X3: option_b] :
            ( P
           != ( produc3989389360737052631tion_b @ X3 @ X3 ) ) ) ).

% IdE
thf(fact_898_in__rel__def,axiom,
    ( fun_in8719054587677919989_b_nat
    = ( ^ [R4: set_Pr6944079702279229084_b_nat,X: produc7781606619528532327tion_b,Y4: nat] : ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ X @ Y4 ) @ R4 ) ) ) ).

% in_rel_def
thf(fact_899_in__rel__def,axiom,
    ( fun_in4608036175315572994tion_b
    = ( ^ [R4: set_Pr258254617715012167tion_b,X: option_b,Y4: option_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ Y4 ) @ R4 ) ) ) ).

% in_rel_def
thf(fact_900_in__rel__def,axiom,
    ( fun_in_rel_b_nat
    = ( ^ [R4: set_Pr1307281990691478580_b_nat,X: b,Y4: nat] : ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ X @ Y4 ) @ R4 ) ) ) ).

% in_rel_def
thf(fact_901_Greatest__equality,axiom,
    ! [P2: nat > $o,X4: nat] :
      ( ( P2 @ X4 )
     => ( ! [Y3: nat] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ X4 ) )
       => ( ( order_Greatest_nat @ P2 )
          = X4 ) ) ) ).

% Greatest_equality
thf(fact_902_GreatestI2__order,axiom,
    ! [P2: nat > $o,X4: nat,Q: nat > $o] :
      ( ( P2 @ X4 )
     => ( ! [Y3: nat] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ X4 ) )
       => ( ! [X3: nat] :
              ( ( P2 @ X3 )
             => ( ! [Y9: nat] :
                    ( ( P2 @ Y9 )
                   => ( ord_less_eq_nat @ Y9 @ X3 ) )
               => ( Q @ X3 ) ) )
         => ( Q @ ( order_Greatest_nat @ P2 ) ) ) ) ) ).

% GreatestI2_order
thf(fact_903_antisym__on__def,axiom,
    ( antisym_on_option_b
    = ( ^ [A8: set_option_b,R2: set_Pr258254617715012167tion_b] :
        ! [X: option_b] :
          ( ( member_option_b @ X @ A8 )
         => ! [Y4: option_b] :
              ( ( member_option_b @ Y4 @ A8 )
             => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ Y4 ) @ R2 )
               => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y4 @ X ) @ R2 )
                 => ( X = Y4 ) ) ) ) ) ) ) ).

% antisym_on_def
thf(fact_904_antisym__onI,axiom,
    ! [A5: set_option_b,R5: set_Pr258254617715012167tion_b] :
      ( ! [X3: option_b,Y3: option_b] :
          ( ( member_option_b @ X3 @ A5 )
         => ( ( member_option_b @ Y3 @ A5 )
           => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X3 @ Y3 ) @ R5 )
             => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y3 @ X3 ) @ R5 )
               => ( X3 = Y3 ) ) ) ) )
     => ( antisym_on_option_b @ A5 @ R5 ) ) ).

% antisym_onI
thf(fact_905_antisym__onD,axiom,
    ! [A5: set_option_b,R5: set_Pr258254617715012167tion_b,X4: option_b,Y5: option_b] :
      ( ( antisym_on_option_b @ A5 @ R5 )
     => ( ( member_option_b @ X4 @ A5 )
       => ( ( member_option_b @ Y5 @ A5 )
         => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ R5 )
           => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y5 @ X4 ) @ R5 )
             => ( X4 = Y5 ) ) ) ) ) ) ).

% antisym_onD
thf(fact_906_antisymp__on__antisym__on__eq,axiom,
    ! [A5: set_option_b,R5: set_Pr258254617715012167tion_b] :
      ( ( antisymp_on_option_b @ A5
        @ ^ [X: option_b,Y4: option_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ Y4 ) @ R5 ) )
      = ( antisym_on_option_b @ A5 @ R5 ) ) ).

% antisymp_on_antisym_on_eq
thf(fact_907_curry__case__prod,axiom,
    ! [F: b > nat > produc1396820459975257318_b_nat] :
      ( ( produc3837372375975176876_b_nat @ ( produc5650155531262253963_b_nat @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_908_curry__case__prod,axiom,
    ! [F: option_b > option_b > produc7781606619528532327tion_b] :
      ( ( produc6534564051900649756tion_b @ ( produc5931753611847401595tion_b @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_909_curry__case__prod,axiom,
    ! [F: produc7781606619528532327tion_b > nat > produc1396820459975257318_b_nat] :
      ( ( produc2697271878510243092_b_nat @ ( produc1170353205628678643_b_nat @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_910_case__prod__curry,axiom,
    ! [F: product_prod_b_nat > produc1396820459975257318_b_nat] :
      ( ( produc5650155531262253963_b_nat @ ( produc3837372375975176876_b_nat @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_911_case__prod__curry,axiom,
    ! [F: produc7781606619528532327tion_b > produc7781606619528532327tion_b] :
      ( ( produc5931753611847401595tion_b @ ( produc6534564051900649756tion_b @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_912_case__prod__curry,axiom,
    ! [F: produc1396820459975257318_b_nat > produc1396820459975257318_b_nat] :
      ( ( produc1170353205628678643_b_nat @ ( produc2697271878510243092_b_nat @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_913_The__split__eq,axiom,
    ! [X4: produc7781606619528532327tion_b,Y5: nat] :
      ( ( the_Pr6254308514702899231_b_nat
        @ ( produc5232674473337540683_nat_o
          @ ^ [X5: produc7781606619528532327tion_b,Y6: nat] :
              ( ( X4 = X5 )
              & ( Y5 = Y6 ) ) ) )
      = ( produc2093246604507004384_b_nat @ X4 @ Y5 ) ) ).

% The_split_eq
thf(fact_914_The__split__eq,axiom,
    ! [X4: option_b,Y5: option_b] :
      ( ( the_Pr8527084792148607342tion_b
        @ ( produc3378454745690301612on_b_o
          @ ^ [X5: option_b,Y6: option_b] :
              ( ( X4 = X5 )
              & ( Y5 = Y6 ) ) ) )
      = ( produc3989389360737052631tion_b @ X4 @ Y5 ) ) ).

% The_split_eq
thf(fact_915_The__split__eq,axiom,
    ! [X4: b,Y5: nat] :
      ( ( the_Pr757299968590014903_b_nat
        @ ( produc795641402153621683_nat_o
          @ ^ [X5: b,Y6: nat] :
              ( ( X4 = X5 )
              & ( Y5 = Y6 ) ) ) )
      = ( product_Pair_b_nat @ X4 @ Y5 ) ) ).

% The_split_eq
thf(fact_916_curryI,axiom,
    ! [F: produc1396820459975257318_b_nat > $o,A: produc7781606619528532327tion_b,B: nat] :
      ( ( F @ ( produc2093246604507004384_b_nat @ A @ B ) )
     => ( produc2920251780219706858_nat_o @ F @ A @ B ) ) ).

% curryI
thf(fact_917_curryI,axiom,
    ! [F: produc7781606619528532327tion_b > $o,A: option_b,B: option_b] :
      ( ( F @ ( produc3989389360737052631tion_b @ A @ B ) )
     => ( produc3448963759890581965on_b_o @ F @ A @ B ) ) ).

% curryI
thf(fact_918_curryI,axiom,
    ! [F: product_prod_b_nat > $o,A: b,B: nat] :
      ( ( F @ ( product_Pair_b_nat @ A @ B ) )
     => ( produc2461434047082304082_nat_o @ F @ A @ B ) ) ).

% curryI
thf(fact_919_not__None__eq,axiom,
    ! [X4: option_b] :
      ( ( X4 != none_b )
      = ( ? [Y4: b] :
            ( X4
            = ( some_b @ Y4 ) ) ) ) ).

% not_None_eq
thf(fact_920_not__Some__eq,axiom,
    ! [X4: option_b] :
      ( ( ! [Y4: b] :
            ( X4
           != ( some_b @ Y4 ) ) )
      = ( X4 = none_b ) ) ).

% not_Some_eq
thf(fact_921_option_Odistinct_I1_J,axiom,
    ! [X2: b] :
      ( none_b
     != ( some_b @ X2 ) ) ).

% option.distinct(1)
thf(fact_922_option_OdiscI,axiom,
    ! [Option: option_b,X2: b] :
      ( ( Option
        = ( some_b @ X2 ) )
     => ( Option != none_b ) ) ).

% option.discI
thf(fact_923_option_Oexhaust,axiom,
    ! [Y5: option_b] :
      ( ( Y5 != none_b )
     => ~ ! [X23: b] :
            ( Y5
           != ( some_b @ X23 ) ) ) ).

% option.exhaust
thf(fact_924_split__option__ex,axiom,
    ( ( ^ [P4: option_b > $o] :
        ? [X7: option_b] : ( P4 @ X7 ) )
    = ( ^ [P3: option_b > $o] :
          ( ( P3 @ none_b )
          | ? [X: b] : ( P3 @ ( some_b @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_925_split__option__all,axiom,
    ( ( ^ [P4: option_b > $o] :
        ! [X7: option_b] : ( P4 @ X7 ) )
    = ( ^ [P3: option_b > $o] :
          ( ( P3 @ none_b )
          & ! [X: b] : ( P3 @ ( some_b @ X ) ) ) ) ) ).

% split_option_all
thf(fact_926_combine__options__cases,axiom,
    ! [X4: option_b,P2: option_b > option_b > $o,Y5: option_b] :
      ( ( ( X4 = none_b )
       => ( P2 @ X4 @ Y5 ) )
     => ( ( ( Y5 = none_b )
         => ( P2 @ X4 @ Y5 ) )
       => ( ! [A4: b,B3: b] :
              ( ( X4
                = ( some_b @ A4 ) )
             => ( ( Y5
                  = ( some_b @ B3 ) )
               => ( P2 @ X4 @ Y5 ) ) )
         => ( P2 @ X4 @ Y5 ) ) ) ) ).

% combine_options_cases
thf(fact_927_antisymp__on__le,axiom,
    ! [A5: set_nat] : ( antisymp_on_nat @ A5 @ ord_less_eq_nat ) ).

% antisymp_on_le
thf(fact_928_curryD,axiom,
    ! [F: produc1396820459975257318_b_nat > $o,A: produc7781606619528532327tion_b,B: nat] :
      ( ( produc2920251780219706858_nat_o @ F @ A @ B )
     => ( F @ ( produc2093246604507004384_b_nat @ A @ B ) ) ) ).

% curryD
thf(fact_929_curryD,axiom,
    ! [F: produc7781606619528532327tion_b > $o,A: option_b,B: option_b] :
      ( ( produc3448963759890581965on_b_o @ F @ A @ B )
     => ( F @ ( produc3989389360737052631tion_b @ A @ B ) ) ) ).

% curryD
thf(fact_930_curryD,axiom,
    ! [F: product_prod_b_nat > $o,A: b,B: nat] :
      ( ( produc2461434047082304082_nat_o @ F @ A @ B )
     => ( F @ ( product_Pair_b_nat @ A @ B ) ) ) ).

% curryD
thf(fact_931_curryE,axiom,
    ! [F: produc1396820459975257318_b_nat > $o,A: produc7781606619528532327tion_b,B: nat] :
      ( ( produc2920251780219706858_nat_o @ F @ A @ B )
     => ( F @ ( produc2093246604507004384_b_nat @ A @ B ) ) ) ).

% curryE
thf(fact_932_curryE,axiom,
    ! [F: produc7781606619528532327tion_b > $o,A: option_b,B: option_b] :
      ( ( produc3448963759890581965on_b_o @ F @ A @ B )
     => ( F @ ( produc3989389360737052631tion_b @ A @ B ) ) ) ).

% curryE
thf(fact_933_curryE,axiom,
    ! [F: product_prod_b_nat > $o,A: b,B: nat] :
      ( ( produc2461434047082304082_nat_o @ F @ A @ B )
     => ( F @ ( product_Pair_b_nat @ A @ B ) ) ) ).

% curryE
thf(fact_934_antisymp__on__ge,axiom,
    ! [A5: set_nat] :
      ( antisymp_on_nat @ A5
      @ ^ [X: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X ) ) ).

% antisymp_on_ge
thf(fact_935_case__optionE,axiom,
    ! [P2: $o,Q: b > $o,X4: option_b] :
      ( ( case_option_o_b @ P2 @ Q @ X4 )
     => ( ( ( X4 = none_b )
         => ~ P2 )
       => ~ ! [Y3: b] :
              ( ( X4
                = ( some_b @ Y3 ) )
             => ~ ( Q @ Y3 ) ) ) ) ).

% case_optionE
thf(fact_936_max__extp__max__ext__eq,axiom,
    ! [R: set_Pr258254617715012167tion_b] :
      ( ( max_extp_option_b
        @ ^ [X: option_b,Y4: option_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ Y4 ) @ R ) )
      = ( ^ [X: set_option_b,Y4: set_option_b] : ( member9052098846246995088tion_b @ ( produc5242204166755002327tion_b @ X @ Y4 ) @ ( max_ext_option_b @ R ) ) ) ) ).

% max_extp_max_ext_eq
thf(fact_937_max__ext__def,axiom,
    ( max_ext_option_b
    = ( ^ [R4: set_Pr258254617715012167tion_b] :
          ( collec6622971782277219538tion_b
          @ ( produc7796829984862323884on_b_o
            @ ( max_extp_option_b
              @ ^ [X: option_b,Y4: option_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ Y4 ) @ R4 ) ) ) ) ) ) ).

% max_ext_def
thf(fact_938_antisymp__antisym__eq,axiom,
    ! [R5: set_Pr258254617715012167tion_b] :
      ( ( antisymp_on_option_b @ top_top_set_option_b
        @ ^ [X: option_b,Y4: option_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ Y4 ) @ R5 ) )
      = ( antisym_on_option_b @ top_top_set_option_b @ R5 ) ) ).

% antisymp_antisym_eq
thf(fact_939_Greatest__def,axiom,
    ( order_Greatest_nat
    = ( ^ [P3: nat > $o] :
          ( the_nat
          @ ^ [X: nat] :
              ( ( P3 @ X )
              & ! [Y4: nat] :
                  ( ( P3 @ Y4 )
                 => ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ) ) ).

% Greatest_def
thf(fact_940_map__option__eq__Some,axiom,
    ! [F: b > b,Xo: option_b,Y5: b] :
      ( ( ( map_option_b_b @ F @ Xo )
        = ( some_b @ Y5 ) )
      = ( ? [Z4: b] :
            ( ( Xo
              = ( some_b @ Z4 ) )
            & ( ( F @ Z4 )
              = Y5 ) ) ) ) ).

% map_option_eq_Some
thf(fact_941_option_Osimps_I9_J,axiom,
    ! [F: b > b,X2: b] :
      ( ( map_option_b_b @ F @ ( some_b @ X2 ) )
      = ( some_b @ ( F @ X2 ) ) ) ).

% option.simps(9)
thf(fact_942_antisymI,axiom,
    ! [R5: set_Pr258254617715012167tion_b] :
      ( ! [X3: option_b,Y3: option_b] :
          ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X3 @ Y3 ) @ R5 )
         => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y3 @ X3 ) @ R5 )
           => ( X3 = Y3 ) ) )
     => ( antisym_on_option_b @ top_top_set_option_b @ R5 ) ) ).

% antisymI
thf(fact_943_antisymD,axiom,
    ! [R5: set_Pr258254617715012167tion_b,X4: option_b,Y5: option_b] :
      ( ( antisym_on_option_b @ top_top_set_option_b @ R5 )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ R5 )
       => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y5 @ X4 ) @ R5 )
         => ( X4 = Y5 ) ) ) ) ).

% antisymD
thf(fact_944_Least__def,axiom,
    ( ord_Least_nat
    = ( ^ [P3: nat > $o] :
          ( the_nat
          @ ^ [X: nat] :
              ( ( P3 @ X )
              & ! [Y4: nat] :
                  ( ( P3 @ Y4 )
                 => ( ord_less_eq_nat @ X @ Y4 ) ) ) ) ) ) ).

% Least_def
thf(fact_945_init__seg__of__def,axiom,
    ( init_seg_of_option_b
    = ( collec1918043184897894738tion_b
      @ ( produc2671211398375001388on_b_o
        @ ^ [R2: set_Pr258254617715012167tion_b,S6: set_Pr258254617715012167tion_b] :
            ( ( ord_le6367397903258896871tion_b @ R2 @ S6 )
            & ! [A6: option_b,B4: option_b,C4: option_b] :
                ( ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A6 @ B4 ) @ S6 )
                  & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ B4 @ C4 ) @ R2 ) )
               => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A6 @ B4 ) @ R2 ) ) ) ) ) ) ).

% init_seg_of_def
thf(fact_946_asymI,axiom,
    ! [R5: set_Pr258254617715012167tion_b] :
      ( ! [X3: option_b,Y3: option_b] :
          ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X3 @ Y3 ) @ R5 )
         => ~ ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y3 @ X3 ) @ R5 ) )
     => ( asym_on_option_b @ top_top_set_option_b @ R5 ) ) ).

% asymI
thf(fact_947_asym__onI,axiom,
    ! [A5: set_option_b,R5: set_Pr258254617715012167tion_b] :
      ( ! [X3: option_b,Y3: option_b] :
          ( ( member_option_b @ X3 @ A5 )
         => ( ( member_option_b @ Y3 @ A5 )
           => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X3 @ Y3 ) @ R5 )
             => ~ ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y3 @ X3 ) @ R5 ) ) ) )
     => ( asym_on_option_b @ A5 @ R5 ) ) ).

% asym_onI
thf(fact_948_top__empty__eq2,axiom,
    ( top_to4726238945551381741_nat_o
    = ( ^ [X: produc7781606619528532327tion_b,Y4: nat] : ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ X @ Y4 ) @ top_to6093440770071985484_b_nat ) ) ) ).

% top_empty_eq2
thf(fact_949_top__empty__eq2,axiom,
    ( top_to8960343510913153662on_b_o
    = ( ^ [X: option_b,Y4: option_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ Y4 ) @ top_to1926935634769518487tion_b ) ) ) ).

% top_empty_eq2
thf(fact_950_top__empty__eq2,axiom,
    ( top_top_b_nat_o
    = ( ^ [X: b,Y4: nat] : ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ X @ Y4 ) @ top_to8949910960566930148_b_nat ) ) ) ).

% top_empty_eq2
thf(fact_951_Least1I,axiom,
    ! [P2: nat > $o] :
      ( ? [X8: nat] :
          ( ( P2 @ X8 )
          & ! [Y3: nat] :
              ( ( P2 @ Y3 )
             => ( ord_less_eq_nat @ X8 @ Y3 ) )
          & ! [Y3: nat] :
              ( ( ( P2 @ Y3 )
                & ! [Ya2: nat] :
                    ( ( P2 @ Ya2 )
                   => ( ord_less_eq_nat @ Y3 @ Ya2 ) ) )
             => ( Y3 = X8 ) ) )
     => ( P2 @ ( ord_Least_nat @ P2 ) ) ) ).

% Least1I
thf(fact_952_Least1__le,axiom,
    ! [P2: nat > $o,Z: nat] :
      ( ? [X8: nat] :
          ( ( P2 @ X8 )
          & ! [Y3: nat] :
              ( ( P2 @ Y3 )
             => ( ord_less_eq_nat @ X8 @ Y3 ) )
          & ! [Y3: nat] :
              ( ( ( P2 @ Y3 )
                & ! [Ya2: nat] :
                    ( ( P2 @ Ya2 )
                   => ( ord_less_eq_nat @ Y3 @ Ya2 ) ) )
             => ( Y3 = X8 ) ) )
     => ( ( P2 @ Z )
       => ( ord_less_eq_nat @ ( ord_Least_nat @ P2 ) @ Z ) ) ) ).

% Least1_le
thf(fact_953_LeastI2__order,axiom,
    ! [P2: nat > $o,X4: nat,Q: nat > $o] :
      ( ( P2 @ X4 )
     => ( ! [Y3: nat] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_nat @ X4 @ Y3 ) )
       => ( ! [X3: nat] :
              ( ( P2 @ X3 )
             => ( ! [Y9: nat] :
                    ( ( P2 @ Y9 )
                   => ( ord_less_eq_nat @ X3 @ Y9 ) )
               => ( Q @ X3 ) ) )
         => ( Q @ ( ord_Least_nat @ P2 ) ) ) ) ) ).

% LeastI2_order
thf(fact_954_Least__equality,axiom,
    ! [P2: nat > $o,X4: nat] :
      ( ( P2 @ X4 )
     => ( ! [Y3: nat] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_nat @ X4 @ Y3 ) )
       => ( ( ord_Least_nat @ P2 )
          = X4 ) ) ) ).

% Least_equality
thf(fact_955_LeastI2__wellorder,axiom,
    ! [P2: nat > $o,A: nat,Q: nat > $o] :
      ( ( P2 @ A )
     => ( ! [A4: nat] :
            ( ( P2 @ A4 )
           => ( ! [B9: nat] :
                  ( ( P2 @ B9 )
                 => ( ord_less_eq_nat @ A4 @ B9 ) )
             => ( Q @ A4 ) ) )
       => ( Q @ ( ord_Least_nat @ P2 ) ) ) ) ).

