%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV114^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n097.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 13:33:45 EDT 2014

% Result   : Timeout 300.03s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV114^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n097.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 08:07:31 CDT 2014
% % CPUTime  : 300.03 
% 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 0xb95098>, <kernel.Type object at 0xb95908>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula ((forall (X2:((a->Prop)->Prop)), ((ex ((a->Prop)->Prop)) (fun (M:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((M Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (M U)) (M V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((M Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(M Xx)))))))))->(forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx))))))))))) of role conjecture named cTHM540_pme
% Conjecture to prove = ((forall (X2:((a->Prop)->Prop)), ((ex ((a->Prop)->Prop)) (fun (M:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((M Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (M U)) (M V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((M Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(M Xx)))))))))->(forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx))))))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['((forall (X2:((a->Prop)->Prop)), ((ex ((a->Prop)->Prop)) (fun (M:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((M Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (M U)) (M V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((M Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(M Xx)))))))))->(forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))))))']
% Parameter a:Type.
% Trying to prove ((forall (X2:((a->Prop)->Prop)), ((ex ((a->Prop)->Prop)) (fun (M:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((M Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (M U)) (M V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((M Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(M Xx)))))))))->(forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))))))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x1:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x1 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x3 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x3 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x3 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x3:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x3 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x3 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x3 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x3 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x1:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x1 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found eq_ref000:=(eq_ref00 (fun (x3:Prop)=> (x1 Xy))):((x1 Xy)->(x1 Xy))
% Found (eq_ref00 (fun (x3:Prop)=> (x1 Xy))) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((eq_ref0 ((R Xy) Xx)) (fun (x3:Prop)=> (x1 Xy))) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (((eq_ref Prop) ((R Xy) Xx)) (fun (x3:Prop)=> (x1 Xy))) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (((eq_ref Prop) ((R Xy) Xx)) (fun (x3:Prop)=> (x1 Xy))) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x2:(x1 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x3:Prop)=> (x1 Xy)))) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x2:(x1 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x3:Prop)=> (x1 Xy)))) as proof of ((x1 Xx)->((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect00 (fun (x2:(x1 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x3:Prop)=> (x1 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect0 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x2:(x1 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x3:Prop)=> (x1 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x2:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x2) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x2:(x1 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x3:Prop)=> (x1 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x1 Xx)) (x1 Xy)))=> (((fun (P:Type) (x2:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x2) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x2:(x1 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x3:Prop)=> (x1 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x1 Xx)) (x1 Xy)))=> (((fun (P:Type) (x2:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x2) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x2:(x1 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x3:Prop)=> (x1 Xy)))))) as proof of (((and (x1 Xx)) (x1 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found x01:(Xy Xx)
% Instantiate: x5:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x5 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x5 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x5 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x5 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x5 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x5 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x5 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x5 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x5 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x5 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x5 Xx)))
