## TPTP Problem File: SEV523^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEV523^1 : TPTP v7.5.0. Released v7.5.0.
% Domain   : Set Theory
% Problem  : ProofGold problem HOSetConstr_20_pos_th0
% Version  : Especial.
% English  :

% Refs     : [Urb20] Urban (2020) Email to Geoff Sutcliffe
% Source   : [Urb20]
% Names    : HOSetConstr_20_pos_th0.p [Urb20]

% Status   : Theorem
% Rating   : 1.00 v7.5.0
% Syntax   : Number of formulae    :  213 (   0 unit; 104 type;   0 defn)
%            Number of atoms       : 1720 ( 135 equality; 878 variable)
%            Maximal formula depth :   36 (   8 average)
%            Number of connectives : 1341 (   0   ~;   0   |;   0   &;1231   @)
%                                         (   0 <=>; 110  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :  697 ( 697   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  106 ( 104   :;   0   =)
%            Number of variables   :  504 (  74 sgn; 129   !;  37   ?; 338   ^)
%                                         ( 504   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_EQU_NAR

%------------------------------------------------------------------------------
thf(c_Eps_i_tp,type,(
c_Eps_i: ( \$i > \$o ) > \$i )).

thf(c_False_tp,type,(
c_False: \$o )).

thf(c_True_tp,type,(
c_True: \$o )).

thf(c_not_tp,type,(
c_not: \$o > \$o )).

thf(c_and_tp,type,(
c_and: \$o > \$o > \$o )).

thf(c_or_tp,type,(
c_or: \$o > \$o > \$o )).

thf(c_iff_tp,type,(
c_iff: \$o > \$o > \$o )).

thf(c_In_tp,type,(
c_In: \$i > \$i > \$o )).

thf(c_Subq_tp,type,(
c_Subq: \$i > \$i > \$o )).

thf(c_Empty_tp,type,(
c_Empty: \$i )).

thf(c_Union_tp,type,(
c_Union: \$i > \$i )).

thf(c_Power_tp,type,(
c_Power: \$i > \$i )).

thf(c_Repl_tp,type,(
c_Repl: \$i > ( \$i > \$i ) > \$i )).

thf(c_TransSet_tp,type,(
c_TransSet: \$i > \$o )).

thf(c_atleast2_tp,type,(
c_atleast2: \$i > \$o )).

thf(c_atleast3_tp,type,(
c_atleast3: \$i > \$o )).

thf(c_atleast4_tp,type,(
c_atleast4: \$i > \$o )).

thf(c_atleast5_tp,type,(
c_atleast5: \$i > \$o )).

thf(c_atleast6_tp,type,(
c_atleast6: \$i > \$o )).

thf(c_exactly2_tp,type,(
c_exactly2: \$i > \$o )).

thf(c_exactly3_tp,type,(
c_exactly3: \$i > \$o )).

thf(c_exactly4_tp,type,(
c_exactly4: \$i > \$o )).

thf(c_exactly5_tp,type,(
c_exactly5: \$i > \$o )).

thf(c_exu_i_tp,type,(
c_exu_i: ( \$i > \$o ) > \$o )).

thf(c_reflexive_i_tp,type,(
c_reflexive_i: ( \$i > \$i > \$o ) > \$o )).

thf(c_irreflexive_i_tp,type,(
c_irreflexive_i: ( \$i > \$i > \$o ) > \$o )).

thf(c_symmetric_i_tp,type,(
c_symmetric_i: ( \$i > \$i > \$o ) > \$o )).

thf(c_antisymmetric_i_tp,type,(
c_antisymmetric_i: ( \$i > \$i > \$o ) > \$o )).

thf(c_transitive_i_tp,type,(
c_transitive_i: ( \$i > \$i > \$o ) > \$o )).

thf(c_eqreln_i_tp,type,(
c_eqreln_i: ( \$i > \$i > \$o ) > \$o )).

thf(c_per_i_tp,type,(
c_per_i: ( \$i > \$i > \$o ) > \$o )).

thf(c_linear_i_tp,type,(
c_linear_i: ( \$i > \$i > \$o ) > \$o )).

thf(c_trichotomous_or_i_tp,type,(
c_trichotomous_or_i: ( \$i > \$i > \$o ) > \$o )).

thf(c_partialorder_i_tp,type,(
c_partialorder_i: ( \$i > \$i > \$o ) > \$o )).

thf(c_totalorder_i_tp,type,(
c_totalorder_i: ( \$i > \$i > \$o ) > \$o )).

thf(c_strictpartialorder_i_tp,type,(
c_strictpartialorder_i: ( \$i > \$i > \$o ) > \$o )).

thf(c_stricttotalorder_i_tp,type,(
c_stricttotalorder_i: ( \$i > \$i > \$o ) > \$o )).

thf(c_If_i_tp,type,(
c_If_i: \$o > \$i > \$i > \$i )).

thf(c_exactly1of2_tp,type,(
c_exactly1of2: \$o > \$o > \$o )).

thf(c_exactly1of3_tp,type,(
c_exactly1of3: \$o > \$o > \$o > \$o )).

thf(c_nIn_tp,type,(
c_nIn: \$i > \$i > \$o )).

thf(c_nSubq_tp,type,(
c_nSubq: \$i > \$i > \$o )).

thf(c_UPair_tp,type,(
c_UPair: \$i > \$i > \$i )).

thf(c_Sing_tp,type,(
c_Sing: \$i > \$i )).

thf(c_binunion_tp,type,(
c_binunion: \$i > \$i > \$i )).

c_SetAdjoin: \$i > \$i > \$i )).

thf(c_famunion_tp,type,(
c_famunion: \$i > ( \$i > \$i ) > \$i )).

thf(c_Sep_tp,type,(
c_Sep: \$i > ( \$i > \$o ) > \$i )).

thf(c_ReplSep_tp,type,(
c_ReplSep: \$i > ( \$i > \$o ) > ( \$i > \$i ) > \$i )).

thf(c_binintersect_tp,type,(
c_binintersect: \$i > \$i > \$i )).

thf(c_setminus_tp,type,(
c_setminus: \$i > \$i > \$i )).

thf(c_inj_tp,type,(
c_inj: \$i > \$i > ( \$i > \$i ) > \$o )).

thf(c_bij_tp,type,(
c_bij: \$i > \$i > ( \$i > \$i ) > \$o )).

thf(c_atleastp_tp,type,(
c_atleastp: \$i > \$i > \$o )).

thf(c_equip_tp,type,(
c_equip: \$i > \$i > \$o )).

thf(c_In_rec_poly_G_i_tp,type,(
c_In_rec_poly_G_i: ( \$i > ( \$i > \$i ) > \$i ) > \$i > \$i > \$o )).

thf(c_In_rec_poly_i_tp,type,(
c_In_rec_poly_i: ( \$i > ( \$i > \$i ) > \$i ) > \$i > \$i )).

thf(c_ordsucc_tp,type,(
c_ordsucc: \$i > \$i )).

thf(c_nat_p_tp,type,(
c_nat_p: \$i > \$o )).

thf(c_nat_primrec_tp,type,(
c_nat_primrec: \$i > ( \$i > \$i > \$i ) > \$i > \$i )).

c_add_nat: \$i > \$i > \$i )).

thf(c_mul_nat_tp,type,(
c_mul_nat: \$i > \$i > \$i )).

thf(c_ordinal_tp,type,(
c_ordinal: \$i > \$o )).

thf(c_V__tp,type,(
c_V_: \$i > \$i )).

thf(c_Inj1_tp,type,(
c_Inj1: \$i > \$i )).

thf(c_Inj0_tp,type,(
c_Inj0: \$i > \$i )).

thf(c_Unj_tp,type,(
c_Unj: \$i > \$i )).

thf(c_combine_funcs_tp,type,(
c_combine_funcs: \$i > \$i > ( \$i > \$i ) > ( \$i > \$i ) > \$i > \$i )).

thf(c_setsum_tp,type,(
c_setsum: \$i > \$i > \$i )).

thf(c_proj0_tp,type,(
c_proj0: \$i > \$i )).

thf(c_proj1_tp,type,(
c_proj1: \$i > \$i )).

thf(c_binrep_tp,type,(
c_binrep: \$i > \$i > \$i )).