% LeastI2_wellorder
thf(fact_956_LeastI2__wellorder__ex,axiom,
    ! [P2: nat > $o,Q: nat > $o] :
      ( ? [X_1: nat] : ( P2 @ X_1 )
     => ( ! [A4: nat] :
            ( ( P2 @ A4 )
           => ( ! [B9: nat] :
                  ( ( P2 @ B9 )
                 => ( ord_less_eq_nat @ A4 @ B9 ) )
             => ( Q @ A4 ) ) )
       => ( Q @ ( ord_Least_nat @ P2 ) ) ) ) ).

% LeastI2_wellorder_ex
thf(fact_957_asym__on__def,axiom,
    ( asym_on_option_b
    = ( ^ [A8: set_option_b,R2: set_Pr258254617715012167tion_b] :
        ! [X: option_b] :
          ( ( member_option_b @ X @ A8 )
         => ! [Y4: option_b] :
              ( ( member_option_b @ Y4 @ A8 )
             => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ Y4 ) @ R2 )
               => ~ ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y4 @ X ) @ R2 ) ) ) ) ) ) ).

% asym_on_def
thf(fact_958_asym__onD,axiom,
    ! [A5: set_option_b,R5: set_Pr258254617715012167tion_b,X4: option_b,Y5: option_b] :
      ( ( asym_on_option_b @ A5 @ R5 )
     => ( ( member_option_b @ X4 @ A5 )
       => ( ( member_option_b @ Y5 @ A5 )
         => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ R5 )
           => ~ ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y5 @ X4 ) @ R5 ) ) ) ) ) ).

% asym_onD
thf(fact_959_Least__le,axiom,
    ! [P2: nat > $o,K: nat] :
      ( ( P2 @ K )
     => ( ord_less_eq_nat @ ( ord_Least_nat @ P2 ) @ K ) ) ).

% Least_le
thf(fact_960_asym__iff,axiom,
    ! [R5: set_Pr258254617715012167tion_b] :
      ( ( asym_on_option_b @ top_top_set_option_b @ R5 )
      = ( ! [X: option_b,Y4: option_b] :
            ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ Y4 ) @ R5 )
           => ~ ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y4 @ X ) @ R5 ) ) ) ) ).

% asym_iff
thf(fact_961_asymD,axiom,
    ! [R5: set_Pr258254617715012167tion_b,X4: option_b,Y5: option_b] :
      ( ( asym_on_option_b @ top_top_set_option_b @ R5 )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ R5 )
       => ~ ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y5 @ X4 ) @ R5 ) ) ) ).

% asymD
thf(fact_962_asymp__asym__eq,axiom,
    ! [R5: set_Pr258254617715012167tion_b] :
      ( ( asymp_on_option_b @ top_top_set_option_b
        @ ^ [X: option_b,Y4: option_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ Y4 ) @ R5 ) )
      = ( asym_on_option_b @ top_top_set_option_b @ R5 ) ) ).

% asymp_asym_eq
thf(fact_963_asymp__on__asym__on__eq,axiom,
    ! [A5: set_option_b,R5: set_Pr258254617715012167tion_b] :
      ( ( asymp_on_option_b @ A5
        @ ^ [X: option_b,Y4: option_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ Y4 ) @ R5 ) )
      = ( asym_on_option_b @ A5 @ R5 ) ) ).

% asymp_on_asym_on_eq
thf(fact_964_relax__pos__lift__root__steps,axiom,
    ! [F2: set_Pr1307281990691478580_b_nat,W: fOR_pos_step,X6: fOR_ext_step,R: set_Pr3134649703833893895term_b] : ( ord_le2408605988743369639term_b @ ( transi8854406251171887185term_b @ ( lift_R8854095231239425056step_b @ F2 @ W @ X6 @ R ) ) @ ( transi8854406251171887185term_b @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PAny @ X6 @ R ) ) ) ).

% relax_pos_lift_root_steps
thf(fact_965_trancl_Ocases,axiom,
    ! [A1: option_b,A22: option_b,R5: set_Pr258254617715012167tion_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A1 @ A22 ) @ ( transi596562996543779814tion_b @ R5 ) )
     => ( ~ ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A1 @ A22 ) @ R5 )
       => ~ ! [B3: option_b] :
              ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A1 @ B3 ) @ ( transi596562996543779814tion_b @ R5 ) )
             => ~ ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ B3 @ A22 ) @ R5 ) ) ) ) ).

% trancl.cases
thf(fact_966_trancl_Osimps,axiom,
    ! [A1: option_b,A22: option_b,R5: set_Pr258254617715012167tion_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A1 @ A22 ) @ ( transi596562996543779814tion_b @ R5 ) )
      = ( ? [A6: option_b,B4: option_b] :
            ( ( A1 = A6 )
            & ( A22 = B4 )
            & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A6 @ B4 ) @ R5 ) )
        | ? [A6: option_b,B4: option_b,C4: option_b] :
            ( ( A1 = A6 )
            & ( A22 = C4 )
            & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A6 @ B4 ) @ ( transi596562996543779814tion_b @ R5 ) )
            & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ B4 @ C4 ) @ R5 ) ) ) ) ).

% trancl.simps
thf(fact_967_trancl_Or__into__trancl,axiom,
    ! [A: option_b,B: option_b,R5: set_Pr258254617715012167tion_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ R5 )
     => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ ( transi596562996543779814tion_b @ R5 ) ) ) ).

% trancl.r_into_trancl
thf(fact_968_tranclE,axiom,
    ! [A: option_b,B: option_b,R5: set_Pr258254617715012167tion_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ ( transi596562996543779814tion_b @ R5 ) )
     => ( ~ ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ R5 )
       => ~ ! [C2: option_b] :
              ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ C2 ) @ ( transi596562996543779814tion_b @ R5 ) )
             => ~ ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ C2 @ B ) @ R5 ) ) ) ) ).

% tranclE
thf(fact_969_trancl__trans,axiom,
    ! [X4: option_b,Y5: option_b,R5: set_Pr258254617715012167tion_b,Z: option_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ ( transi596562996543779814tion_b @ R5 ) )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y5 @ Z ) @ ( transi596562996543779814tion_b @ R5 ) )
       => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Z ) @ ( transi596562996543779814tion_b @ R5 ) ) ) ) ).

% trancl_trans
thf(fact_970_trancl__induct,axiom,
    ! [A: option_b,B: option_b,R5: set_Pr258254617715012167tion_b,P2: option_b > $o] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ ( transi596562996543779814tion_b @ R5 ) )
     => ( ! [Y3: option_b] :
            ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ Y3 ) @ R5 )
           => ( P2 @ Y3 ) )
       => ( ! [Y3: option_b,Z2: option_b] :
              ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ Y3 ) @ ( transi596562996543779814tion_b @ R5 ) )
             => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y3 @ Z2 ) @ R5 )
               => ( ( P2 @ Y3 )
                 => ( P2 @ Z2 ) ) ) )
         => ( P2 @ B ) ) ) ) ).

% trancl_induct
thf(fact_971_r__r__into__trancl,axiom,
    ! [A: option_b,B: option_b,R: set_Pr258254617715012167tion_b,C: option_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ R )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ B @ C ) @ R )
       => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ C ) @ ( transi596562996543779814tion_b @ R ) ) ) ) ).

% r_r_into_trancl
thf(fact_972_converse__tranclE,axiom,
    ! [X4: option_b,Z: option_b,R5: set_Pr258254617715012167tion_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Z ) @ ( transi596562996543779814tion_b @ R5 ) )
     => ( ~ ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Z ) @ R5 )
       => ~ ! [Y3: option_b] :
              ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y3 ) @ R5 )
             => ~ ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y3 @ Z ) @ ( transi596562996543779814tion_b @ R5 ) ) ) ) ) ).

% converse_tranclE
thf(fact_973_irrefl__trancl__rD,axiom,
    ! [R5: set_Pr258254617715012167tion_b,X4: option_b,Y5: option_b] :
      ( ! [X3: option_b] :
          ~ ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X3 @ X3 ) @ ( transi596562996543779814tion_b @ R5 ) )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ R5 )
       => ( X4 != Y5 ) ) ) ).

% irrefl_trancl_rD
thf(fact_974_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: option_b,B: option_b,R5: set_Pr258254617715012167tion_b,C: option_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ ( transi596562996543779814tion_b @ R5 ) )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ B @ C ) @ R5 )
       => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ C ) @ ( transi596562996543779814tion_b @ R5 ) ) ) ) ).

% Transitive_Closure.trancl_into_trancl
thf(fact_975_trancl__into__trancl2,axiom,
    ! [A: option_b,B: option_b,R5: set_Pr258254617715012167tion_b,C: option_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ R5 )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ B @ C ) @ ( transi596562996543779814tion_b @ R5 ) )
       => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ C ) @ ( transi596562996543779814tion_b @ R5 ) ) ) ) ).

% trancl_into_trancl2
thf(fact_976_trancl__trans__induct,axiom,
    ! [X4: option_b,Y5: option_b,R5: set_Pr258254617715012167tion_b,P2: option_b > option_b > $o] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ ( transi596562996543779814tion_b @ R5 ) )
     => ( ! [X3: option_b,Y3: option_b] :
            ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X3 @ Y3 ) @ R5 )
           => ( P2 @ X3 @ Y3 ) )
       => ( ! [X3: option_b,Y3: option_b,Z2: option_b] :
              ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X3 @ Y3 ) @ ( transi596562996543779814tion_b @ R5 ) )
             => ( ( P2 @ X3 @ Y3 )
               => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y3 @ Z2 ) @ ( transi596562996543779814tion_b @ R5 ) )
                 => ( ( P2 @ Y3 @ Z2 )
                   => ( P2 @ X3 @ Z2 ) ) ) ) )
         => ( P2 @ X4 @ Y5 ) ) ) ) ).

% trancl_trans_induct
thf(fact_977_converse__trancl__induct,axiom,
    ! [A: option_b,B: option_b,R5: set_Pr258254617715012167tion_b,P2: option_b > $o] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ ( transi596562996543779814tion_b @ R5 ) )
     => ( ! [Y3: option_b] :
            ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y3 @ B ) @ R5 )
           => ( P2 @ Y3 ) )
       => ( ! [Y3: option_b,Z2: option_b] :
              ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y3 @ Z2 ) @ R5 )
             => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Z2 @ B ) @ ( transi596562996543779814tion_b @ R5 ) )
               => ( ( P2 @ Z2 )
                 => ( P2 @ Y3 ) ) ) )
         => ( P2 @ A ) ) ) ) ).

% converse_trancl_induct
thf(fact_978_refl__onD,axiom,
    ! [A5: set_option_b,R5: set_Pr258254617715012167tion_b,A: option_b] :
      ( ( refl_on_option_b @ A5 @ R5 )
     => ( ( member_option_b @ A @ A5 )
       => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ A ) @ R5 ) ) ) ).

% refl_onD
thf(fact_979_refl__onD1,axiom,
    ! [A5: set_option_b,R5: set_Pr258254617715012167tion_b,X4: option_b,Y5: option_b] :
      ( ( refl_on_option_b @ A5 @ R5 )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ R5 )
       => ( member_option_b @ X4 @ A5 ) ) ) ).

% refl_onD1
thf(fact_980_refl__onD2,axiom,
    ! [A5: set_option_b,R5: set_Pr258254617715012167tion_b,X4: option_b,Y5: option_b] :
      ( ( refl_on_option_b @ A5 @ R5 )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ R5 )
       => ( member_option_b @ Y5 @ A5 ) ) ) ).

% refl_onD2
thf(fact_981_reflD,axiom,
    ! [R5: set_Pr258254617715012167tion_b,A: option_b] :
      ( ( refl_on_option_b @ top_top_set_option_b @ R5 )
     => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ A ) @ R5 ) ) ).

% reflD
thf(fact_982_reflI,axiom,
    ! [R5: set_Pr258254617715012167tion_b] :
      ( ! [X3: option_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X3 @ X3 ) @ R5 )
     => ( refl_on_option_b @ top_top_set_option_b @ R5 ) ) ).

% reflI
thf(fact_983_refl__on__domain,axiom,
    ! [A5: set_option_b,R5: set_Pr258254617715012167tion_b,A: option_b,B: option_b] :
      ( ( refl_on_option_b @ A5 @ R5 )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ R5 )
       => ( ( member_option_b @ A @ A5 )
          & ( member_option_b @ B @ A5 ) ) ) ) ).

% refl_on_domain
thf(fact_984_trancl__map,axiom,
    ! [R5: set_Pr258254617715012167tion_b,F: option_b > option_b,S4: set_Pr258254617715012167tion_b,X4: option_b,Y5: option_b] :
      ( ! [X3: option_b,Y3: option_b] :
          ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X3 @ Y3 ) @ R5 )
         => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ ( F @ X3 ) @ ( F @ Y3 ) ) @ S4 ) )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ ( transi596562996543779814tion_b @ R5 ) )
       => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ ( F @ X4 ) @ ( F @ Y5 ) ) @ ( transi596562996543779814tion_b @ S4 ) ) ) ) ).

% trancl_map
thf(fact_985_trancl__insert2,axiom,
    ! [A: option_b,B: option_b,R5: set_Pr258254617715012167tion_b] :
      ( ( transi596562996543779814tion_b @ ( insert5518505416251236151tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ R5 ) )
      = ( sup_su3110286242275913755tion_b @ ( transi596562996543779814tion_b @ R5 )
        @ ( collec8407377912107443410tion_b
          @ ( produc3378454745690301612on_b_o
            @ ^ [X: option_b,Y4: option_b] :
                ( ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ A ) @ ( transi596562996543779814tion_b @ R5 ) )
                  | ( X = A ) )
                & ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ B @ Y4 ) @ ( transi596562996543779814tion_b @ R5 ) )
                  | ( Y4 = B ) ) ) ) ) ) ) ).

% trancl_insert2
thf(fact_986_SigmaI,axiom,
    ! [A: produc7781606619528532327tion_b,A5: set_Pr258254617715012167tion_b,B: nat,B5: produc7781606619528532327tion_b > set_nat] :
      ( ( member547026996284216464tion_b @ A @ A5 )
     => ( ( member_nat @ B @ ( B5 @ A ) )
       => ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ A @ B ) @ ( produc7925964276780242415_b_nat @ A5 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_987_SigmaI,axiom,
    ! [A: option_b,A5: set_option_b,B: option_b,B5: option_b > set_option_b] :
      ( ( member_option_b @ A @ A5 )
     => ( ( member_option_b @ B @ ( B5 @ A ) )
       => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ ( produc4470504317345894024tion_b @ A5 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_988_SigmaI,axiom,
    ! [A: b,A5: set_b,B: nat,B5: b > set_nat] :
      ( ( member_b @ A @ A5 )
     => ( ( member_nat @ B @ ( B5 @ A ) )
       => ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A @ B ) @ ( product_Sigma_b_nat @ A5 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_989_mem__Sigma__iff,axiom,
    ! [A: produc7781606619528532327tion_b,B: nat,A5: set_Pr258254617715012167tion_b,B5: produc7781606619528532327tion_b > set_nat] :
      ( ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ A @ B ) @ ( produc7925964276780242415_b_nat @ A5 @ B5 ) )
      = ( ( member547026996284216464tion_b @ A @ A5 )
        & ( member_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_990_mem__Sigma__iff,axiom,
    ! [A: option_b,B: option_b,A5: set_option_b,B5: option_b > set_option_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ ( produc4470504317345894024tion_b @ A5 @ B5 ) )
      = ( ( member_option_b @ A @ A5 )
        & ( member_option_b @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_991_mem__Sigma__iff,axiom,
    ! [A: b,B: nat,A5: set_b,B5: b > set_nat] :
      ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A @ B ) @ ( product_Sigma_b_nat @ A5 @ B5 ) )
      = ( ( member_b @ A @ A5 )
        & ( member_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_992_insert__Times__insert,axiom,
    ! [A: produc7781606619528532327tion_b,A5: set_Pr258254617715012167tion_b,B: nat,B5: set_nat] :
      ( ( produc7925964276780242415_b_nat @ ( insert5518505416251236151tion_b @ A @ A5 )
        @ ^ [Uu: produc7781606619528532327tion_b] : ( insert_nat @ B @ B5 ) )
      = ( insert4981606947241745174_b_nat @ ( produc2093246604507004384_b_nat @ A @ B )
        @ ( sup_su934465365883254728_b_nat
          @ ( produc7925964276780242415_b_nat @ A5
            @ ^ [Uu: produc7781606619528532327tion_b] : ( insert_nat @ B @ B5 ) )
          @ ( produc7925964276780242415_b_nat @ ( insert5518505416251236151tion_b @ A @ A5 )
            @ ^ [Uu: produc7781606619528532327tion_b] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_993_insert__Times__insert,axiom,
    ! [A: option_b,A5: set_option_b,B: option_b,B5: set_option_b] :
      ( ( produc4470504317345894024tion_b @ ( insert_option_b @ A @ A5 )
        @ ^ [Uu: option_b] : ( insert_option_b @ B @ B5 ) )
      = ( insert5518505416251236151tion_b @ ( produc3989389360737052631tion_b @ A @ B )
        @ ( sup_su3110286242275913755tion_b
          @ ( produc4470504317345894024tion_b @ A5
            @ ^ [Uu: option_b] : ( insert_option_b @ B @ B5 ) )
          @ ( produc4470504317345894024tion_b @ ( insert_option_b @ A @ A5 )
            @ ^ [Uu: option_b] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_994_insert__Times__insert,axiom,
    ! [A: b,A5: set_b,B: nat,B5: set_nat] :
      ( ( product_Sigma_b_nat @ ( insert_b @ A @ A5 )
        @ ^ [Uu: b] : ( insert_nat @ B @ B5 ) )
      = ( insert66675715695368366_b_nat @ ( product_Pair_b_nat @ A @ B )
        @ ( sup_su6056130500584125792_b_nat
          @ ( product_Sigma_b_nat @ A5
            @ ^ [Uu: b] : ( insert_nat @ B @ B5 ) )
          @ ( product_Sigma_b_nat @ ( insert_b @ A @ A5 )
            @ ^ [Uu: b] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_995_SigmaE,axiom,
    ! [C: produc1396820459975257318_b_nat,A5: set_Pr258254617715012167tion_b,B5: produc7781606619528532327tion_b > set_nat] :
      ( ( member7289896697870905981_b_nat @ C @ ( produc7925964276780242415_b_nat @ A5 @ B5 ) )
     => ~ ! [X3: produc7781606619528532327tion_b] :
            ( ( member547026996284216464tion_b @ X3 @ A5 )
           => ! [Y3: nat] :
                ( ( member_nat @ Y3 @ ( B5 @ X3 ) )
               => ( C
                 != ( produc2093246604507004384_b_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_996_SigmaE,axiom,
    ! [C: produc7781606619528532327tion_b,A5: set_option_b,B5: option_b > set_option_b] :
      ( ( member547026996284216464tion_b @ C @ ( produc4470504317345894024tion_b @ A5 @ B5 ) )
     => ~ ! [X3: option_b] :
            ( ( member_option_b @ X3 @ A5 )
           => ! [Y3: option_b] :
                ( ( member_option_b @ Y3 @ ( B5 @ X3 ) )
               => ( C
                 != ( produc3989389360737052631tion_b @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_997_SigmaE,axiom,
    ! [C: product_prod_b_nat,A5: set_b,B5: b > set_nat] :
      ( ( member6959632917342813205_b_nat @ C @ ( product_Sigma_b_nat @ A5 @ B5 ) )
     => ~ ! [X3: b] :
            ( ( member_b @ X3 @ A5 )
           => ! [Y3: nat] :
                ( ( member_nat @ Y3 @ ( B5 @ X3 ) )
               => ( C
                 != ( product_Pair_b_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_998_SigmaD1,axiom,
    ! [A: produc7781606619528532327tion_b,B: nat,A5: set_Pr258254617715012167tion_b,B5: produc7781606619528532327tion_b > set_nat] :
      ( ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ A @ B ) @ ( produc7925964276780242415_b_nat @ A5 @ B5 ) )
     => ( member547026996284216464tion_b @ A @ A5 ) ) ).

% SigmaD1
thf(fact_999_SigmaD1,axiom,
    ! [A: option_b,B: option_b,A5: set_option_b,B5: option_b > set_option_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ ( produc4470504317345894024tion_b @ A5 @ B5 ) )
     => ( member_option_b @ A @ A5 ) ) ).

% SigmaD1
thf(fact_1000_SigmaD1,axiom,
    ! [A: b,B: nat,A5: set_b,B5: b > set_nat] :
      ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A @ B ) @ ( product_Sigma_b_nat @ A5 @ B5 ) )
     => ( member_b @ A @ A5 ) ) ).

