%------------------------------------------------------------------------------
% File : SEU881^5 : TPTP v9.0.0. Released v4.0.0.
% Domain : Set Theory (Finite sets)
% Problem : TPS problem from SET-TOPOLOGY-THMS
% Version : Especial.
% English :
% Refs : [Bro09] Brown (2009), Email to Geoff Sutcliffe
% Source : [Bro09]
% Names : tps_1259 [Bro09]
% Status : Theorem
% Rating : 1.00 v6.2.0, 0.86 v6.0.0, 1.00 v4.0.0
% Syntax : Number of formulae : 5 ( 0 unt; 4 typ; 0 def)
% Number of atoms : 158 ( 29 equ; 0 cnn)
% Maximal formula atoms : 154 ( 158 avg)
% Number of connectives : 795 ( 0 ~; 20 |; 160 &; 400 @)
% ( 5 <=>; 210 =>; 0 <=; 0 <~>)
% Maximal formula depth : 54 ( 54 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 477 ( 477 >; 0 *; 0 +; 0 <<)
% Number of symbols : 5 ( 3 usr; 1 con; 0-2 aty)
% Number of variables : 297 ( 56 ^; 209 !; 32 ?; 297 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This problem is from the TPS library. Copyright (c) 2009 The TPS
% project in the Department of Mathematical Sciences at Carnegie
% Mellon University. Distributed under the Creative Commons copyleft
% license: http://creativecommons.org/licenses/by-sa/3.0/
%------------------------------------------------------------------------------
thf(a_type,type,
a: $tType ).
thf(cA,type,
cA: ( ( a > $o ) > $o ) > $o ).
thf(cO,type,
cO: ( a > $o ) > $o ).
thf(cT,type,
cT: a > $o ).
thf(cDOMTHM6_pme,conjecture,
( ( ! [Y: a > $o] :
( ( cO @ Y )
=> ! [Xx: a] :
( ( Y @ Xx )
=> ( cT @ Xx ) ) )
& ( cO @ cT )
& ( cO
@ ^ [Xx: a] : $false )
& ! [K: ( a > $o ) > $o] :
( ! [Xx: a > $o] :
( ( K @ Xx )
=> ( cO @ Xx ) )
=> ( cO
@ ^ [Xx: a] :
? [S: a > $o] :
( ( K @ S )
& ( S @ Xx ) ) ) )
& ! [Y: a > $o,Z: a > $o] :
( ( ( cO @ Y )
& ( cO @ Z ) )
=> ( cO
@ ^ [Xx: a] :
( ( Y @ Xx )
& ( Z @ Xx ) ) ) )
& ! [Xx: a] :
( ( cT @ Xx )
=> ! [Xy: a] :
( ( cT @ Xy )
=> ( ! [Y: a > $o] :
( ( cO @ Y )
=> ( ( Y @ Xx )
<=> ( Y @ Xy ) ) )
=> ( Xx = Xy ) ) ) )
& ( cA
= ( ^ [V: ( a > $o ) > $o] :
? [Xx: a] :
( ( cT @ Xx )
& ( V
= ( ^ [U: a > $o] :
( ( cO @ U )
& ( U @ Xx ) ) ) ) ) ) ) )
=> ? [Xf: a > ( a > $o ) > $o,Xg: ( ( a > $o ) > $o ) > a] :
( ! [Y: a > $o] :
( ( cO @ Y )
=> ! [Xx: a] :
( ( Y @ Xx )
=> ( cT @ Xx ) ) )
& ( cO @ cT )
& ( cO
@ ^ [Xx: a] : $false )
& ! [K: ( a > $o ) > $o] :