thf(c_lam_tp,type,(
c_lam: \$i > ( \$i > \$i ) > \$i )).

thf(c_setprod_tp,type,(
c_setprod: \$i > \$i > \$i )).

thf(c_ap_tp,type,(
c_ap: \$i > \$i > \$i )).

thf(c_setsum_p_tp,type,(
c_setsum_p: \$i > \$o )).

thf(c_tuple_p_tp,type,(
c_tuple_p: \$i > \$i > \$o )).

thf(c_Pi_tp,type,(
c_Pi: \$i > ( \$i > \$i ) > \$i )).

thf(c_setexp_tp,type,(
c_setexp: \$i > \$i > \$i )).

thf(c_Sep2_tp,type,(
c_Sep2: \$i > ( \$i > \$i ) > ( \$i > \$i > \$o ) > \$i )).

thf(c_set_of_pairs_tp,type,(
c_set_of_pairs: \$i > \$o )).

thf(c_lam2_tp,type,(
c_lam2: \$i > ( \$i > \$i ) > ( \$i > \$i > \$i ) > \$i )).

thf(c_PNoEq__tp,type,(
c_PNoEq_: \$i > ( \$i > \$o ) > ( \$i > \$o ) > \$o )).

thf(c_PNoLt__tp,type,(
c_PNoLt_: \$i > ( \$i > \$o ) > ( \$i > \$o ) > \$o )).

thf(c_PNoLt_tp,type,(
c_PNoLt: \$i > ( \$i > \$o ) > \$i > ( \$i > \$o ) > \$o )).

thf(c_PNoLe_tp,type,(
c_PNoLe: \$i > ( \$i > \$o ) > \$i > ( \$i > \$o ) > \$o )).

thf(c_PNo_downc_tp,type,(
c_PNo_downc: ( \$i > ( \$i > \$o ) > \$o ) > \$i > ( \$i > \$o ) > \$o )).

thf(c_PNo_upc_tp,type,(
c_PNo_upc: ( \$i > ( \$i > \$o ) > \$o ) > \$i > ( \$i > \$o ) > \$o )).

thf(c_SNoElts__tp,type,(
c_SNoElts_: \$i > \$i )).

thf(c_SNo__tp,type,(
c_SNo_: \$i > \$i > \$o )).

thf(c_PSNo_tp,type,(
c_PSNo: \$i > ( \$i > \$o ) > \$i )).

thf(c_SNo_tp,type,(
c_SNo: \$i > \$o )).

thf(c_SNoLev_tp,type,(
c_SNoLev: \$i > \$i )).

thf(c_SNoEq__tp,type,(
c_SNoEq_: \$i > \$i > \$i > \$o )).

thf(c_SNoLt_tp,type,(
c_SNoLt: \$i > \$i > \$o )).

thf(c_SNoLe_tp,type,(
c_SNoLe: \$i > \$i > \$o )).

thf(c_binop_on_tp,type,(
c_binop_on: \$i > ( \$i > \$i > \$i ) > \$o )).

thf(c_Loop_tp,type,(
c_Loop: \$i > ( \$i > \$i > \$i ) > ( \$i > \$i > \$i ) > ( \$i > \$i > \$i ) > \$i > \$o )).

thf(c_Loop_with_defs_tp,type,(
c_Loop_with_defs: \$i > ( \$i > \$i > \$i ) > ( \$i > \$i > \$i ) > ( \$i > \$i > \$i ) > \$i > ( \$i > \$i > \$i ) > ( \$i > \$i > \$i > \$i ) > ( \$i > \$i > \$i ) > ( \$i > \$i > \$i > \$i ) > ( \$i > \$i > \$i > \$i ) > ( \$i > \$i > \$i ) > ( \$i > \$i > \$i ) > ( \$i > \$i > \$i ) > ( \$i > \$i > \$i ) > \$o )).

thf(c_Loop_with_defs_cex1_tp,type,(
c_Loop_with_defs_cex1: \$i > ( \$i > \$i > \$i ) > ( \$i > \$i > \$i ) > ( \$i > \$i > \$i ) > \$i > ( \$i > \$i > \$i ) > ( \$i > \$i > \$i > \$i ) > ( \$i > \$i > \$i ) > ( \$i > \$i > \$i > \$i ) > ( \$i > \$i > \$i > \$i ) > ( \$i > \$i > \$i ) > ( \$i > \$i > \$i ) > ( \$i > \$i > \$i ) > ( \$i > \$i > \$i ) > \$o )).

thf(c_Loop_with_defs_cex2_tp,type,(
c_Loop_with_defs_cex2: \$i > ( \$i > \$i > \$i ) > ( \$i > \$i > \$i ) > ( \$i > \$i > \$i ) > \$i > ( \$i > \$i > \$i ) > ( \$i > \$i > \$i > \$i ) > ( \$i > \$i > \$i ) > ( \$i > \$i > \$i > \$i ) > ( \$i > \$i > \$i > \$i ) > ( \$i > \$i > \$i ) > ( \$i > \$i > \$i ) > ( \$i > \$i > \$i ) > ( \$i > \$i > \$i ) > \$o )).

thf(c_combinator_tp,type,(
c_combinator: \$i > \$o )).

thf(c_combinator_equiv_tp,type,(
c_combinator_equiv: \$i > \$i > \$o )).

thf(c_equip_mod_tp,type,(
c_equip_mod: \$i > \$i > \$i > \$o )).

thf(ax1,axiom,(
! [X0: \$i > \$o,X1: \$i] :
( ( X0 @ X1 )
=> ( X0 @ ( c_Eps_i @ X0 ) ) ) )).

thf(ax2,axiom,(
! [X0: \$o] :
( ( c_not @ ( c_not @ X0 ) )
=> X0 ) )).

thf(ax3,axiom,(
! [X0: \$o,X1: \$o] :
( ( c_iff @ X0 @ X1 )
=> ( X0 = X1 ) ) )).

thf(ax4,axiom,(
! [X0: \$i,X1: \$i] :
( ( c_Subq @ X0 @ X1 )
=> ( ( c_Subq @ X1 @ X0 )
=> ( X0 = X1 ) ) ) )).

thf(ax5,axiom,
( c_not
@ ? [X0: \$i] :
( c_In @ X0 @ c_Empty ) )).

thf(ax6,axiom,(
! [X0: \$i,X1: \$i] :
( c_iff @ ( c_In @ X1 @ ( c_Union @ X0 ) )
@ ? [X2: \$i] :
( c_and @ ( c_In @ X1 @ X2 ) @ ( c_In @ X2 @ X0 ) ) ) )).

thf(ax7,axiom,(
! [X0: \$i,X1: \$i] :
( c_iff @ ( c_In @ X1 @ ( c_Power @ X0 ) ) @ ( c_Subq @ X1 @ X0 ) ) )).

thf(ax8,axiom,(
! [X0: \$i,X1: \$i > \$i,X2: \$i] :
( c_iff
@ ( c_In @ X2
@ ( c_Repl @ X0
@ ^ [X3: \$i] :
( X1 @ X3 ) ) )
@ ? [X3: \$i] :
( c_and @ ( c_In @ X3 @ X0 )
@ ( X2
= ( X1 @ X3 ) ) ) ) )).

thf(ax9,axiom,(
! [X0: \$i > \$o] :
( ! [X1: \$i] :
( ( X0 @ X1 )
=> ! [X2: \$i] :
( ( c_In @ X2 @ X1 )
=> ( X0 @ X2 ) ) )
=> ( ( X0 @ c_Empty )
=> ( ! [X1: \$i] :
( ( X0 @ X1 )
=> ( X0 @ ( c_Union @ X1 ) ) )
=> ( ! [X1: \$i] :
( ( X0 @ X1 )
=> ( X0 @ ( c_Power @ X1 ) ) )
=> ( ! [X1: \$i] :
( ( X0 @ X1 )
=> ! [X2: \$i > \$i] :
( ! [X3: \$i] :
( ( c_In @ X3 @ X1 )
=> ( X0 @ ( X2 @ X3 ) ) )
=> ( X0
@ ( c_Repl @ X1
@ ^ [X3: \$i] :
( X2 @ X3 ) ) ) ) )
=> ! [X1: \$i] :
( X0 @ X1 ) ) ) ) ) ) )).