% SigmaD1
thf(fact_1001_SigmaD2,axiom,
    ! [A: produc7781606619528532327tion_b,B: nat,A5: set_Pr258254617715012167tion_b,B5: produc7781606619528532327tion_b > set_nat] :
      ( ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ A @ B ) @ ( produc7925964276780242415_b_nat @ A5 @ B5 ) )
     => ( member_nat @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_1002_SigmaD2,axiom,
    ! [A: option_b,B: option_b,A5: set_option_b,B5: option_b > set_option_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ ( produc4470504317345894024tion_b @ A5 @ B5 ) )
     => ( member_option_b @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_1003_SigmaD2,axiom,
    ! [A: b,B: nat,A5: set_b,B5: b > set_nat] :
      ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A @ B ) @ ( product_Sigma_b_nat @ A5 @ B5 ) )
     => ( member_nat @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_1004_SigmaE2,axiom,
    ! [A: produc7781606619528532327tion_b,B: nat,A5: set_Pr258254617715012167tion_b,B5: produc7781606619528532327tion_b > set_nat] :
      ( ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ A @ B ) @ ( produc7925964276780242415_b_nat @ A5 @ B5 ) )
     => ~ ( ( member547026996284216464tion_b @ A @ A5 )
         => ~ ( member_nat @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1005_SigmaE2,axiom,
    ! [A: option_b,B: option_b,A5: set_option_b,B5: option_b > set_option_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ ( produc4470504317345894024tion_b @ A5 @ B5 ) )
     => ~ ( ( member_option_b @ A @ A5 )
         => ~ ( member_option_b @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1006_SigmaE2,axiom,
    ! [A: b,B: nat,A5: set_b,B5: b > set_nat] :
      ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A @ B ) @ ( product_Sigma_b_nat @ A5 @ B5 ) )
     => ~ ( ( member_b @ A @ A5 )
         => ~ ( member_nat @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1007_trancl__induct2,axiom,
    ! [Ax: produc7781606619528532327tion_b,Ay: nat,Bx: produc7781606619528532327tion_b,By: nat,R5: set_Pr8202489598513264781_b_nat,P2: produc7781606619528532327tion_b > nat > $o] :
      ( ( member4316259128586992598_b_nat @ ( produc7802874649797638501_b_nat @ ( produc2093246604507004384_b_nat @ Ax @ Ay ) @ ( produc2093246604507004384_b_nat @ Bx @ By ) ) @ ( transi4339739544926745349_b_nat @ R5 ) )
     => ( ! [A4: produc7781606619528532327tion_b,B3: nat] :
            ( ( member4316259128586992598_b_nat @ ( produc7802874649797638501_b_nat @ ( produc2093246604507004384_b_nat @ Ax @ Ay ) @ ( produc2093246604507004384_b_nat @ A4 @ B3 ) ) @ R5 )
           => ( P2 @ A4 @ B3 ) )
       => ( ! [A4: produc7781606619528532327tion_b,B3: nat,Aa2: produc7781606619528532327tion_b,Ba: nat] :
              ( ( member4316259128586992598_b_nat @ ( produc7802874649797638501_b_nat @ ( produc2093246604507004384_b_nat @ Ax @ Ay ) @ ( produc2093246604507004384_b_nat @ A4 @ B3 ) ) @ ( transi4339739544926745349_b_nat @ R5 ) )
             => ( ( member4316259128586992598_b_nat @ ( produc7802874649797638501_b_nat @ ( produc2093246604507004384_b_nat @ A4 @ B3 ) @ ( produc2093246604507004384_b_nat @ Aa2 @ Ba ) ) @ R5 )
               => ( ( P2 @ A4 @ B3 )
                 => ( P2 @ Aa2 @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_1008_trancl__induct2,axiom,
    ! [Ax: option_b,Ay: option_b,Bx: option_b,By: option_b,R5: set_Pr5366877986981747911tion_b,P2: option_b > option_b > $o] :
      ( ( member6435960643780680464tion_b @ ( produc7954421581189286743tion_b @ ( produc3989389360737052631tion_b @ Ax @ Ay ) @ ( produc3989389360737052631tion_b @ Bx @ By ) ) @ ( transi5393868930814811656tion_b @ R5 ) )
     => ( ! [A4: option_b,B3: option_b] :
            ( ( member6435960643780680464tion_b @ ( produc7954421581189286743tion_b @ ( produc3989389360737052631tion_b @ Ax @ Ay ) @ ( produc3989389360737052631tion_b @ A4 @ B3 ) ) @ R5 )
           => ( P2 @ A4 @ B3 ) )
       => ( ! [A4: option_b,B3: option_b,Aa2: option_b,Ba: option_b] :
              ( ( member6435960643780680464tion_b @ ( produc7954421581189286743tion_b @ ( produc3989389360737052631tion_b @ Ax @ Ay ) @ ( produc3989389360737052631tion_b @ A4 @ B3 ) ) @ ( transi5393868930814811656tion_b @ R5 ) )
             => ( ( member6435960643780680464tion_b @ ( produc7954421581189286743tion_b @ ( produc3989389360737052631tion_b @ A4 @ B3 ) @ ( produc3989389360737052631tion_b @ Aa2 @ Ba ) ) @ R5 )
               => ( ( P2 @ A4 @ B3 )
                 => ( P2 @ Aa2 @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_1009_trancl__induct2,axiom,
    ! [Ax: b,Ay: nat,Bx: b,By: nat,R5: set_Pr5241429435540800445_b_nat,P2: b > nat > $o] :
      ( ( member4966892972527255302_b_nat @ ( produc2793586353817733269_b_nat @ ( product_Pair_b_nat @ Ax @ Ay ) @ ( product_Pair_b_nat @ Bx @ By ) ) @ ( transi8041030836719688861_b_nat @ R5 ) )
     => ( ! [A4: b,B3: nat] :
            ( ( member4966892972527255302_b_nat @ ( produc2793586353817733269_b_nat @ ( product_Pair_b_nat @ Ax @ Ay ) @ ( product_Pair_b_nat @ A4 @ B3 ) ) @ R5 )
           => ( P2 @ A4 @ B3 ) )
       => ( ! [A4: b,B3: nat,Aa2: b,Ba: nat] :
              ( ( member4966892972527255302_b_nat @ ( produc2793586353817733269_b_nat @ ( product_Pair_b_nat @ Ax @ Ay ) @ ( product_Pair_b_nat @ A4 @ B3 ) ) @ ( transi8041030836719688861_b_nat @ R5 ) )
             => ( ( member4966892972527255302_b_nat @ ( produc2793586353817733269_b_nat @ ( product_Pair_b_nat @ A4 @ B3 ) @ ( product_Pair_b_nat @ Aa2 @ Ba ) ) @ R5 )
               => ( ( P2 @ A4 @ B3 )
                 => ( P2 @ Aa2 @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_1010_refl__on__def,axiom,
    ( refl_on_option_b
    = ( ^ [A8: set_option_b,R2: set_Pr258254617715012167tion_b] :
          ( ( ord_le6367397903258896871tion_b @ R2
            @ ( produc4470504317345894024tion_b @ A8
              @ ^ [Uu: option_b] : A8 ) )
          & ! [X: option_b] :
              ( ( member_option_b @ X @ A8 )
             => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ X ) @ R2 ) ) ) ) ) ).

% refl_on_def
thf(fact_1011_refl__onI,axiom,
    ! [R5: set_Pr258254617715012167tion_b,A5: set_option_b] :
      ( ( ord_le6367397903258896871tion_b @ R5
        @ ( produc4470504317345894024tion_b @ A5
          @ ^ [Uu: option_b] : A5 ) )
     => ( ! [X3: option_b] :
            ( ( member_option_b @ X3 @ A5 )
           => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X3 @ X3 ) @ R5 ) )
       => ( refl_on_option_b @ A5 @ R5 ) ) ) ).

% refl_onI
thf(fact_1012_lift__root__step__StrictParallel__seq,axiom,
    ! [R: set_Pr3134649703833893895term_b,F2: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le2408605988743369639term_b @ R
        @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ F2 )
          @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ F2 ) ) )
     => ( ord_le2408605988743369639term_b @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PAny @ fOR_ex5070955383060788274rallel @ R ) @ ( transi8854406251171887185term_b @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PAny @ fOR_ext_ESingle @ R ) ) ) ) ).

% lift_root_step_StrictParallel_seq
thf(fact_1013_relation__of__def,axiom,
    ( order_7817789072647864851tion_b
    = ( ^ [P3: option_b > option_b > $o,A8: set_option_b] :
          ( collec8407377912107443410tion_b
          @ ( produc3378454745690301612on_b_o
            @ ^ [A6: option_b,B4: option_b] :
                ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A6 @ B4 )
                  @ ( produc4470504317345894024tion_b @ A8
                    @ ^ [Uu: option_b] : A8 ) )
                & ( P3 @ A6 @ B4 ) ) ) ) ) ) ).

% relation_of_def
thf(fact_1014_trancl__insert,axiom,
    ! [Y5: option_b,X4: option_b,R5: set_Pr258254617715012167tion_b] :
      ( ( transi596562996543779814tion_b @ ( insert5518505416251236151tion_b @ ( produc3989389360737052631tion_b @ Y5 @ X4 ) @ R5 ) )
      = ( sup_su3110286242275913755tion_b @ ( transi596562996543779814tion_b @ R5 )
        @ ( collec8407377912107443410tion_b
          @ ( produc3378454745690301612on_b_o
            @ ^ [A6: option_b,B4: option_b] :
                ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A6 @ Y5 ) @ ( transi330218195068109384tion_b @ R5 ) )
                & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ B4 ) @ ( transi330218195068109384tion_b @ R5 ) ) ) ) ) ) ) ).

% trancl_insert
thf(fact_1015_rtrancl_Ocases,axiom,
    ! [A1: option_b,A22: option_b,R5: set_Pr258254617715012167tion_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A1 @ A22 ) @ ( transi330218195068109384tion_b @ R5 ) )
     => ( ( A22 != A1 )
       => ~ ! [B3: option_b] :
              ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A1 @ B3 ) @ ( transi330218195068109384tion_b @ R5 ) )
             => ~ ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ B3 @ A22 ) @ R5 ) ) ) ) ).

% rtrancl.cases
thf(fact_1016_rtrancl_Osimps,axiom,
    ! [A1: option_b,A22: option_b,R5: set_Pr258254617715012167tion_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A1 @ A22 ) @ ( transi330218195068109384tion_b @ R5 ) )
      = ( ? [A6: option_b] :
            ( ( A1 = A6 )
            & ( A22 = A6 ) )
        | ? [A6: option_b,B4: option_b,C4: option_b] :
            ( ( A1 = A6 )
            & ( A22 = C4 )
            & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A6 @ B4 ) @ ( transi330218195068109384tion_b @ R5 ) )
            & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ B4 @ C4 ) @ R5 ) ) ) ) ).

% rtrancl.simps
thf(fact_1017_rtrancl_Ortrancl__refl,axiom,
    ! [A: option_b,R5: set_Pr258254617715012167tion_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ A ) @ ( transi330218195068109384tion_b @ R5 ) ) ).

% rtrancl.rtrancl_refl
thf(fact_1018_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: option_b,B: option_b,R5: set_Pr258254617715012167tion_b,C: option_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ ( transi330218195068109384tion_b @ R5 ) )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ B @ C ) @ R5 )
       => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ C ) @ ( transi330218195068109384tion_b @ R5 ) ) ) ) ).

% rtrancl.rtrancl_into_rtrancl
thf(fact_1019_rtranclE,axiom,
    ! [A: option_b,B: option_b,R5: set_Pr258254617715012167tion_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ ( transi330218195068109384tion_b @ R5 ) )
     => ( ( A != B )
       => ~ ! [Y3: option_b] :
              ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ Y3 ) @ ( transi330218195068109384tion_b @ R5 ) )
             => ~ ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y3 @ B ) @ R5 ) ) ) ) ).

% rtranclE
thf(fact_1020_rtrancl__trans,axiom,
    ! [X4: option_b,Y5: option_b,R5: set_Pr258254617715012167tion_b,Z: option_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ ( transi330218195068109384tion_b @ R5 ) )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y5 @ Z ) @ ( transi330218195068109384tion_b @ R5 ) )
       => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Z ) @ ( transi330218195068109384tion_b @ R5 ) ) ) ) ).

% rtrancl_trans
thf(fact_1021_rtrancl__induct,axiom,
    ! [A: option_b,B: option_b,R5: set_Pr258254617715012167tion_b,P2: option_b > $o] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ ( transi330218195068109384tion_b @ R5 ) )
     => ( ( P2 @ A )
       => ( ! [Y3: option_b,Z2: option_b] :
              ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ Y3 ) @ ( transi330218195068109384tion_b @ R5 ) )
             => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y3 @ Z2 ) @ R5 )
               => ( ( P2 @ Y3 )
                 => ( P2 @ Z2 ) ) ) )
         => ( P2 @ B ) ) ) ) ).

% rtrancl_induct
thf(fact_1022_converse__rtranclE,axiom,
    ! [X4: option_b,Z: option_b,R5: set_Pr258254617715012167tion_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Z ) @ ( transi330218195068109384tion_b @ R5 ) )
     => ( ( X4 != Z )
       => ~ ! [Y3: option_b] :
              ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y3 ) @ R5 )
             => ~ ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y3 @ Z ) @ ( transi330218195068109384tion_b @ R5 ) ) ) ) ) ).

% converse_rtranclE
thf(fact_1023_converse__rtrancl__induct,axiom,
    ! [A: option_b,B: option_b,R5: set_Pr258254617715012167tion_b,P2: option_b > $o] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ ( transi330218195068109384tion_b @ R5 ) )
     => ( ( P2 @ B )
       => ( ! [Y3: option_b,Z2: option_b] :
              ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y3 @ Z2 ) @ R5 )
             => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Z2 @ B ) @ ( transi330218195068109384tion_b @ R5 ) )
               => ( ( P2 @ Z2 )
                 => ( P2 @ Y3 ) ) ) )
         => ( P2 @ A ) ) ) ) ).

% converse_rtrancl_induct
thf(fact_1024_converse__rtrancl__into__rtrancl,axiom,
    ! [A: option_b,B: option_b,R5: set_Pr258254617715012167tion_b,C: option_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ R5 )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ B @ C ) @ ( transi330218195068109384tion_b @ R5 ) )
       => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ C ) @ ( transi330218195068109384tion_b @ R5 ) ) ) ) ).

% converse_rtrancl_into_rtrancl
thf(fact_1025_trancl__rtrancl__trancl,axiom,
    ! [A: option_b,B: option_b,R5: set_Pr258254617715012167tion_b,C: option_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ ( transi596562996543779814tion_b @ R5 ) )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ B @ C ) @ ( transi330218195068109384tion_b @ R5 ) )
       => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ C ) @ ( transi596562996543779814tion_b @ R5 ) ) ) ) ).

% trancl_rtrancl_trancl
thf(fact_1026_rtrancl__trancl__trancl,axiom,
    ! [X4: option_b,Y5: option_b,R5: set_Pr258254617715012167tion_b,Z: option_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ ( transi330218195068109384tion_b @ R5 ) )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y5 @ Z ) @ ( transi596562996543779814tion_b @ R5 ) )
       => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Z ) @ ( transi596562996543779814tion_b @ R5 ) ) ) ) ).

% rtrancl_trancl_trancl
thf(fact_1027_rtrancl__into__trancl2,axiom,
    ! [A: option_b,B: option_b,R5: set_Pr258254617715012167tion_b,C: option_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ R5 )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ B @ C ) @ ( transi330218195068109384tion_b @ R5 ) )
       => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ C ) @ ( transi596562996543779814tion_b @ R5 ) ) ) ) ).

% rtrancl_into_trancl2
thf(fact_1028_rtrancl__into__trancl1,axiom,
    ! [A: option_b,B: option_b,R5: set_Pr258254617715012167tion_b,C: option_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ ( transi330218195068109384tion_b @ R5 ) )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ B @ C ) @ R5 )
       => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ C ) @ ( transi596562996543779814tion_b @ R5 ) ) ) ) ).

% rtrancl_into_trancl1
thf(fact_1029_rtrancl__eq__or__trancl,axiom,
    ! [X4: option_b,Y5: option_b,R: set_Pr258254617715012167tion_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ ( transi330218195068109384tion_b @ R ) )
      = ( ( X4 = Y5 )
        | ( ( X4 != Y5 )
          & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ ( transi596562996543779814tion_b @ R ) ) ) ) ) ).

% rtrancl_eq_or_trancl
thf(fact_1030_trancl__into__rtrancl,axiom,
    ! [A: option_b,B: option_b,R5: set_Pr258254617715012167tion_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ ( transi596562996543779814tion_b @ R5 ) )
     => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ ( transi330218195068109384tion_b @ R5 ) ) ) ).

% trancl_into_rtrancl
thf(fact_1031_tranclD2,axiom,
    ! [X4: option_b,Y5: option_b,R: set_Pr258254617715012167tion_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ ( transi596562996543779814tion_b @ R ) )
     => ? [Z2: option_b] :
          ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Z2 ) @ ( transi330218195068109384tion_b @ R ) )
          & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Z2 @ Y5 ) @ R ) ) ) ).

% tranclD2
thf(fact_1032_rtranclD,axiom,
    ! [A: option_b,B: option_b,R: set_Pr258254617715012167tion_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ ( transi330218195068109384tion_b @ R ) )
     => ( ( A = B )
        | ( ( A != B )
          & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ ( transi596562996543779814tion_b @ R ) ) ) ) ) ).

% rtranclD
thf(fact_1033_tranclD,axiom,
    ! [X4: option_b,Y5: option_b,R: set_Pr258254617715012167tion_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ ( transi596562996543779814tion_b @ R ) )
     => ? [Z2: option_b] :
          ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Z2 ) @ R )
          & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Z2 @ Y5 ) @ ( transi330218195068109384tion_b @ R ) ) ) ) ).

% tranclD
thf(fact_1034_rtrancl__Un__separatorE,axiom,
    ! [A: option_b,B: option_b,P2: set_Pr258254617715012167tion_b,Q: set_Pr258254617715012167tion_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ ( transi330218195068109384tion_b @ ( sup_su3110286242275913755tion_b @ P2 @ Q ) ) )
     => ( ! [X3: option_b,Y3: option_b] :
            ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ X3 ) @ ( transi330218195068109384tion_b @ P2 ) )
           => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X3 @ Y3 ) @ Q )
             => ( X3 = Y3 ) ) )
       => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ ( transi330218195068109384tion_b @ P2 ) ) ) ) ).

% rtrancl_Un_separatorE
thf(fact_1035_rtrancl__Un__separator__converseE,axiom,
    ! [A: option_b,B: option_b,P2: set_Pr258254617715012167tion_b,Q: set_Pr258254617715012167tion_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ ( transi330218195068109384tion_b @ ( sup_su3110286242275913755tion_b @ P2 @ Q ) ) )
     => ( ! [X3: option_b,Y3: option_b] :
            ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X3 @ B ) @ ( transi330218195068109384tion_b @ P2 ) )
           => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y3 @ X3 ) @ Q )
             => ( Y3 = X3 ) ) )
       => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ ( transi330218195068109384tion_b @ P2 ) ) ) ) ).