( ! [Xx: a > $o] :
( ( K @ Xx )
=> ( cO @ Xx ) )
=> ( cO
@ ^ [Xx: a] :
? [S: a > $o] :
( ( K @ S )
& ( S @ Xx ) ) ) )
& ! [Y: a > $o,Z: a > $o] :
( ( ( cO @ Y )
& ( cO @ Z ) )
=> ( cO
@ ^ [Xx: a] :
( ( Y @ Xx )
& ( Z @ Xx ) ) ) )
& ! [Xx: a] :
( ( cT @ Xx )
=> ! [Xy: a] :
( ( cT @ Xy )
=> ( ! [Y: a > $o] :
( ( cO @ Y )
=> ( ( Y @ Xx )
<=> ( Y @ Xy ) ) )
=> ( Xx = Xy ) ) ) )
& ! [Y: ( ( a > $o ) > $o ) > $o] :
( ( ! [Xx: ( a > $o ) > $o] :
( ( Y @ Xx )
=> ( cA @ Xx ) )
& ! [Xx: ( a > $o ) > $o] :
( ( Y @ Xx )
=> ? [Xe: ( a > $o ) > $o] :
( ! [X: ( ( a > $o ) > $o ) > $o] :
( ( ( X
@ ^ [Xy: a > $o] : $false )
& ! [Xx0: ( a > $o ) > $o] :
( ( X @ Xx0 )
=> ! [Xt: a > $o] :
( ( Xe @ Xt )
=> ( X
@ ^ [Xz: a > $o] :
( ( Xx0 @ Xz )
| ( Xt = Xz ) ) ) ) ) )
=> ( X @ Xe ) )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xx @ Xx0 ) )
& ! [Xy: ( a > $o ) > $o] :
( ( ( cA @ Xy )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xy @ Xx0 ) ) )
=> ( Y @ Xy ) ) ) ) )
=> ! [Xx: ( a > $o ) > $o] :
( ( Y @ Xx )
=> ( cA @ Xx ) ) )
& ! [Xx: ( a > $o ) > $o] :
( ( cA @ Xx )
=> ( cA @ Xx ) )
& ! [Xx: ( a > $o ) > $o] :
( ( cA @ Xx )
=> ? [Xe: ( a > $o ) > $o] :
( ! [X: ( ( a > $o ) > $o ) > $o] :
( ( ( X
@ ^ [Xy: a > $o] : $false )
& ! [Xx0: ( a > $o ) > $o] :
( ( X @ Xx0 )
=> ! [Xt: a > $o] :
( ( Xe @ Xt )
=> ( X
@ ^ [Xz: a > $o] :
( ( Xx0 @ Xz )
| ( Xt = Xz ) ) ) ) ) )
=> ( X @ Xe ) )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xx @ Xx0 ) )
& ! [Xy: ( a > $o ) > $o] :
( ( ( cA @ Xy )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xy @ Xx0 ) ) )
=> ( cA @ Xy ) ) ) )
& ! [Xx: ( a > $o ) > $o] :
( $false
=> ( cA @ Xx ) )
& ! [Xx: ( a > $o ) > $o] :
( $false
=> ? [Xe: ( a > $o ) > $o] :
( ! [X: ( ( a > $o ) > $o ) > $o] :
( ( ( X
@ ^ [Xy: a > $o] : $false )
& ! [Xx0: ( a > $o ) > $o] :
( ( X @ Xx0 )
=> ! [Xt: a > $o] :
( ( Xe @ Xt )
=> ( X
@ ^ [Xz: a > $o] :
( ( Xx0 @ Xz )
| ( Xt = Xz ) ) ) ) ) )
=> ( X @ Xe ) )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xx @ Xx0 ) )
& ! [Xy: ( a > $o ) > $o] :
( ( ( cA @ Xy )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xy @ Xx0 ) ) )
=> $false ) ) )
& ! [K: ( ( ( a > $o ) > $o ) > $o ) > $o] :
( ! [Xx: ( ( a > $o ) > $o ) > $o] :
( ( K @ Xx )
=> ( ! [Xx0: ( a > $o ) > $o] :
( ( Xx @ Xx0 )
=> ( cA @ Xx0 ) )