thf(ax10,axiom,(
! [X0: \$i > \$o] :
( ! [X1: \$i] :
( ! [X2: \$i] :
( ( c_In @ X2 @ X1 )
=> ( X0 @ X2 ) )
=> ( X0 @ X1 ) )
=> ! [X1: \$i] :
( X0 @ X1 ) ) )).

thf(ax11,axiom,
( c_False
= ( ! [X0: \$o] : X0 ) )).

thf(ax12,axiom,
( c_True
= ( ! [X0: \$o] :
( X0
=> X0 ) ) )).

thf(ax13,axiom,
( c_not
= ( ^ [X0: \$o] :
( X0
=> c_False ) ) )).

thf(ax14,axiom,
( c_and
= ( ^ [X0: \$o,X1: \$o] :
! [X2: \$o] :
( ( X0
=> ( X1
=> X2 ) )
=> X2 ) ) )).

thf(ax15,axiom,
( c_or
= ( ^ [X0: \$o,X1: \$o] :
! [X2: \$o] :
( ( X0
=> X2 )
=> ( ( X1
=> X2 )
=> X2 ) ) ) )).

thf(ax16,axiom,
( c_iff
= ( ^ [X0: \$o,X1: \$o] :
( c_and
@ ( X0
=> X1 )
@ ( X1
=> X0 ) ) ) )).

thf(ax17,axiom,
( c_Subq
= ( ^ [X0: \$i,X1: \$i] :
! [X2: \$i] :
( ( c_In @ X2 @ X0 )
=> ( c_In @ X2 @ X1 ) ) ) )).

thf(ax18,axiom,
( c_TransSet
= ( ^ [X0: \$i] :
! [X1: \$i] :
( ( c_In @ X1 @ X0 )
=> ( c_Subq @ X1 @ X0 ) ) ) )).

thf(ax19,axiom,
( c_atleast2
= ( ^ [X0: \$i] :
? [X1: \$i] :
( c_and @ ( c_In @ X1 @ X0 ) @ ( c_not @ ( c_Subq @ X0 @ ( c_Power @ X1 ) ) ) ) ) )).

thf(ax20,axiom,
( c_atleast3
= ( ^ [X0: \$i] :
? [X1: \$i] :
( c_and @ ( c_Subq @ X1 @ X0 ) @ ( c_and @ ( c_not @ ( c_Subq @ X0 @ X1 ) ) @ ( c_atleast2 @ X1 ) ) ) ) )).

thf(ax21,axiom,
( c_atleast4
= ( ^ [X0: \$i] :
? [X1: \$i] :
( c_and @ ( c_Subq @ X1 @ X0 ) @ ( c_and @ ( c_not @ ( c_Subq @ X0 @ X1 ) ) @ ( c_atleast3 @ X1 ) ) ) ) )).

thf(ax22,axiom,
( c_atleast5
= ( ^ [X0: \$i] :
? [X1: \$i] :
( c_and @ ( c_Subq @ X1 @ X0 ) @ ( c_and @ ( c_not @ ( c_Subq @ X0 @ X1 ) ) @ ( c_atleast4 @ X1 ) ) ) ) )).

thf(ax23,axiom,
( c_atleast6
= ( ^ [X0: \$i] :
? [X1: \$i] :
( c_and @ ( c_Subq @ X1 @ X0 ) @ ( c_and @ ( c_not @ ( c_Subq @ X0 @ X1 ) ) @ ( c_atleast5 @ X1 ) ) ) ) )).

thf(ax24,axiom,
( c_exactly2
= ( ^ [X0: \$i] :
( c_and @ ( c_atleast2 @ X0 ) @ ( c_not @ ( c_atleast3 @ X0 ) ) ) ) )).

thf(ax25,axiom,
( c_exactly3
= ( ^ [X0: \$i] :
( c_and @ ( c_atleast3 @ X0 ) @ ( c_not @ ( c_atleast4 @ X0 ) ) ) ) )).

thf(ax26,axiom,
( c_exactly4
= ( ^ [X0: \$i] :
( c_and @ ( c_atleast4 @ X0 ) @ ( c_not @ ( c_atleast5 @ X0 ) ) ) ) )).

thf(ax27,axiom,
( c_exactly5
= ( ^ [X0: \$i] :
( c_and @ ( c_atleast5 @ X0 ) @ ( c_not @ ( c_atleast6 @ X0 ) ) ) ) )).

thf(ax28,axiom,
( c_exu_i
= ( ^ [X0: \$i > \$o] :
( c_and
@ ? [X1: \$i] :
( X0 @ X1 )
@ ! [X1: \$i,X2: \$i] :
( ( X0 @ X1 )
=> ( ( X0 @ X2 )
=> ( X1 = X2 ) ) ) ) ) )).

thf(ax29,axiom,
( c_reflexive_i
= ( ^ [X0: \$i > \$i > \$o] :
! [X1: \$i] :
( X0 @ X1 @ X1 ) ) )).

thf(ax30,axiom,
( c_irreflexive_i
= ( ^ [X0: \$i > \$i > \$o] :
! [X1: \$i] :
( c_not @ ( X0 @ X1 @ X1 ) ) ) )).

thf(ax31,axiom,
( c_symmetric_i
= ( ^ [X0: \$i > \$i > \$o] :
! [X1: \$i,X2: \$i] :
( ( X0 @ X1 @ X2 )
=> ( X0 @ X2 @ X1 ) ) ) )).

thf(ax32,axiom,
( c_antisymmetric_i
= ( ^ [X0: \$i > \$i > \$o] :
! [X1: \$i,X2: \$i] :
( ( X0 @ X1 @ X2 )
=> ( ( X0 @ X2 @ X1 )
=> ( X1 = X2 ) ) ) ) )).

thf(ax33,axiom,
( c_transitive_i
= ( ^ [X0: \$i > \$i > \$o] :
! [X1: \$i,X2: \$i,X3: \$i] :
( ( X0 @ X1 @ X2 )
=> ( ( X0 @ X2 @ X3 )
=> ( X0 @ X1 @ X3 ) ) ) ) )).

thf(ax34,axiom,
( c_eqreln_i
= ( ^ [X0: \$i > \$i > \$o] :
( c_and @ ( c_and @ ( c_reflexive_i @ X0 ) @ ( c_symmetric_i @ X0 ) ) @ ( c_transitive_i @ X0 ) ) ) )).

thf(ax35,axiom,
( c_per_i
= ( ^ [X0: \$i > \$i > \$o] :
( c_and @ ( c_symmetric_i @ X0 ) @ ( c_transitive_i @ X0 ) ) ) )).

thf(ax36,axiom,
( c_linear_i
= ( ^ [X0: \$i > \$i > \$o] :
! [X1: \$i,X2: \$i] :
( c_or @ ( X0 @ X1 @ X2 ) @ ( X0 @ X2 @ X1 ) ) ) )).

thf(ax37,axiom,
( c_trichotomous_or_i
= ( ^ [X0: \$i > \$i > \$o] :
! [X1: \$i,X2: \$i] :
( c_or @ ( c_or @ ( X0 @ X1 @ X2 ) @ ( X1 = X2 ) ) @ ( X0 @ X2 @ X1 ) ) ) )).

thf(ax38,axiom,
( c_partialorder_i
= ( ^ [X0: \$i > \$i > \$o] :
( c_and @ ( c_and @ ( c_reflexive_i @ X0 ) @ ( c_antisymmetric_i @ X0 ) ) @ ( c_transitive_i @ X0 ) ) ) )).

thf(ax39,axiom,
( c_totalorder_i
= ( ^ [X0: \$i > \$i > \$o] :
( c_and @ ( c_partialorder_i @ X0 ) @ ( c_linear_i @ X0 ) ) ) )).

thf(ax40,axiom,
( c_strictpartialorder_i
= ( ^ [X0: \$i > \$i > \$o] :
( c_and @ ( c_irreflexive_i @ X0 ) @ ( c_transitive_i @ X0 ) ) ) )).

thf(ax41,axiom,
( c_stricttotalorder_i
= ( ^ [X0: \$i > \$i > \$o] :
( c_and @ ( c_strictpartialorder_i @ X0 ) @ ( c_trichotomous_or_i @ X0 ) ) ) )).