% rtrancl_Un_separator_converseE
thf(fact_1036_rtrancl__induct2,axiom,
    ! [Ax: produc7781606619528532327tion_b,Ay: nat,Bx: produc7781606619528532327tion_b,By: nat,R5: set_Pr8202489598513264781_b_nat,P2: produc7781606619528532327tion_b > nat > $o] :
      ( ( member4316259128586992598_b_nat @ ( produc7802874649797638501_b_nat @ ( produc2093246604507004384_b_nat @ Ax @ Ay ) @ ( produc2093246604507004384_b_nat @ Bx @ By ) ) @ ( transi4479307401911045923_b_nat @ R5 ) )
     => ( ( P2 @ Ax @ Ay )
       => ( ! [A4: produc7781606619528532327tion_b,B3: nat,Aa2: produc7781606619528532327tion_b,Ba: nat] :
              ( ( member4316259128586992598_b_nat @ ( produc7802874649797638501_b_nat @ ( produc2093246604507004384_b_nat @ Ax @ Ay ) @ ( produc2093246604507004384_b_nat @ A4 @ B3 ) ) @ ( transi4479307401911045923_b_nat @ R5 ) )
             => ( ( member4316259128586992598_b_nat @ ( produc7802874649797638501_b_nat @ ( produc2093246604507004384_b_nat @ A4 @ B3 ) @ ( produc2093246604507004384_b_nat @ Aa2 @ Ba ) ) @ R5 )
               => ( ( P2 @ A4 @ B3 )
                 => ( P2 @ Aa2 @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_1037_rtrancl__induct2,axiom,
    ! [Ax: option_b,Ay: option_b,Bx: option_b,By: option_b,R5: set_Pr5366877986981747911tion_b,P2: option_b > option_b > $o] :
      ( ( member6435960643780680464tion_b @ ( produc7954421581189286743tion_b @ ( produc3989389360737052631tion_b @ Ax @ Ay ) @ ( produc3989389360737052631tion_b @ Bx @ By ) ) @ ( transi2849677033159065450tion_b @ R5 ) )
     => ( ( P2 @ Ax @ Ay )
       => ( ! [A4: option_b,B3: option_b,Aa2: option_b,Ba: option_b] :
              ( ( member6435960643780680464tion_b @ ( produc7954421581189286743tion_b @ ( produc3989389360737052631tion_b @ Ax @ Ay ) @ ( produc3989389360737052631tion_b @ A4 @ B3 ) ) @ ( transi2849677033159065450tion_b @ R5 ) )
             => ( ( member6435960643780680464tion_b @ ( produc7954421581189286743tion_b @ ( produc3989389360737052631tion_b @ A4 @ B3 ) @ ( produc3989389360737052631tion_b @ Aa2 @ Ba ) ) @ R5 )
               => ( ( P2 @ A4 @ B3 )
                 => ( P2 @ Aa2 @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_1038_rtrancl__induct2,axiom,
    ! [Ax: b,Ay: nat,Bx: b,By: nat,R5: set_Pr5241429435540800445_b_nat,P2: b > nat > $o] :
      ( ( member4966892972527255302_b_nat @ ( produc2793586353817733269_b_nat @ ( product_Pair_b_nat @ Ax @ Ay ) @ ( product_Pair_b_nat @ Bx @ By ) ) @ ( transi3961590246294786235_b_nat @ R5 ) )
     => ( ( P2 @ Ax @ Ay )
       => ( ! [A4: b,B3: nat,Aa2: b,Ba: nat] :
              ( ( member4966892972527255302_b_nat @ ( produc2793586353817733269_b_nat @ ( product_Pair_b_nat @ Ax @ Ay ) @ ( product_Pair_b_nat @ A4 @ B3 ) ) @ ( transi3961590246294786235_b_nat @ R5 ) )
             => ( ( member4966892972527255302_b_nat @ ( produc2793586353817733269_b_nat @ ( product_Pair_b_nat @ A4 @ B3 ) @ ( product_Pair_b_nat @ Aa2 @ Ba ) ) @ R5 )
               => ( ( P2 @ A4 @ B3 )
                 => ( P2 @ Aa2 @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_1039_converse__rtranclE2,axiom,
    ! [Xa: produc7781606619528532327tion_b,Xb: nat,Za2: produc7781606619528532327tion_b,Zb: nat,R5: set_Pr8202489598513264781_b_nat] :
      ( ( member4316259128586992598_b_nat @ ( produc7802874649797638501_b_nat @ ( produc2093246604507004384_b_nat @ Xa @ Xb ) @ ( produc2093246604507004384_b_nat @ Za2 @ Zb ) ) @ ( transi4479307401911045923_b_nat @ R5 ) )
     => ( ( ( produc2093246604507004384_b_nat @ Xa @ Xb )
         != ( produc2093246604507004384_b_nat @ Za2 @ Zb ) )
       => ~ ! [A4: produc7781606619528532327tion_b,B3: nat] :
              ( ( member4316259128586992598_b_nat @ ( produc7802874649797638501_b_nat @ ( produc2093246604507004384_b_nat @ Xa @ Xb ) @ ( produc2093246604507004384_b_nat @ A4 @ B3 ) ) @ R5 )
             => ~ ( member4316259128586992598_b_nat @ ( produc7802874649797638501_b_nat @ ( produc2093246604507004384_b_nat @ A4 @ B3 ) @ ( produc2093246604507004384_b_nat @ Za2 @ Zb ) ) @ ( transi4479307401911045923_b_nat @ R5 ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_1040_converse__rtranclE2,axiom,
    ! [Xa: option_b,Xb: option_b,Za2: option_b,Zb: option_b,R5: set_Pr5366877986981747911tion_b] :
      ( ( member6435960643780680464tion_b @ ( produc7954421581189286743tion_b @ ( produc3989389360737052631tion_b @ Xa @ Xb ) @ ( produc3989389360737052631tion_b @ Za2 @ Zb ) ) @ ( transi2849677033159065450tion_b @ R5 ) )
     => ( ( ( produc3989389360737052631tion_b @ Xa @ Xb )
         != ( produc3989389360737052631tion_b @ Za2 @ Zb ) )
       => ~ ! [A4: option_b,B3: option_b] :
              ( ( member6435960643780680464tion_b @ ( produc7954421581189286743tion_b @ ( produc3989389360737052631tion_b @ Xa @ Xb ) @ ( produc3989389360737052631tion_b @ A4 @ B3 ) ) @ R5 )
             => ~ ( member6435960643780680464tion_b @ ( produc7954421581189286743tion_b @ ( produc3989389360737052631tion_b @ A4 @ B3 ) @ ( produc3989389360737052631tion_b @ Za2 @ Zb ) ) @ ( transi2849677033159065450tion_b @ R5 ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_1041_converse__rtranclE2,axiom,
    ! [Xa: b,Xb: nat,Za2: b,Zb: nat,R5: set_Pr5241429435540800445_b_nat] :
      ( ( member4966892972527255302_b_nat @ ( produc2793586353817733269_b_nat @ ( product_Pair_b_nat @ Xa @ Xb ) @ ( product_Pair_b_nat @ Za2 @ Zb ) ) @ ( transi3961590246294786235_b_nat @ R5 ) )
     => ( ( ( product_Pair_b_nat @ Xa @ Xb )
         != ( product_Pair_b_nat @ Za2 @ Zb ) )
       => ~ ! [A4: b,B3: nat] :
              ( ( member4966892972527255302_b_nat @ ( produc2793586353817733269_b_nat @ ( product_Pair_b_nat @ Xa @ Xb ) @ ( product_Pair_b_nat @ A4 @ B3 ) ) @ R5 )
             => ~ ( member4966892972527255302_b_nat @ ( produc2793586353817733269_b_nat @ ( product_Pair_b_nat @ A4 @ B3 ) @ ( product_Pair_b_nat @ Za2 @ Zb ) ) @ ( transi3961590246294786235_b_nat @ R5 ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_1042_converse__rtrancl__induct2,axiom,
    ! [Ax: produc7781606619528532327tion_b,Ay: nat,Bx: produc7781606619528532327tion_b,By: nat,R5: set_Pr8202489598513264781_b_nat,P2: produc7781606619528532327tion_b > nat > $o] :
      ( ( member4316259128586992598_b_nat @ ( produc7802874649797638501_b_nat @ ( produc2093246604507004384_b_nat @ Ax @ Ay ) @ ( produc2093246604507004384_b_nat @ Bx @ By ) ) @ ( transi4479307401911045923_b_nat @ R5 ) )
     => ( ( P2 @ Bx @ By )
       => ( ! [A4: produc7781606619528532327tion_b,B3: nat,Aa2: produc7781606619528532327tion_b,Ba: nat] :
              ( ( member4316259128586992598_b_nat @ ( produc7802874649797638501_b_nat @ ( produc2093246604507004384_b_nat @ A4 @ B3 ) @ ( produc2093246604507004384_b_nat @ Aa2 @ Ba ) ) @ R5 )
             => ( ( member4316259128586992598_b_nat @ ( produc7802874649797638501_b_nat @ ( produc2093246604507004384_b_nat @ Aa2 @ Ba ) @ ( produc2093246604507004384_b_nat @ Bx @ By ) ) @ ( transi4479307401911045923_b_nat @ R5 ) )
               => ( ( P2 @ Aa2 @ Ba )
                 => ( P2 @ A4 @ B3 ) ) ) )
         => ( P2 @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_1043_converse__rtrancl__induct2,axiom,
    ! [Ax: option_b,Ay: option_b,Bx: option_b,By: option_b,R5: set_Pr5366877986981747911tion_b,P2: option_b > option_b > $o] :
      ( ( member6435960643780680464tion_b @ ( produc7954421581189286743tion_b @ ( produc3989389360737052631tion_b @ Ax @ Ay ) @ ( produc3989389360737052631tion_b @ Bx @ By ) ) @ ( transi2849677033159065450tion_b @ R5 ) )
     => ( ( P2 @ Bx @ By )
       => ( ! [A4: option_b,B3: option_b,Aa2: option_b,Ba: option_b] :
              ( ( member6435960643780680464tion_b @ ( produc7954421581189286743tion_b @ ( produc3989389360737052631tion_b @ A4 @ B3 ) @ ( produc3989389360737052631tion_b @ Aa2 @ Ba ) ) @ R5 )
             => ( ( member6435960643780680464tion_b @ ( produc7954421581189286743tion_b @ ( produc3989389360737052631tion_b @ Aa2 @ Ba ) @ ( produc3989389360737052631tion_b @ Bx @ By ) ) @ ( transi2849677033159065450tion_b @ R5 ) )
               => ( ( P2 @ Aa2 @ Ba )
                 => ( P2 @ A4 @ B3 ) ) ) )
         => ( P2 @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_1044_converse__rtrancl__induct2,axiom,
    ! [Ax: b,Ay: nat,Bx: b,By: nat,R5: set_Pr5241429435540800445_b_nat,P2: b > nat > $o] :
      ( ( member4966892972527255302_b_nat @ ( produc2793586353817733269_b_nat @ ( product_Pair_b_nat @ Ax @ Ay ) @ ( product_Pair_b_nat @ Bx @ By ) ) @ ( transi3961590246294786235_b_nat @ R5 ) )
     => ( ( P2 @ Bx @ By )
       => ( ! [A4: b,B3: nat,Aa2: b,Ba: nat] :
              ( ( member4966892972527255302_b_nat @ ( produc2793586353817733269_b_nat @ ( product_Pair_b_nat @ A4 @ B3 ) @ ( product_Pair_b_nat @ Aa2 @ Ba ) ) @ R5 )
             => ( ( member4966892972527255302_b_nat @ ( produc2793586353817733269_b_nat @ ( product_Pair_b_nat @ Aa2 @ Ba ) @ ( product_Pair_b_nat @ Bx @ By ) ) @ ( transi3961590246294786235_b_nat @ R5 ) )
               => ( ( P2 @ Aa2 @ Ba )
                 => ( P2 @ A4 @ B3 ) ) ) )
         => ( P2 @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_1045_lift__root__step__sig,axiom,
    ! [R3: set_Pr3134649703833893895term_b,F2: set_Pr1307281990691478580_b_nat,W: fOR_pos_step,X6: fOR_ext_step] :
      ( ( ord_le2408605988743369639term_b @ R3
        @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ F2 )
          @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ F2 ) ) )
     => ( ord_le2408605988743369639term_b @ ( lift_R8854095231239425056step_b @ F2 @ W @ X6 @ R3 )
        @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ F2 )
          @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ F2 ) ) ) ) ).

% lift_root_step_sig
thf(fact_1046_trancl__subset__Sigma__aux,axiom,
    ! [A: option_b,B: option_b,R5: set_Pr258254617715012167tion_b,A5: set_option_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ ( transi330218195068109384tion_b @ R5 ) )
     => ( ( ord_le6367397903258896871tion_b @ R5
          @ ( produc4470504317345894024tion_b @ A5
            @ ^ [Uu: option_b] : A5 ) )
       => ( ( A = B )
          | ( member_option_b @ A @ A5 ) ) ) ) ).

% trancl_subset_Sigma_aux
thf(fact_1047_lift__root__step__sig_H,axiom,
    ! [R3: set_Pr3134649703833893895term_b,G2: set_Pr1307281990691478580_b_nat,H2: set_Pr1307281990691478580_b_nat,F2: set_Pr1307281990691478580_b_nat,W: fOR_pos_step,X6: fOR_ext_step] :
      ( ( ord_le2408605988743369639term_b @ R3
        @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ G2 )
          @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ H2 ) ) )
     => ( ( ord_le5038853854344477844_b_nat @ F2 @ G2 )
       => ( ( ord_le5038853854344477844_b_nat @ F2 @ H2 )
         => ( ord_le2408605988743369639term_b @ ( lift_R8854095231239425056step_b @ F2 @ W @ X6 @ R3 )
            @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ G2 )
              @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ H2 ) ) ) ) ) ) ).

% lift_root_step_sig'
thf(fact_1048_rtrancl__insert,axiom,
    ! [A: option_b,B: option_b,R5: set_Pr258254617715012167tion_b] :
      ( ( transi330218195068109384tion_b @ ( insert5518505416251236151tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ R5 ) )
      = ( sup_su3110286242275913755tion_b @ ( transi330218195068109384tion_b @ R5 )
        @ ( collec8407377912107443410tion_b
          @ ( produc3378454745690301612on_b_o
            @ ^ [X: option_b,Y4: option_b] :
                ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ A ) @ ( transi330218195068109384tion_b @ R5 ) )
                & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ B @ Y4 ) @ ( transi330218195068109384tion_b @ R5 ) ) ) ) ) ) ) ).

% rtrancl_insert
thf(fact_1049_strongly__confluentI,axiom,
    ! [R5: set_Pr258254617715012167tion_b] :
      ( ! [X3: option_b,Y3: option_b,Z2: option_b] :
          ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X3 @ Y3 ) @ R5 )
         => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X3 @ Z2 ) @ R5 )
           => ? [U: option_b] :
                ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y3 @ U ) @ ( transi330218195068109384tion_b @ R5 ) )
                & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Z2 @ U ) @ ( sup_su3110286242275913755tion_b @ R5 @ id_option_b ) ) ) ) )
     => ( abstra1975357486212897574tion_b @ R5 @ top_top_set_option_b ) ) ).

% strongly_confluentI
thf(fact_1050_lift__root__step__Parallels__single,axiom,
    ! [R: set_Pr3134649703833893895term_b,F2: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le2408605988743369639term_b @ R
        @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ F2 )
          @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ F2 ) ) )
     => ( ( transi8854406251171887185term_b @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PAny @ fOR_ext_EParallel @ R ) )
        = ( sup_su1163039302778497499term_b @ ( transi8854406251171887185term_b @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PAny @ fOR_ext_ESingle @ R ) )
          @ ( inf_in8391146210534189557term_b @ id_Ground_gterm_b
            @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ F2 )
              @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ F2 ) ) ) ) ) ) ).

% lift_root_step_Parallels_single
thf(fact_1051_lift__root__step__Parallel__seq,axiom,
    ! [R: set_Pr3134649703833893895term_b,F2: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le2408605988743369639term_b @ R
        @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ F2 )
          @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ F2 ) ) )
     => ( ord_le2408605988743369639term_b @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PAny @ fOR_ext_EParallel @ R )
        @ ( sup_su1163039302778497499term_b @ ( transi8854406251171887185term_b @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PAny @ fOR_ext_ESingle @ R ) )
          @ ( inf_in8391146210534189557term_b @ id_Ground_gterm_b
            @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ F2 )
              @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ F2 ) ) ) ) ) ) ).

% lift_root_step_Parallel_seq
thf(fact_1052_strongly__confluent__on__def,axiom,
    ( abstra1975357486212897574tion_b
    = ( ^ [R2: set_Pr258254617715012167tion_b,A8: set_option_b] :
        ! [X: option_b] :
          ( ( member_option_b @ X @ A8 )
         => ! [Y4: option_b,Z4: option_b] :
              ( ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ Y4 ) @ R2 )
                & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ Z4 ) @ R2 ) )
             => ? [U2: option_b] :
                  ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y4 @ U2 ) @ ( transi330218195068109384tion_b @ R2 ) )
                  & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Z4 @ U2 ) @ ( sup_su3110286242275913755tion_b @ R2 @ id_option_b ) ) ) ) ) ) ) ).

% strongly_confluent_on_def
thf(fact_1053_inf_Obounded__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
      = ( ( ord_less_eq_nat @ A @ B )
        & ( ord_less_eq_nat @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_1054_le__inf__iff,axiom,
    ! [X4: nat,Y5: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X4 @ ( inf_inf_nat @ Y5 @ Z ) )
      = ( ( ord_less_eq_nat @ X4 @ Y5 )
        & ( ord_less_eq_nat @ X4 @ Z ) ) ) ).

% le_inf_iff
thf(fact_1055_inf__sup__ord_I2_J,axiom,
    ! [X4: nat,Y5: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X4 @ Y5 ) @ Y5 ) ).

% inf_sup_ord(2)
thf(fact_1056_inf__sup__ord_I1_J,axiom,
    ! [X4: nat,Y5: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X4 @ Y5 ) @ X4 ) ).

% inf_sup_ord(1)
thf(fact_1057_inf__le1,axiom,
    ! [X4: nat,Y5: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X4 @ Y5 ) @ X4 ) ).

% inf_le1
thf(fact_1058_inf__le2,axiom,
    ! [X4: nat,Y5: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X4 @ Y5 ) @ Y5 ) ).

% inf_le2
thf(fact_1059_le__infE,axiom,
    ! [X4: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X4 @ ( inf_inf_nat @ A @ B ) )
     => ~ ( ( ord_less_eq_nat @ X4 @ A )
         => ~ ( ord_less_eq_nat @ X4 @ B ) ) ) ).

% le_infE
thf(fact_1060_le__infI,axiom,
    ! [X4: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X4 @ A )
     => ( ( ord_less_eq_nat @ X4 @ B )
       => ( ord_less_eq_nat @ X4 @ ( inf_inf_nat @ A @ B ) ) ) ) ).

% le_infI
thf(fact_1061_inf__mono,axiom,
    ! [A: nat,C: nat,B: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ D2 )
       => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_1062_le__infI1,axiom,
    ! [A: nat,X4: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ X4 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X4 ) ) ).

% le_infI1
thf(fact_1063_le__infI2,axiom,
    ! [B: nat,X4: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ X4 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X4 ) ) ).

% le_infI2
thf(fact_1064_inf_OorderE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( A
        = ( inf_inf_nat @ A @ B ) ) ) ).

% inf.orderE
thf(fact_1065_inf_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( inf_inf_nat @ A @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% inf.orderI
thf(fact_1066_inf__unique,axiom,
    ! [F: nat > nat > nat,X4: nat,Y5: nat] :
      ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y3 ) @ X3 )
     => ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y3 ) @ Y3 )
       => ( ! [X3: nat,Y3: nat,Z2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ( ord_less_eq_nat @ X3 @ Z2 )
               => ( ord_less_eq_nat @ X3 @ ( F @ Y3 @ Z2 ) ) ) )
         => ( ( inf_inf_nat @ X4 @ Y5 )
            = ( F @ X4 @ Y5 ) ) ) ) ) ).

% inf_unique
thf(fact_1067_le__iff__inf,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y4: nat] :
          ( ( inf_inf_nat @ X @ Y4 )
          = X ) ) ) ).

% le_iff_inf
thf(fact_1068_inf_Oabsorb1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( inf_inf_nat @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_1069_inf_Oabsorb2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( inf_inf_nat @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_1070_inf__absorb1,axiom,
    ! [X4: nat,Y5: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y5 )
     => ( ( inf_inf_nat @ X4 @ Y5 )
        = X4 ) ) ).

% inf_absorb1
thf(fact_1071_inf__absorb2,axiom,
    ! [Y5: nat,X4: nat] :
      ( ( ord_less_eq_nat @ Y5 @ X4 )
     => ( ( inf_inf_nat @ X4 @ Y5 )
        = Y5 ) ) ).

% inf_absorb2
thf(fact_1072_inf_OboundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
     => ~ ( ( ord_less_eq_nat @ A @ B )
         => ~ ( ord_less_eq_nat @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_1073_inf_OboundedI,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ C )
       => ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_1074_inf__greatest,axiom,
    ! [X4: nat,Y5: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y5 )
     => ( ( ord_less_eq_nat @ X4 @ Z )
       => ( ord_less_eq_nat @ X4 @ ( inf_inf_nat @ Y5 @ Z ) ) ) ) ).

% inf_greatest
thf(fact_1075_inf_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [A6: nat,B4: nat] :
          ( A6
          = ( inf_inf_nat @ A6 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_1076_inf_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_1077_inf_Ocobounded2,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_1078_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [A6: nat,B4: nat] :
          ( ( inf_inf_nat @ A6 @ B4 )
          = A6 ) ) ) ).

% inf.absorb_iff1
thf(fact_1079_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A6: nat] :
          ( ( inf_inf_nat @ A6 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_1080_inf_OcoboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_1081_inf_OcoboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ C )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_1082_distrib__sup__le,axiom,
    ! [X4: nat,Y5: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X4 @ ( inf_inf_nat @ Y5 @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X4 @ Y5 ) @ ( sup_sup_nat @ X4 @ Z ) ) ) ).

% distrib_sup_le
thf(fact_1083_distrib__inf__le,axiom,
    ! [X4: nat,Y5: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X4 @ Y5 ) @ ( inf_inf_nat @ X4 @ Z ) ) @ ( inf_inf_nat @ X4 @ ( sup_sup_nat @ Y5 @ Z ) ) ) ).

% distrib_inf_le
thf(fact_1084_rtrancl__Restr,axiom,
    ! [X4: option_b,Y5: option_b,R5: set_Pr258254617715012167tion_b,A5: set_option_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 )
        @ ( transi330218195068109384tion_b
          @ ( inf_in9047234346837383221tion_b @ R5
            @ ( produc4470504317345894024tion_b @ A5
              @ ^ [Uu: option_b] : A5 ) ) ) )
     => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ ( transi330218195068109384tion_b @ R5 ) ) ) ).

% rtrancl_Restr
thf(fact_1085_lift__root__step__Parallel__conv,axiom,
    ! [F2: set_Pr1307281990691478580_b_nat,W: fOR_pos_step,R3: set_Pr3134649703833893895term_b] :
      ( ( lift_R8854095231239425056step_b @ F2 @ W @ fOR_ext_EParallel @ R3 )
      = ( sup_su1163039302778497499term_b @ ( lift_R8854095231239425056step_b @ F2 @ W @ fOR_ex5070955383060788274rallel @ R3 )
        @ ( inf_in8391146210534189557term_b @ id_Ground_gterm_b
          @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ F2 )
            @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ F2 ) ) ) ) ) ).

