%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV040^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n115.star.cs.uiowa.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory   : 32286.75MB
% OS       : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:38 EDT 2014

% Result   : Timeout 300.04s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV040^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n115.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 07:40:46 CDT 2014
% % CPUTime  : 300.04 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x206f710>, <kernel.Type object at 0x206f908>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula ((and ((and (forall (Xx:(a->(a->Prop))) (Xy:(a->(a->Prop))), (((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))) (((eq (a->(a->Prop))) Xx) Xy))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))) (((eq (a->(a->Prop))) Xy) Xx))))) (forall (Xx:(a->(a->Prop))) (Xy:(a->(a->Prop))) (Xz:(a->(a->Prop))), (((and ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz)))))) (((eq ((a->(a->Prop))->((a->(a->Prop))->Prop))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq))))) of role conjecture named cTHM515_pme
% Conjecture to prove = ((and ((and (forall (Xx:(a->(a->Prop))) (Xy:(a->(a->Prop))), (((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))) (((eq (a->(a->Prop))) Xx) Xy))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))) (((eq (a->(a->Prop))) Xy) Xx))))) (forall (Xx:(a->(a->Prop))) (Xy:(a->(a->Prop))) (Xz:(a->(a->Prop))), (((and ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz)))))) (((eq ((a->(a->Prop))->((a->(a->Prop))->Prop))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['((and ((and (forall (Xx:(a->(a->Prop))) (Xy:(a->(a->Prop))), (((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))) (((eq (a->(a->Prop))) Xx) Xy))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))) (((eq (a->(a->Prop))) Xy) Xx))))) (forall (Xx:(a->(a->Prop))) (Xy:(a->(a->Prop))) (Xz:(a->(a->Prop))), (((and ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz)))))) (((eq ((a->(a->Prop))->((a->(a->Prop))->Prop))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))))']
% Parameter a:Type.
% Trying to prove ((and ((and (forall (Xx:(a->(a->Prop))) (Xy:(a->(a->Prop))), (((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))) (((eq (a->(a->Prop))) Xx) Xy))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))) (((eq (a->(a->Prop))) Xy) Xx))))) (forall (Xx:(a->(a->Prop))) (Xy:(a->(a->Prop))) (Xz:(a->(a->Prop))), (((and ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz)))))) (((eq ((a->(a->Prop))->((a->(a->Prop))->Prop))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))):(((eq ((a->(a->Prop))->((a->(a->Prop))->Prop))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))) (fun (x:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((x Xx) Xy)->((x Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (((eq (a->(a->Prop))) x) Xq))))
% Found (eta_expansion00 (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))) as proof of (((eq ((a->(a->Prop))->((a->(a->Prop))->Prop))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq))))
% Found ((eta_expansion0 ((a->(a->Prop))->Prop)) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))) as proof of (((eq ((a->(a->Prop))->((a->(a->Prop))->Prop))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq))))
% Found (((eta_expansion (a->(a->Prop))) ((a->(a->Prop))->Prop)) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))) as proof of (((eq ((a->(a->Prop))->((a->(a->Prop))->Prop))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq))))
% Found (((eta_expansion (a->(a->Prop))) ((a->(a->Prop))->Prop)) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))) as proof of (((eq ((a->(a->Prop))->((a->(a->Prop))->Prop))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq))))
% Found eq_ref00:=(eq_ref0 (((eq ((a->(a->Prop))->((a->(a->Prop))->Prop))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq))))):(((eq Prop) (((eq ((a->(a->Prop))->((a->(a->Prop))->Prop))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq))))) (((eq ((a->(a->Prop))->((a->(a->Prop))->Prop))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))))
% Found (eq_ref0 (((eq ((a->(a->Prop))->((a->(a->Prop))->Prop))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq))))) as proof of (((eq Prop) (((eq ((a->(a->Prop))->((a->(a->Prop))->Prop))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq))))) b)
% Found ((eq_ref Prop) (((eq ((a->(a->Prop))->((a->(a->Prop))->Prop))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq))))) as proof of (((eq Prop) (((eq ((a->(a->Prop))->((a->(a->Prop))->Prop))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq))))) b)
% Found ((eq_ref Prop) (((eq ((a->(a->Prop))->((a->(a->Prop))->Prop))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq))))) as proof of (((eq Prop) (((eq ((a->(a->Prop))->((a->(a->Prop))->Prop))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq))))) b)
% Found ((eq_ref Prop) (((eq ((a->(a->Prop))->((a->(a->Prop))->Prop))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq))))) as proof of (((eq Prop) (((eq ((a->(a->Prop))->((a->(a->Prop))->Prop))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->(a->Prop))) (Xy:(a->(a->Prop))) (Xz:(a->(a->Prop))), (((and ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))))):(((eq Prop) (forall (Xx:(a->(a->Prop))) (Xy:(a->(a->Prop))) (Xz:(a->(a->Prop))), (((and ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))))) (forall (Xx:(a->(a->Prop))) (Xy:(a->(a->Prop))) (Xz:(a->(a->Prop))), (((and ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz)))))
% Found (eq_ref0 (forall (Xx:(a->(a->Prop))) (Xy:(a->(a->Prop))) (Xz:(a->(a->Prop))), (((and ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))))) as proof of (((eq Prop) (forall (Xx:(a->(a->Prop))) (Xy:(a->(a->Prop))) (Xz:(a->(a->Prop))), (((and ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->(a->Prop))) (Xy:(a->(a->Prop))) (Xz:(a->(a->Prop))), (((and ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))))) as proof of (((eq Prop) (forall (Xx:(a->(a->Prop))) (Xy:(a->(a->Prop))) (Xz:(a->(a->Prop))), (((and ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->(a->Prop))) (Xy:(a->(a->Prop))) (Xz:(a->(a->Prop))), (((and ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))))) as proof of (((eq Prop) (forall (Xx:(a->(a->Prop))) (Xy:(a->(a->Prop))) (Xz:(a->(a->Prop))), (((and ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->(a->Prop))) (Xy:(a->(a->Prop))) (Xz:(a->(a->Prop))), (((and ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))))) as proof of (((eq Prop) (forall (Xx:(a->(a->Prop))) (Xy:(a->(a->Prop))) (Xz:(a->(a->Prop))), (((and ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))))) b)