thf(ax42,axiom,
( c_If_i
= ( ^ [X0: \$o,X1: \$i,X2: \$i] :
( c_Eps_i
@ ^ [X3: \$i] :
( c_or @ ( c_and @ X0 @ ( X3 = X1 ) ) @ ( c_and @ ( c_not @ X0 ) @ ( X3 = X2 ) ) ) ) ) )).

thf(ax43,axiom,
( c_exactly1of2
= ( ^ [X0: \$o,X1: \$o] :
( c_or @ ( c_and @ X0 @ ( c_not @ X1 ) ) @ ( c_and @ ( c_not @ X0 ) @ X1 ) ) ) )).

thf(ax44,axiom,
( c_exactly1of3
= ( ^ [X0: \$o,X1: \$o,X2: \$o] :
( c_or @ ( c_and @ ( c_exactly1of2 @ X0 @ X1 ) @ ( c_not @ X2 ) ) @ ( c_and @ ( c_and @ ( c_not @ X0 ) @ ( c_not @ X1 ) ) @ X2 ) ) ) )).

thf(ax45,axiom,
( c_nIn
= ( ^ [X0: \$i,X1: \$i] :
( c_not @ ( c_In @ X0 @ X1 ) ) ) )).

thf(ax46,axiom,
( c_nSubq
= ( ^ [X0: \$i,X1: \$i] :
( c_not @ ( c_Subq @ X0 @ X1 ) ) ) )).

thf(ax47,axiom,
( c_UPair
= ( ^ [X0: \$i,X1: \$i] :
( c_Repl @ ( c_Power @ ( c_Power @ c_Empty ) )
@ ^ [X2: \$i] :
( c_If_i @ ( c_In @ c_Empty @ X2 ) @ X0 @ X1 ) ) ) )).

thf(ax48,axiom,
( c_Sing
= ( ^ [X0: \$i] :
( c_UPair @ X0 @ X0 ) ) )).

thf(ax49,axiom,
( c_binunion
= ( ^ [X0: \$i,X1: \$i] :
( c_Union @ ( c_UPair @ X0 @ X1 ) ) ) )).

thf(ax50,axiom,
= ( ^ [X0: \$i,X1: \$i] :
( c_binunion @ X0 @ ( c_Sing @ X1 ) ) ) )).

thf(ax51,axiom,
( c_famunion
= ( ^ [X0: \$i,X1: \$i > \$i] :
( c_Union
@ ( c_Repl @ X0
@ ^ [X2: \$i] :
( X1 @ X2 ) ) ) ) )).

thf(ax52,axiom,
( c_Sep
= ( ^ [X0: \$i,X1: \$i > \$o] :
( c_If_i
@ ? [X2: \$i] :
( c_and @ ( c_In @ X2 @ X0 ) @ ( X1 @ X2 ) )
@ ( c_Repl @ X0
@ ^ [X2: \$i] :
( ^ [X3: \$i] :
( c_If_i @ ( X1 @ X3 ) @ X3
@ ( c_Eps_i
@ ^ [X4: \$i] :
( c_and @ ( c_In @ X4 @ X0 ) @ ( X1 @ X4 ) ) ) )
@ X2 ) )
@ c_Empty ) ) )).

thf(ax53,axiom,
( c_ReplSep
= ( ^ [X0: \$i,X1: \$i > \$o,X2: \$i > \$i] :
( c_Repl
@ ( c_Sep @ X0
@ ^ [X3: \$i] :
( X1 @ X3 ) )
@ ^ [X3: \$i] :
( X2 @ X3 ) ) ) )).

thf(ax54,axiom,
( c_binintersect
= ( ^ [X0: \$i,X1: \$i] :
( c_Sep @ X0
@ ^ [X2: \$i] :
( c_In @ X2 @ X1 ) ) ) )).

thf(ax55,axiom,
( c_setminus
= ( ^ [X0: \$i,X1: \$i] :
( c_Sep @ X0
@ ^ [X2: \$i] :
( c_nIn @ X2 @ X1 ) ) ) )).

thf(ax56,axiom,
( c_inj
= ( ^ [X0: \$i,X1: \$i,X2: \$i > \$i] :
( c_and
@ ! [X3: \$i] :
( ( c_In @ X3 @ X0 )
=> ( c_In @ ( X2 @ X3 ) @ X1 ) )
@ ! [X3: \$i] :
( ( c_In @ X3 @ X0 )
=> ! [X4: \$i] :
( ( c_In @ X4 @ X0 )
=> ( ( ( X2 @ X3 )
= ( X2 @ X4 ) )
=> ( X3 = X4 ) ) ) ) ) ) )).

thf(ax57,axiom,
( c_bij
= ( ^ [X0: \$i,X1: \$i,X2: \$i > \$i] :
( c_and @ ( c_inj @ X0 @ X1 @ X2 )
@ ! [X3: \$i] :
( ( c_In @ X3 @ X1 )
=> ? [X4: \$i] :
( c_and @ ( c_In @ X4 @ X0 )
@ ( ( X2 @ X4 )
= X3 ) ) ) ) ) )).

thf(ax58,axiom,
( c_atleastp
= ( ^ [X0: \$i,X1: \$i] :
? [X2: \$i > \$i] :
( c_inj @ X0 @ X1 @ X2 ) ) )).

thf(ax59,axiom,
( c_equip
= ( ^ [X0: \$i,X1: \$i] :
? [X2: \$i > \$i] :
( c_bij @ X0 @ X1 @ X2 ) ) )).

thf(ax60,axiom,
( c_In_rec_poly_G_i
= ( ^ [X0: \$i > ( \$i > \$i ) > \$i,X1: \$i,X2: \$i] :
! [X3: \$i > \$i > \$o] :
( ! [X4: \$i,X5: \$i > \$i] :
( ! [X6: \$i] :
( ( c_In @ X6 @ X4 )
=> ( X3 @ X6 @ ( X5 @ X6 ) ) )
=> ( X3 @ X4 @ ( X0 @ X4 @ X5 ) ) )
=> ( X3 @ X1 @ X2 ) ) ) )).

thf(ax61,axiom,
( c_In_rec_poly_i
= ( ^ [X0: \$i > ( \$i > \$i ) > \$i,X1: \$i] :
( c_Eps_i
@ ^ [X2: \$i] :
( c_In_rec_poly_G_i @ X0 @ X1 @ X2 ) ) ) )).

thf(ax62,axiom,
( c_ordsucc
= ( ^ [X0: \$i] :
( c_binunion @ X0 @ ( c_Sing @ X0 ) ) ) )).

thf(ax63,axiom,
( c_nat_p
= ( ^ [X0: \$i] :
! [X1: \$i > \$o] :
( ( X1 @ c_Empty )
=> ( ! [X2: \$i] :
( ( X1 @ X2 )
=> ( X1 @ ( c_ordsucc @ X2 ) ) )
=> ( X1 @ X0 ) ) ) ) )).

thf(ax64,axiom,
( c_nat_primrec
= ( ^ [X0: \$i,X1: \$i > \$i > \$i] :
( c_In_rec_poly_i
@ ^ [X2: \$i,X3: \$i > \$i] :
( c_If_i @ ( c_In @ ( c_Union @ X2 ) @ X2 ) @ ( X1 @ ( c_Union @ X2 ) @ ( X3 @ ( c_Union @ X2 ) ) ) @ X0 ) ) ) )).

thf(ax65,axiom,
= ( ^ [X0: \$i,X1: \$i] :
( c_nat_primrec @ X0
@ ^ [X2: \$i,X3: \$i] :
( c_ordsucc @ X3 )
@ X1 ) ) )).

thf(ax66,axiom,
( c_mul_nat
= ( ^ [X0: \$i,X1: \$i] :
( c_nat_primrec @ c_Empty
@ ^ [X2: \$i,X3: \$i] :
( c_add_nat @ X0 @ X3 )
@ X1 ) ) )).

thf(ax67,axiom,
( c_ordinal
= ( ^ [X0: \$i] :
( c_and @ ( c_TransSet @ X0 )
@ ! [X1: \$i] :
( ( c_In @ X1 @ X0 )
=> ( c_TransSet @ X1 ) ) ) ) )).