% lift_root_step_Parallel_conv
thf(fact_1086_lift__root__step_Osimps_I9_J,axiom,
    ! [F2: set_Pr1307281990691478580_b_nat,R3: set_Pr3134649703833893895term_b] :
      ( ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PRoot @ fOR_ext_EParallel @ R3 )
      = ( sup_su1163039302778497499term_b @ R3
        @ ( inf_in8391146210534189557term_b @ id_Ground_gterm_b
          @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ F2 )
            @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ F2 ) ) ) ) ) ).

% lift_root_step.simps(9)
thf(fact_1087_strongly__confluent__on__E11,axiom,
    ! [R5: set_Pr258254617715012167tion_b,A5: set_option_b,X4: option_b,Y5: option_b,Z: option_b] :
      ( ( abstra1975357486212897574tion_b @ R5 @ A5 )
     => ( ( member_option_b @ X4 @ A5 )
       => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ R5 )
         => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Z ) @ R5 )
           => ? [U3: option_b] :
                ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y5 @ U3 ) @ ( transi330218195068109384tion_b @ R5 ) )
                & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Z @ U3 ) @ ( sup_su3110286242275913755tion_b @ R5 @ id_option_b ) ) ) ) ) ) ) ).

% strongly_confluent_on_E11
thf(fact_1088_inter__fset,axiom,
    ! [X4: fset_P2289935846317411886_b_nat,Xa: fset_P2289935846317411886_b_nat] :
      ( ( fset_P7858563055952297295_b_nat @ ( inf_in2594289221948343744_b_nat @ X4 @ Xa ) )
      = ( inf_in185880566399614150_b_nat @ ( fset_P7858563055952297295_b_nat @ X4 ) @ ( fset_P7858563055952297295_b_nat @ Xa ) ) ) ).

% inter_fset
thf(fact_1089_fimage__finter__fsubset,axiom,
    ! [F: product_prod_b_nat > produc1396820459975257318_b_nat,A5: fset_P2289935846317411886_b_nat,B5: fset_P2289935846317411886_b_nat] : ( ord_le1476955799578339574_b_nat @ ( fimage4266613729464776675_b_nat @ F @ ( inf_in2594289221948343744_b_nat @ A5 @ B5 ) ) @ ( inf_in1949489464691279912_b_nat @ ( fimage4266613729464776675_b_nat @ F @ A5 ) @ ( fimage4266613729464776675_b_nat @ F @ B5 ) ) ) ).

% fimage_finter_fsubset
thf(fact_1090_inf__Int__eq2,axiom,
    ! [R: set_Pr6944079702279229084_b_nat,S: set_Pr6944079702279229084_b_nat] :
      ( ( inf_in3225216818913299151_nat_o
        @ ^ [X: produc7781606619528532327tion_b,Y4: nat] : ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ X @ Y4 ) @ R )
        @ ^ [X: produc7781606619528532327tion_b,Y4: nat] : ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ X @ Y4 ) @ S ) )
      = ( ^ [X: produc7781606619528532327tion_b,Y4: nat] : ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ X @ Y4 ) @ ( inf_in6491300177809868590_b_nat @ R @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_1091_inf__Int__eq2,axiom,
    ! [R: set_Pr258254617715012167tion_b,S: set_Pr258254617715012167tion_b] :
      ( ( inf_in5776535092010985244on_b_o
        @ ^ [X: option_b,Y4: option_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ Y4 ) @ R )
        @ ^ [X: option_b,Y4: option_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ Y4 ) @ S ) )
      = ( ^ [X: option_b,Y4: option_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ Y4 ) @ ( inf_in9047234346837383221tion_b @ R @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_1092_inf__Int__eq2,axiom,
    ! [R: set_Pr1307281990691478580_b_nat,S: set_Pr1307281990691478580_b_nat] :
      ( ( inf_inf_b_nat_o
        @ ^ [X: b,Y4: nat] : ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ X @ Y4 ) @ R )
        @ ^ [X: b,Y4: nat] : ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ X @ Y4 ) @ S ) )
      = ( ^ [X: b,Y4: nat] : ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ X @ Y4 ) @ ( inf_in185880566399614150_b_nat @ R @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_1093_meetI,axiom,
    ! [A: option_b,B: option_b,A5: set_Pr258254617715012167tion_b,C: option_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ ( transi330218195068109384tion_b @ A5 ) )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ C ) @ ( transi330218195068109384tion_b @ A5 ) )
       => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ B @ C ) @ ( abstra6838757939696345090tion_b @ A5 ) ) ) ) ).

% meetI
thf(fact_1094_meetD,axiom,
    ! [B: option_b,C: option_b,A5: set_Pr258254617715012167tion_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ B @ C ) @ ( abstra6838757939696345090tion_b @ A5 ) )
     => ? [A4: option_b] :
          ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A4 @ B ) @ ( transi330218195068109384tion_b @ A5 ) )
          & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A4 @ C ) @ ( transi330218195068109384tion_b @ A5 ) ) ) ) ).

% meetD
thf(fact_1095_meetE,axiom,
    ! [B: option_b,C: option_b,A5: set_Pr258254617715012167tion_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ B @ C ) @ ( abstra6838757939696345090tion_b @ A5 ) )
     => ~ ! [A4: option_b] :
            ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A4 @ B ) @ ( transi330218195068109384tion_b @ A5 ) )
           => ~ ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A4 @ C ) @ ( transi330218195068109384tion_b @ A5 ) ) ) ) ).

% meetE
thf(fact_1096_strongly__confluent__E1n,axiom,
    ! [R5: set_Pr258254617715012167tion_b,X4: option_b,Y5: option_b,Z: option_b,N: nat] :
      ( ( abstra1975357486212897574tion_b @ R5 @ top_top_set_option_b )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ ( sup_su3110286242275913755tion_b @ R5 @ id_option_b ) )
       => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Z ) @ ( compow3776960924738975994tion_b @ N @ R5 ) )
         => ? [U3: option_b] :
              ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y5 @ U3 ) @ ( transi330218195068109384tion_b @ R5 ) )
              & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Z @ U3 ) @ ( sup_su3110286242275913755tion_b @ R5 @ id_option_b ) ) ) ) ) ) ).

% strongly_confluent_E1n
thf(fact_1097_gpar__rstepD_Hp__gpar__rstepD_H__eq,axiom,
    ! [F2: set_Pr6944079702279229084_b_nat,R3: set_Pr1711769230977380871tion_b] :
      ( ( lift_R6104920336080474628tion_b
        @ ^ [X: produc7781606619528532327tion_b,Y4: nat] : ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ X @ Y4 ) @ F2 )
        @ ^ [X: ground7210675428589265304tion_b,Y4: ground7210675428589265304tion_b] : ( member4891326893515703376tion_b @ ( produc6695998013266469143tion_b @ X @ Y4 ) @ R3 ) )
      = ( ^ [X: ground7210675428589265304tion_b,Y4: ground7210675428589265304tion_b] : ( member4891326893515703376tion_b @ ( produc6695998013266469143tion_b @ X @ Y4 ) @ ( lift_R3029990548839263256tion_b @ F2 @ R3 ) ) ) ) ).

% gpar_rstepD'p_gpar_rstepD'_eq
thf(fact_1098_gpar__rstepD_Hp__gpar__rstepD_H__eq,axiom,
    ! [F2: set_Pr1307281990691478580_b_nat,R3: set_Pr3134649703833893895term_b] :
      ( ( lift_R7250500093386647452pD_p_b
        @ ^ [X: b,Y4: nat] : ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ X @ Y4 ) @ F2 )
        @ ^ [X: ground_gterm_b,Y4: ground_gterm_b] : ( member7340517243570908752term_b @ ( produc1113259529098229527term_b @ X @ Y4 ) @ R3 ) )
      = ( ^ [X: ground_gterm_b,Y4: ground_gterm_b] : ( member7340517243570908752term_b @ ( produc1113259529098229527term_b @ X @ Y4 ) @ ( lift_R3469022553741414832tepD_b @ F2 @ R3 ) ) ) ) ).

% gpar_rstepD'p_gpar_rstepD'_eq
thf(fact_1099_gpar__rstepD_H__def,axiom,
    ( lift_R3029990548839263256tion_b
    = ( ^ [F4: set_Pr6944079702279229084_b_nat,R6: set_Pr1711769230977380871tion_b] :
          ( collec9012883117494140050tion_b
          @ ( produc5811763897571003244on_b_o
            @ ( lift_R6104920336080474628tion_b
              @ ^ [X: produc7781606619528532327tion_b,Y4: nat] : ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ X @ Y4 ) @ F4 )
              @ ^ [X: ground7210675428589265304tion_b,Y4: ground7210675428589265304tion_b] : ( member4891326893515703376tion_b @ ( produc6695998013266469143tion_b @ X @ Y4 ) @ R6 ) ) ) ) ) ) ).

% gpar_rstepD'_def
thf(fact_1100_gpar__rstepD_H__def,axiom,
    ( lift_R3469022553741414832tepD_b
    = ( ^ [F4: set_Pr1307281990691478580_b_nat,R6: set_Pr3134649703833893895term_b] :
          ( collec478437889140176530term_b
          @ ( produc1717101583779059052rm_b_o
            @ ( lift_R7250500093386647452pD_p_b
              @ ^ [X: b,Y4: nat] : ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ X @ Y4 ) @ F4 )
              @ ^ [X: ground_gterm_b,Y4: ground_gterm_b] : ( member7340517243570908752term_b @ ( produc1113259529098229527term_b @ X @ Y4 ) @ R6 ) ) ) ) ) ) ).

% gpar_rstepD'_def
thf(fact_1101_rtrancl__len__E,axiom,
    ! [X4: option_b,Y5: option_b,R5: set_Pr258254617715012167tion_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ ( transi330218195068109384tion_b @ R5 ) )
     => ~ ! [N2: nat] :
            ~ ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ ( compow3776960924738975994tion_b @ N2 @ R5 ) ) ) ).

% rtrancl_len_E
thf(fact_1102_relpow__refl__mono,axiom,
    ! [Rel: set_Pr258254617715012167tion_b,M: nat,N: nat,A: option_b,B: option_b] :
      ( ! [X3: option_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X3 @ X3 ) @ Rel )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ ( compow3776960924738975994tion_b @ M @ Rel ) )
         => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ ( compow3776960924738975994tion_b @ N @ Rel ) ) ) ) ) ).

% relpow_refl_mono
thf(fact_1103_relpow__image,axiom,
    ! [R5: set_Pr258254617715012167tion_b,F: option_b > option_b,R7: set_Pr258254617715012167tion_b,S4: option_b,T: option_b,N: nat] :
      ( ! [S7: option_b,T3: option_b] :
          ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ S7 @ T3 ) @ R5 )
         => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ ( F @ S7 ) @ ( F @ T3 ) ) @ R7 ) )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ S4 @ T ) @ ( compow3776960924738975994tion_b @ N @ R5 ) )
       => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ ( F @ S4 ) @ ( F @ T ) ) @ ( compow3776960924738975994tion_b @ N @ R7 ) ) ) ) ).

% relpow_image
thf(fact_1104_relpow__Suc__D2_H,axiom,
    ! [N: nat,R: set_Pr258254617715012167tion_b,X8: option_b,Y9: option_b,Z5: option_b] :
      ( ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X8 @ Y9 ) @ ( compow3776960924738975994tion_b @ N @ R ) )
        & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y9 @ Z5 ) @ R ) )
     => ? [W2: option_b] :
          ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X8 @ W2 ) @ R )
          & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ W2 @ Z5 ) @ ( compow3776960924738975994tion_b @ N @ R ) ) ) ) ).

% relpow_Suc_D2'
thf(fact_1105_trancl__steps__relpow,axiom,
    ! [A: set_Pr258254617715012167tion_b,B: set_Pr258254617715012167tion_b,X4: option_b,Y5: option_b,N: nat] :
      ( ( ord_le6367397903258896871tion_b @ A @ ( transi596562996543779814tion_b @ B ) )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ ( compow3776960924738975994tion_b @ N @ A ) )
       => ? [M2: nat] :
            ( ( ord_less_eq_nat @ N @ M2 )
            & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ ( compow3776960924738975994tion_b @ M2 @ B ) ) ) ) ) ).

% trancl_steps_relpow
thf(fact_1106_acyclic__insert,axiom,
    ! [Y5: option_b,X4: option_b,R5: set_Pr258254617715012167tion_b] :
      ( ( transi5850314681905841172tion_b @ ( insert5518505416251236151tion_b @ ( produc3989389360737052631tion_b @ Y5 @ X4 ) @ R5 ) )
      = ( ( transi5850314681905841172tion_b @ R5 )
        & ~ ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ ( transi330218195068109384tion_b @ R5 ) ) ) ) ).

% acyclic_insert
thf(fact_1107_acyclicI,axiom,
    ! [R5: set_Pr258254617715012167tion_b] :
      ( ! [X3: option_b] :
          ~ ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X3 @ X3 ) @ ( transi596562996543779814tion_b @ R5 ) )
     => ( transi5850314681905841172tion_b @ R5 ) ) ).

% acyclicI
thf(fact_1108_acyclic__def,axiom,
    ( transi5850314681905841172tion_b
    = ( ^ [R2: set_Pr258254617715012167tion_b] :
        ! [X: option_b] :
          ~ ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X @ X ) @ ( transi596562996543779814tion_b @ R2 ) ) ) ) ).

% acyclic_def
thf(fact_1109_lift__root__step_Osimps_I1_J,axiom,
    ! [F2: set_Pr1307281990691478580_b_nat,R3: set_Pr3134649703833893895term_b] :
      ( ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PAny @ fOR_ext_ESingle @ R3 )
      = ( contex6947914739494133533_onp_b
        @ ^ [C5: ground_gctxt_b] : ( ord_le5038853854344477844_b_nat @ ( ground_funas_gctxt_b @ C5 ) @ F2 )
        @ R3 ) ) ).

% lift_root_step.simps(1)
thf(fact_1110_lift__root__step_Osimps_I4_J,axiom,
    ! [F2: set_Pr1307281990691478580_b_nat,R3: set_Pr3134649703833893895term_b] :
      ( ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PNonRoot @ fOR_ext_ESingle @ R3 )
      = ( contex6947914739494133533_onp_b
        @ ^ [C5: ground_gctxt_b] :
            ( ( ord_le5038853854344477844_b_nat @ ( ground_funas_gctxt_b @ C5 ) @ F2 )
            & ( C5 != ground_GHole_b ) )
        @ R3 ) ) ).

% lift_root_step.simps(4)
thf(fact_1111_comp__rtrancl__trancl,axiom,
    ! [R: set_Pr258254617715012167tion_b,S: set_Pr258254617715012167tion_b,S4: option_b,T: option_b] :
      ( ( ord_le6367397903258896871tion_b @ ( relcom7330294423875589572tion_b @ R @ S ) @ S )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ S4 @ T ) @ ( relcom7330294423875589572tion_b @ ( transi330218195068109384tion_b @ ( sup_su3110286242275913755tion_b @ R @ S ) ) @ S ) )
       => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ S4 @ T ) @ ( transi596562996543779814tion_b @ S ) ) ) ) ).

% comp_rtrancl_trancl
thf(fact_1112_steps__map,axiom,
    ! [P2: option_b > $o,Q: set_Pr258254617715012167tion_b > $o,F: option_b > option_b,G: set_Pr258254617715012167tion_b > set_Pr258254617715012167tion_b,T: option_b,R: set_Pr258254617715012167tion_b,S: set_Pr258254617715012167tion_b,U4: option_b] :
      ( ! [T3: option_b,U3: option_b,R8: set_Pr258254617715012167tion_b] :
          ( ( P2 @ T3 )
         => ( ( Q @ R8 )
           => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ T3 @ U3 ) @ R8 )
             => ( ( P2 @ U3 )
                & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ ( F @ T3 ) @ ( F @ U3 ) ) @ ( G @ R8 ) ) ) ) ) )
     => ( ( P2 @ T )
       => ( ( Q @ R )
         => ( ( Q @ S )
           => ( ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ T @ U4 ) @ ( transi330218195068109384tion_b @ R ) )
               => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ ( F @ T ) @ ( F @ U4 ) ) @ ( transi330218195068109384tion_b @ ( G @ R ) ) ) )
              & ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ T @ U4 ) @ ( relcom7330294423875589572tion_b @ ( transi330218195068109384tion_b @ R ) @ ( relcom7330294423875589572tion_b @ S @ ( transi330218195068109384tion_b @ R ) ) ) )
               => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ ( F @ T ) @ ( F @ U4 ) ) @ ( relcom7330294423875589572tion_b @ ( transi330218195068109384tion_b @ ( G @ R ) ) @ ( relcom7330294423875589572tion_b @ ( G @ S ) @ ( transi330218195068109384tion_b @ ( G @ R ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_1113_relcomp_Ocases,axiom,
    ! [A1: produc7781606619528532327tion_b,A22: nat,R5: set_Pr5366877986981747911tion_b,S4: set_Pr6944079702279229084_b_nat] :
      ( ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ A1 @ A22 ) @ ( relcom2340228640715659537_b_nat @ R5 @ S4 ) )
     => ~ ! [B3: produc7781606619528532327tion_b] :
            ( ( member6435960643780680464tion_b @ ( produc7954421581189286743tion_b @ A1 @ B3 ) @ R5 )
           => ~ ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ B3 @ A22 ) @ S4 ) ) ) ).

% relcomp.cases
thf(fact_1114_relcomp_Ocases,axiom,
    ! [A1: produc7781606619528532327tion_b,A22: nat,R5: set_Pr7804114732869508703on_b_b,S4: set_Pr1307281990691478580_b_nat] :
      ( ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ A1 @ A22 ) @ ( relcom2865295403206704809_b_nat @ R5 @ S4 ) )
     => ~ ! [B3: b] :
            ( ( member5793571475316979880on_b_b @ ( produc3415191419167169775on_b_b @ A1 @ B3 ) @ R5 )
           => ~ ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ B3 @ A22 ) @ S4 ) ) ) ).

% relcomp.cases
thf(fact_1115_relcomp_Ocases,axiom,
    ! [A1: produc7781606619528532327tion_b,A22: nat,R5: set_Pr6944079702279229084_b_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ A1 @ A22 ) @ ( relcom8278593728572068040at_nat @ R5 @ S4 ) )
     => ~ ! [B3: nat] :
            ( ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ A1 @ B3 ) @ R5 )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A22 ) @ S4 ) ) ) ).

% relcomp.cases
thf(fact_1116_relcomp_Ocases,axiom,
    ! [A1: option_b,A22: option_b,R5: set_Pr258254617715012167tion_b,S4: set_Pr258254617715012167tion_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A1 @ A22 ) @ ( relcom7330294423875589572tion_b @ R5 @ S4 ) )
     => ~ ! [B3: option_b] :
            ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A1 @ B3 ) @ R5 )
           => ~ ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ B3 @ A22 ) @ S4 ) ) ) ).

% relcomp.cases
thf(fact_1117_relcomp_Ocases,axiom,
    ! [A1: b,A22: nat,R5: set_Pr5422417554659304751tion_b,S4: set_Pr6944079702279229084_b_nat] :
      ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A1 @ A22 ) @ ( relcom8300095267181474169_b_nat @ R5 @ S4 ) )
     => ~ ! [B3: produc7781606619528532327tion_b] :
            ( ( member8570655873678976888tion_b @ ( produc5187340250978832319tion_b @ A1 @ B3 ) @ R5 )
           => ~ ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ B3 @ A22 ) @ S4 ) ) ) ).

% relcomp.cases
thf(fact_1118_relcomp_Ocases,axiom,
    ! [A1: b,A22: nat,R5: set_Product_prod_b_b,S4: set_Pr1307281990691478580_b_nat] :
      ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A1 @ A22 ) @ ( relcomp_b_b_nat @ R5 @ S4 ) )
     => ~ ! [B3: b] :
            ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A1 @ B3 ) @ R5 )
           => ~ ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ B3 @ A22 ) @ S4 ) ) ) ).

% relcomp.cases
thf(fact_1119_relcomp_Ocases,axiom,
    ! [A1: b,A22: nat,R5: set_Pr1307281990691478580_b_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A1 @ A22 ) @ ( relcomp_b_nat_nat @ R5 @ S4 ) )
     => ~ ! [B3: nat] :
            ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A1 @ B3 ) @ R5 )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A22 ) @ S4 ) ) ) ).

% relcomp.cases
thf(fact_1120_relcomp_Osimps,axiom,
    ! [A1: produc7781606619528532327tion_b,A22: nat,R5: set_Pr5366877986981747911tion_b,S4: set_Pr6944079702279229084_b_nat] :
      ( ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ A1 @ A22 ) @ ( relcom2340228640715659537_b_nat @ R5 @ S4 ) )
      = ( ? [A6: produc7781606619528532327tion_b,B4: produc7781606619528532327tion_b,C4: nat] :
            ( ( A1 = A6 )
            & ( A22 = C4 )
            & ( member6435960643780680464tion_b @ ( produc7954421581189286743tion_b @ A6 @ B4 ) @ R5 )
            & ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ B4 @ C4 ) @ S4 ) ) ) ) ).

% relcomp.simps
thf(fact_1121_relcomp_Osimps,axiom,
    ! [A1: produc7781606619528532327tion_b,A22: nat,R5: set_Pr7804114732869508703on_b_b,S4: set_Pr1307281990691478580_b_nat] :
      ( ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ A1 @ A22 ) @ ( relcom2865295403206704809_b_nat @ R5 @ S4 ) )
      = ( ? [A6: produc7781606619528532327tion_b,B4: b,C4: nat] :
            ( ( A1 = A6 )
            & ( A22 = C4 )
            & ( member5793571475316979880on_b_b @ ( produc3415191419167169775on_b_b @ A6 @ B4 ) @ R5 )
            & ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ B4 @ C4 ) @ S4 ) ) ) ) ).