% Found eq_sym000:=(eq_sym00 Xy):((((eq (a->(a->Prop))) Xx) Xy)->(((eq (a->(a->Prop))) Xy) Xx))
% Found (eq_sym00 Xy) as proof of ((((eq (a->(a->Prop))) Xx) Xy)->(((eq (a->(a->Prop))) Xy) Xx))
% Found ((eq_sym0 Xx) Xy) as proof of ((((eq (a->(a->Prop))) Xx) Xy)->(((eq (a->(a->Prop))) Xy) Xx))
% Found (((eq_sym (a->(a->Prop))) Xx) Xy) as proof of ((((eq (a->(a->Prop))) Xx) Xy)->(((eq (a->(a->Prop))) Xy) Xx))
% Found (fun (x0:((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz)))))=> (((eq_sym (a->(a->Prop))) Xx) Xy)) as proof of ((((eq (a->(a->Prop))) Xx) Xy)->(((eq (a->(a->Prop))) Xy) Xx))
% Found (fun (x0:((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz)))))=> (((eq_sym (a->(a->Prop))) Xx) Xy)) as proof of (((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))->((((eq (a->(a->Prop))) Xx) Xy)->(((eq (a->(a->Prop))) Xy) Xx)))
% Found (and_rect00 (fun (x0:((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz)))))=> (((eq_sym (a->(a->Prop))) Xx) Xy))) as proof of (((eq (a->(a->Prop))) Xy) Xx)
% Found ((and_rect0 (((eq (a->(a->Prop))) Xy) Xx)) (fun (x0:((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz)))))=> (((eq_sym (a->(a->Prop))) Xx) Xy))) as proof of (((eq (a->(a->Prop))) Xy) Xx)
% Found (((fun (P:Type) (x0:(((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))->((((eq (a->(a->Prop))) Xx) Xy)->P)))=> (((((and_rect ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))) (((eq (a->(a->Prop))) Xx) Xy)) P) x0) x)) (((eq (a->(a->Prop))) Xy) Xx)) (fun (x0:((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz)))))=> (((eq_sym (a->(a->Prop))) Xx) Xy))) as proof of (((eq (a->(a->Prop))) Xy) Xx)
% Found (((fun (P:Type) (x0:(((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))->((((eq (a->(a->Prop))) Xx) Xy)->P)))=> (((((and_rect ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))) (((eq (a->(a->Prop))) Xx) Xy)) P) x0) x)) (((eq (a->(a->Prop))) Xy) Xx)) (fun (x0:((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz)))))=> (((eq_sym (a->(a->Prop))) Xx) Xy))) as proof of (((eq (a->(a->Prop))) Xy) Xx)
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->(a->Prop))) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->(a->Prop))) Xx) Xx)
% Found ((eq_ref (a->(a->Prop))) Xx) as proof of (((eq (a->(a->Prop))) Xx) Xx)
% Found ((eq_ref (a->(a->Prop))) Xx) as proof of (((eq (a->(a->Prop))) Xx) Xx)
% Found (x10 ((eq_ref (a->(a->Prop))) Xx)) as proof of (((eq (a->(a->Prop))) Xy) Xx)
% Found ((x1 (fun (x3:(a->(a->Prop)))=> (((eq (a->(a->Prop))) x3) Xx))) ((eq_ref (a->(a->Prop))) Xx)) as proof of (((eq (a->(a->Prop))) Xy) Xx)
% Found ((x1 (fun (x3:(a->(a->Prop)))=> (((eq (a->(a->Prop))) x3) Xx))) ((eq_ref (a->(a->Prop))) Xx)) as proof of (((eq (a->(a->Prop))) Xy) Xx)
% Found x100:=(x10 x4):((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))
% Found (x10 x4) as proof of ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))
% Found ((x1 (fun (x7:(a->(a->Prop)))=> ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) x7)))) x4) as proof of ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))
% Found (fun (x5:(forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))=> ((x1 (fun (x7:(a->(a->Prop)))=> ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) x7)))) x4)) as proof of ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))
% Found (fun (x4:((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (x5:(forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))=> ((x1 (fun (x7:(a->(a->Prop)))=> ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) x7)))) x4)) as proof of ((forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz)))
% Found (fun (x4:((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (x5:(forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))=> ((x1 (fun (x7:(a->(a->Prop)))=> ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) x7)))) x4)) as proof of (((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))->((forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))))
% Found (and_rect20 (fun (x4:((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (x5:(forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))=> ((x1 (fun (x7:(a->(a->Prop)))=> ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) x7)))) x4))) as proof of ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))
% Found ((and_rect2 ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x4:((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (x5:(forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))=> ((x1 (fun (x7:(a->(a->Prop)))=> ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) x7)))) x4))) as proof of ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))
% Found (((fun (P:Type) (x4:(((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))->((forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))->P)))=> (((((and_rect ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) P) x4) x2)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x4:((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (x5:(forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))=> ((x1 (fun (x7:(a->(a->Prop)))=> ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) x7)))) x4))) as proof of ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))
% Found (fun (x3:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))=> (((fun (P:Type) (x4:(((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))->((forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))->P)))=> (((((and_rect ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) P) x4) x2)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x4:((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (x5:(forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))=> ((x1 (fun (x7:(a->(a->Prop)))=> ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) x7)))) x4)))) as proof of ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))
% Found (fun (x2:((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (x3:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))=> (((fun (P:Type) (x4:(((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))->((forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))->P)))=> (((((and_rect ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) P) x4) x2)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x4:((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (x5:(forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))=> ((x1 (fun (x7:(a->(a->Prop)))=> ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) x7)))) x4)))) as proof of ((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0)))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz)))