& ! [Xx0: ( a > $o ) > $o] :
( ( Xx @ Xx0 )
=> ? [Xe: ( a > $o ) > $o] :
( ! [X: ( ( a > $o ) > $o ) > $o] :
( ( ( X
@ ^ [Xy: a > $o] : $false )
& ! [Xx1: ( a > $o ) > $o] :
( ( X @ Xx1 )
=> ! [Xt: a > $o] :
( ( Xe @ Xt )
=> ( X
@ ^ [Xz: a > $o] :
( ( Xx1 @ Xz )
| ( Xt = Xz ) ) ) ) ) )
=> ( X @ Xe ) )
& ! [Xx1: a > $o] :
( ( Xe @ Xx1 )
=> ( Xx0 @ Xx1 ) )
& ! [Xy: ( a > $o ) > $o] :
( ( ( cA @ Xy )
& ! [Xx1: a > $o] :
( ( Xe @ Xx1 )
=> ( Xy @ Xx1 ) ) )
=> ( Xx @ Xy ) ) ) ) ) )
=> ( ! [Xx: ( a > $o ) > $o] :
( ? [S: ( ( a > $o ) > $o ) > $o] :
( ( K @ S )
& ( S @ Xx ) )
=> ( cA @ Xx ) )
& ! [Xx: ( a > $o ) > $o] :
( ? [S: ( ( a > $o ) > $o ) > $o] :
( ( K @ S )
& ( S @ Xx ) )
=> ? [Xe: ( a > $o ) > $o] :
( ! [X: ( ( a > $o ) > $o ) > $o] :
( ( ( X
@ ^ [Xy: a > $o] : $false )
& ! [Xx0: ( a > $o ) > $o] :
( ( X @ Xx0 )
=> ! [Xt: a > $o] :
( ( Xe @ Xt )
=> ( X
@ ^ [Xz: a > $o] :
( ( Xx0 @ Xz )
| ( Xt = Xz ) ) ) ) ) )
=> ( X @ Xe ) )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xx @ Xx0 ) )
& ! [Xy: ( a > $o ) > $o] :
( ( ( cA @ Xy )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xy @ Xx0 ) ) )
=> ? [S: ( ( a > $o ) > $o ) > $o] :
( ( K @ S )
& ( S @ Xy ) ) ) ) ) ) )
& ! [Y: ( ( a > $o ) > $o ) > $o,Z: ( ( a > $o ) > $o ) > $o] :
( ( ! [Xx: ( a > $o ) > $o] :
( ( Y @ Xx )
=> ( cA @ Xx ) )
& ! [Xx: ( a > $o ) > $o] :
( ( Y @ Xx )
=> ? [Xe: ( a > $o ) > $o] :
( ! [X: ( ( a > $o ) > $o ) > $o] :
( ( ( X
@ ^ [Xy: a > $o] : $false )
& ! [Xx0: ( a > $o ) > $o] :
( ( X @ Xx0 )
=> ! [Xt: a > $o] :
( ( Xe @ Xt )
=> ( X
@ ^ [Xz: a > $o] :
( ( Xx0 @ Xz )
| ( Xt = Xz ) ) ) ) ) )
=> ( X @ Xe ) )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xx @ Xx0 ) )
& ! [Xy: ( a > $o ) > $o] :
( ( ( cA @ Xy )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xy @ Xx0 ) ) )
=> ( Y @ Xy ) ) ) )
& ! [Xx: ( a > $o ) > $o] :
( ( Z @ Xx )
=> ( cA @ Xx ) )
& ! [Xx: ( a > $o ) > $o] :
( ( Z @ Xx )
=> ? [Xe: ( a > $o ) > $o] :
( ! [X: ( ( a > $o ) > $o ) > $o] :
( ( ( X
@ ^ [Xy: a > $o] : $false )
& ! [Xx0: ( a > $o ) > $o] :
( ( X @ Xx0 )
=> ! [Xt: a > $o] :
( ( Xe @ Xt )
=> ( X
@ ^ [Xz: a > $o] :
( ( Xx0 @ Xz )
| ( Xt = Xz ) ) ) ) ) )
=> ( X @ Xe ) )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xx @ Xx0 ) )
& ! [Xy: ( a > $o ) > $o] :