% Found eq_ref00:=(eq_ref0 (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx))))))))
% Found (eq_ref0 (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) b)
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x1:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x1 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x3 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x3 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x3 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x3:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x3 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x3 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x3 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x3 Xx)))
% Found or_introl00:=(or_introl0 ((R Xy) Xx)):((x3 Xy)->((or (x3 Xy)) ((R Xy) Xx)))
% Found (or_introl0 ((R Xy) Xx)) as proof of ((x3 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_introl (x3 Xy)) ((R Xy) Xx)) as proof of ((x3 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_introl (x3 Xy)) ((R Xy) Xx)) as proof of ((x3 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x4:(x3 Xx))=> ((or_introl (x3 Xy)) ((R Xy) Xx))) as proof of ((x3 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x4:(x3 Xx))=> ((or_introl (x3 Xy)) ((R Xy) Xx))) as proof of ((x3 Xx)->((x3 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect10 (fun (x4:(x3 Xx))=> ((or_introl (x3 Xy)) ((R Xy) Xx)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect1 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x4:(x3 Xx))=> ((or_introl (x3 Xy)) ((R Xy) Xx)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x4:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x4) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x4:(x3 Xx))=> ((or_introl (x3 Xy)) ((R Xy) Xx)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x3 Xx)) (x3 Xy)))=> (((fun (P:Type) (x4:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x4) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x4:(x3 Xx))=> ((or_introl (x3 Xy)) ((R Xy) Xx))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x3 Xx)) (x3 Xy)))=> (((fun (P:Type) (x4:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x4) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x4:(x3 Xx))=> ((or_introl (x3 Xy)) ((R Xy) Xx))))) as proof of (((and (x3 Xx)) (x3 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) (fun (x:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (x Xx)) (x Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(x Xx))))))))
% Found (eta_expansion_dep00 (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) b)
% Found ((eta_expansion_dep0 (fun (x4:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) b)
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> (x1 Xy))):((x1 Xy)->(x1 Xy))
% Found (eq_ref00 (fun (x5:Prop)=> (x1 Xy))) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((eq_ref0 ((R Xy) Xx)) (fun (x5:Prop)=> (x1 Xy))) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (((eq_ref Prop) ((R Xy) Xx)) (fun (x5:Prop)=> (x1 Xy))) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (((eq_ref Prop) ((R Xy) Xx)) (fun (x5:Prop)=> (x1 Xy))) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x4:(x1 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x5:Prop)=> (x1 Xy)))) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x4:(x1 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x5:Prop)=> (x1 Xy)))) as proof of ((x1 Xx)->((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect10 (fun (x4:(x1 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x5:Prop)=> (x1 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect1 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x4:(x1 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x5:Prop)=> (x1 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x4:(x1 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x5:Prop)=> (x1 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x1 Xx)) (x1 Xy)))=> (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x4:(x1 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x5:Prop)=> (x1 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x1 Xx)) (x1 Xy)))=> (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x4:(x1 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x5:Prop)=> (x1 Xy)))))) as proof of (((and (x1 Xx)) (x1 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found eq_ref000:=(eq_ref00 (fun (x3:a)=> (x1 Xy))):((x1 Xy)->(x1 Xy))
% Found (eq_ref00 (fun (x3:a)=> (x1 Xy))) as proof of ((x1 Xy)->((R Xy) Xx))
% Found ((eq_ref0 Xx) (fun (x3:a)=> (x1 Xy))) as proof of ((x1 Xy)->((R Xy) Xx))
% Found (((eq_ref a) Xx) (fun (x3:a)=> (x1 Xy))) as proof of ((x1 Xy)->((R Xy) Xx))
% Found (((eq_ref a) Xx) (fun (x3:a)=> (x1 Xy))) as proof of ((x1 Xy)->((R Xy) Xx))
% Found (fun (x2:(x1 Xx))=> (((eq_ref a) Xx) (fun (x3:a)=> (x1 Xy)))) as proof of ((x1 Xy)->((R Xy) Xx))
% Found (fun (x2:(x1 Xx))=> (((eq_ref a) Xx) (fun (x3:a)=> (x1 Xy)))) as proof of ((x1 Xx)->((x1 Xy)->((R Xy) Xx)))
% Found (and_rect00 (fun (x2:(x1 Xx))=> (((eq_ref a) Xx) (fun (x3:a)=> (x1 Xy))))) as proof of ((R Xy) Xx)
% Found ((and_rect0 ((R Xy) Xx)) (fun (x2:(x1 Xx))=> (((eq_ref a) Xx) (fun (x3:a)=> (x1 Xy))))) as proof of ((R Xy) Xx)
% Found (((fun (P:Type) (x2:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x2) x00)) ((R Xy) Xx)) (fun (x2:(x1 Xx))=> (((eq_ref a) Xx) (fun (x3:a)=> (x1 Xy))))) as proof of ((R Xy) Xx)