% Found (fun (x2:((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (x3:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))=> (((fun (P:Type) (x4:(((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))->((forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))->P)))=> (((((and_rect ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) P) x4) x2)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x4:((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (x5:(forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))=> ((x1 (fun (x7:(a->(a->Prop)))=> ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) x7)))) x4)))) as proof of (((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0)))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))))
% Found (and_rect10 (fun (x2:((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (x3:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))=> (((fun (P:Type) (x4:(((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))->((forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))->P)))=> (((((and_rect ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) P) x4) x2)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x4:((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (x5:(forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))=> ((x1 (fun (x7:(a->(a->Prop)))=> ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) x7)))) x4))))) as proof of ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))
% Found ((and_rect1 ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x2:((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (x3:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))=> (((fun (P:Type) (x4:(((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))->((forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))->P)))=> (((((and_rect ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) P) x4) x2)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x4:((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (x5:(forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))=> ((x1 (fun (x7:(a->(a->Prop)))=> ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) x7)))) x4))))) as proof of ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))
% Found (((fun (P:Type) (x2:(((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0)))->P)))=> (((((and_rect ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0)))) P) x2) x0)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x2:((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (x3:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))=> (((fun (P:Type) (x4:(((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))->((forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))->P)))=> (((((and_rect ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) P) x4) x2)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x4:((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (x5:(forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))=> ((x1 (fun (x7:(a->(a->Prop)))=> ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) x7)))) x4))))) as proof of ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))
% Found (fun (x1:(((eq (a->(a->Prop))) Xy) Xz))=> (((fun (P:Type) (x2:(((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0)))->P)))=> (((((and_rect ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0)))) P) x2) x0)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x2:((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (x3:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))=> (((fun (P:Type) (x4:(((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))->((forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))->P)))=> (((((and_rect ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) P) x4) x2)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x4:((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (x5:(forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))=> ((x1 (fun (x7:(a->(a->Prop)))=> ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) x7)))) x4)))))) as proof of ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))
% Found (fun (x0:((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (x1:(((eq (a->(a->Prop))) Xy) Xz))=> (((fun (P:Type) (x2:(((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0)))->P)))=> (((((and_rect ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0)))) P) x2) x0)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x2:((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (x3:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))=> (((fun (P:Type) (x4:(((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))->((forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))->P)))=> (((((and_rect ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) P) x4) x2)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x4:((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (x5:(forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))=> ((x1 (fun (x7:(a->(a->Prop)))=> ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) x7)))) x4)))))) as proof of ((((eq (a->(a->Prop))) Xy) Xz)->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz)))
% Found (fun (x0:((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (x1:(((eq (a->(a->Prop))) Xy) Xz))=> (((fun (P:Type) (x2:(((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0)))->P)))=> (((((and_rect ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0)))) P) x2) x0)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x2:((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (x3:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))=> (((fun (P:Type) (x4:(((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))->((forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))->P)))=> (((((and_rect ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) P) x4) x2)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x4:((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (x5:(forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))=> ((x1 (fun (x7:(a->(a->Prop)))=> ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) x7)))) x4)))))) as proof of (((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))->((((eq (a->(a->Prop))) Xy) Xz)->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))))