( ( ( cA @ Xy )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xy @ Xx0 ) ) )
=> ( Z @ Xy ) ) ) ) )
=> ( ! [Xx: ( a > $o ) > $o] :
( ( ( Y @ Xx )
& ( Z @ Xx ) )
=> ( cA @ Xx ) )
& ! [Xx: ( a > $o ) > $o] :
( ( ( Y @ Xx )
& ( Z @ Xx ) )
=> ? [Xe: ( a > $o ) > $o] :
( ! [X: ( ( a > $o ) > $o ) > $o] :
( ( ( X
@ ^ [Xy: a > $o] : $false )
& ! [Xx0: ( a > $o ) > $o] :
( ( X @ Xx0 )
=> ! [Xt: a > $o] :
( ( Xe @ Xt )
=> ( X
@ ^ [Xz: a > $o] :
( ( Xx0 @ Xz )
| ( Xt = Xz ) ) ) ) ) )
=> ( X @ Xe ) )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xx @ Xx0 ) )
& ! [Xy: ( a > $o ) > $o] :
( ( ( cA @ Xy )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xy @ Xx0 ) ) )
=> ( ( Y @ Xy )
& ( Z @ Xy ) ) ) ) ) ) )
& ! [Xx: ( a > $o ) > $o] :
( ( cA @ Xx )
=> ! [Xy: ( a > $o ) > $o] :
( ( cA @ Xy )
=> ( ! [Y: ( ( a > $o ) > $o ) > $o] :
( ( ! [Xx0: ( a > $o ) > $o] :
( ( Y @ Xx0 )
=> ( cA @ Xx0 ) )
& ! [Xx0: ( a > $o ) > $o] :
( ( Y @ Xx0 )
=> ? [Xe: ( a > $o ) > $o] :
( ! [X: ( ( a > $o ) > $o ) > $o] :
( ( ( X
@ ^ [Xy0: a > $o] : $false )
& ! [Xx1: ( a > $o ) > $o] :
( ( X @ Xx1 )
=> ! [Xt: a > $o] :
( ( Xe @ Xt )
=> ( X
@ ^ [Xz: a > $o] :
( ( Xx1 @ Xz )
| ( Xt = Xz ) ) ) ) ) )
=> ( X @ Xe ) )
& ! [Xx1: a > $o] :
( ( Xe @ Xx1 )
=> ( Xx0 @ Xx1 ) )
& ! [Xy0: ( a > $o ) > $o] :
( ( ( cA @ Xy0 )
& ! [Xx1: a > $o] :
( ( Xe @ Xx1 )
=> ( Xy0 @ Xx1 ) ) )
=> ( Y @ Xy0 ) ) ) ) )
=> ( ( Y @ Xx )
<=> ( Y @ Xy ) ) )
=> ( Xx = Xy ) ) ) )
& ! [Xx: a] :
( ( cT @ Xx )
=> ( cA @ ( Xf @ Xx ) ) )
& ! [W: ( ( a > $o ) > $o ) > $o] :
( ( ! [Xx: ( a > $o ) > $o] :
( ( W @ Xx )
=> ( cA @ Xx ) )
& ! [Xx: ( a > $o ) > $o] :
( ( W @ Xx )
=> ? [Xe: ( a > $o ) > $o] :
( ! [X: ( ( a > $o ) > $o ) > $o] :
( ( ( X
@ ^ [Xy: a > $o] : $false )
& ! [Xx0: ( a > $o ) > $o] :
( ( X @ Xx0 )
=> ! [Xt: a > $o] :
( ( Xe @ Xt )
=> ( X
@ ^ [Xz: a > $o] :
( ( Xx0 @ Xz )
| ( Xt = Xz ) ) ) ) ) )
=> ( X @ Xe ) )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xx @ Xx0 ) )
& ! [Xy: ( a > $o ) > $o] :
( ( ( cA @ Xy )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xy @ Xx0 ) ) )
=> ( W @ Xy ) ) ) ) )
=> ( cO
@ ^ [Xx: a] : ( W @ ( Xf @ Xx ) ) ) )
& ! [Y: ( ( a > $o ) > $o ) > $o] :
( ( ! [Xx: ( a > $o ) > $o] :
( ( Y @ Xx )
=> ( cA @ Xx ) )
& ! [Xx: ( a > $o ) > $o] :
( ( Y @ Xx )
=> ? [Xe: ( a > $o ) > $o] :
( ! [X: ( ( a > $o ) > $o ) > $o] :