% Found (((fun (P:Type) (x2:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x2) x00)) ((R Xy) Xx)) (fun (x2:(x1 Xx))=> (((eq_ref a) Xx) (fun (x3:a)=> (x1 Xy))))) as proof of ((R Xy) Xx)
% Found (or_intror00 (((fun (P:Type) (x2:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x2) x00)) ((R Xy) Xx)) (fun (x2:(x1 Xx))=> (((eq_ref a) Xx) (fun (x3:a)=> (x1 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((or_intror0 ((R Xy) Xx)) (((fun (P:Type) (x2:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x2) x00)) ((R Xy) Xx)) (fun (x2:(x1 Xx))=> (((eq_ref a) Xx) (fun (x3:a)=> (x1 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((or_intror ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x2:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x2) x00)) ((R Xy) Xx)) (fun (x2:(x1 Xx))=> (((eq_ref a) Xx) (fun (x3:a)=> (x1 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x1 Xx)) (x1 Xy)))=> (((or_intror ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x2:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x2) x00)) ((R Xy) Xx)) (fun (x2:(x1 Xx))=> (((eq_ref a) Xx) (fun (x3:a)=> (x1 Xy))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x1 Xx)) (x1 Xy)))=> (((or_intror ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x2:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x2) x00)) ((R Xy) Xx)) (fun (x2:(x1 Xx))=> (((eq_ref a) Xx) (fun (x3:a)=> (x1 Xy))))))) as proof of (((and (x1 Xx)) (x1 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found x3:(x1 Xy)
% Instantiate: x1:=(R Xx):(a->Prop)
% Found (fun (x3:(x1 Xy))=> x3) as proof of ((R Xx) Xy)
% Found (fun (x2:(x1 Xx)) (x3:(x1 Xy))=> x3) as proof of ((x1 Xy)->((R Xx) Xy))
% Found (fun (x2:(x1 Xx)) (x3:(x1 Xy))=> x3) as proof of ((x1 Xx)->((x1 Xy)->((R Xx) Xy)))
% Found (and_rect00 (fun (x2:(x1 Xx)) (x3:(x1 Xy))=> x3)) as proof of ((R Xx) Xy)
% Found ((and_rect0 ((R Xx) Xy)) (fun (x2:(x1 Xx)) (x3:(x1 Xy))=> x3)) as proof of ((R Xx) Xy)
% Found (((fun (P:Type) (x2:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x2) x00)) ((R Xx) Xy)) (fun (x2:(x1 Xx)) (x3:(x1 Xy))=> x3)) as proof of ((R Xx) Xy)
% Found (((fun (P:Type) (x2:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x2) x00)) ((R Xx) Xy)) (fun (x2:(x1 Xx)) (x3:(x1 Xy))=> x3)) as proof of ((R Xx) Xy)
% Found (or_introl00 (((fun (P:Type) (x2:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x2) x00)) ((R Xx) Xy)) (fun (x2:(x1 Xx)) (x3:(x1 Xy))=> x3))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((or_introl0 ((R Xy) Xx)) (((fun (P:Type) (x2:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x2) x00)) ((R Xx) Xy)) (fun (x2:(x1 Xx)) (x3:(x1 Xy))=> x3))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((or_introl ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x2:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x2) x00)) ((R Xx) Xy)) (fun (x2:(x1 Xx)) (x3:(x1 Xy))=> x3))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x1 Xx)) (x1 Xy)))=> (((or_introl ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x2:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x2) x00)) ((R Xx) Xy)) (fun (x2:(x1 Xx)) (x3:(x1 Xy))=> x3)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x1 Xx)) (x1 Xy)))=> (((or_introl ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x2:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x2) x00)) ((R Xx) Xy)) (fun (x2:(x1 Xx)) (x3:(x1 Xy))=> x3)))) as proof of (((and (x1 Xx)) (x1 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found eq_ref000:=(eq_ref00 (fun (x3:Prop)=> (x1 Xy))):((x1 Xy)->(x1 Xy))
% Found (eq_ref00 (fun (x3:Prop)=> (x1 Xy))) as proof of ((x1 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found ((eq_ref0 ((R Xx) Xy)) (fun (x3:Prop)=> (x1 Xy))) as proof of ((x1 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found (((eq_ref Prop) ((R Xx) Xy)) (fun (x3:Prop)=> (x1 Xy))) as proof of ((x1 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found (((eq_ref Prop) ((R Xx) Xy)) (fun (x3:Prop)=> (x1 Xy))) as proof of ((x1 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found (fun (x2:(x1 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x3:Prop)=> (x1 Xy)))) as proof of ((x1 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found (fun (x2:(x1 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x3:Prop)=> (x1 Xy)))) as proof of ((x1 Xx)->((x1 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy))))
% Found (and_rect00 (fun (x2:(x1 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x3:Prop)=> (x1 Xy))))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found ((and_rect0 ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x2:(x1 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x3:Prop)=> (x1 Xy))))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found (((fun (P:Type) (x2:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x2) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x2:(x1 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x3:Prop)=> (x1 Xy))))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found (((fun (P:Type) (x2:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x2) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x2:(x1 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x3:Prop)=> (x1 Xy))))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found (or_comm_i00 (((fun (P:Type) (x2:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x2) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x2:(x1 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x3:Prop)=> (x1 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((or_comm_i0 ((R