% Found (and_rect00 (fun (x0:((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (x1:(((eq (a->(a->Prop))) Xy) Xz))=> (((fun (P:Type) (x2:(((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0)))->P)))=> (((((and_rect ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0)))) P) x2) x0)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x2:((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (x3:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))=> (((fun (P:Type) (x4:(((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))->((forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))->P)))=> (((((and_rect ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) P) x4) x2)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x4:((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (x5:(forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))=> ((x1 (fun (x7:(a->(a->Prop)))=> ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) x7)))) x4))))))) as proof of ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))
% Found ((and_rect0 ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x0:((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (x1:(((eq (a->(a->Prop))) Xy) Xz))=> (((fun (P:Type) (x2:(((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0)))->P)))=> (((((and_rect ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0)))) P) x2) x0)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x2:((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (x3:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))=> (((fun (P:Type) (x4:(((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))->((forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))->P)))=> (((((and_rect ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) P) x4) x2)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x4:((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (x5:(forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))=> ((x1 (fun (x7:(a->(a->Prop)))=> ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) x7)))) x4))))))) as proof of ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))
% Found (((fun (P:Type) (x0:(((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))->((((eq (a->(a->Prop))) Xy) Xz)->P)))=> (((((and_rect ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz)) P) x0) x)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x0:((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (x1:(((eq (a->(a->Prop))) Xy) Xz))=> (((fun (P:Type) (x2:(((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0)))->P)))=> (((((and_rect ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0)))) P) x2) x0)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x2:((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (x3:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))=> (((fun (P:Type) (x4:(((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))->((forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))->P)))=> (((((and_rect ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) P) x4) x2)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x4:((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (x5:(forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))=> ((x1 (fun (x7:(a->(a->Prop)))=> ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) x7)))) x4))))))) as proof of ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))
% Found (fun (x:((and ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz)))=> (((fun (P:Type) (x0:(((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))->((((eq (a->(a->Prop))) Xy) Xz)->P)))=> (((((and_rect ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz)) P) x0) x)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x0:((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (x1:(((eq (a->(a->Prop))) Xy) Xz))=> (((fun (P:Type) (x2:(((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0)))->P)))=> (((((and_rect ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0)))) P) x2) x0)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x2:((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (x3:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))=> (((fun (P:Type) (x4:(((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))->((forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))->P)))=> (((((and_rect ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) P) x4) x2)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x4:((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (x5:(forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))=> ((x1 (fun (x7:(a->(a->Prop)))=> ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) x7)))) x4)))))))) as proof of ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))
% Found (fun (Xz:(a->(a->Prop))) (x:((and ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz)))=> (((fun (P:Type) (x0:(((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))->((((eq (a->(a->Prop))) Xy) Xz)->P)))=> (((((and_rect ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz)) P) x0) x)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x0:((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (x1:(((eq (a->(a->Prop))) Xy) Xz))=> (((fun (P:Type) (x2:(((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0)))->P)))=> (((((and_rect ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0)))) P) x2) x0)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x2:((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (x3:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))=> (((fun (P:Type) (x4:(((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))->((forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))->P)))=> (((((and_rect ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) P) x4) x2)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x4:((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (x5:(forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))=> ((x1 (fun (x7:(a->(a->Prop)))=> ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) x7)))) x4)))))))) as proof of (((and ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz)))