( ( ( X
@ ^ [Xy: a > $o] : $false )
& ! [Xx0: ( a > $o ) > $o] :
( ( X @ Xx0 )
=> ! [Xt: a > $o] :
( ( Xe @ Xt )
=> ( X
@ ^ [Xz: a > $o] :
( ( Xx0 @ Xz )
| ( Xt = Xz ) ) ) ) ) )
=> ( X @ Xe ) )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xx @ Xx0 ) )
& ! [Xy: ( a > $o ) > $o] :
( ( ( cA @ Xy )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xy @ Xx0 ) ) )
=> ( Y @ Xy ) ) ) ) )
=> ! [Xx: ( a > $o ) > $o] :
( ( Y @ Xx )
=> ( cA @ Xx ) ) )
& ! [Xx: ( a > $o ) > $o] :
( ( cA @ Xx )
=> ( cA @ Xx ) )
& ! [Xx: ( a > $o ) > $o] :
( ( cA @ Xx )
=> ? [Xe: ( a > $o ) > $o] :
( ! [X: ( ( a > $o ) > $o ) > $o] :
( ( ( X
@ ^ [Xy: a > $o] : $false )
& ! [Xx0: ( a > $o ) > $o] :
( ( X @ Xx0 )
=> ! [Xt: a > $o] :
( ( Xe @ Xt )
=> ( X
@ ^ [Xz: a > $o] :
( ( Xx0 @ Xz )
| ( Xt = Xz ) ) ) ) ) )
=> ( X @ Xe ) )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xx @ Xx0 ) )
& ! [Xy: ( a > $o ) > $o] :
( ( ( cA @ Xy )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xy @ Xx0 ) ) )
=> ( cA @ Xy ) ) ) )
& ! [Xx: ( a > $o ) > $o] :
( $false
=> ( cA @ Xx ) )
& ! [Xx: ( a > $o ) > $o] :
( $false
=> ? [Xe: ( a > $o ) > $o] :
( ! [X: ( ( a > $o ) > $o ) > $o] :
( ( ( X
@ ^ [Xy: a > $o] : $false )
& ! [Xx0: ( a > $o ) > $o] :
( ( X @ Xx0 )
=> ! [Xt: a > $o] :
( ( Xe @ Xt )
=> ( X
@ ^ [Xz: a > $o] :
( ( Xx0 @ Xz )
| ( Xt = Xz ) ) ) ) ) )
=> ( X @ Xe ) )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xx @ Xx0 ) )
& ! [Xy: ( a > $o ) > $o] :
( ( ( cA @ Xy )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xy @ Xx0 ) ) )
=> $false ) ) )
& ! [K: ( ( ( a > $o ) > $o ) > $o ) > $o] :
( ! [Xx: ( ( a > $o ) > $o ) > $o] :
( ( K @ Xx )
=> ( ! [Xx0: ( a > $o ) > $o] :
( ( Xx @ Xx0 )
=> ( cA @ Xx0 ) )
& ! [Xx0: ( a > $o ) > $o] :
( ( Xx @ Xx0 )
=> ? [Xe: ( a > $o ) > $o] :
( ! [X: ( ( a > $o ) > $o ) > $o] :
( ( ( X
@ ^ [Xy: a > $o] : $false )
& ! [Xx1: ( a > $o ) > $o] :
( ( X @ Xx1 )
=> ! [Xt: a > $o] :
( ( Xe @ Xt )
=> ( X
@ ^ [Xz: a > $o] :
( ( Xx1 @ Xz )
| ( Xt = Xz ) ) ) ) ) )
=> ( X @ Xe ) )
& ! [Xx1: a > $o] :
( ( Xe @ Xx1 )
=> ( Xx0 @ Xx1 ) )
& ! [Xy: ( a > $o ) > $o] :
( ( ( cA @ Xy )
& ! [Xx1: a > $o] :
( ( Xe @ Xx1 )
=> ( Xy @ Xx1 ) ) )
=> ( Xx @ Xy ) ) ) ) ) )
=> ( ! [Xx: ( a > $o ) > $o] :
( ? [S: ( ( a > $o ) > $o ) > $o] :
( ( K @ S )