% relcomp.simps
thf(fact_1122_relcomp_Osimps,axiom,
    ! [A1: produc7781606619528532327tion_b,A22: nat,R5: set_Pr6944079702279229084_b_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ A1 @ A22 ) @ ( relcom8278593728572068040at_nat @ R5 @ S4 ) )
      = ( ? [A6: produc7781606619528532327tion_b,B4: nat,C4: nat] :
            ( ( A1 = A6 )
            & ( A22 = C4 )
            & ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ A6 @ B4 ) @ R5 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ C4 ) @ S4 ) ) ) ) ).

% relcomp.simps
thf(fact_1123_relcomp_Osimps,axiom,
    ! [A1: option_b,A22: option_b,R5: set_Pr258254617715012167tion_b,S4: set_Pr258254617715012167tion_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A1 @ A22 ) @ ( relcom7330294423875589572tion_b @ R5 @ S4 ) )
      = ( ? [A6: option_b,B4: option_b,C4: option_b] :
            ( ( A1 = A6 )
            & ( A22 = C4 )
            & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A6 @ B4 ) @ R5 )
            & ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ B4 @ C4 ) @ S4 ) ) ) ) ).

% relcomp.simps
thf(fact_1124_relcomp_Osimps,axiom,
    ! [A1: b,A22: nat,R5: set_Pr5422417554659304751tion_b,S4: set_Pr6944079702279229084_b_nat] :
      ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A1 @ A22 ) @ ( relcom8300095267181474169_b_nat @ R5 @ S4 ) )
      = ( ? [A6: b,B4: produc7781606619528532327tion_b,C4: nat] :
            ( ( A1 = A6 )
            & ( A22 = C4 )
            & ( member8570655873678976888tion_b @ ( produc5187340250978832319tion_b @ A6 @ B4 ) @ R5 )
            & ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ B4 @ C4 ) @ S4 ) ) ) ) ).

% relcomp.simps
thf(fact_1125_relcomp_Osimps,axiom,
    ! [A1: b,A22: nat,R5: set_Product_prod_b_b,S4: set_Pr1307281990691478580_b_nat] :
      ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A1 @ A22 ) @ ( relcomp_b_b_nat @ R5 @ S4 ) )
      = ( ? [A6: b,B4: b,C4: nat] :
            ( ( A1 = A6 )
            & ( A22 = C4 )
            & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A6 @ B4 ) @ R5 )
            & ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ B4 @ C4 ) @ S4 ) ) ) ) ).

% relcomp.simps
thf(fact_1126_relcomp_Osimps,axiom,
    ! [A1: b,A22: nat,R5: set_Pr1307281990691478580_b_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A1 @ A22 ) @ ( relcomp_b_nat_nat @ R5 @ S4 ) )
      = ( ? [A6: b,B4: nat,C4: nat] :
            ( ( A1 = A6 )
            & ( A22 = C4 )
            & ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A6 @ B4 ) @ R5 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ C4 ) @ S4 ) ) ) ) ).

% relcomp.simps
thf(fact_1127_relcomp_OrelcompI,axiom,
    ! [A: produc7781606619528532327tion_b,B: produc7781606619528532327tion_b,R5: set_Pr5366877986981747911tion_b,C: nat,S4: set_Pr6944079702279229084_b_nat] :
      ( ( member6435960643780680464tion_b @ ( produc7954421581189286743tion_b @ A @ B ) @ R5 )
     => ( ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ B @ C ) @ S4 )
       => ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ A @ C ) @ ( relcom2340228640715659537_b_nat @ R5 @ S4 ) ) ) ) ).

% relcomp.relcompI
thf(fact_1128_relcomp_OrelcompI,axiom,
    ! [A: b,B: produc7781606619528532327tion_b,R5: set_Pr5422417554659304751tion_b,C: nat,S4: set_Pr6944079702279229084_b_nat] :
      ( ( member8570655873678976888tion_b @ ( produc5187340250978832319tion_b @ A @ B ) @ R5 )
     => ( ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ B @ C ) @ S4 )
       => ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A @ C ) @ ( relcom8300095267181474169_b_nat @ R5 @ S4 ) ) ) ) ).

% relcomp.relcompI
thf(fact_1129_relcomp_OrelcompI,axiom,
    ! [A: produc7781606619528532327tion_b,B: b,R5: set_Pr7804114732869508703on_b_b,C: nat,S4: set_Pr1307281990691478580_b_nat] :
      ( ( member5793571475316979880on_b_b @ ( produc3415191419167169775on_b_b @ A @ B ) @ R5 )
     => ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ B @ C ) @ S4 )
       => ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ A @ C ) @ ( relcom2865295403206704809_b_nat @ R5 @ S4 ) ) ) ) ).

% relcomp.relcompI
thf(fact_1130_relcomp_OrelcompI,axiom,
    ! [A: b,B: b,R5: set_Product_prod_b_b,C: nat,S4: set_Pr1307281990691478580_b_nat] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ R5 )
     => ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ B @ C ) @ S4 )
       => ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A @ C ) @ ( relcomp_b_b_nat @ R5 @ S4 ) ) ) ) ).

% relcomp.relcompI
thf(fact_1131_relcomp_OrelcompI,axiom,
    ! [A: produc7781606619528532327tion_b,B: nat,R5: set_Pr6944079702279229084_b_nat,C: nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ A @ B ) @ R5 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ S4 )
       => ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ A @ C ) @ ( relcom8278593728572068040at_nat @ R5 @ S4 ) ) ) ) ).

% relcomp.relcompI
thf(fact_1132_relcomp_OrelcompI,axiom,
    ! [A: option_b,B: option_b,R5: set_Pr258254617715012167tion_b,C: option_b,S4: set_Pr258254617715012167tion_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B ) @ R5 )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ B @ C ) @ S4 )
       => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ C ) @ ( relcom7330294423875589572tion_b @ R5 @ S4 ) ) ) ) ).

% relcomp.relcompI
thf(fact_1133_relcomp_OrelcompI,axiom,
    ! [A: b,B: nat,R5: set_Pr1307281990691478580_b_nat,C: nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A @ B ) @ R5 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ S4 )
       => ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A @ C ) @ ( relcomp_b_nat_nat @ R5 @ S4 ) ) ) ) ).

% relcomp.relcompI
thf(fact_1134_relcompE,axiom,
    ! [Xz: produc1396820459975257318_b_nat,R5: set_Pr5366877986981747911tion_b,S4: set_Pr6944079702279229084_b_nat] :
      ( ( member7289896697870905981_b_nat @ Xz @ ( relcom2340228640715659537_b_nat @ R5 @ S4 ) )
     => ~ ! [X3: produc7781606619528532327tion_b,Y3: produc7781606619528532327tion_b,Z2: nat] :
            ( ( Xz
              = ( produc2093246604507004384_b_nat @ X3 @ Z2 ) )
           => ( ( member6435960643780680464tion_b @ ( produc7954421581189286743tion_b @ X3 @ Y3 ) @ R5 )
             => ~ ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ Y3 @ Z2 ) @ S4 ) ) ) ) ).

% relcompE
thf(fact_1135_relcompE,axiom,
    ! [Xz: produc1396820459975257318_b_nat,R5: set_Pr7804114732869508703on_b_b,S4: set_Pr1307281990691478580_b_nat] :
      ( ( member7289896697870905981_b_nat @ Xz @ ( relcom2865295403206704809_b_nat @ R5 @ S4 ) )
     => ~ ! [X3: produc7781606619528532327tion_b,Y3: b,Z2: nat] :
            ( ( Xz
              = ( produc2093246604507004384_b_nat @ X3 @ Z2 ) )
           => ( ( member5793571475316979880on_b_b @ ( produc3415191419167169775on_b_b @ X3 @ Y3 ) @ R5 )
             => ~ ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ Y3 @ Z2 ) @ S4 ) ) ) ) ).

% relcompE
thf(fact_1136_relcompE,axiom,
    ! [Xz: produc1396820459975257318_b_nat,R5: set_Pr6944079702279229084_b_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( member7289896697870905981_b_nat @ Xz @ ( relcom8278593728572068040at_nat @ R5 @ S4 ) )
     => ~ ! [X3: produc7781606619528532327tion_b,Y3: nat,Z2: nat] :
            ( ( Xz
              = ( produc2093246604507004384_b_nat @ X3 @ Z2 ) )
           => ( ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ X3 @ Y3 ) @ R5 )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ S4 ) ) ) ) ).

% relcompE
thf(fact_1137_relcompE,axiom,
    ! [Xz: produc7781606619528532327tion_b,R5: set_Pr258254617715012167tion_b,S4: set_Pr258254617715012167tion_b] :
      ( ( member547026996284216464tion_b @ Xz @ ( relcom7330294423875589572tion_b @ R5 @ S4 ) )
     => ~ ! [X3: option_b,Y3: option_b,Z2: option_b] :
            ( ( Xz
              = ( produc3989389360737052631tion_b @ X3 @ Z2 ) )
           => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X3 @ Y3 ) @ R5 )
             => ~ ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y3 @ Z2 ) @ S4 ) ) ) ) ).

% relcompE
thf(fact_1138_relcompE,axiom,
    ! [Xz: product_prod_b_nat,R5: set_Pr5422417554659304751tion_b,S4: set_Pr6944079702279229084_b_nat] :
      ( ( member6959632917342813205_b_nat @ Xz @ ( relcom8300095267181474169_b_nat @ R5 @ S4 ) )
     => ~ ! [X3: b,Y3: produc7781606619528532327tion_b,Z2: nat] :
            ( ( Xz
              = ( product_Pair_b_nat @ X3 @ Z2 ) )
           => ( ( member8570655873678976888tion_b @ ( produc5187340250978832319tion_b @ X3 @ Y3 ) @ R5 )
             => ~ ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ Y3 @ Z2 ) @ S4 ) ) ) ) ).

% relcompE
thf(fact_1139_relcompE,axiom,
    ! [Xz: product_prod_b_nat,R5: set_Product_prod_b_b,S4: set_Pr1307281990691478580_b_nat] :
      ( ( member6959632917342813205_b_nat @ Xz @ ( relcomp_b_b_nat @ R5 @ S4 ) )
     => ~ ! [X3: b,Y3: b,Z2: nat] :
            ( ( Xz
              = ( product_Pair_b_nat @ X3 @ Z2 ) )
           => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X3 @ Y3 ) @ R5 )
             => ~ ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ Y3 @ Z2 ) @ S4 ) ) ) ) ).

% relcompE
thf(fact_1140_relcompE,axiom,
    ! [Xz: product_prod_b_nat,R5: set_Pr1307281990691478580_b_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( member6959632917342813205_b_nat @ Xz @ ( relcomp_b_nat_nat @ R5 @ S4 ) )
     => ~ ! [X3: b,Y3: nat,Z2: nat] :
            ( ( Xz
              = ( product_Pair_b_nat @ X3 @ Z2 ) )
           => ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ X3 @ Y3 ) @ R5 )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ S4 ) ) ) ) ).

% relcompE
thf(fact_1141_relcompEpair,axiom,
    ! [A: produc7781606619528532327tion_b,C: nat,R5: set_Pr5366877986981747911tion_b,S4: set_Pr6944079702279229084_b_nat] :
      ( ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ A @ C ) @ ( relcom2340228640715659537_b_nat @ R5 @ S4 ) )
     => ~ ! [B3: produc7781606619528532327tion_b] :
            ( ( member6435960643780680464tion_b @ ( produc7954421581189286743tion_b @ A @ B3 ) @ R5 )
           => ~ ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ B3 @ C ) @ S4 ) ) ) ).

% relcompEpair
thf(fact_1142_relcompEpair,axiom,
    ! [A: produc7781606619528532327tion_b,C: nat,R5: set_Pr7804114732869508703on_b_b,S4: set_Pr1307281990691478580_b_nat] :
      ( ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ A @ C ) @ ( relcom2865295403206704809_b_nat @ R5 @ S4 ) )
     => ~ ! [B3: b] :
            ( ( member5793571475316979880on_b_b @ ( produc3415191419167169775on_b_b @ A @ B3 ) @ R5 )
           => ~ ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ B3 @ C ) @ S4 ) ) ) ).

% relcompEpair
thf(fact_1143_relcompEpair,axiom,
    ! [A: produc7781606619528532327tion_b,C: nat,R5: set_Pr6944079702279229084_b_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ A @ C ) @ ( relcom8278593728572068040at_nat @ R5 @ S4 ) )
     => ~ ! [B3: nat] :
            ( ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ A @ B3 ) @ R5 )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ C ) @ S4 ) ) ) ).

% relcompEpair
thf(fact_1144_relcompEpair,axiom,
    ! [A: option_b,C: option_b,R5: set_Pr258254617715012167tion_b,S4: set_Pr258254617715012167tion_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ C ) @ ( relcom7330294423875589572tion_b @ R5 @ S4 ) )
     => ~ ! [B3: option_b] :
            ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A @ B3 ) @ R5 )
           => ~ ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ B3 @ C ) @ S4 ) ) ) ).

% relcompEpair
thf(fact_1145_relcompEpair,axiom,
    ! [A: b,C: nat,R5: set_Pr5422417554659304751tion_b,S4: set_Pr6944079702279229084_b_nat] :
      ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A @ C ) @ ( relcom8300095267181474169_b_nat @ R5 @ S4 ) )
     => ~ ! [B3: produc7781606619528532327tion_b] :
            ( ( member8570655873678976888tion_b @ ( produc5187340250978832319tion_b @ A @ B3 ) @ R5 )
           => ~ ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ B3 @ C ) @ S4 ) ) ) ).

% relcompEpair
thf(fact_1146_relcompEpair,axiom,
    ! [A: b,C: nat,R5: set_Product_prod_b_b,S4: set_Pr1307281990691478580_b_nat] :
      ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A @ C ) @ ( relcomp_b_b_nat @ R5 @ S4 ) )
     => ~ ! [B3: b] :
            ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B3 ) @ R5 )
           => ~ ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ B3 @ C ) @ S4 ) ) ) ).

% relcompEpair
thf(fact_1147_relcompEpair,axiom,
    ! [A: b,C: nat,R5: set_Pr1307281990691478580_b_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A @ C ) @ ( relcomp_b_nat_nat @ R5 @ S4 ) )
     => ~ ! [B3: nat] :
            ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A @ B3 ) @ R5 )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ C ) @ S4 ) ) ) ).

% relcompEpair
thf(fact_1148_relcomp3__I,axiom,
    ! [T: nat,U4: produc7781606619528532327tion_b,A5: set_Pr4178509052509721666tion_b,S4: produc7781606619528532327tion_b,B5: set_Pr6944079702279229084_b_nat,V: nat] :
      ( ( member3694883655523461923tion_b @ ( produc7510438065615382334tion_b @ T @ U4 ) @ A5 )
     => ( ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ S4 @ T ) @ B5 )
       => ( ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ U4 @ V ) @ B5 )
         => ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ S4 @ V ) @ ( relcom8278593728572068040at_nat @ B5 @ ( relcom3041546249966780966_b_nat @ A5 @ B5 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_1149_relcomp3__I,axiom,
    ! [T: nat,U4: b,A5: set_Pr4264375888882495962_nat_b,S4: b,B5: set_Pr1307281990691478580_b_nat,V: nat] :
      ( ( member8962352056413324475_nat_b @ ( product_Pair_nat_b @ T @ U4 ) @ A5 )
     => ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ S4 @ T ) @ B5 )
       => ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ U4 @ V ) @ B5 )
         => ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ S4 @ V ) @ ( relcomp_b_nat_nat @ B5 @ ( relcomp_nat_b_nat @ A5 @ B5 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_1150_relcomp3__I,axiom,
    ! [T: produc7781606619528532327tion_b,U4: nat,A5: set_Pr6944079702279229084_b_nat,S4: nat,B5: set_Pr4178509052509721666tion_b,V: produc7781606619528532327tion_b] :
      ( ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ T @ U4 ) @ A5 )
     => ( ( member3694883655523461923tion_b @ ( produc7510438065615382334tion_b @ S4 @ T ) @ B5 )
       => ( ( member3694883655523461923tion_b @ ( produc7510438065615382334tion_b @ U4 @ V ) @ B5 )
         => ( member3694883655523461923tion_b @ ( produc7510438065615382334tion_b @ S4 @ V ) @ ( relcom8466619888906355857tion_b @ B5 @ ( relcom7757420101824037487tion_b @ A5 @ B5 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_1151_relcomp3__I,axiom,
    ! [T: option_b,U4: option_b,A5: set_Pr258254617715012167tion_b,S4: option_b,B5: set_Pr258254617715012167tion_b,V: option_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ T @ U4 ) @ A5 )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ S4 @ T ) @ B5 )
       => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ U4 @ V ) @ B5 )
         => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ S4 @ V ) @ ( relcom7330294423875589572tion_b @ B5 @ ( relcom7330294423875589572tion_b @ A5 @ B5 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_1152_relcomp3__I,axiom,
    ! [T: b,U4: nat,A5: set_Pr1307281990691478580_b_nat,S4: nat,B5: set_Pr4264375888882495962_nat_b,V: b] :
      ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ T @ U4 ) @ A5 )
     => ( ( member8962352056413324475_nat_b @ ( product_Pair_nat_b @ S4 @ T ) @ B5 )
       => ( ( member8962352056413324475_nat_b @ ( product_Pair_nat_b @ U4 @ V ) @ B5 )
         => ( member8962352056413324475_nat_b @ ( product_Pair_nat_b @ S4 @ V ) @ ( relcomp_nat_b_b @ B5 @ ( relcomp_b_nat_b @ A5 @ B5 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_1153_non__strict__into__strict,axiom,
    ! [NS: set_Pr5366877986981747911tion_b,S: set_Pr6944079702279229084_b_nat,S4: produc7781606619528532327tion_b,T: nat] :
      ( ( ord_le3933197595962519292_b_nat @ ( relcom2340228640715659537_b_nat @ NS @ S ) @ S )
     => ( ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ S4 @ T ) @ ( relcom2340228640715659537_b_nat @ ( transi2849677033159065450tion_b @ NS ) @ S ) )
       => ( member7289896697870905981_b_nat @ ( produc2093246604507004384_b_nat @ S4 @ T ) @ S ) ) ) ).

% non_strict_into_strict
thf(fact_1154_non__strict__into__strict,axiom,
    ! [NS: set_Pr258254617715012167tion_b,S: set_Pr258254617715012167tion_b,S4: option_b,T: option_b] :
      ( ( ord_le6367397903258896871tion_b @ ( relcom7330294423875589572tion_b @ NS @ S ) @ S )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ S4 @ T ) @ ( relcom7330294423875589572tion_b @ ( transi330218195068109384tion_b @ NS ) @ S ) )
       => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ S4 @ T ) @ S ) ) ) ).

% non_strict_into_strict
thf(fact_1155_non__strict__into__strict,axiom,
    ! [NS: set_Product_prod_b_b,S: set_Pr1307281990691478580_b_nat,S4: b,T: nat] :
      ( ( ord_le5038853854344477844_b_nat @ ( relcomp_b_b_nat @ NS @ S ) @ S )
     => ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ S4 @ T ) @ ( relcomp_b_b_nat @ ( transitive_rtrancl_b @ NS ) @ S ) )
       => ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ S4 @ T ) @ S ) ) ) ).

% non_strict_into_strict
thf(fact_1156_firstStep,axiom,
    ! [L2: set_Pr258254617715012167tion_b,S: set_Pr258254617715012167tion_b,R: set_Pr258254617715012167tion_b,X4: option_b,Y5: option_b] :
      ( ( L2
        = ( sup_su3110286242275913755tion_b @ S @ R ) )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ ( transi330218195068109384tion_b @ L2 ) )
       => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ ( transi330218195068109384tion_b @ R ) )
          | ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ ( relcom7330294423875589572tion_b @ ( transi330218195068109384tion_b @ R ) @ ( relcom7330294423875589572tion_b @ S @ ( transi330218195068109384tion_b @ L2 ) ) ) ) ) ) ) ).

% firstStep
thf(fact_1157_first__step,axiom,
    ! [C3: set_Pr258254617715012167tion_b,A5: set_Pr258254617715012167tion_b,B5: set_Pr258254617715012167tion_b,X4: option_b,Y5: option_b,Z: option_b] :
      ( ( C3
        = ( sup_su3110286242275913755tion_b @ A5 @ B5 ) )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ ( transi330218195068109384tion_b @ C3 ) )
       => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ Y5 @ Z ) @ B5 )
         => ? [Y3: option_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y3 ) @ ( relcom7330294423875589572tion_b @ ( transi330218195068109384tion_b @ A5 ) @ B5 ) ) ) ) ) ).