% Found (fun (Xy:(a->(a->Prop))) (Xz:(a->(a->Prop))) (x:((and ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz)))=> (((fun (P:Type) (x0:(((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))->((((eq (a->(a->Prop))) Xy) Xz)->P)))=> (((((and_rect ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz)) P) x0) x)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x0:((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (x1:(((eq (a->(a->Prop))) Xy) Xz))=> (((fun (P:Type) (x2:(((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0)))->P)))=> (((((and_rect ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0)))) P) x2) x0)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x2:((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (x3:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))=> (((fun (P:Type) (x4:(((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))->((forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))->P)))=> (((((and_rect ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) P) x4) x2)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x4:((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (x5:(forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))=> ((x1 (fun (x7:(a->(a->Prop)))=> ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) x7)))) x4)))))))) as proof of (forall (Xz:(a->(a->Prop))), (((and ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))))
% Found (fun (Xx:(a->(a->Prop))) (Xy:(a->(a->Prop))) (Xz:(a->(a->Prop))) (x:((and ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz)))=> (((fun (P:Type) (x0:(((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))->((((eq (a->(a->Prop))) Xy) Xz)->P)))=> (((((and_rect ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz)) P) x0) x)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x0:((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (x1:(((eq (a->(a->Prop))) Xy) Xz))=> (((fun (P:Type) (x2:(((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0)))->P)))=> (((((and_rect ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0)))) P) x2) x0)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x2:((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (x3:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))=> (((fun (P:Type) (x4:(((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))->((forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))->P)))=> (((((and_rect ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) P) x4) x2)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x4:((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (x5:(forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))=> ((x1 (fun (x7:(a->(a->Prop)))=> ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) x7)))) x4)))))))) as proof of (forall (Xy:(a->(a->Prop))) (Xz:(a->(a->Prop))), (((and ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))))
% Found (fun (Xx:(a->(a->Prop))) (Xy:(a->(a->Prop))) (Xz:(a->(a->Prop))) (x:((and ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz)))=> (((fun (P:Type) (x0:(((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))->((((eq (a->(a->Prop))) Xy) Xz)->P)))=> (((((and_rect ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz)) P) x0) x)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x0:((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (x1:(((eq (a->(a->Prop))) Xy) Xz))=> (((fun (P:Type) (x2:(((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0)))->P)))=> (((((and_rect ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0)))) P) x2) x0)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x2:((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (x3:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))=> (((fun (P:Type) (x4:(((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))->((forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))->P)))=> (((((and_rect ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) P) x4) x2)) ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))) (fun (x4:((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (x5:(forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))=> ((x1 (fun (x7:(a->(a->Prop)))=> ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) x7)))) x4)))))))) as proof of (forall (Xx:(a->(a->Prop))) (Xy:(a->(a->Prop))) (Xz:(a->(a->Prop))), (((and ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->(a->Prop))) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->(a->Prop))) Xx) Xx)
% Found ((eq_ref (a->(a->Prop))) Xx) as proof of (((eq (a->(a->Prop))) Xx) Xx)
% Found ((eq_ref (a->(a->Prop))) Xx) as proof of (((eq (a->(a->Prop))) Xx) Xx)
% Found (x10 ((eq_ref (a->(a->Prop))) Xx)) as proof of (((eq (a->(a->Prop))) Xy) Xx)
% Found ((x1 (fun (x5:(a->(a->Prop)))=> (((eq (a->(a->Prop))) x5) Xx))) ((eq_ref (a->(a->Prop))) Xx)) as proof of (((eq (a->(a->Prop))) Xy) Xx)
% Found ((x1 (fun (x5:(a->(a->Prop)))=> (((eq (a->(a->Prop))) x5) Xx))) ((eq_ref (a->(a->Prop))) Xx)) as proof of (((eq (a->(a->Prop))) Xy) Xx)
% Found eq_ref00:=(eq_ref0 (((eq (a->(a->Prop))) Xy) Xx)):(((eq Prop) (((eq (a->(a->Prop))) Xy) Xx)) (((eq (a->(a->Prop))) Xy) Xx))
% Found (eq_ref0 (((eq (a->(a->Prop))) Xy) Xx)) as proof of (((eq Prop) (((eq (a->(a->Prop))) Xy) Xx)) b)
% Found ((eq_ref Prop) (((eq (a->(a->Prop))) Xy) Xx)) as proof of (((eq Prop) (((eq (a->(a->Prop))) Xy) Xx)) b)
% Found ((eq_ref Prop) (((eq (a->(a->Prop))) Xy) Xx)) as proof of (((eq Prop) (((eq (a->(a->Prop))) Xy) Xx)) b)
% Found ((eq_ref Prop) (((eq (a->(a->Prop))) Xy) Xx)) as proof of (((eq Prop) (((eq (a->(a->Prop))) Xy) Xx)) b)
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found eq_ref00:=(eq_ref0 (((eq (a->(a->Prop))) Xy) Xx)):(((eq Prop) (((eq (a->(a->Prop))) Xy) Xx)) (((eq (a->(a->Prop))) Xy) Xx))
% Found (eq_ref0 (((eq (a->(a->Prop))) Xy) Xx)) as proof of (((eq Prop) (((eq (a->(a->Prop))) Xy) Xx)) b)
% Found ((eq_ref Prop) (((eq (a->(a->Prop))) Xy) Xx)) as proof of (((eq Prop) (((eq (a->(a->Prop))) Xy) Xx)) b)
% Found ((eq_ref Prop) (((eq (a->(a->Prop))) Xy) Xx)) as proof of (((eq Prop) (((eq (a->(a->Prop))) Xy) Xx)) b)
% Found ((eq_ref Prop) (((eq (a->(a->Prop))) Xy) Xx)) as proof of (((eq Prop) (((eq (a->(a->Prop))) Xy) Xx)) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))):(((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))
% Found (eq_ref0 (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) b)
% Found conj2:=(conj ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))):(forall (B:Prop), (((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))->(B->((and ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))) B))))
% Instantiate: b:=(forall (B:Prop), (((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))->(B->((and ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))) B)))):Prop
% Found conj2 as proof of b
% Found eta_expansion:=(fun (A:Type) (B:Type)=> ((eta_expansion_dep A) (fun (x1:A)=> B))):(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x))))
% Instantiate: b:=(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x)))):Prop
% Found eta_expansion as proof of b
% Found eq_ref00:=(eq_ref0 (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))):(((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))
% Found (eq_ref0 (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) b)