& ( S @ Xx ) )
=> ( cA @ Xx ) )
& ! [Xx: ( a > $o ) > $o] :
( ? [S: ( ( a > $o ) > $o ) > $o] :
( ( K @ S )
& ( S @ Xx ) )
=> ? [Xe: ( a > $o ) > $o] :
( ! [X: ( ( a > $o ) > $o ) > $o] :
( ( ( X
@ ^ [Xy: a > $o] : $false )
& ! [Xx0: ( a > $o ) > $o] :
( ( X @ Xx0 )
=> ! [Xt: a > $o] :
( ( Xe @ Xt )
=> ( X
@ ^ [Xz: a > $o] :
( ( Xx0 @ Xz )
| ( Xt = Xz ) ) ) ) ) )
=> ( X @ Xe ) )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xx @ Xx0 ) )
& ! [Xy: ( a > $o ) > $o] :
( ( ( cA @ Xy )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xy @ Xx0 ) ) )
=> ? [S: ( ( a > $o ) > $o ) > $o] :
( ( K @ S )
& ( S @ Xy ) ) ) ) ) ) )
& ! [Y: ( ( a > $o ) > $o ) > $o,Z: ( ( a > $o ) > $o ) > $o] :
( ( ! [Xx: ( a > $o ) > $o] :
( ( Y @ Xx )
=> ( cA @ Xx ) )
& ! [Xx: ( a > $o ) > $o] :
( ( Y @ Xx )
=> ? [Xe: ( a > $o ) > $o] :
( ! [X: ( ( a > $o ) > $o ) > $o] :
( ( ( X
@ ^ [Xy: a > $o] : $false )
& ! [Xx0: ( a > $o ) > $o] :
( ( X @ Xx0 )
=> ! [Xt: a > $o] :
( ( Xe @ Xt )
=> ( X
@ ^ [Xz: a > $o] :
( ( Xx0 @ Xz )
| ( Xt = Xz ) ) ) ) ) )
=> ( X @ Xe ) )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xx @ Xx0 ) )
& ! [Xy: ( a > $o ) > $o] :
( ( ( cA @ Xy )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xy @ Xx0 ) ) )
=> ( Y @ Xy ) ) ) )
& ! [Xx: ( a > $o ) > $o] :
( ( Z @ Xx )
=> ( cA @ Xx ) )
& ! [Xx: ( a > $o ) > $o] :
( ( Z @ Xx )
=> ? [Xe: ( a > $o ) > $o] :
( ! [X: ( ( a > $o ) > $o ) > $o] :
( ( ( X
@ ^ [Xy: a > $o] : $false )
& ! [Xx0: ( a > $o ) > $o] :
( ( X @ Xx0 )
=> ! [Xt: a > $o] :
( ( Xe @ Xt )
=> ( X
@ ^ [Xz: a > $o] :
( ( Xx0 @ Xz )
| ( Xt = Xz ) ) ) ) ) )
=> ( X @ Xe ) )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xx @ Xx0 ) )
& ! [Xy: ( a > $o ) > $o] :
( ( ( cA @ Xy )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xy @ Xx0 ) ) )
=> ( Z @ Xy ) ) ) ) )
=> ( ! [Xx: ( a > $o ) > $o] :
( ( ( Y @ Xx )
& ( Z @ Xx ) )
=> ( cA @ Xx ) )
& ! [Xx: ( a > $o ) > $o] :
( ( ( Y @ Xx )
& ( Z @ Xx ) )
=> ? [Xe: ( a > $o ) > $o] :
( ! [X: ( ( a > $o ) > $o ) > $o] :
( ( ( X
@ ^ [Xy: a > $o] : $false )
& ! [Xx0: ( a > $o ) > $o] :
( ( X @ Xx0 )
=> ! [Xt: a > $o] :
( ( Xe @ Xt )
=> ( X
@ ^ [Xz: a > $o] :
( ( Xx0 @ Xz )
| ( Xt = Xz ) ) ) ) ) )
=> ( X @ Xe ) )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xx @ Xx0 ) )
& ! [Xy: ( a > $o ) > $o] :