% first_step
thf(fact_1158_first__step__O,axiom,
    ! [C3: set_Pr258254617715012167tion_b,A5: set_Pr258254617715012167tion_b,B5: set_Pr258254617715012167tion_b,X4: option_b,Y5: option_b] :
      ( ( C3
        = ( sup_su3110286242275913755tion_b @ A5 @ B5 ) )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ ( relcom7330294423875589572tion_b @ ( transi330218195068109384tion_b @ C3 ) @ B5 ) )
       => ? [Y3: option_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y3 ) @ ( relcom7330294423875589572tion_b @ ( transi330218195068109384tion_b @ A5 ) @ B5 ) ) ) ) ).

% first_step_O
thf(fact_1159_right__comp__S,axiom,
    ! [X4: option_b,Y5: option_b,S: set_Pr258254617715012167tion_b,NS: set_Pr258254617715012167tion_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ ( relcom7330294423875589572tion_b @ S @ ( sup_su3110286242275913755tion_b @ ( relcom7330294423875589572tion_b @ S @ ( relcom7330294423875589572tion_b @ ( transi330218195068109384tion_b @ S ) @ ( transi330218195068109384tion_b @ NS ) ) ) @ ( transi330218195068109384tion_b @ NS ) ) ) )
     => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ ( relcom7330294423875589572tion_b @ S @ ( relcom7330294423875589572tion_b @ ( transi330218195068109384tion_b @ S ) @ ( transi330218195068109384tion_b @ NS ) ) ) ) ) ).

% right_comp_S
thf(fact_1160_compatible__rtrancl__split,axiom,
    ! [NS: set_Pr258254617715012167tion_b,S: set_Pr258254617715012167tion_b,X4: option_b,Y5: option_b] :
      ( ( ord_le6367397903258896871tion_b @ ( relcom7330294423875589572tion_b @ NS @ S ) @ S )
     => ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ ( transi330218195068109384tion_b @ ( sup_su3110286242275913755tion_b @ NS @ S ) ) )
       => ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ ( sup_su3110286242275913755tion_b @ ( relcom7330294423875589572tion_b @ S @ ( relcom7330294423875589572tion_b @ ( transi330218195068109384tion_b @ S ) @ ( transi330218195068109384tion_b @ NS ) ) ) @ ( transi330218195068109384tion_b @ NS ) ) ) ) ) ).

% compatible_rtrancl_split
thf(fact_1161_Lift__Root__Step_Ogcomp__rel,axiom,
    ! [F2: set_Pr1307281990691478580_b_nat,R3: set_Pr3134649703833893895term_b,S3: set_Pr3134649703833893895term_b] :
      ( ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PAny @ fOR_ext_EParallel @ ( ground_gcomp_rel_b @ F2 @ R3 @ S3 ) )
      = ( relcom4795790519912162031term_b @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PAny @ fOR_ext_EParallel @ R3 ) @ ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PAny @ fOR_ext_EParallel @ S3 ) ) ) ).

% Lift_Root_Step.gcomp_rel
thf(fact_1162_rr2__compositon,axiom,
    ! [R3: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat,LL: set_Pr3134649703833893895term_b,A5: tree_r4611138855727183398tion_b,B5: tree_r4611138855727183398tion_b] :
      ( ( ord_le2408605988743369639term_b @ R3
        @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) )
          @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) ) ) )
     => ( ( ord_le2408605988743369639term_b @ LL
          @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) )
            @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) ) ) )
       => ( ( rRn_RR2_spec_a_b_b @ A5 @ R3 )
         => ( ( rRn_RR2_spec_a_b_b @ B5 @ LL )
           => ( rRn_RR1611153016866580401_a_b_b @ ( rRn_rr1616545004011552257_b_a_a @ F2 @ A5 @ B5 ) @ ( relcom4795790519912162031term_b @ R3 @ LL ) ) ) ) ) ) ).

% rr2_compositon
thf(fact_1163_rr2__compositon,axiom,
    ! [R3: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat,LL: set_Pr3134649703833893895term_b,A5: tree_r4611138855727183398tion_b,B5: tree_r6993104598208160480tion_b] :
      ( ( ord_le2408605988743369639term_b @ R3
        @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) )
          @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) ) ) )
     => ( ( ord_le2408605988743369639term_b @ LL
          @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) )
            @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) ) ) )
       => ( ( rRn_RR2_spec_a_b_b @ A5 @ R3 )
         => ( ( rRn_RR5637769166088770792_a_b_b @ B5 @ LL )
           => ( rRn_RR4170495842214901495_a_b_b @ ( rRn_rr8804827536813319111tion_a @ F2 @ A5 @ B5 ) @ ( relcom4795790519912162031term_b @ R3 @ LL ) ) ) ) ) ) ).

% rr2_compositon
thf(fact_1164_rr2__compositon,axiom,
    ! [R3: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat,LL: set_Pr3134649703833893895term_b,A5: tree_r6993104598208160480tion_b,B5: tree_r4611138855727183398tion_b] :
      ( ( ord_le2408605988743369639term_b @ R3
        @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) )
          @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) ) ) )
     => ( ( ord_le2408605988743369639term_b @ LL
          @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) )
            @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) ) ) )
       => ( ( rRn_RR5637769166088770792_a_b_b @ A5 @ R3 )
         => ( ( rRn_RR2_spec_a_b_b @ B5 @ LL )
           => ( rRn_RR5872896495431105655_a_b_b @ ( rRn_rr1803969381497338427on_a_a @ F2 @ A5 @ B5 ) @ ( relcom4795790519912162031term_b @ R3 @ LL ) ) ) ) ) ) ).

% rr2_compositon
thf(fact_1165_rr2__compositon,axiom,
    ! [R3: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat,LL: set_Pr3134649703833893895term_b,A5: tree_r2154205730188421652tion_b,B5: tree_r4611138855727183398tion_b] :
      ( ( ord_le2408605988743369639term_b @ R3
        @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) )
          @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) ) ) )
     => ( ( ord_le2408605988743369639term_b @ LL
          @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) )
            @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) ) ) )
       => ( ( rRn_RR7260272242312503694es_b_b @ A5 @ R3 )
         => ( ( rRn_RR2_spec_a_b_b @ B5 @ LL )
           => ( rRn_RR3583459555964036135_a_b_b @ ( rRn_rr6991515034359556489ates_a @ F2 @ A5 @ B5 ) @ ( relcom4795790519912162031term_b @ R3 @ LL ) ) ) ) ) ) ).

% rr2_compositon
thf(fact_1166_rr2__compositon,axiom,
    ! [R3: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat,LL: set_Pr3134649703833893895term_b,A5: tree_r4611138855727183398tion_b,B5: tree_r2154205730188421652tion_b] :
      ( ( ord_le2408605988743369639term_b @ R3
        @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) )
          @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) ) ) )
     => ( ( ord_le2408605988743369639term_b @ LL
          @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) )
            @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) ) ) )
       => ( ( rRn_RR2_spec_a_b_b @ A5 @ R3 )
         => ( ( rRn_RR7260272242312503694es_b_b @ B5 @ LL )
           => ( rRn_RR1666436325986766849es_b_b @ ( rRn_rr246032541601434479states @ F2 @ A5 @ B5 ) @ ( relcom4795790519912162031term_b @ R3 @ LL ) ) ) ) ) ) ).

% rr2_compositon
thf(fact_1167_rr2__compositon,axiom,
    ! [R3: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat,LL: set_Pr3134649703833893895term_b,A5: tree_r4611138855727183398tion_b,B5: tree_r7504422106042656583tion_b] :
      ( ( ord_le2408605988743369639term_b @ R3
        @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) )
          @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) ) ) )
     => ( ( ord_le2408605988743369639term_b @ LL
          @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) )
            @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) ) ) )
       => ( ( rRn_RR2_spec_a_b_b @ A5 @ R3 )
         => ( ( rRn_RR5856091920061875867_o_b_b @ B5 @ LL )
           => ( rRn_RR6682877704082549902_o_b_b @ ( rRn_rr4900109824242877564um_a_o @ F2 @ A5 @ B5 ) @ ( relcom4795790519912162031term_b @ R3 @ LL ) ) ) ) ) ) ).

% rr2_compositon
thf(fact_1168_rr2__compositon,axiom,
    ! [R3: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat,LL: set_Pr3134649703833893895term_b,A5: tree_r7504422106042656583tion_b,B5: tree_r4611138855727183398tion_b] :
      ( ( ord_le2408605988743369639term_b @ R3
        @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) )
          @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) ) ) )
     => ( ( ord_le2408605988743369639term_b @ LL
          @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) )
            @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) ) ) )
       => ( ( rRn_RR5856091920061875867_o_b_b @ A5 @ R3 )
         => ( ( rRn_RR2_spec_a_b_b @ B5 @ LL )
           => ( rRn_RR4825814222543937204_a_b_b @ ( rRn_rr1882487269650480700_a_o_a @ F2 @ A5 @ B5 ) @ ( relcom4795790519912162031term_b @ R3 @ LL ) ) ) ) ) ) ).

% rr2_compositon
thf(fact_1169_rr2__compositon,axiom,
    ! [R3: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat,LL: set_Pr3134649703833893895term_b,A5: tree_r6993104598208160480tion_b,B5: tree_r6993104598208160480tion_b] :
      ( ( ord_le2408605988743369639term_b @ R3
        @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) )
          @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) ) ) )
     => ( ( ord_le2408605988743369639term_b @ LL
          @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) )
            @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) ) ) )
       => ( ( rRn_RR5637769166088770792_a_b_b @ A5 @ R3 )
         => ( ( rRn_RR5637769166088770792_a_b_b @ B5 @ LL )
           => ( rRn_RR7840990895237753533_a_b_b @ ( rRn_rr6424448918311796993tion_a @ F2 @ A5 @ B5 ) @ ( relcom4795790519912162031term_b @ R3 @ LL ) ) ) ) ) ) ).

% rr2_compositon
thf(fact_1170_rr2__compositon,axiom,
    ! [R3: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat,LL: set_Pr3134649703833893895term_b,A5: tree_r2154205730188421652tion_b,B5: tree_r6993104598208160480tion_b] :
      ( ( ord_le2408605988743369639term_b @ R3
        @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) )
          @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) ) ) )
     => ( ( ord_le2408605988743369639term_b @ LL
          @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) )
            @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) ) ) )
       => ( ( rRn_RR7260272242312503694es_b_b @ A5 @ R3 )
         => ( ( rRn_RR5637769166088770792_a_b_b @ B5 @ LL )
           => ( rRn_RR3319046072957078125_a_b_b @ ( rRn_rr605321073299645263tion_a @ F2 @ A5 @ B5 ) @ ( relcom4795790519912162031term_b @ R3 @ LL ) ) ) ) ) ) ).

% rr2_compositon
thf(fact_1171_rr2__compositon,axiom,
    ! [R3: set_Pr3134649703833893895term_b,F2: fset_P2289935846317411886_b_nat,LL: set_Pr3134649703833893895term_b,A5: tree_r4611138855727183398tion_b,B5: tree_r3013368800753843012tion_b] :
      ( ( ord_le2408605988743369639term_b @ R3
        @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) )
          @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) ) ) )
     => ( ( ord_le2408605988743369639term_b @ LL
          @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) )
            @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ ( fset_P7858563055952297295_b_nat @ F2 ) ) ) )
       => ( ( rRn_RR2_spec_a_b_b @ A5 @ R3 )
         => ( ( rRn_RR225467092841841886es_b_b @ B5 @ LL )
           => ( rRn_RR7397758064740657233es_b_b @ ( rRn_rr2486298437868761407states @ F2 @ A5 @ B5 ) @ ( relcom4795790519912162031term_b @ R3 @ LL ) ) ) ) ) ) ).

% rr2_compositon
thf(fact_1172_lift__root__step_Osimps_I3_J,axiom,
    ! [F2: set_Pr1307281990691478580_b_nat,R3: set_Pr3134649703833893895term_b] :
      ( ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PAny @ fOR_ext_EParallel @ R3 )
      = ( contex3086340098647126762_onp_b
        @ ^ [C5: ground_gmctxt_b] : ( ord_le5038853854344477844_b_nat @ ( ground4323624445104842081ctxt_b @ C5 ) @ F2 )
        @ R3 ) ) ).

% lift_root_step.simps(3)
thf(fact_1173_lift__root__step_Osimps_I6_J,axiom,
    ! [F2: set_Pr1307281990691478580_b_nat,R3: set_Pr3134649703833893895term_b] :
      ( ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PNonRoot @ fOR_ext_EParallel @ R3 )
      = ( contex3086340098647126762_onp_b
        @ ^ [C5: ground_gmctxt_b] :
            ( ( ord_le5038853854344477844_b_nat @ ( ground4323624445104842081ctxt_b @ C5 ) @ F2 )
            & ( C5 != ground_GMHole_b ) )
        @ R3 ) ) ).

% lift_root_step.simps(6)
thf(fact_1174_lift__root__step_Oelims,axiom,
    ! [X4: set_Pr1307281990691478580_b_nat,Xa: fOR_pos_step,Xb: fOR_ext_step,Xc: set_Pr3134649703833893895term_b,Y5: set_Pr3134649703833893895term_b] :
      ( ( ( lift_R8854095231239425056step_b @ X4 @ Xa @ Xb @ Xc )
        = Y5 )
     => ( ( ( Xa = fOR_pos_PAny )
         => ( ( Xb = fOR_ext_ESingle )
           => ( Y5
             != ( contex6947914739494133533_onp_b
                @ ^ [C5: ground_gctxt_b] : ( ord_le5038853854344477844_b_nat @ ( ground_funas_gctxt_b @ C5 ) @ X4 )
                @ Xc ) ) ) )
       => ( ( ( Xa = fOR_pos_PAny )
           => ( ( Xb = fOR_ex5070955383060788274rallel )
             => ( Y5
               != ( contex3086340098647126762_onp_b
                  @ ^ [C5: ground_gmctxt_b] :
                      ( ( ord_less_nat @ zero_zero_nat @ ( ground_num_gholes_b @ C5 ) )
                      & ( ord_le5038853854344477844_b_nat @ ( ground4323624445104842081ctxt_b @ C5 ) @ X4 ) )
                  @ Xc ) ) ) )
         => ( ( ( Xa = fOR_pos_PAny )
             => ( ( Xb = fOR_ext_EParallel )
               => ( Y5
                 != ( contex3086340098647126762_onp_b
                    @ ^ [C5: ground_gmctxt_b] : ( ord_le5038853854344477844_b_nat @ ( ground4323624445104842081ctxt_b @ C5 ) @ X4 )
                    @ Xc ) ) ) )
           => ( ( ( Xa = fOR_pos_PNonRoot )
               => ( ( Xb = fOR_ext_ESingle )
                 => ( Y5
                   != ( contex6947914739494133533_onp_b
                      @ ^ [C5: ground_gctxt_b] :
                          ( ( ord_le5038853854344477844_b_nat @ ( ground_funas_gctxt_b @ C5 ) @ X4 )
                          & ( C5 != ground_GHole_b ) )
                      @ Xc ) ) ) )
             => ( ( ( Xa = fOR_pos_PNonRoot )
                 => ( ( Xb = fOR_ex5070955383060788274rallel )
                   => ( Y5
                     != ( contex3086340098647126762_onp_b
                        @ ^ [C5: ground_gmctxt_b] :
                            ( ( ord_less_nat @ zero_zero_nat @ ( ground_num_gholes_b @ C5 ) )
                            & ( ord_le5038853854344477844_b_nat @ ( ground4323624445104842081ctxt_b @ C5 ) @ X4 )
                            & ( C5 != ground_GMHole_b ) )
                        @ Xc ) ) ) )
               => ( ( ( Xa = fOR_pos_PNonRoot )
                   => ( ( Xb = fOR_ext_EParallel )
                     => ( Y5
                       != ( contex3086340098647126762_onp_b
                          @ ^ [C5: ground_gmctxt_b] :
                              ( ( ord_le5038853854344477844_b_nat @ ( ground4323624445104842081ctxt_b @ C5 ) @ X4 )
                              & ( C5 != ground_GMHole_b ) )
                          @ Xc ) ) ) )
                 => ( ( ( Xa = fOR_pos_PRoot )
                     => ( ( Xb = fOR_ext_ESingle )
                       => ( Y5 != Xc ) ) )
                   => ( ( ( Xa = fOR_pos_PRoot )
                       => ( ( Xb = fOR_ex5070955383060788274rallel )
                         => ( Y5 != Xc ) ) )
                     => ~ ( ( Xa = fOR_pos_PRoot )
                         => ( ( Xb = fOR_ext_EParallel )
                           => ( Y5
                             != ( sup_su1163039302778497499term_b @ Xc
                                @ ( inf_in8391146210534189557term_b @ id_Ground_gterm_b
                                  @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ X4 )
                                    @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% lift_root_step.elims
thf(fact_1175_lift__root__step_Opelims,axiom,
    ! [X4: set_Pr1307281990691478580_b_nat,Xa: fOR_pos_step,Xb: fOR_ext_step,Xc: set_Pr3134649703833893895term_b,Y5: set_Pr3134649703833893895term_b] :
      ( ( ( lift_R8854095231239425056step_b @ X4 @ Xa @ Xb @ Xc )
        = Y5 )
     => ( ( accp_P323873057692896796term_b @ lift_R1576223848067188451_rel_b @ ( produc4640078800242017687term_b @ X4 @ ( produc5352714453822653076term_b @ Xa @ ( produc3933249678690294607term_b @ Xb @ Xc ) ) ) )
       => ( ( ( Xa = fOR_pos_PAny )
           => ( ( Xb = fOR_ext_ESingle )
             => ( ( Y5
                  = ( contex6947914739494133533_onp_b
                    @ ^ [C5: ground_gctxt_b] : ( ord_le5038853854344477844_b_nat @ ( ground_funas_gctxt_b @ C5 ) @ X4 )
                    @ Xc ) )
               => ~ ( accp_P323873057692896796term_b @ lift_R1576223848067188451_rel_b @ ( produc4640078800242017687term_b @ X4 @ ( produc5352714453822653076term_b @ fOR_pos_PAny @ ( produc3933249678690294607term_b @ fOR_ext_ESingle @ Xc ) ) ) ) ) ) )
         => ( ( ( Xa = fOR_pos_PAny )
             => ( ( Xb = fOR_ex5070955383060788274rallel )
               => ( ( Y5
                    = ( contex3086340098647126762_onp_b
                      @ ^ [C5: ground_gmctxt_b] :
                          ( ( ord_less_nat @ zero_zero_nat @ ( ground_num_gholes_b @ C5 ) )
                          & ( ord_le5038853854344477844_b_nat @ ( ground4323624445104842081ctxt_b @ C5 ) @ X4 ) )
                      @ Xc ) )
                 => ~ ( accp_P323873057692896796term_b @ lift_R1576223848067188451_rel_b @ ( produc4640078800242017687term_b @ X4 @ ( produc5352714453822653076term_b @ fOR_pos_PAny @ ( produc3933249678690294607term_b @ fOR_ex5070955383060788274rallel @ Xc ) ) ) ) ) ) )
           => ( ( ( Xa = fOR_pos_PAny )
               => ( ( Xb = fOR_ext_EParallel )
                 => ( ( Y5
                      = ( contex3086340098647126762_onp_b
                        @ ^ [C5: ground_gmctxt_b] : ( ord_le5038853854344477844_b_nat @ ( ground4323624445104842081ctxt_b @ C5 ) @ X4 )
                        @ Xc ) )
                   => ~ ( accp_P323873057692896796term_b @ lift_R1576223848067188451_rel_b @ ( produc4640078800242017687term_b @ X4 @ ( produc5352714453822653076term_b @ fOR_pos_PAny @ ( produc3933249678690294607term_b @ fOR_ext_EParallel @ Xc ) ) ) ) ) ) )
             => ( ( ( Xa = fOR_pos_PNonRoot )
                 => ( ( Xb = fOR_ext_ESingle )
                   => ( ( Y5
                        = ( contex6947914739494133533_onp_b
                          @ ^ [C5: ground_gctxt_b] :
                              ( ( ord_le5038853854344477844_b_nat @ ( ground_funas_gctxt_b @ C5 ) @ X4 )
                              & ( C5 != ground_GHole_b ) )
                          @ Xc ) )
                     => ~ ( accp_P323873057692896796term_b @ lift_R1576223848067188451_rel_b @ ( produc4640078800242017687term_b @ X4 @ ( produc5352714453822653076term_b @ fOR_pos_PNonRoot @ ( produc3933249678690294607term_b @ fOR_ext_ESingle @ Xc ) ) ) ) ) ) )
               => ( ( ( Xa = fOR_pos_PNonRoot )
                   => ( ( Xb = fOR_ex5070955383060788274rallel )
                     => ( ( Y5
                          = ( contex3086340098647126762_onp_b
                            @ ^ [C5: ground_gmctxt_b] :
                                ( ( ord_less_nat @ zero_zero_nat @ ( ground_num_gholes_b @ C5 ) )
                                & ( ord_le5038853854344477844_b_nat @ ( ground4323624445104842081ctxt_b @ C5 ) @ X4 )
                                & ( C5 != ground_GMHole_b ) )
                            @ Xc ) )
                       => ~ ( accp_P323873057692896796term_b @ lift_R1576223848067188451_rel_b @ ( produc4640078800242017687term_b @ X4 @ ( produc5352714453822653076term_b @ fOR_pos_PNonRoot @ ( produc3933249678690294607term_b @ fOR_ex5070955383060788274rallel @ Xc ) ) ) ) ) ) )
                 => ( ( ( Xa = fOR_pos_PNonRoot )
                     => ( ( Xb = fOR_ext_EParallel )
                       => ( ( Y5
                            = ( contex3086340098647126762_onp_b
                              @ ^ [C5: ground_gmctxt_b] :
                                  ( ( ord_le5038853854344477844_b_nat @ ( ground4323624445104842081ctxt_b @ C5 ) @ X4 )
                                  & ( C5 != ground_GMHole_b ) )
                              @ Xc ) )
                         => ~ ( accp_P323873057692896796term_b @ lift_R1576223848067188451_rel_b @ ( produc4640078800242017687term_b @ X4 @ ( produc5352714453822653076term_b @ fOR_pos_PNonRoot @ ( produc3933249678690294607term_b @ fOR_ext_EParallel @ Xc ) ) ) ) ) ) )
                   => ( ( ( Xa = fOR_pos_PRoot )
                       => ( ( Xb = fOR_ext_ESingle )
                         => ( ( Y5 = Xc )
                           => ~ ( accp_P323873057692896796term_b @ lift_R1576223848067188451_rel_b @ ( produc4640078800242017687term_b @ X4 @ ( produc5352714453822653076term_b @ fOR_pos_PRoot @ ( produc3933249678690294607term_b @ fOR_ext_ESingle @ Xc ) ) ) ) ) ) )
                     => ( ( ( Xa = fOR_pos_PRoot )
                         => ( ( Xb = fOR_ex5070955383060788274rallel )
                           => ( ( Y5 = Xc )
                             => ~ ( accp_P323873057692896796term_b @ lift_R1576223848067188451_rel_b @ ( produc4640078800242017687term_b @ X4 @ ( produc5352714453822653076term_b @ fOR_pos_PRoot @ ( produc3933249678690294607term_b @ fOR_ex5070955383060788274rallel @ Xc ) ) ) ) ) ) )
                       => ~ ( ( Xa = fOR_pos_PRoot )
                           => ( ( Xb = fOR_ext_EParallel )
                             => ( ( Y5
                                  = ( sup_su1163039302778497499term_b @ Xc
                                    @ ( inf_in8391146210534189557term_b @ id_Ground_gterm_b
                                      @ ( produc3596519815479089608term_b @ ( ground_T_G_b @ X4 )
                                        @ ^ [Uu: ground_gterm_b] : ( ground_T_G_b @ X4 ) ) ) ) )
                               => ~ ( accp_P323873057692896796term_b @ lift_R1576223848067188451_rel_b @ ( produc4640078800242017687term_b @ X4 @ ( produc5352714453822653076term_b @ fOR_pos_PRoot @ ( produc3933249678690294607term_b @ fOR_ext_EParallel @ Xc ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% lift_root_step.pelims
thf(fact_1176_less__option__Some,axiom,
    ! [X4: nat,Y5: nat] :
      ( ( ord_less_option_nat @ ( some_nat @ X4 ) @ ( some_nat @ Y5 ) )
      = ( ord_less_nat @ X4 @ Y5 ) ) ).