% Found conj2:=(conj ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))):(forall (B:Prop), (((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))->(B->((and ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))) B))))
% Instantiate: b:=(forall (B:Prop), (((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))->(B->((and ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))) B)))):Prop
% Found conj2 as proof of b
% Found eq_ref00:=(eq_ref0 (((eq (a->(a->Prop))) Xy) Xx)):(((eq Prop) (((eq (a->(a->Prop))) Xy) Xx)) (((eq (a->(a->Prop))) Xy) Xx))
% Found (eq_ref0 (((eq (a->(a->Prop))) Xy) Xx)) as proof of (((eq Prop) (((eq (a->(a->Prop))) Xy) Xx)) b)
% Found ((eq_ref Prop) (((eq (a->(a->Prop))) Xy) Xx)) as proof of (((eq Prop) (((eq (a->(a->Prop))) Xy) Xx)) b)
% Found ((eq_ref Prop) (((eq (a->(a->Prop))) Xy) Xx)) as proof of (((eq Prop) (((eq (a->(a->Prop))) Xy) Xx)) b)
% Found ((eq_ref Prop) (((eq (a->(a->Prop))) Xy) Xx)) as proof of (((eq Prop) (((eq (a->(a->Prop))) Xy) Xx)) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))):(((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))
% Found (eq_ref0 (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) b)
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b
% Found conj2:=(conj ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))):(forall (B:Prop), (((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))->(B->((and ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))) B))))
% Instantiate: b:=(forall (B:Prop), (((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))->(B->((and ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))) B)))):Prop
% Found conj2 as proof of b
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b
% Found eq_sym100:=(eq_sym10 Xy):((((eq (a->(a->Prop))) Xx) Xy)->(((eq (a->(a->Prop))) Xy) Xx))
% Found (eq_sym10 Xy) as proof of ((((eq (a->(a->Prop))) Xx) Xy)->b)
% Found ((eq_sym1 Xx) Xy) as proof of ((((eq (a->(a->Prop))) Xx) Xy)->b)
% Found (((eq_sym (a->(a->Prop))) Xx) Xy) as proof of ((((eq (a->(a->Prop))) Xx) Xy)->b)
% Found (fun (x0:((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz)))))=> (((eq_sym (a->(a->Prop))) Xx) Xy)) as proof of ((((eq (a->(a->Prop))) Xx) Xy)->b)
% Found (fun (x0:((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz)))))=> (((eq_sym (a->(a->Prop))) Xx) Xy)) as proof of (((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))->((((eq (a->(a->Prop))) Xx) Xy)->b))
% Found (and_rect00 (fun (x0:((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz)))))=> (((eq_sym (a->(a->Prop))) Xx) Xy))) as proof of b
% Found ((and_rect0 b) (fun (x0:((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz)))))=> (((eq_sym (a->(a->Prop))) Xx) Xy))) as proof of b
% Found (((fun (P0:Type) (x0:(((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))->((((eq (a->(a->Prop))) Xx) Xy)->P0)))=> (((((and_rect ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))) (((eq (a->(a->Prop))) Xx) Xy)) P0) x0) x)) b) (fun (x0:((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz)))))=> (((eq_sym (a->(a->Prop))) Xx) Xy))) as proof of b
% Found (((fun (P0:Type) (x0:(((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))->((((eq (a->(a->Prop))) Xx) Xy)->P0)))=> (((((and_rect ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))) (((eq (a->(a->Prop))) Xx) Xy)) P0) x0) x)) b) (fun (x0:((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz)))))=> (((eq_sym (a->(a->Prop))) Xx) Xy))) as proof of b
% Found eq_ref00:=(eq_ref0 (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))):(((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))
% Found (eq_ref0 (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) b)
% Found conj2:=(conj ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))):(forall (B:Prop), (((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))->(B->((and ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))) B))))
% Instantiate: b:=(forall (B:Prop), (((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))->(B->((and ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))) B)))):Prop
% Found conj2 as proof of b
% Found conj2:=(conj ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))):(forall (B:Prop), (((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))->(B->((and ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))) B))))
% Instantiate: b:=(forall (B:Prop), (((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))->(B->((and ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))) B)))):Prop
% Found conj2 as proof of b
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b
% Found eq_ref00:=(eq_ref0 (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))):(((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))
% Found (eq_ref0 (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))):(((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))
% Found (eq_ref0 (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz)))) b)
% Found eq_sym1000:=(eq_sym100 x1):(((eq (a->(a->Prop))) Xy) Xx)
% Found (eq_sym100 x1) as proof of b
% Found ((eq_sym10 Xy) x1) as proof of b
% Found (((eq_sym1 Xx) Xy) x1) as proof of b
% Found ((((eq_sym (a->(a->Prop))) Xx) Xy) x1) as proof of b
% Found ((((eq_sym (a->(a->Prop))) Xx) Xy) x1) as proof of b
% Found eq_sym1000:=(eq_sym100 x1):(((eq (a->(a->Prop))) Xy) Xx)
% Found (eq_sym100 x1) as proof of b
% Found ((eq_sym10 Xy) x1) as proof of b
% Found (((eq_sym1 Xx) Xy) x1) as proof of b
% Found ((((eq_sym (a->(a->Prop))) Xx) Xy) x1) as proof of b
% Found ((((eq_sym (a->(a->Prop))) Xx) Xy) x1) as proof of b
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of b
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of b
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of b
% Found eq_ref00:=(eq_ref0 Xx0):(((eq a) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq ((a->(a->Prop))->((a->(a->Prop))->Prop))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq ((a->(a->Prop))->((a->(a->Prop))->Prop))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq ((a->(a->Prop))->((a->(a->Prop))->Prop))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq ((a->(a->Prop))->((a->(a->Prop))->Prop))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))) (fun (Xp:(a->(a->Prop))) (Xq:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (((eq (a->(a->Prop))) Xp) Xq)))))
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of b
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of b
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of b
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found eq_sym1000:=(eq_sym100 x1):(((eq (a->(a->Prop))) Xy) Xx)
% Found (eq_sym100 x1) as proof of b
% Found ((eq_sym10 Xy) x1) as proof of b