Xx) Xy)) (((fun (P:Type) (x2:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x2) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x2:(x1 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x3:Prop)=> (x1 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((or_comm_i ((R Xy) Xx)) ((R Xx) Xy)) (((fun (P:Type) (x2:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x2) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x2:(x1 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x3:Prop)=> (x1 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x1 Xx)) (x1 Xy)))=> (((or_comm_i ((R Xy) Xx)) ((R Xx) Xy)) (((fun (P:Type) (x2:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x2) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x2:(x1 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x3:Prop)=> (x1 Xy))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x1 Xx)) (x1 Xy)))=> (((or_comm_i ((R Xy) Xx)) ((R Xx) Xy)) (((fun (P:Type) (x2:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x2) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x2:(x1 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x3:Prop)=> (x1 Xy))))))) as proof of (((and (x1 Xx)) (x1 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) (fun (x:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (x Xx)) (x Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(x Xx))))))))
% Found (eta_expansion_dep00 (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) b)
% Found ((eta_expansion_dep0 (fun (x6:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x6:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x6:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x6:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) b)
% Found or_intror00:=(or_intror0 (x5 Xy)):((x5 Xy)->((or ((R Xx) Xy)) (x5 Xy)))
% Found (or_intror0 (x5 Xy)) as proof of ((x5 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_intror ((R Xx) Xy)) (x5 Xy)) as proof of ((x5 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_intror ((R Xx) Xy)) (x5 Xy)) as proof of ((x5 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x6:(x5 Xx))=> ((or_intror ((R Xx) Xy)) (x5 Xy))) as proof of ((x5 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x6:(x5 Xx))=> ((or_intror ((R Xx) Xy)) (x5 Xy))) as proof of ((x5 Xx)->((x5 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect20 (fun (x6:(x5 Xx))=> ((or_intror ((R Xx) Xy)) (x5 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect2 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x5 Xx))=> ((or_intror ((R Xx) Xy)) (x5 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x6:((x5 Xx)->((x5 Xy)->P)))=> (((((and_rect (x5 Xx)) (x5 Xy)) P) x6) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x5 Xx))=> ((or_intror ((R Xx) Xy)) (x5 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x5 Xx)) (x5 Xy)))=> (((fun (P:Type) (x6:((x5 Xx)->((x5 Xy)->P)))=> (((((and_rect (x5 Xx)) (x5 Xy)) P) x6) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x5 Xx))=> ((or_intror ((R Xx) Xy)) (x5 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x5 Xx)) (x5 Xy)))=> (((fun (P:Type) (x6:((x5 Xx)->((x5 Xy)->P)))=> (((((and_rect (x5 Xx)) (x5 Xy)) P) x6) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x5 Xx))=> ((or_intror ((R Xx) Xy)) (x5 Xy))))) as proof of (((and (x5 Xx)) (x5 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found or_intror00:=(or_intror0 (x1 Xy)):((x1 Xy)->((or ((R Xx) Xy)) (x1 Xy)))
% Found (or_intror0 (x1 Xy)) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_intror ((R Xx) Xy)) (x1 Xy)) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_intror ((R Xx) Xy)) (x1 Xy)) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x6:(x1 Xx))=> ((or_intror ((R Xx) Xy)) (x1 Xy))) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x6:(x1 Xx))=> ((or_intror ((R Xx) Xy)) (x1 Xy))) as proof of ((x1 Xx)->((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect20 (fun (x6:(x1 Xx))=> ((or_intror ((R Xx) Xy)) (x1 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect2 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x1 Xx))=> ((or_intror ((R Xx) Xy)) (x1 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x1 Xx))=> ((or_intror ((R Xx) Xy)) (x1 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x1 Xx)) (x1 Xy)))=> (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x1 Xx))=> ((or_intror ((R Xx) Xy)) (x1 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x1 Xx)) (x1 Xy)))=> (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x1 Xx))=> ((or_intror ((R Xx) Xy)) (x1 Xy))))) as proof of (((and (x1 Xx)) (x1 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found or_intror00:=(or_intror0 (x3 Xy)):((x3 Xy)->((or ((R Xx) Xy)) (x3 Xy)))