thf(ax68,axiom,
( c_V_
= ( c_In_rec_poly_i
@ ^ [X0: \$i,X1: \$i > \$i] :
( c_famunion @ X0
@ ^ [X2: \$i] :
( c_Power @ ( X1 @ X2 ) ) ) ) )).

thf(ax69,axiom,
( c_Inj1
= ( c_In_rec_poly_i
@ ^ [X0: \$i,X1: \$i > \$i] :
( c_binunion @ ( c_Sing @ c_Empty )
@ ( c_Repl @ X0
@ ^ [X2: \$i] :
( X1 @ X2 ) ) ) ) )).

thf(ax70,axiom,
( c_Inj0
= ( ^ [X0: \$i] :
( c_Repl @ X0
@ ^ [X1: \$i] :
( c_Inj1 @ X1 ) ) ) )).

thf(ax71,axiom,
( c_Unj
= ( c_In_rec_poly_i
@ ^ [X0: \$i,X1: \$i > \$i] :
( c_Repl @ ( c_setminus @ X0 @ ( c_Sing @ c_Empty ) )
@ ^ [X2: \$i] :
( X1 @ X2 ) ) ) )).

thf(ax72,axiom,
( c_combine_funcs
= ( ^ [X0: \$i,X1: \$i,X2: \$i > \$i,X3: \$i > \$i,X4: \$i] :
( c_If_i
@ ( X4
= ( c_Inj0 @ ( c_Unj @ X4 ) ) )
@ ( X2 @ ( c_Unj @ X4 ) )
@ ( X3 @ ( c_Unj @ X4 ) ) ) ) )).

thf(ax73,axiom,
( c_setsum
= ( ^ [X0: \$i,X1: \$i] :
( c_binunion
@ ( c_Repl @ X0
@ ^ [X2: \$i] :
( c_Inj0 @ X2 ) )
@ ( c_Repl @ X1
@ ^ [X2: \$i] :
( c_Inj1 @ X2 ) ) ) ) )).

thf(ax74,axiom,
( c_proj0
= ( ^ [X0: \$i] :
( c_ReplSep @ X0
@ ^ [X1: \$i] :
? [X2: \$i] :
( ( c_Inj0 @ X2 )
= X1 )
@ ^ [X1: \$i] :
( c_Unj @ X1 ) ) ) )).

thf(ax75,axiom,
( c_proj1
= ( ^ [X0: \$i] :
( c_ReplSep @ X0
@ ^ [X1: \$i] :
? [X2: \$i] :
( ( c_Inj1 @ X2 )
= X1 )
@ ^ [X1: \$i] :
( c_Unj @ X1 ) ) ) )).

thf(ax76,axiom,
( c_binrep
= ( ^ [X0: \$i,X1: \$i] :
( c_setsum @ X0 @ ( c_Power @ X1 ) ) ) )).

thf(ax77,axiom,
( c_lam
= ( ^ [X0: \$i,X1: \$i > \$i] :
( c_famunion @ X0
@ ^ [X2: \$i] :
( c_Repl @ ( X1 @ X2 )
@ ^ [X3: \$i] :
( c_setsum @ X2 @ X3 ) ) ) ) )).

thf(ax78,axiom,
( c_setprod
= ( ^ [X0: \$i,X1: \$i] :
( c_lam @ X0
@ ^ [X2: \$i] : X1 ) ) )).

thf(ax79,axiom,
( c_ap
= ( ^ [X0: \$i,X1: \$i] :
( c_ReplSep @ X0
@ ^ [X2: \$i] :
? [X3: \$i] :
( X2
= ( c_setsum @ X1 @ X3 ) )
@ ^ [X2: \$i] :
( c_proj1 @ X2 ) ) ) )).

thf(ax80,axiom,
( c_setsum_p
= ( ^ [X0: \$i] :
( ( c_setsum @ ( c_ap @ X0 @ c_Empty ) @ ( c_ap @ X0 @ ( c_ordsucc @ c_Empty ) ) )
= X0 ) ) )).

thf(ax81,axiom,
( c_tuple_p
= ( ^ [X0: \$i,X1: \$i] :
! [X2: \$i] :
( ( c_In @ X2 @ X1 )
=> ? [X3: \$i] :
( c_and @ ( c_In @ X3 @ X0 )
@ ? [X4: \$i] :
( X2
= ( c_setsum @ X3 @ X4 ) ) ) ) ) )).

thf(ax82,axiom,
( c_Pi
= ( ^ [X0: \$i,X1: \$i > \$i] :
( c_Sep
@ ( c_Power
@ ( c_lam @ X0
@ ^ [X2: \$i] :
( c_Union @ ( X1 @ X2 ) ) ) )
@ ^ [X2: \$i] :
! [X3: \$i] :
( ( c_In @ X3 @ X0 )
=> ( c_In @ ( c_ap @ X2 @ X3 ) @ ( X1 @ X3 ) ) ) ) ) )).

thf(ax83,axiom,
( c_setexp
= ( ^ [X0: \$i,X1: \$i] :
( c_Pi @ X1
@ ^ [X2: \$i] : X0 ) ) )).

thf(ax84,axiom,
( c_Sep2
= ( ^ [X0: \$i,X1: \$i > \$i,X2: \$i > \$i > \$o] :
( c_Sep
@ ( c_lam @ X0
@ ^ [X3: \$i] :
( X1 @ X3 ) )
@ ^ [X3: \$i] :
( X2 @ ( c_ap @ X3 @ c_Empty ) @ ( c_ap @ X3 @ ( c_ordsucc @ c_Empty ) ) ) ) ) )).

thf(ax85,axiom,
( c_set_of_pairs
= ( ^ [X0: \$i] :
! [X1: \$i] :
( ( c_In @ X1 @ X0 )
=> ? [X2: \$i,X3: \$i] :
( X1
= ( c_lam @ ( c_ordsucc @ ( c_ordsucc @ c_Empty ) )
@ ^ [X4: \$i] :
( c_If_i @ ( X4 = c_Empty ) @ X2 @ X3 ) ) ) ) ) )).

thf(ax86,axiom,
( c_lam2
= ( ^ [X0: \$i,X1: \$i > \$i,X2: \$i > \$i > \$i] :
( c_lam @ X0
@ ^ [X3: \$i] :
( c_lam @ ( X1 @ X3 )
@ ^ [X4: \$i] :
( X2 @ X3 @ X4 ) ) ) ) )).

thf(ax87,axiom,
( c_PNoEq_
= ( ^ [X0: \$i,X1: \$i > \$o,X2: \$i > \$o] :
! [X3: \$i] :
( ( c_In @ X3 @ X0 )
=> ( c_iff @ ( X1 @ X3 ) @ ( X2 @ X3 ) ) ) ) )).

thf(ax88,axiom,
( c_PNoLt_
= ( ^ [X0: \$i,X1: \$i > \$o,X2: \$i > \$o] :
? [X3: \$i] :
( c_and @ ( c_In @ X3 @ X0 ) @ ( c_and @ ( c_and @ ( c_PNoEq_ @ X3 @ X1 @ X2 ) @ ( c_not @ ( X1 @ X3 ) ) ) @ ( X2 @ X3 ) ) ) ) )).

thf(ax89,axiom,
( c_PNoLt
= ( ^ [X0: \$i,X1: \$i > \$o,X2: \$i,X3: \$i > \$o] :
( c_or @ ( c_or @ ( c_PNoLt_ @ ( c_binintersect @ X0 @ X2 ) @ X1 @ X3 ) @ ( c_and @ ( c_and @ ( c_In @ X0 @ X2 ) @ ( c_PNoEq_ @ X0 @ X1 @ X3 ) ) @ ( X3 @ X0 ) ) ) @ ( c_and @ ( c_and @ ( c_In @ X2 @ X0 ) @ ( c_PNoEq_ @ X2 @ X1 @ X3 ) ) @ ( c_not @ ( X1 @ X2 ) ) ) ) ) )).

thf(ax90,axiom,
( c_PNoLe
= ( ^ [X0: \$i,X1: \$i > \$o,X2: \$i,X3: \$i > \$o] :
( c_or @ ( c_PNoLt @ X0 @ X1 @ X2 @ X3 ) @ ( c_and @ ( X0 = X2 ) @ ( c_PNoEq_ @ X0 @ X1 @ X3 ) ) ) ) )).