( ( ( cA @ Xy )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xy @ Xx0 ) ) )
=> ( ( Y @ Xy )
& ( Z @ Xy ) ) ) ) ) ) )
& ! [Xx: ( a > $o ) > $o] :
( ( cA @ Xx )
=> ! [Xy: ( a > $o ) > $o] :
( ( cA @ Xy )
=> ( ! [Y: ( ( a > $o ) > $o ) > $o] :
( ( ! [Xx0: ( a > $o ) > $o] :
( ( Y @ Xx0 )
=> ( cA @ Xx0 ) )
& ! [Xx0: ( a > $o ) > $o] :
( ( Y @ Xx0 )
=> ? [Xe: ( a > $o ) > $o] :
( ! [X: ( ( a > $o ) > $o ) > $o] :
( ( ( X
@ ^ [Xy0: a > $o] : $false )
& ! [Xx1: ( a > $o ) > $o] :
( ( X @ Xx1 )
=> ! [Xt: a > $o] :
( ( Xe @ Xt )
=> ( X
@ ^ [Xz: a > $o] :
( ( Xx1 @ Xz )
| ( Xt = Xz ) ) ) ) ) )
=> ( X @ Xe ) )
& ! [Xx1: a > $o] :
( ( Xe @ Xx1 )
=> ( Xx0 @ Xx1 ) )
& ! [Xy0: ( a > $o ) > $o] :
( ( ( cA @ Xy0 )
& ! [Xx1: a > $o] :
( ( Xe @ Xx1 )
=> ( Xy0 @ Xx1 ) ) )
=> ( Y @ Xy0 ) ) ) ) )
=> ( ( Y @ Xx )
<=> ( Y @ Xy ) ) )
=> ( Xx = Xy ) ) ) )
& ! [Y: a > $o] :
( ( cO @ Y )
=> ! [Xx: a] :
( ( Y @ Xx )
=> ( cT @ Xx ) ) )
& ( cO @ cT )
& ( cO
@ ^ [Xx: a] : $false )
& ! [K: ( a > $o ) > $o] :
( ! [Xx: a > $o] :
( ( K @ Xx )
=> ( cO @ Xx ) )
=> ( cO
@ ^ [Xx: a] :
? [S: a > $o] :
( ( K @ S )
& ( S @ Xx ) ) ) )
& ! [Y: a > $o,Z: a > $o] :
( ( ( cO @ Y )
& ( cO @ Z ) )
=> ( cO
@ ^ [Xx: a] :
( ( Y @ Xx )
& ( Z @ Xx ) ) ) )
& ! [Xx: a] :
( ( cT @ Xx )
=> ! [Xy: a] :
( ( cT @ Xy )
=> ( ! [Y: a > $o] :
( ( cO @ Y )
=> ( ( Y @ Xx )
<=> ( Y @ Xy ) ) )
=> ( Xx = Xy ) ) ) )
& ! [Xx: ( a > $o ) > $o] :
( ( cA @ Xx )
=> ( cT @ ( Xg @ Xx ) ) )
& ! [W: a > $o] :
( ( cO @ W )
=> ( ! [Xx: ( a > $o ) > $o] :
( ( W @ ( Xg @ Xx ) )
=> ( cA @ Xx ) )
& ! [Xx: ( a > $o ) > $o] :
( ( W @ ( Xg @ Xx ) )
=> ? [Xe: ( a > $o ) > $o] :
( ! [X: ( ( a > $o ) > $o ) > $o] :
( ( ( X
@ ^ [Xy: a > $o] : $false )
& ! [Xx0: ( a > $o ) > $o] :
( ( X @ Xx0 )
=> ! [Xt: a > $o] :
( ( Xe @ Xt )
=> ( X
@ ^ [Xz: a > $o] :
( ( Xx0 @ Xz )
| ( Xt = Xz ) ) ) ) ) )
=> ( X @ Xe ) )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xx @ Xx0 ) )
& ! [Xy: ( a > $o ) > $o] :
( ( ( cA @ Xy )
& ! [Xx0: a > $o] :
( ( Xe @ Xx0 )
=> ( Xy @ Xx0 ) ) )
=> ( W @ ( Xg @ Xy ) ) ) ) ) ) )
& ( ( ^ [Xx: ( a > $o ) > $o] : ( Xf @ ( Xg @ Xx ) ) )
= ( ^ [Xy: ( a > $o ) > $o] : Xy ) )
& ( ( ^ [Xx: a] : ( Xg @ ( Xf @ Xx ) ) )
= ( ^ [Xx: a] : Xx ) ) ) ) ).
%------------------------------------------------------------------------------