% Found (or_intror0 (x3 Xy)) as proof of ((x3 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_intror ((R Xx) Xy)) (x3 Xy)) as proof of ((x3 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_intror ((R Xx) Xy)) (x3 Xy)) as proof of ((x3 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x6:(x3 Xx))=> ((or_intror ((R Xx) Xy)) (x3 Xy))) as proof of ((x3 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x6:(x3 Xx))=> ((or_intror ((R Xx) Xy)) (x3 Xy))) as proof of ((x3 Xx)->((x3 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect20 (fun (x6:(x3 Xx))=> ((or_intror ((R Xx) Xy)) (x3 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect2 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x3 Xx))=> ((or_intror ((R Xx) Xy)) (x3 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x3 Xx))=> ((or_intror ((R Xx) Xy)) (x3 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x3 Xx)) (x3 Xy)))=> (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x3 Xx))=> ((or_intror ((R Xx) Xy)) (x3 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x3 Xx)) (x3 Xy)))=> (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x3 Xx))=> ((or_intror ((R Xx) Xy)) (x3 Xy))))) as proof of (((and (x3 Xx)) (x3 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found and_rect00:=(and_rect0 ((or ((R Xx) Xy)) ((R Xy) Xx))):((((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))->((forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0)))->((or ((R Xx) Xy)) ((R Xy) Xx))))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Instantiate: x1:=(fun (x4:a)=> (((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))->((forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0)))->((or ((R Xx) x4)) ((R x4) Xx))))):(a->Prop)
% Found (fun (x4:(x1 Xx))=> and_rect00) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x4:(x1 Xx))=> and_rect00) as proof of ((x1 Xx)->((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect10 (fun (x4:(x1 Xx))=> and_rect00)) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect1 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x4:(x1 Xx))=> and_rect00)) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x4:(x1 Xx))=> and_rect00)) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x3:(forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x4:(x1 Xx))=> and_rect00))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x2:((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))) (x3:(forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x4:(x1 Xx))=> and_rect00))) as proof of ((forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0)))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x2:((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))) (x3:(forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x4:(x1 Xx))=> and_rect00))) as proof of (((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))->((forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0)))->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect00 (fun (x2:((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))) (x3:(forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x4:(x1 Xx))=> and_rect00)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect0 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x2:((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))) (x3:(forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x4:(x1 Xx))=> (and_rect0 ((or ((R Xx) Xy)) ((R Xy) Xx))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x2:(((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))->P)))=> (((((and_rect ((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) P) x2) x0)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x2:((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))) (x3:(forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x4:(x1 Xx))=> ((fun (P:Type) (x2:(((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))->P)))=> (((((and_rect ((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) P) x2) x0)) ((or ((R Xx) Xy)) ((R Xy) Xx))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x1 Xx)) (x1 Xy)))=> (((fun (P:Type) (x2:(((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))->P)))=> (((((and_rect ((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) P) x2) x0)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x2:((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))) (x3:(forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x4:(x1 Xx))=> ((fun (P:Type) (x2:(((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))->P)))=> (((((and_rect ((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) P) x2) x0)) ((or ((R Xx) Xy)) ((R Xy) Xx)))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x1 Xx)) (x1 Xy)))=> (((fun (P:Type) (x2:(((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))->P)))=> (((((and_rect ((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) P) x2) x0)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x2:((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))) (x3:(forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x4:(x1 Xx))=> ((fun (P:Type) (x2:(((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))->P)))=> (((((and_rect ((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) P) x2) x0)) ((or ((R Xx) Xy)) ((R Xy) Xx)))))))) as proof of (((and (x1 Xx)) (x1 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found x01:(Xy Xx)
% Instantiate: x4:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x4 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x4 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x4 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x4 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x4 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x4 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x4 Xx)))
% Found x5:(x3 Xy)
% Instantiate: x3:=(R Xx):(a->Prop)
% Found (fun (x5:(x3 Xy))=> x5) as proof of ((R Xx) Xy)
% Found (fun (x4:(x3 Xx)) (x5:(x3 Xy))=> x5) as proof of ((x3 Xy)->((R Xx) Xy))
% Found (fun (x4:(x3 Xx)) (x5:(x3 Xy))=> x5) as proof of ((x3 Xx)->((x3 Xy)->((R Xx) Xy)))
% Found (and_rect10 (fun (x4:(x3 Xx)) (x5:(x3 Xy))=> x5)) as proof of ((R Xx) Xy)
% Found ((and_rect1 ((R Xx) Xy)) (fun (x4:(x3 Xx)) (x5:(x3 Xy))=> x5)) as proof of ((R Xx) Xy)