thf(ax91,axiom,
( c_PNo_downc
= ( ^ [X0: \$i > ( \$i > \$o ) > \$o,X1: \$i,X2: \$i > \$o] :
? [X3: \$i] :
( c_and @ ( c_ordinal @ X3 )
@ ? [X4: \$i > \$o] :
( c_and @ ( X0 @ X3 @ X4 ) @ ( c_PNoLe @ X1 @ X2 @ X3 @ X4 ) ) ) ) )).

thf(ax92,axiom,
( c_PNo_upc
= ( ^ [X0: \$i > ( \$i > \$o ) > \$o,X1: \$i,X2: \$i > \$o] :
? [X3: \$i] :
( c_and @ ( c_ordinal @ X3 )
@ ? [X4: \$i > \$o] :
( c_and @ ( X0 @ X3 @ X4 ) @ ( c_PNoLe @ X3 @ X4 @ X1 @ X2 ) ) ) ) )).

thf(ax93,axiom,
( c_SNoElts_
= ( ^ [X0: \$i] :
( c_binunion @ X0
@ ( c_Repl @ X0
@ ^ [X1: \$i] :
( ^ [X2: \$i] :
( c_SetAdjoin @ X2 @ ( c_Sing @ ( c_ordsucc @ c_Empty ) ) )
@ X1 ) ) ) ) )).

thf(ax94,axiom,
( c_SNo_
= ( ^ [X0: \$i,X1: \$i] :
( c_and @ ( c_Subq @ X1 @ ( c_SNoElts_ @ X0 ) )
@ ! [X2: \$i] :
( ( c_In @ X2 @ X0 )
=> ( c_exactly1of2
@ ( c_In
@ ( ^ [X3: \$i] :
( c_SetAdjoin @ X3 @ ( c_Sing @ ( c_ordsucc @ c_Empty ) ) )
@ X2 )
@ X1 )
@ ( c_In @ X2 @ X1 ) ) ) ) ) )).

thf(ax95,axiom,
( c_PSNo
= ( ^ [X0: \$i,X1: \$i > \$o] :
( c_binunion
@ ( c_Sep @ X0
@ ^ [X2: \$i] :
( X1 @ X2 ) )
@ ( c_ReplSep @ X0
@ ^ [X2: \$i] :
( c_not @ ( X1 @ X2 ) )
@ ^ [X2: \$i] :
( ^ [X3: \$i] :
( c_SetAdjoin @ X3 @ ( c_Sing @ ( c_ordsucc @ c_Empty ) ) )
@ X2 ) ) ) ) )).

thf(ax96,axiom,
( c_SNo
= ( ^ [X0: \$i] :
? [X1: \$i] :
( c_and @ ( c_ordinal @ X1 ) @ ( c_SNo_ @ X1 @ X0 ) ) ) )).

thf(ax97,axiom,
( c_SNoLev
= ( ^ [X0: \$i] :
( c_Eps_i
@ ^ [X1: \$i] :
( c_and @ ( c_ordinal @ X1 ) @ ( c_SNo_ @ X1 @ X0 ) ) ) ) )).

thf(ax98,axiom,
( c_SNoEq_
= ( ^ [X0: \$i,X1: \$i,X2: \$i] :
( c_PNoEq_ @ X0
@ ^ [X3: \$i] :
( c_In @ X3 @ X1 )
@ ^ [X3: \$i] :
( c_In @ X3 @ X2 ) ) ) )).

thf(ax99,axiom,
( c_SNoLt
= ( ^ [X0: \$i,X1: \$i] :
( c_PNoLt @ ( c_SNoLev @ X0 )
@ ^ [X2: \$i] :
( c_In @ X2 @ X0 )
@ ( c_SNoLev @ X1 )
@ ^ [X2: \$i] :
( c_In @ X2 @ X1 ) ) ) )).

thf(ax100,axiom,
( c_SNoLe
= ( ^ [X0: \$i,X1: \$i] :
( c_PNoLe @ ( c_SNoLev @ X0 )
@ ^ [X2: \$i] :
( c_In @ X2 @ X0 )
@ ( c_SNoLev @ X1 )
@ ^ [X2: \$i] :
( c_In @ X2 @ X1 ) ) ) )).

thf(ax101,axiom,
( c_binop_on
= ( ^ [X0: \$i,X1: \$i > \$i > \$i] :
! [X2: \$i] :
( ( c_In @ X2 @ X0 )
=> ! [X3: \$i] :
( ( c_In @ X3 @ X0 )
=> ( c_In @ ( X1 @ X2 @ X3 ) @ X0 ) ) ) ) )).

thf(ax102,axiom,
( c_Loop
= ( ^ [X0: \$i,X1: \$i > \$i > \$i,X2: \$i > \$i > \$i,X3: \$i > \$i > \$i,X4: \$i] :
( c_and
@ ( c_and @ ( c_and @ ( c_and @ ( c_binop_on @ X0 @ X1 ) @ ( c_binop_on @ X0 @ X2 ) ) @ ( c_binop_on @ X0 @ X3 ) )
@ ! [X5: \$i] :
( ( c_In @ X5 @ X0 )
=> ( c_and
@ ( ( X1 @ X4 @ X5 )
= X5 )
@ ( ( X1 @ X5 @ X4 )
= X5 ) ) ) )
@ ! [X5: \$i] :
( ( c_In @ X5 @ X0 )
=> ! [X6: \$i] :
( ( c_In @ X6 @ X0 )
=> ( c_and
@ ( c_and
@ ( c_and
@ ( ( X2 @ X5 @ ( X1 @ X5 @ X6 ) )
= X6 )
@ ( ( X1 @ X5 @ ( X2 @ X5 @ X6 ) )
= X6 ) )
@ ( ( X3 @ ( X1 @ X5 @ X6 ) @ X6 )
= X5 ) )
@ ( ( X1 @ ( X3 @ X5 @ X6 ) @ X6 )
= X5 ) ) ) ) ) ) )).

thf(ax103,axiom,
( c_Loop_with_defs
= ( ^ [X0: \$i,X1: \$i > \$i > \$i,X2: \$i > \$i > \$i,X3: \$i > \$i > \$i,X4: \$i,X5: \$i > \$i > \$i,X6: \$i > \$i > \$i > \$i,X7: \$i > \$i > \$i,X8: \$i > \$i > \$i > \$i,X9: \$i > \$i > \$i > \$i,X10: \$i > \$i > \$i,X11: \$i > \$i > \$i,X12: \$i > \$i > \$i,X13: \$i > \$i > \$i] :
( c_and
@ ( c_and
@ ( c_and
@ ( c_and @ ( c_Loop @ X0 @ X1 @ X2 @ X3 @ X4 )
@ ! [X14: \$i] :
( ( c_In @ X14 @ X0 )
=> ! [X15: \$i] :
( ( c_In @ X15 @ X0 )
=> ( ( X5 @ X14 @ X15 )
= ( X2 @ ( X1 @ X15 @ X14 ) @ ( X1 @ X14 @ X15 ) ) ) ) ) )
@ ! [X14: \$i] :
( ( c_In @ X14 @ X0 )
=> ! [X15: \$i] :
( ( c_In @ X15 @ X0 )
=> ! [X16: \$i] :
( ( c_In @ X16 @ X0 )
=> ( ( X6 @ X14 @ X15 @ X16 )
= ( X2 @ ( X1 @ X14 @ ( X1 @ X15 @ X16 ) ) @ ( X1 @ ( X1 @ X14 @ X15 ) @ X16 ) ) ) ) ) ) )
@ ! [X14: \$i] :
( ( c_In @ X14 @ X0 )
=> ! [X15: \$i] :
( ( c_In @ X15 @ X0 )
=> ( c_and
@ ( c_and
@ ( c_and
@ ( c_and
@ ( ( X7 @ X14 @ X15 )
= ( X2 @ X14 @ ( X1 @ X15 @ X14 ) ) )
@ ( ( X10 @ X14 @ X15 )
= ( X1 @ X14 @ ( X1 @ X15 @ ( X2 @ X14 @ X4 ) ) ) ) )
@ ( ( X11 @ X14 @ X15 )
= ( X1 @ ( X1 @ ( X3 @ X4 @ X14 ) @ X15 ) @ X14 ) ) )
@ ( ( X12 @ X14 @ X15 )
= ( X1 @ ( X2 @ X14 @ X15 ) @ ( X2 @ ( X2 @ X14 @ X4 ) @ X4 ) ) ) )
@ ( ( X13 @ X14 @ X15 )
= ( X1 @ ( X3 @ X4 @ ( X3 @ X4 @ X14 ) ) @ ( X3 @ X15 @ X14 ) ) ) ) ) ) )
@ ! [X14: \$i] :
( ( c_In @ X14 @ X0 )
=> ! [X15: \$i] :
( ( c_In @ X15 @ X0 )
=> ! [X16: \$i] :
( ( c_In @ X16 @ X0 )
=> ( c_and
@ ( ( X8 @ X14 @ X15 @ X16 )
= ( X2 @ ( X1 @ X15 @ X14 ) @ ( X1 @ X15 @ ( X1 @ X14 @ X16 ) ) ) )
@ ( ( X9 @ X14 @ X15 @ X16 )
= ( X3 @ ( X1 @ ( X1 @ X16 @ X14 ) @ X15 ) @ ( X1 @ X14 @ X15 ) ) ) ) ) ) ) ) ) )).