% less_option_Some
thf(fact_1177_less__prod__simp,axiom,
    ! [X1: nat,Y1: nat,X2: nat,Y2: nat] :
      ( ( ord_le1203424502768444845at_nat @ ( product_Pair_nat_nat @ X1 @ Y1 ) @ ( product_Pair_nat_nat @ X2 @ Y2 ) )
      = ( ( ord_less_nat @ X1 @ X2 )
        | ( ( ord_less_eq_nat @ X1 @ X2 )
          & ( ord_less_nat @ Y1 @ Y2 ) ) ) ) ).

% less_prod_simp
thf(fact_1178_less__eq__prod__simp,axiom,
    ! [X1: nat,Y1: nat,X2: nat,Y2: nat] :
      ( ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ X1 @ Y1 ) @ ( product_Pair_nat_nat @ X2 @ Y2 ) )
      = ( ( ord_less_nat @ X1 @ X2 )
        | ( ( ord_less_eq_nat @ X1 @ X2 )
          & ( ord_less_eq_nat @ Y1 @ Y2 ) ) ) ) ).

% less_eq_prod_simp
thf(fact_1179_lift__root__step_Osimps_I2_J,axiom,
    ! [F2: set_Pr1307281990691478580_b_nat,R3: set_Pr3134649703833893895term_b] :
      ( ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PAny @ fOR_ex5070955383060788274rallel @ R3 )
      = ( contex3086340098647126762_onp_b
        @ ^ [C5: ground_gmctxt_b] :
            ( ( ord_less_nat @ zero_zero_nat @ ( ground_num_gholes_b @ C5 ) )
            & ( ord_le5038853854344477844_b_nat @ ( ground4323624445104842081ctxt_b @ C5 ) @ F2 ) )
        @ R3 ) ) ).

% lift_root_step.simps(2)
thf(fact_1180_lift__root__step_Osimps_I5_J,axiom,
    ! [F2: set_Pr1307281990691478580_b_nat,R3: set_Pr3134649703833893895term_b] :
      ( ( lift_R8854095231239425056step_b @ F2 @ fOR_pos_PNonRoot @ fOR_ex5070955383060788274rallel @ R3 )
      = ( contex3086340098647126762_onp_b
        @ ^ [C5: ground_gmctxt_b] :
            ( ( ord_less_nat @ zero_zero_nat @ ( ground_num_gholes_b @ C5 ) )
            & ( ord_le5038853854344477844_b_nat @ ( ground4323624445104842081ctxt_b @ C5 ) @ F2 )
            & ( C5 != ground_GMHole_b ) )
        @ R3 ) ) ).

% lift_root_step.simps(5)
thf(fact_1181_less__supI1,axiom,
    ! [X4: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ X4 @ A )
     => ( ord_less_nat @ X4 @ ( sup_sup_nat @ A @ B ) ) ) ).

% less_supI1
thf(fact_1182_less__supI2,axiom,
    ! [X4: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ X4 @ B )
     => ( ord_less_nat @ X4 @ ( sup_sup_nat @ A @ B ) ) ) ).

% less_supI2
thf(fact_1183_sup_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( sup_sup_nat @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_1184_sup_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( sup_sup_nat @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_1185_sup_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_nat @ B @ A )
         => ~ ( ord_less_nat @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_1186_sup_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A6: nat] :
          ( ( A6
            = ( sup_sup_nat @ A6 @ B4 ) )
          & ( A6 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_1187_sup_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ C @ A )
     => ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_1188_sup_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ C @ B )
     => ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_1189_leD,axiom,
    ! [Y5: nat,X4: nat] :
      ( ( ord_less_eq_nat @ Y5 @ X4 )
     => ~ ( ord_less_nat @ X4 @ Y5 ) ) ).

% leD
thf(fact_1190_leI,axiom,
    ! [X4: nat,Y5: nat] :
      ( ~ ( ord_less_nat @ X4 @ Y5 )
     => ( ord_less_eq_nat @ Y5 @ X4 ) ) ).

% leI
thf(fact_1191_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1192_antisym__conv1,axiom,
    ! [X4: nat,Y5: nat] :
      ( ~ ( ord_less_nat @ X4 @ Y5 )
     => ( ( ord_less_eq_nat @ X4 @ Y5 )
        = ( X4 = Y5 ) ) ) ).

% antisym_conv1
thf(fact_1193_antisym__conv2,axiom,
    ! [X4: nat,Y5: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y5 )
     => ( ( ~ ( ord_less_nat @ X4 @ Y5 ) )
        = ( X4 = Y5 ) ) ) ).

% antisym_conv2
thf(fact_1194_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X @ Y4 )
          & ~ ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).

% less_le_not_le
thf(fact_1195_not__le__imp__less,axiom,
    ! [Y5: nat,X4: nat] :
      ( ~ ( ord_less_eq_nat @ Y5 @ X4 )
     => ( ord_less_nat @ X4 @ Y5 ) ) ).

% not_le_imp_less
thf(fact_1196_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A6: nat,B4: nat] :
          ( ( ord_less_nat @ A6 @ B4 )
          | ( A6 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1197_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A6: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A6 @ B4 )
          & ( A6 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1198_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1199_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1200_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A6: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A6 @ B4 )
          & ~ ( ord_less_eq_nat @ B4 @ A6 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1201_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A6: nat] :
          ( ( ord_less_nat @ B4 @ A6 )
          | ( A6 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1202_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A6: nat] :
          ( ( ord_less_eq_nat @ B4 @ A6 )
          & ( A6 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1203_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1204_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1205_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A6: nat] :
          ( ( ord_less_eq_nat @ B4 @ A6 )
          & ~ ( ord_less_eq_nat @ A6 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1206_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1207_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1208_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y4: nat] :
          ( ( ord_less_nat @ X @ Y4 )
          | ( X = Y4 ) ) ) ) ).

% order_le_less
thf(fact_1209_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X @ Y4 )
          & ( X != Y4 ) ) ) ) ).

% order_less_le
thf(fact_1210_linorder__not__le,axiom,
    ! [X4: nat,Y5: nat] :
      ( ( ~ ( ord_less_eq_nat @ X4 @ Y5 ) )
      = ( ord_less_nat @ Y5 @ X4 ) ) ).

% linorder_not_le
thf(fact_1211_linorder__not__less,axiom,
    ! [X4: nat,Y5: nat] :
      ( ( ~ ( ord_less_nat @ X4 @ Y5 ) )
      = ( ord_less_eq_nat @ Y5 @ X4 ) ) ).

% linorder_not_less
thf(fact_1212_order__less__imp__le,axiom,
    ! [X4: nat,Y5: nat] :
      ( ( ord_less_nat @ X4 @ Y5 )
     => ( ord_less_eq_nat @ X4 @ Y5 ) ) ).

% order_less_imp_le
thf(fact_1213_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1214_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1215_order__le__less__trans,axiom,
    ! [X4: nat,Y5: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y5 )
     => ( ( ord_less_nat @ Y5 @ Z )
       => ( ord_less_nat @ X4 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1216_order__less__le__trans,axiom,
    ! [X4: nat,Y5: nat,Z: nat] :
      ( ( ord_less_nat @ X4 @ Y5 )
     => ( ( ord_less_eq_nat @ Y5 @ Z )
       => ( ord_less_nat @ X4 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1217_order__le__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1218_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1219_order__less__le__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1220_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1221_linorder__le__less__linear,axiom,
    ! [X4: nat,Y5: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y5 )
      | ( ord_less_nat @ Y5 @ X4 ) ) ).

% linorder_le_less_linear
thf(fact_1222_order__le__imp__less__or__eq,axiom,
    ! [X4: nat,Y5: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y5 )
     => ( ( ord_less_nat @ X4 @ Y5 )
        | ( X4 = Y5 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1223_nat__descend__induct,axiom,
    ! [N: nat,P2: nat > $o,M: nat] :
      ( ! [K2: nat] :
          ( ( ord_less_nat @ N @ K2 )
         => ( P2 @ K2 ) )
     => ( ! [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
           => ( ! [I2: nat] :
                  ( ( ord_less_nat @ K2 @ I2 )
                 => ( P2 @ I2 ) )
             => ( P2 @ K2 ) ) )
       => ( P2 @ M ) ) ) ).

% nat_descend_induct
thf(fact_1224_not__less__Least,axiom,
    ! [K: nat,P2: nat > $o] :
      ( ( ord_less_nat @ K @ ( ord_Least_nat @ P2 ) )
     => ~ ( P2 @ K ) ) ).

% not_less_Least
thf(fact_1225_antisymp__on__greater,axiom,
    ! [A5: set_nat] :
      ( antisymp_on_nat @ A5
      @ ^ [X: nat,Y4: nat] : ( ord_less_nat @ Y4 @ X ) ) ).

% antisymp_on_greater
thf(fact_1226_asymp__on__greater,axiom,
    ! [A5: set_nat] :
      ( asymp_on_nat @ A5
      @ ^ [X: nat,Y4: nat] : ( ord_less_nat @ Y4 @ X ) ) ).

% asymp_on_greater
thf(fact_1227_antisymp__on__less,axiom,
    ! [A5: set_nat] : ( antisymp_on_nat @ A5 @ ord_less_nat ) ).

% antisymp_on_less
thf(fact_1228_asymp__on__less,axiom,
    ! [A5: set_nat] : ( asymp_on_nat @ A5 @ ord_less_nat ) ).

% asymp_on_less
thf(fact_1229_less__option__def,axiom,
    ( ord_less_option_nat
    = ( ^ [X: option_nat] :
          ( case_option_o_nat @ $false
          @ ^ [Y4: nat] :
              ( case_option_o_nat @ $true
              @ ^ [Z4: nat] : ( ord_less_nat @ Z4 @ Y4 )
              @ X ) ) ) ) ).

% less_option_def
thf(fact_1230_order__less__imp__not__less,axiom,
    ! [X4: nat,Y5: nat] :
      ( ( ord_less_nat @ X4 @ Y5 )
     => ~ ( ord_less_nat @ Y5 @ X4 ) ) ).

% order_less_imp_not_less
thf(fact_1231_order__less__imp__not__eq2,axiom,
    ! [X4: nat,Y5: nat] :
      ( ( ord_less_nat @ X4 @ Y5 )
     => ( Y5 != X4 ) ) ).

% order_less_imp_not_eq2
thf(fact_1232_order__less__imp__not__eq,axiom,
    ! [X4: nat,Y5: nat] :
      ( ( ord_less_nat @ X4 @ Y5 )
     => ( X4 != Y5 ) ) ).

% order_less_imp_not_eq
thf(fact_1233_linorder__less__linear,axiom,
    ! [X4: nat,Y5: nat] :
      ( ( ord_less_nat @ X4 @ Y5 )
      | ( X4 = Y5 )
      | ( ord_less_nat @ Y5 @ X4 ) ) ).

% linorder_less_linear
thf(fact_1234_order__less__imp__triv,axiom,
    ! [X4: nat,Y5: nat,P2: $o] :
      ( ( ord_less_nat @ X4 @ Y5 )
     => ( ( ord_less_nat @ Y5 @ X4 )
       => P2 ) ) ).

% order_less_imp_triv
thf(fact_1235_order__less__not__sym,axiom,
    ! [X4: nat,Y5: nat] :
      ( ( ord_less_nat @ X4 @ Y5 )
     => ~ ( ord_less_nat @ Y5 @ X4 ) ) ).

% order_less_not_sym
thf(fact_1236_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1237_order__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1238_order__less__irrefl,axiom,
    ! [X4: nat] :
      ~ ( ord_less_nat @ X4 @ X4 ) ).

% order_less_irrefl
thf(fact_1239_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1240_ord__eq__less__subst,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1241_order__less__trans,axiom,
    ! [X4: nat,Y5: nat,Z: nat] :
      ( ( ord_less_nat @ X4 @ Y5 )
     => ( ( ord_less_nat @ Y5 @ Z )
       => ( ord_less_nat @ X4 @ Z ) ) ) ).

% order_less_trans
thf(fact_1242_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_1243_linorder__neq__iff,axiom,
    ! [X4: nat,Y5: nat] :
      ( ( X4 != Y5 )
      = ( ( ord_less_nat @ X4 @ Y5 )
        | ( ord_less_nat @ Y5 @ X4 ) ) ) ).

% linorder_neq_iff
thf(fact_1244_order__less__asym,axiom,
    ! [X4: nat,Y5: nat] :
      ( ( ord_less_nat @ X4 @ Y5 )
     => ~ ( ord_less_nat @ Y5 @ X4 ) ) ).

% order_less_asym
thf(fact_1245_linorder__neqE,axiom,
    ! [X4: nat,Y5: nat] :
      ( ( X4 != Y5 )
     => ( ~ ( ord_less_nat @ X4 @ Y5 )
       => ( ord_less_nat @ Y5 @ X4 ) ) ) ).

% linorder_neqE
thf(fact_1246_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_1247_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_1248_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_1249_not__less__iff__gr__or__eq,axiom,
    ! [X4: nat,Y5: nat] :
      ( ( ~ ( ord_less_nat @ X4 @ Y5 ) )
      = ( ( ord_less_nat @ Y5 @ X4 )
        | ( X4 = Y5 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_1250_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_1251_linorder__less__wlog,axiom,
    ! [P2: nat > nat > $o,A: nat,B: nat] :
      ( ! [A4: nat,B3: nat] :
          ( ( ord_less_nat @ A4 @ B3 )
         => ( P2 @ A4 @ B3 ) )
     => ( ! [A4: nat] : ( P2 @ A4 @ A4 )
       => ( ! [A4: nat,B3: nat] :
              ( ( P2 @ B3 @ A4 )
             => ( P2 @ A4 @ B3 ) )
         => ( P2 @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_1252_exists__least__iff,axiom,
    ( ( ^ [P4: nat > $o] :
        ? [X7: nat] : ( P4 @ X7 ) )
    = ( ^ [P3: nat > $o] :
        ? [N3: nat] :
          ( ( P3 @ N3 )
          & ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N3 )
             => ~ ( P3 @ M3 ) ) ) ) ) ).

% exists_least_iff
thf(fact_1253_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_1254_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_1255_linorder__cases,axiom,
    ! [X4: nat,Y5: nat] :
      ( ~ ( ord_less_nat @ X4 @ Y5 )
     => ( ( X4 != Y5 )
       => ( ord_less_nat @ Y5 @ X4 ) ) ) ).

% linorder_cases
thf(fact_1256_antisym__conv3,axiom,
    ! [Y5: nat,X4: nat] :
      ( ~ ( ord_less_nat @ Y5 @ X4 )
     => ( ( ~ ( ord_less_nat @ X4 @ Y5 ) )
        = ( X4 = Y5 ) ) ) ).

% antisym_conv3
thf(fact_1257_less__induct,axiom,
    ! [P2: nat > $o,A: nat] :
      ( ! [X3: nat] :
          ( ! [Y9: nat] :
              ( ( ord_less_nat @ Y9 @ X3 )
             => ( P2 @ Y9 ) )
         => ( P2 @ X3 ) )
     => ( P2 @ A ) ) ).

% less_induct
thf(fact_1258_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_1259_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_1260_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_1261_less__imp__neq,axiom,
    ! [X4: nat,Y5: nat] :
      ( ( ord_less_nat @ X4 @ Y5 )
     => ( X4 != Y5 ) ) ).

% less_imp_neq
thf(fact_1262_gt__ex,axiom,
    ! [X4: nat] :
    ? [X_12: nat] : ( ord_less_nat @ X4 @ X_12 ) ).

% gt_ex
thf(fact_1263_inf_Ostrict__coboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_nat @ B @ C )
     => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_1264_inf_Ostrict__coboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ A @ C )
     => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_1265_inf_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [A6: nat,B4: nat] :
          ( ( A6
            = ( inf_inf_nat @ A6 @ B4 ) )
          & ( A6 != B4 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_1266_inf_Ostrict__boundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( inf_inf_nat @ B @ C ) )
     => ~ ( ( ord_less_nat @ A @ B )
         => ~ ( ord_less_nat @ A @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_1267_inf_Oabsorb4,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( inf_inf_nat @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_1268_inf_Oabsorb3,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( inf_inf_nat @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_1269_less__infI2,axiom,
    ! [B: nat,X4: nat,A: nat] :
      ( ( ord_less_nat @ B @ X4 )
     => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X4 ) ) ).

% less_infI2
thf(fact_1270_less__infI1,axiom,
    ! [A: nat,X4: nat,B: nat] :
      ( ( ord_less_nat @ A @ X4 )
     => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X4 ) ) ).

% less_infI1
thf(fact_1271_acyclicI__order,axiom,
    ! [R5: set_Pr258254617715012167tion_b,F: option_b > nat] :
      ( ! [A4: option_b,B3: option_b] :
          ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ A4 @ B3 ) @ R5 )
         => ( ord_less_nat @ ( F @ B3 ) @ ( F @ A4 ) ) )
     => ( transi5850314681905841172tion_b @ R5 ) ) ).

% acyclicI_order
thf(fact_1272_relpow__0__I,axiom,
    ! [X4: option_b,R: set_Pr258254617715012167tion_b] : ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ X4 ) @ ( compow3776960924738975994tion_b @ zero_zero_nat @ R ) ) ).

% relpow_0_I
thf(fact_1273_relpow__0__E,axiom,
    ! [X4: option_b,Y5: option_b,R: set_Pr258254617715012167tion_b] :
      ( ( member547026996284216464tion_b @ ( produc3989389360737052631tion_b @ X4 @ Y5 ) @ ( compow3776960924738975994tion_b @ zero_zero_nat @ R ) )
     => ( X4 = Y5 ) ) ).

% relpow_0_E
thf(fact_1274_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_1275_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum

% Conjectures (1)
thf(conj_0,conjecture,
    ( rRn_RR7260272242312503694es_b_b
    @ ( tA_Clo384869186672131716on_b_a
      @ ( fimage4266613729464776675_b_nat
        @ ( produc5650155531262253963_b_nat
          @ ^ [F3: b] : ( produc2093246604507004384_b_nat @ ( produc3989389360737052631tion_b @ ( some_b @ F3 ) @ ( some_b @ F3 ) ) ) )
        @ f )
      @ a )
    @ ( lift_R8854095231239425056step_b @ ( fset_P7858563055952297295_b_nat @ f ) @ fOR_pos_PAny @ fOR_ex5070955383060788274rallel @ r ) ) ).

%------------------------------------------------------------------------------