% Found (((fun (P:Type) (x4:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x4) x00)) ((R Xx) Xy)) (fun (x4:(x3 Xx)) (x5:(x3 Xy))=> x5)) as proof of ((R Xx) Xy)
% Found (((fun (P:Type) (x4:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x4) x00)) ((R Xx) Xy)) (fun (x4:(x3 Xx)) (x5:(x3 Xy))=> x5)) as proof of ((R Xx) Xy)
% Found (or_introl00 (((fun (P:Type) (x4:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x4) x00)) ((R Xx) Xy)) (fun (x4:(x3 Xx)) (x5:(x3 Xy))=> x5))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((or_introl0 ((R Xy) Xx)) (((fun (P:Type) (x4:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x4) x00)) ((R Xx) Xy)) (fun (x4:(x3 Xx)) (x5:(x3 Xy))=> x5))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((or_introl ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x4:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x4) x00)) ((R Xx) Xy)) (fun (x4:(x3 Xx)) (x5:(x3 Xy))=> x5))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x3 Xx)) (x3 Xy)))=> (((or_introl ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x4:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x4) x00)) ((R Xx) Xy)) (fun (x4:(x3 Xx)) (x5:(x3 Xy))=> x5)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x3 Xx)) (x3 Xy)))=> (((or_introl ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x4:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x4) x00)) ((R Xx) Xy)) (fun (x4:(x3 Xx)) (x5:(x3 Xy))=> x5)))) as proof of (((and (x3 Xx)) (x3 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found x4:(x3 Xx)
% Instantiate: x3:=(R Xy):(a->Prop)
% Found (fun (x5:(x3 Xy))=> x4) as proof of ((R Xy) Xx)
% Found (fun (x4:(x3 Xx)) (x5:(x3 Xy))=> x4) as proof of ((x3 Xy)->((R Xy) Xx))
% Found (fun (x4:(x3 Xx)) (x5:(x3 Xy))=> x4) as proof of ((x3 Xx)->((x3 Xy)->((R Xy) Xx)))
% Found (and_rect10 (fun (x4:(x3 Xx)) (x5:(x3 Xy))=> x4)) as proof of ((R Xy) Xx)
% Found ((and_rect1 ((R Xy) Xx)) (fun (x4:(x3 Xx)) (x5:(x3 Xy))=> x4)) as proof of ((R Xy) Xx)
% Found (((fun (P:Type) (x4:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x4) x00)) ((R Xy) Xx)) (fun (x4:(x3 Xx)) (x5:(x3 Xy))=> x4)) as proof of ((R Xy) Xx)
% Found (((fun (P:Type) (x4:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x4) x00)) ((R Xy) Xx)) (fun (x4:(x3 Xx)) (x5:(x3 Xy))=> x4)) as proof of ((R Xy) Xx)
% Found (or_intror00 (((fun (P:Type) (x4:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x4) x00)) ((R Xy) Xx)) (fun (x4:(x3 Xx)) (x5:(x3 Xy))=> x4))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((or_intror0 ((R Xy) Xx)) (((fun (P:Type) (x4:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x4) x00)) ((R Xy) Xx)) (fun (x4:(x3 Xx)) (x5:(x3 Xy))=> x4))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((or_intror ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x4:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x4) x00)) ((R Xy) Xx)) (fun (x4:(x3 Xx)) (x5:(x3 Xy))=> x4))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x3 Xx)) (x3 Xy)))=> (((or_intror ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x4:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x4) x00)) ((R Xy) Xx)) (fun (x4:(x3 Xx)) (x5:(x3 Xy))=> x4)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found or_introl00:=(or_introl0 ((R Xx) Xy)):((x3 Xy)->((or (x3 Xy)) ((R Xx) Xy)))
% Found (or_introl0 ((R Xx) Xy)) as proof of ((x3 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found ((or_introl (x3 Xy)) ((R Xx) Xy)) as proof of ((x3 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found ((or_introl (x3 Xy)) ((R Xx) Xy)) as proof of ((x3 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found (fun (x4:(x3 Xx))=> ((or_introl (x3 Xy)) ((R Xx) Xy))) as proof of ((x3 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found (fun (x4:(x3 Xx))=> ((or_introl (x3 Xy)) ((R Xx) Xy))) as proof of ((x3 Xx)->((x3 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy))))
% Found (and_rect10 (fun (x4:(x3 Xx))=> ((or_introl (x3 Xy)) ((R Xx) Xy)))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found ((and_rect1 ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x4:(x3 Xx))=> ((or_introl (x3 Xy)) ((R Xx) Xy)))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found (((fun (P:Type) (x4:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x4) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x4:(x3 Xx))=> ((or_introl (x3 Xy)) ((R Xx) Xy)))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found (((fun (P:Type) (x4:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x4) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x4:(x3 Xx))=> ((or_introl (x3 Xy)) ((R Xx) Xy)))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found (or_comm_i00 (((fun (P:Type) (x4:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x4) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x4:(x3 Xx))=> ((or_introl (x3 Xy)) ((R Xx) Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((or_comm_i0 ((R Xx) Xy)) (((fun (P:Type) (x4:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x4) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x4:(x3 Xx))=> ((or_introl (x3 Xy)) ((R Xx) Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((or_comm_i ((R Xy) Xx)) ((R Xx) Xy)) (((fun (P:Type) (x4:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x4) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x4:(x3 Xx))=> ((or_introl (x3 Xy)) ((R Xx) Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x3 Xx)) (x3 Xy)))=> (((or_comm_i ((R Xy) Xx)) ((R Xx) Xy)) (((fun (P:Type) (x4:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x4) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x4:(x3 Xx))=> ((or_introl (x3 Xy)) ((R Xx) Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x3 Xx)) (x3 Xy)))=> (((or_comm_i ((R Xy) Xx)) ((R Xx) Xy)) (((fun (P:Type) (x4:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x4) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x4:(x3 Xx))=> ((or_introl (x3 Xy)) ((R Xx) Xy)))))) as proof of (((and (x3 Xx)) (x3 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x1:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x1 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found eq_ref000:=(eq_ref00 x1):((x1 Xy)->(x1 Xy))