thf(ax104,axiom,
( c_Loop_with_defs_cex1
= ( ^ [X0: \$i,X1: \$i > \$i > \$i,X2: \$i > \$i > \$i,X3: \$i > \$i > \$i,X4: \$i,X5: \$i > \$i > \$i,X6: \$i > \$i > \$i > \$i,X7: \$i > \$i > \$i,X8: \$i > \$i > \$i > \$i,X9: \$i > \$i > \$i > \$i,X10: \$i > \$i > \$i,X11: \$i > \$i > \$i,X12: \$i > \$i > \$i,X13: \$i > \$i > \$i] :
( c_and @ ( c_Loop_with_defs @ X0 @ X1 @ X2 @ X3 @ X4 @ X5 @ X6 @ X7 @ X8 @ X9 @ X10 @ X11 @ X12 @ X13 )
@ ? [X14: \$i] :
( c_and @ ( c_In @ X14 @ X0 )
@ ? [X15: \$i] :
( c_and @ ( c_In @ X15 @ X0 )
@ ? [X16: \$i] :
( c_and @ ( c_In @ X16 @ X0 )
@ ? [X17: \$i] :
( c_and @ ( c_In @ X17 @ X0 )
@ ( c_not
@ ( ( X5 @ ( X1 @ ( X2 @ ( X8 @ X15 @ X16 @ X14 ) @ X4 ) @ X14 ) @ X17 )
= X4 ) ) ) ) ) ) ) ) )).

thf(ax105,axiom,
( c_Loop_with_defs_cex2
= ( ^ [X0: \$i,X1: \$i > \$i > \$i,X2: \$i > \$i > \$i,X3: \$i > \$i > \$i,X4: \$i,X5: \$i > \$i > \$i,X6: \$i > \$i > \$i > \$i,X7: \$i > \$i > \$i,X8: \$i > \$i > \$i > \$i,X9: \$i > \$i > \$i > \$i,X10: \$i > \$i > \$i,X11: \$i > \$i > \$i,X12: \$i > \$i > \$i,X13: \$i > \$i > \$i] :
( c_and @ ( c_Loop_with_defs @ X0 @ X1 @ X2 @ X3 @ X4 @ X5 @ X6 @ X7 @ X8 @ X9 @ X10 @ X11 @ X12 @ X13 )
@ ? [X14: \$i] :
( c_and @ ( c_In @ X14 @ X0 )
@ ? [X15: \$i] :
( c_and @ ( c_In @ X15 @ X0 )
@ ? [X16: \$i] :
( c_and @ ( c_In @ X16 @ X0 )
@ ? [X17: \$i] :
( c_and @ ( c_In @ X17 @ X0 )
@ ? [X18: \$i] :
( c_and @ ( c_In @ X18 @ X0 )
@ ( c_not
@ ( ( X6 @ X18 @ ( X1 @ ( X3 @ X4 @ X14 ) @ ( X9 @ X15 @ X16 @ X14 ) ) @ X17 )
= X4 ) ) ) ) ) ) ) ) ) )).

thf(ax106,axiom,
( c_combinator
= ( ^ [X0: \$i] :
! [X1: \$i > \$o] :
( ( X1 @ ( c_Inj0 @ c_Empty ) )
=> ( ( X1 @ ( c_Inj0 @ ( c_Power @ c_Empty ) ) )
=> ( ! [X2: \$i,X3: \$i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X1 @ ( c_Inj1 @ ( c_setsum @ X2 @ X3 ) ) ) ) )
=> ( X1 @ X0 ) ) ) ) ) )).

thf(ax107,axiom,
( c_combinator_equiv
= ( ^ [X0: \$i,X1: \$i] :
! [X2: \$i > \$i > \$o] :
( ^ [X3: \$i,X4: \$i,X5: \$i > \$i > \$i] :
( ( c_per_i @ X2 )
=> ( ! [X6: \$i] :
( ( c_combinator @ X6 )
=> ( X2 @ X6 @ X6 ) )
=> ( ! [X6: \$i,X7: \$i,X8: \$i,X9: \$i] :
( ( c_combinator @ X6 )
=> ( ( c_combinator @ X7 )
=> ( ( c_combinator @ X8 )
=> ( ( c_combinator @ X9 )
=> ( ( X2 @ X6 @ X8 )
=> ( ( X2 @ X7 @ X9 )
=> ( X2 @ ( X5 @ X6 @ X7 ) @ ( X5 @ X8 @ X9 ) ) ) ) ) ) ) )
=> ( ! [X6: \$i,X7: \$i] :
( X2 @ ( X5 @ ( X5 @ X3 @ X6 ) @ X7 ) @ X6 )
=> ( ! [X6: \$i,X7: \$i,X8: \$i] :
( X2 @ ( X5 @ ( X5 @ ( X5 @ X4 @ X6 ) @ X7 ) @ X8 ) @ ( X5 @ ( X5 @ X6 @ X8 ) @ ( X5 @ X7 @ X8 ) ) )
=> ( X2 @ X0 @ X1 ) ) ) ) ) )
@ ( c_Inj0 @ c_Empty )
@ ( c_Inj0 @ ( c_Power @ c_Empty ) )
@ ^ [X3: \$i,X4: \$i] :
( c_Inj1 @ ( c_setsum @ X3 @ X4 ) ) ) ) )).

thf(ax108,axiom,
( c_equip_mod
= ( ^ [X0: \$i,X1: \$i,X2: \$i] :
? [X3: \$i,X4: \$i] :
( c_or @ ( c_and @ ( c_equip @ ( c_setsum @ X0 @ X3 ) @ X1 ) @ ( c_equip @ ( c_setprod @ X4 @ X3 ) @ X2 ) ) @ ( c_and @ ( c_equip @ ( c_setsum @ X1 @ X3 ) @ X0 ) @ ( c_equip @ ( c_setprod @ X4 @ X3 ) @ X2 ) ) ) ) )).