% Found (((eq_sym1 Xx) Xy) x1) as proof of b
% Found ((((eq_sym (a->(a->Prop))) Xx) Xy) x1) as proof of b
% Found ((((eq_sym (a->(a->Prop))) Xx) Xy) x1) as proof of b
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx:(a->(a->Prop))) (Xy:(a->(a->Prop))) (Xz:(a->(a->Prop))), (((and ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:(a->(a->Prop))) (Xy:(a->(a->Prop))) (Xz:(a->(a->Prop))), (((and ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:(a->(a->Prop))) (Xy:(a->(a->Prop))) (Xz:(a->(a->Prop))), (((and ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:(a->(a->Prop))) (Xy:(a->(a->Prop))) (Xz:(a->(a->Prop))), (((and ((and ((and ((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xy))) (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz0))->((Xy Xx0) Xz0))))) (((eq (a->(a->Prop))) Xy) Xz))->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz0))->((Xx Xx0) Xz0))))) (((eq (a->(a->Prop))) Xx) Xz)))))
% Found eq_sym1000:=(eq_sym100 x1):(((eq (a->(a->Prop))) Xy) Xx)
% Found (eq_sym100 x1) as proof of b
% Found ((eq_sym10 Xy) x1) as proof of b
% Found (((eq_sym1 Xx) Xy) x1) as proof of b
% Found ((((eq_sym (a->(a->Prop))) Xx) Xy) x1) as proof of b
% Found ((((eq_sym (a->(a->Prop))) Xx) Xy) x1) as proof of b
% Found eq_sym1000:=(eq_sym100 x1):(((eq (a->(a->Prop))) Xy) Xx)
% Found (eq_sym100 x1) as proof of b
% Found ((eq_sym10 Xy) x1) as proof of b
% Found (((eq_sym1 Xx) Xy) x1) as proof of b
% Found ((((eq_sym (a->(a->Prop))) Xx) Xy) x1) as proof of b
% Found ((((eq_sym (a->(a->Prop))) Xx) Xy) x1) as proof of b
% Found eq_ref00:=(eq_ref0 Xx0):(((eq a) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found eq_ref00:=(eq_ref0 Xx0):(((eq a) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq (a->(a->Prop))) Xy) Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->(a->Prop))) Xy) Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->(a->Prop))) Xy) Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->(a->Prop))) Xy) Xx))
% Found x1:((Xy Xx0) Xy0)
% Instantiate: b:=Xy0:a
% Found x1 as proof of (P b)
% Found x1:((Xy Xx0) Xy0)
% Instantiate: b:=Xy0:a
% Found (fun (x2:((Xy Xy0) Xz))=> x1) as proof of (P b)
% Found (fun (x1:((Xy Xx0) Xy0)) (x2:((Xy Xy0) Xz))=> x1) as proof of (((Xy Xy0) Xz)->(P b))
% Found (fun (x1:((Xy Xx0) Xy0)) (x2:((Xy Xy0) Xz))=> x1) as proof of (((Xy Xx0) Xy0)->(((Xy Xy0) Xz)->(P b)))
% Found (and_rect00 (fun (x1:((Xy Xx0) Xy0)) (x2:((Xy Xy0) Xz))=> x1)) as proof of (P b)
% Found ((and_rect0 (P b)) (fun (x1:((Xy Xx0) Xy0)) (x2:((Xy Xy0) Xz))=> x1)) as proof of (P b)
% Found (((fun (P0:Type) (x1:(((Xy Xx0) Xy0)->(((Xy Xy0) Xz)->P0)))=> (((((and_rect ((Xy Xx0) Xy0)) ((Xy Xy0) Xz)) P0) x1) x0)) (P b)) (fun (x1:((Xy Xx0) Xy0)) (x2:((Xy Xy0) Xz))=> x1)) as proof of (P b)
% Found (((fun (P0:Type) (x1:(((Xy Xx0) Xy0)->(((Xy Xy0) Xz)->P0)))=> (((((and_rect ((Xy Xx0) Xy0)) ((Xy Xy0) Xz)) P0) x1) x0)) (P b)) (fun (x1:((Xy Xx0) Xy0)) (x2:((Xy Xy0) Xz))=> x1)) as proof of (P b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found eta_expansion:=(fun (A:Type) (B:Type)=> ((eta_expansion_dep A) (fun (x1:A)=> B))):(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x))))
% Instantiate: a0:=(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x)))):Prop
% Found eta_expansion as proof of a0
% Found eq_ref00:=(eq_ref0 Xx0):(((eq a) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found eq_ref00:=(eq_ref0 Xx0):(((eq a) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found eq_ref00:=(eq_ref0 Xx0):(((eq a) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found eq_ref00:=(eq_ref0 Xx0):(((eq a) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq (a->(a->Prop))) Xy) Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->(a->Prop))) Xy) Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->(a->Prop))) Xy) Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->(a->Prop))) Xy) Xx))
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))
% Found eq_ref00:=(eq_ref0 Xx0):(((eq a) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found eq_ref00:=(eq_ref0 Xx0):(((eq a) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: a0:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of a0
% Found conj2:=(conj ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))):(forall (B:Prop), (((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))->(B->((and ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))) B))))
% Instantiate: a0:=(forall (B:Prop), (((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))->(B->((and ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))) B)))):Prop
% Found conj2 as proof of a0
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found x1:(((eq (a->(a->Prop))) Xx) Xy)
% Instantiate: b:=(forall (P:((a->(a->Prop))->Prop)), ((P Xx)->(P Xy))):Prop
% Found x1 as proof of a0
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found x3:((Xy Xx0) Xy0)
% Instantiate: b:=Xy0:a
% Found x3 as proof of (P b)
% Found x3:((Xy Xx0) Xy0)
% Instantiate: b:=Xy0:a
% Found (fun (x4:((Xy Xy0) Xz))=> x3) as proof of (P b)
% Found (fun (x3:((Xy Xx0) Xy0)) (x4:((Xy Xy0) Xz))=> x3) as proof of (((Xy Xy0) Xz)->(P b))
% Found (fun (x3:((Xy Xx0) Xy0)) (x4:((Xy Xy0) Xz))=> x3) as proof of (((Xy Xx0) Xy0)->(((Xy Xy0) Xz)->(P b)))
% Found (and_rect10 (fun (x3:((Xy Xx0) Xy0)) (x4:((Xy Xy0) Xz))=> x3)) as proof of (P b)
% Found ((and_rect1 (P b)) (fun (x3:((Xy Xx0) Xy0)) (x4:((Xy Xy0) Xz))=> x3)) as proof of (P b)
% Found (((fun (P0:Type) (x3:(((Xy Xx0) Xy0)->(((Xy Xy0) Xz)->P0)))=> (((((and_rect ((Xy Xx0) Xy0)) ((Xy Xy0) Xz)) P0) x3) x2)) (P b)) (fun (x3:((Xy Xx0) Xy0)) (x4:((Xy Xy0) Xz))=> x3)) as proof of (P b)
% Found (((fun (P0:Type) (x3:(((Xy Xx0) Xy0)->(((Xy Xy0) Xz)->P0)))=> (((((and_rect ((Xy Xx0) Xy0)) ((Xy Xy0) Xz)) P0) x3) x2)) (P b)) (fun (x3:((Xy Xx0) Xy0)) (x4:((Xy Xy0) Xz))=> x3)) as proof of (P b)
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: a0:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of a0
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found x3:((Xy Xx0) Xy0)
% Instantiate: b:=Xy0:a
% Found x3 as proof of (P b)
% Found x1:(((eq (a->(a->Prop))) Xx) Xy)
% Instantiate: b:=(forall (P:((a->(a->Prop))->Prop)), ((P Xx)->(P Xy))):Prop
% Found x1 as proof of a0
% Found x3:((Xy Xx0) Xy0)
% Instantiate: b:=Xy0:a
% Found (fun (x4:((Xy Xy0) Xz))=> x3) as proof of (P b)
% Found (fun (x3:((Xy Xx0) Xy0)) (x4:((Xy Xy0) Xz))=> x3) as proof of (((Xy Xy0) Xz)->(P b))
% Found (fun (x3:((Xy Xx0) Xy0)) (x4:((Xy Xy0) Xz))=> x3) as proof of (((Xy Xx0) Xy0)->(((Xy Xy0) Xz)->(P b)))
% Found (and_rect10 (fun (x3:((Xy Xx0) Xy0)) (x4:((Xy Xy0) Xz))=> x3)) as proof of (P b)