% Found (eq_ref00 x1) as proof of ((x1 Xy)->((R Xx) Xy))
% Found ((eq_ref0 Xy) x1) as proof of ((x1 Xy)->((R Xx) Xy))
% Found (((eq_ref a) Xy) x1) as proof of ((x1 Xy)->((R Xx) Xy))
% Found (((eq_ref a) Xy) x1) as proof of ((x1 Xy)->((R Xx) Xy))
% Found (fun (x4:(x1 Xx))=> (((eq_ref a) Xy) x1)) as proof of ((x1 Xy)->((R Xx) Xy))
% Found (fun (x4:(x1 Xx))=> (((eq_ref a) Xy) x1)) as proof of ((x1 Xx)->((x1 Xy)->((R Xx) Xy)))
% Found (and_rect10 (fun (x4:(x1 Xx))=> (((eq_ref a) Xy) x1))) as proof of ((R Xx) Xy)
% Found ((and_rect1 ((R Xx) Xy)) (fun (x4:(x1 Xx))=> (((eq_ref a) Xy) x1))) as proof of ((R Xx) Xy)
% Found (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((R Xx) Xy)) (fun (x4:(x1 Xx))=> (((eq_ref a) Xy) x1))) as proof of ((R Xx) Xy)
% Found (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((R Xx) Xy)) (fun (x4:(x1 Xx))=> (((eq_ref a) Xy) x1))) as proof of ((R Xx) Xy)
% Found (or_introl00 (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((R Xx) Xy)) (fun (x4:(x1 Xx))=> (((eq_ref a) Xy) x1)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((or_introl0 ((R Xy) Xx)) (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((R Xx) Xy)) (fun (x4:(x1 Xx))=> (((eq_ref a) Xy) x1)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((or_introl ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((R Xx) Xy)) (fun (x4:(x1 Xx))=> (((eq_ref a) Xy) x1)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x1 Xx)) (x1 Xy)))=> (((or_introl ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((R Xx) Xy)) (fun (x4:(x1 Xx))=> (((eq_ref a) Xy) x1))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x1 Xx)) (x1 Xy)))=> (((or_introl ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((R Xx) Xy)) (fun (x4:(x1 Xx))=> (((eq_ref a) Xy) x1))))) as proof of (((and (x1 Xx)) (x1 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found x4:(x1 Xx)
% Instantiate: x1:=(R Xy):(a->Prop)
% Found (fun (x5:(x1 Xy))=> x4) as proof of ((R Xy) Xx)
% Found (fun (x4:(x1 Xx)) (x5:(x1 Xy))=> x4) as proof of ((x1 Xy)->((R Xy) Xx))
% Found (fun (x4:(x1 Xx)) (x5:(x1 Xy))=> x4) as proof of ((x1 Xx)->((x1 Xy)->((R Xy) Xx)))
% Found (and_rect10 (fun (x4:(x1 Xx)) (x5:(x1 Xy))=> x4)) as proof of ((R Xy) Xx)
% Found ((and_rect1 ((R Xy) Xx)) (fun (x4:(x1 Xx)) (x5:(x1 Xy))=> x4)) as proof of ((R Xy) Xx)
% Found (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((R Xy) Xx)) (fun (x4:(x1 Xx)) (x5:(x1 Xy))=> x4)) as proof of ((R Xy) Xx)
% Found (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((R Xy) Xx)) (fun (x4:(x1 Xx)) (x5:(x1 Xy))=> x4)) as proof of ((R Xy) Xx)
% Found (or_intror00 (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((R Xy) Xx)) (fun (x4:(x1 Xx)) (x5:(x1 Xy))=> x4))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((or_intror0 ((R Xy) Xx)) (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((R Xy) Xx)) (fun (x4:(x1 Xx)) (x5:(x1 Xy))=> x4))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((or_intror ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((R Xy) Xx)) (fun (x4:(x1 Xx)) (x5:(x1 Xy))=> x4))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x1 Xx)) (x1 Xy)))=> (((or_intror ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((R Xy) Xx)) (fun (x4:(x1 Xx)) (x5:(x1 Xy))=> x4)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found eq_ref000:=(eq_ref00 (fun (x5:Prop)=> (x1 Xy))):((x1 Xy)->(x1 Xy))
% Found (eq_ref00 (fun (x5:Prop)=> (x1 Xy))) as proof of ((x1 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found ((eq_ref0 ((R Xx) Xy)) (fun (x5:Prop)=> (x1 Xy))) as proof of ((x1 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found (((eq_ref Prop) ((R Xx) Xy)) (fun (x5:Prop)=> (x1 Xy))) as proof of ((x1 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found (((eq_ref Prop) ((R Xx) Xy)) (fun (x5:Prop)=> (x1 Xy))) as proof of ((x1 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found (fun (x4:(x1 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x5:Prop)=> (x1 Xy)))) as proof of ((x1 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found (fun (x4:(x1 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x5:Prop)=> (x1 Xy)))) as proof of ((x1 Xx)->((x1 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy))))