thf(conj,conjecture,(
! [X0: ( \$i > \$i > ( ( \$i > \$i ) > \$i > \$i ) > \$i ) > \$i > \$i > \$o,X1: ( \$i > ( ( \$i > \$i ) > \$i ) > \$i ) > \$i > \$o,X2: ( \$i > \$i ) > ( ( ( ( \$i > \$i ) > \$i ) > \$i ) > ( \$i > \$i > \$i ) > \$i > \$i > \$i ) > \$i > \$o,X3: ( \$i > ( \$i > \$i > \$i ) > \$i ) > \$i > \$i > \$o] :
( ! [X4: ( ( \$i > \$i > \$i ) > ( \$i > \$i ) > \$i > \$i ) > \$i,X5: \$i,X6: \$i,X7: \$i] :
( ( c_In @ ( c_Inj0 @ X6 ) @ ( c_setsum @ ( c_setsum @ c_Empty @ c_Empty ) @ ( c_setsum @ X6 @ ( c_Inj0 @ ( c_setsum @ c_Empty @ c_Empty ) ) ) ) )
=> ( ( X2
@ ^ [X8: \$i] :
( c_Inj0 @ ( c_Inj0 @ c_Empty ) )
@ ^ [X8: ( ( \$i > \$i ) > \$i ) > \$i,X9: \$i > \$i > \$i,X10: \$i,X11: \$i] : X10
@ ( c_setsum
@ ( X4
@ ^ [X8: \$i > \$i > \$i,X9: \$i > \$i,X10: \$i] :
( c_Inj0 @ ( c_setsum @ c_Empty @ c_Empty ) ) )
@ c_Empty ) )
=> ( X3
@ ^ [X8: \$i,X9: \$i > \$i > \$i] :
( c_Inj0 @ X8 )
@ X5
@ ( c_setsum @ ( c_setsum @ ( c_setsum @ c_Empty @ ( c_setsum @ c_Empty @ c_Empty ) ) @ ( c_Inj0 @ c_Empty ) ) @ c_Empty ) ) ) )
=> ( ! [X4: ( ( ( \$i > \$i ) > \$i > \$i ) > \$i ) > \$i,X5: \$i > \$i,X6: \$i,X7: \$i] :
( ( X3
@ ^ [X8: \$i,X9: \$i > \$i > \$i] : c_Empty
@ X7
@ ( c_setsum @ ( c_setsum @ ( c_setsum @ ( c_setsum @ c_Empty @ c_Empty ) @ c_Empty ) @ ( c_setsum @ ( c_setsum @ c_Empty @ c_Empty ) @ ( c_Inj0 @ c_Empty ) ) ) @ ( c_setsum @ c_Empty @ ( c_Inj1 @ ( c_setsum @ c_Empty @ c_Empty ) ) ) ) )
=> ( X3
@ ^ [X8: \$i,X9: \$i > \$i > \$i] : X7
@ ( c_setsum
@ ( X4
@ ^ [X8: ( \$i > \$i ) > \$i > \$i] : X7 )
@ X6 )
@ ( c_Inj0 @ ( c_Inj1 @ ( c_setsum @ ( c_setsum @ c_Empty @ c_Empty ) @ ( c_setsum @ c_Empty @ c_Empty ) ) ) ) ) )
=> ( ! [X4: \$i,X5: \$i > ( ( \$i > \$i ) > \$i > \$i ) > \$i,X6: ( ( ( \$i > \$i ) > \$i > \$i ) > ( \$i > \$i ) > \$i > \$i ) > ( \$i > \$i > \$i ) > \$i,X7: \$i] :
( ( c_In @ ( c_Inj0 @ c_Empty )
@ ( c_setsum @ c_Empty
@ ( X5 @ ( c_Inj0 @ ( c_setsum @ c_Empty @ c_Empty ) )
@ ^ [X8: \$i > \$i,X9: \$i] :
( c_Inj0 @ c_Empty ) ) ) )
=> ( ( X0
@ ^ [X8: \$i,X9: \$i,X10: ( \$i > \$i ) > \$i > \$i] : X8
@ X7
@ ( X6
@ ^ [X8: ( \$i > \$i ) > \$i > \$i,X9: \$i > \$i,X10: \$i] :
( c_Inj1 @ c_Empty )
@ ^ [X8: \$i,X9: \$i] : c_Empty ) )
=> ( X2
@ ^ [X8: \$i] : X8
@ ^ [X8: ( ( \$i > \$i ) > \$i ) > \$i,X9: \$i > \$i > \$i,X10: \$i,X11: \$i] : X10
@ ( c_setsum @ ( c_Inj0 @ c_Empty )
@ ( X6
@ ^ [X8: ( \$i > \$i ) > \$i > \$i,X9: \$i > \$i,X10: \$i] :
( c_Inj0 @ c_Empty )
@ ^ [X8: \$i,X9: \$i] : X9 ) ) ) ) )
=> ( ! [X4: \$i > \$i > \$i,X5: \$i > \$i,X6: \$i,X7: \$i > \$i] :
( ( c_In @ ( c_Inj0 @ ( c_Inj1 @ c_Empty ) ) @ ( c_setsum @ ( c_setsum @ ( c_Inj0 @ c_Empty ) @ ( c_setsum @ ( c_setsum @ c_Empty @ c_Empty ) @ ( c_setsum @ c_Empty @ c_Empty ) ) ) @ ( c_setsum @ c_Empty @ ( c_setsum @ ( c_Inj0 @ c_Empty ) @ ( X4 @ c_Empty @ c_Empty ) ) ) ) )
=> ( ( X2
@ ^ [X8: \$i] : c_Empty
@ ^ [X8: ( ( \$i > \$i ) > \$i ) > \$i,X9: \$i > \$i > \$i,X10: \$i,X11: \$i] : c_Empty
@ ( c_setsum @ ( c_setsum @ c_Empty @ ( c_setsum @ ( c_setsum @ c_Empty @ c_Empty ) @ ( c_Inj0 @ c_Empty ) ) ) @ ( c_Inj1 @ X6 ) ) )
=> ( X1
@ ^ [X8: \$i,X9: ( \$i > \$i ) > \$i] :
( c_Inj0 @ c_Empty )
@ ( X4 @ c_Empty @ ( X5 @ ( c_setsum @ ( c_Inj1 @ c_Empty ) @ c_Empty ) ) ) ) ) )
=> ( ! [X4: \$i,X5: \$i > \$i,X6: \$i > \$i,X7: \$i] :
( ( X3
@ ^ [X8: \$i,X9: \$i > \$i > \$i] : c_Empty
@ c_Empty
@ c_Empty )
=> ( X1
@ ^ [X8: \$i,X9: ( \$i > \$i ) > \$i] :
( X9
@ ^ [X10: \$i] :
( c_setsum @ ( c_Inj1 @ ( c_Inj0 @ c_Empty ) ) @ ( c_setsum @ c_Empty @ c_Empty ) ) )
@ c_Empty ) )
=> ( ! [X4: \$i,X5: \$i,X6: \$i,X7: \$i] :
( ( c_In @ ( c_Inj0 @ X6 ) @ ( c_setsum @ ( c_setsum @ X4 @ ( c_setsum @ X5 @ ( c_setsum @ c_Empty @ c_Empty ) ) ) @ ( c_Inj1 @ ( c_setsum @ X5 @ c_Empty ) ) ) )
=> ( ( X1
@ ^ [X8: \$i,X9: ( \$i > \$i ) > \$i] : X7
@ c_Empty )
=> ( X2
@ ^ [X8: \$i] : c_Empty
@ ^ [X8: ( ( \$i > \$i ) > \$i ) > \$i,X9: \$i > \$i > \$i,X10: \$i,X11: \$i] :
( c_setsum @ c_Empty @ c_Empty )
@ X5 ) ) )
=> ( ! [X4: ( ( ( \$i > \$i ) > \$i > \$i ) > \$i ) > \$i,X5: \$i,X6: \$i,X7: \$i] :
( ( c_In @ ( c_Inj1 @ c_Empty ) @ X5 )
=> ( ( X1
@ ^ [X8: \$i,X9: ( \$i > \$i ) > \$i] : c_Empty
@ c_Empty )
=> ( X0
@ ^ [X8: \$i,X9: \$i,X10: ( \$i > \$i ) > \$i > \$i] :
( c_Inj1 @ c_Empty )
@ ( c_Inj1
@ ( c_Inj1
@ ( X4
@ ^ [X8: ( \$i > \$i ) > \$i > \$i] : c_Empty ) ) )
@ ( c_Inj1 @ ( c_setsum @ ( c_setsum @ ( c_Inj0 @ c_Empty ) @ c_Empty ) @ ( c_Inj1 @ ( c_Inj0 @ c_Empty ) ) ) ) ) ) )
=> ( ! [X4: \$i > \$i,X5: \$i,X6: \$i > \$i,X7: \$i] :
( ( X0
@ ^ [X8: \$i,X9: \$i,X10: ( \$i > \$i ) > \$i > \$i] : c_Empty
@ ( X4 @ c_Empty )
@ X7 )
=> c_False )
=> c_False ) ) ) ) ) ) ) ) )).

%------------------------------------------------------------------------------
```