% Found ((and_rect1 (P b)) (fun (x3:((Xy Xx0) Xy0)) (x4:((Xy Xy0) Xz))=> x3)) as proof of (P b)
% Found (((fun (P0:Type) (x3:(((Xy Xx0) Xy0)->(((Xy Xy0) Xz)->P0)))=> (((((and_rect ((Xy Xx0) Xy0)) ((Xy Xy0) Xz)) P0) x3) x2)) (P b)) (fun (x3:((Xy Xx0) Xy0)) (x4:((Xy Xy0) Xz))=> x3)) as proof of (P b)
% Found (((fun (P0:Type) (x3:(((Xy Xx0) Xy0)->(((Xy Xy0) Xz)->P0)))=> (((((and_rect ((Xy Xx0) Xy0)) ((Xy Xy0) Xz)) P0) x3) x2)) (P b)) (fun (x3:((Xy Xx0) Xy0)) (x4:((Xy Xy0) Xz))=> x3)) as proof of (P b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eta_expansion:=(fun (A:Type) (B:Type)=> ((eta_expansion_dep A) (fun (x1:A)=> B))):(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x))))
% Instantiate: b:=(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x)))):Prop
% Found eta_expansion as proof of b
% Found eq_ref00:=(eq_ref0 Xx0):(((eq a) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq (a->(a->Prop))) Xy) Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->(a->Prop))) Xy) Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->(a->Prop))) Xy) Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->(a->Prop))) Xy) Xx))
% Found eq_ref00:=(eq_ref0 Xx0):(((eq a) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found conj2:=(conj ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))):(forall (B:Prop), (((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))->(B->((and ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))) B))))
% Instantiate: a0:=(forall (B:Prop), (((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))->(B->((and ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))) B)))):Prop
% Found conj2 as proof of a0
% Found eq_ref00:=(eq_ref0 Xx0):(((eq a) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found eq_ref00:=(eq_ref0 Xx0):(((eq a) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found eq_ref00:=(eq_ref0 Xx0):(((eq a) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found eq_ref00:=(eq_ref0 Xx0):(((eq a) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found ((eq_ref a) Xx0) as proof of (((eq a) Xx0) b)
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of a0
% Found conj2:=(conj ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))):(forall (B:Prop), (((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))->(B->((and ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))) B))))
% Instantiate: a0:=(forall (B:Prop), (((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))->(B->((and ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))) B)))):Prop
% Found conj2 as proof of a0
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found x1:(((eq (a->(a->Prop))) Xx) Xy)
% Instantiate: b:=(forall (P:((a->(a->Prop))->Prop)), ((P Xx)->(P Xy))):Prop
% Found x1 as proof of a0
% Found conj2:=(conj ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))):(forall (B:Prop), (((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))->(B->((and ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))) B))))
% Instantiate: a0:=(forall (B:Prop), (((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))->(B->((and ((and (forall (Xx0:a) (Xy0:a), (((Xy Xx0) Xy0)->((Xy Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xy Xx0) Xy0)) ((Xy Xy0) Xz))->((Xy Xx0) Xz))))) B)))):Prop
% Found conj2 as proof of a0
% Found conj30:=(conj3 (((eq (a->(a->Prop))) Xx) Xy)):(((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))->((((eq (a->(a->Prop))) Xx) Xy)->((and ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))) (((eq (a->(a->Prop))) Xx) Xy))))
% Found (conj3 (((eq (a->(a->Prop))) Xx) Xy)) as proof of (((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))->((((eq (a->(a->Prop))) Xx) Xy)->a0))
% Found ((conj ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))) (((eq (a->(a->Prop))) Xx) Xy)) as proof of (((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))->((((eq (a->(a->Prop))) Xx) Xy)->a0))
% Found ((conj ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))) (((eq (a->(a->Prop))) Xx) Xy)) as proof of (((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))->((((eq (a->(a->Prop))) Xx) Xy)->a0))
% Found ((conj ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))) (((eq (a->(a->Prop))) Xx) Xy)) as proof of (((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))->((((eq (a->(a->Prop))) Xx) Xy)->a0))
% Found (and_rect00 ((conj ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))) (((eq (a->(a->Prop))) Xx) Xy))) as proof of a0
% Found ((and_rect0 a0) ((conj ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))) (((eq (a->(a->Prop))) Xx) Xy))) as proof of a0
% Found (((fun (P0:Type) (x0:(((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))->((((eq (a->(a->Prop))) Xx) Xy)->P0)))=> (((((and_rect ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))) (((eq (a->(a->Prop))) Xx) Xy)) P0) x0) x)) a0) ((conj ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))) (((eq (a->(a->Prop))) Xx) Xy))) as proof of a0
% Found (((fun (P0:Type) (x0:(((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))->((((eq (a->(a->Prop))) Xx) Xy)->P0)))=> (((((and_rect ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))) (((eq (a->(a->Prop))) Xx) Xy)) P0) x0) x)) a0) ((conj ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))) (((eq (a->(a->Prop))) Xx) Xy))) as proof of a0
% Found eq_sym000:=(eq_sym00 Xy):((((eq (a->(a->Prop))) Xx) Xy)->(((eq (a->(a->Prop))) Xy) Xx))
% Found (eq_sym00 Xy) as proof of ((((eq (a->(a->Prop))) Xx) Xy)->a0)
% Found ((eq_sym0 Xx) Xy) as proof of ((((eq (a->(a->Prop))) Xx) Xy)->a0)
% Found (((eq_sym (a->(a->Prop))) Xx) Xy) as proof of ((((eq (a->(a->Prop))) Xx) Xy)->a0)
% Found (fun (x0:((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz)))))=> (((eq_sym (a->(a->Prop))) Xx) Xy)) as proof of ((((eq (a->(a->Prop))) Xx) Xy)->a0)
% Found (fun (x0:((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz)))))=> (((eq_sym (a->(a->Prop))) Xx) Xy)) as proof of (((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))->((((eq (a->(a->Prop))) Xx) Xy)->a0))
% Found (and_rect00 (fun (x0:((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz)))))=> (((eq_sym (a->(a->Prop))) Xx) Xy))) as proof of a0
% Found ((and_rect0 a0) (fun (x0:((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz)))))=> (((eq_sym (a->(a->Prop))) Xx) Xy))) as proof of a0
% Found (((fun (P0:Type) (x0:(((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((Xx Xy0) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xx Xx0) Xy0)) ((Xx Xy0) Xz))->((Xx Xx0) Xz))))->((((eq (a->(a->Prop))) Xx) Xy)->P0)))=> (((((and_rect ((and (forall (Xx0:a) (Xy0:a), (((Xx Xx0) Xy0)->((X
% EOF
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