% Found (and_rect10 (fun (x4:(x1 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x5:Prop)=> (x1 Xy))))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found ((and_rect1 ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x4:(x1 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x5:Prop)=> (x1 Xy))))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x4:(x1 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x5:Prop)=> (x1 Xy))))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x4:(x1 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x5:Prop)=> (x1 Xy))))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found (or_comm_i00 (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x4:(x1 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x5:Prop)=> (x1 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((or_comm_i0 ((R Xx) Xy)) (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x4:(x1 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x5:Prop)=> (x1 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((or_comm_i ((R Xy) Xx)) ((R Xx) Xy)) (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x4:(x1 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x5:Prop)=> (x1 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x1 Xx)) (x1 Xy)))=> (((or_comm_i ((R Xy) Xx)) ((R Xx) Xy)) (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x4:(x1 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x5:Prop)=> (x1 Xy))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x1 Xx)) (x1 Xy)))=> (((or_comm_i ((R Xy) Xx)) ((R Xx) Xy)) (((fun (P:Type) (x4:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x4) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x4:(x1 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x5:Prop)=> (x1 Xy))))))) as proof of (((and (x1 Xx)) (x1 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found and_rect10:=(and_rect1 ((or ((R Xx) Xy)) ((R Xy) Xx))):(((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))->((forall (Xx0:a), ((R Xx0) Xx0))->((or ((R Xx) Xy)) ((R Xy) Xx))))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Instantiate: x1:=(fun (x6:a)=> ((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))->((forall (Xx0:a), ((R Xx0) Xx0))->((or ((R Xx) x6)) ((R x6) Xx))))):(a->Prop)
% Found (fun (x6:(x1 Xx))=> and_rect10) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x6:(x1 Xx))=> and_rect10) as proof of ((x1 Xx)->((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect20 (fun (x6:(x1 Xx))=> and_rect10)) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect2 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x1 Xx))=> and_rect10)) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x1 Xx))=> and_rect10)) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x5:(forall (Xx0:a), ((R Xx0) Xx0)))=> (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x1 Xx))=> and_rect10))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x4:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (x5:(forall (Xx0:a), ((R Xx0) Xx0)))=> (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x1 Xx))=> and_rect10))) as proof of ((forall (Xx0:a), ((R Xx0) Xx0))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x4:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (x5:(forall (Xx0:a), ((R Xx0) Xx0)))=> (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x1 Xx))=> and_rect10))) as proof of ((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))->((forall (Xx0:a), ((R Xx0) Xx0))->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect10 (fun (x4:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (x5:(forall (Xx0:a), ((R Xx0) Xx0)))=> (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x1 Xx))=> and_rect10)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect1 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x4:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (x5:(forall (Xx0:a), ((R Xx0) Xx0)))=> (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x1 Xx))=> (and_rect1 ((or ((R Xx) Xy)) ((R Xy) Xx))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x4:((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->((forall (Xx:a), ((R Xx) Xx))->P)))=> (((((and_rect (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx))) P) x4) x2)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x4:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (x5:(forall (Xx0:a), ((R Xx0) Xx0)))=> (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x1 Xx))=> ((fun (P:Type) (x4:((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->((forall (Xx:a), ((R Xx) Xx))->P)))=> (((((and_rect (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx))) P) x4) x2)) ((or ((R Xx) Xy)) ((R Xy) Xx))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x1 Xx)) (x1 Xy)))=> (((fun (P:Type) (x4:((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->((forall (Xx:a), ((R Xx) Xx))->P)))=> (((((and_rect (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx))) P) x4) x2)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x4:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (x5:(forall (Xx0:a), ((R Xx0) Xx0)))=> (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x1 Xx))=> ((fun (P:Type) (x4:((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->((forall (Xx:a), ((R Xx) Xx))->P)))=> (((((and_rect (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx))) P) x4) x2)) ((or ((R Xx) Xy)) ((R Xy) Xx)))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x1 Xx)) (x1 Xy)))=> (((fun (P:Type) (x4:((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->((forall (Xx:a), ((R Xx) Xx))->P)))=> (((((and_rect (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx))) P) x4) x2)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x4:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (x5:(forall (Xx0:a), ((R Xx0) Xx0)))=> (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x1 Xx))=> ((fun (P:Type) (x4:((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->((forall (Xx:a), ((R Xx) Xx))->P)))=> (((((and_rect (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx))) P) x4) x2)) ((or ((R Xx) Xy)) ((R Xy) Xx)))))))) as proof of (((and (x1 Xx)) (x1 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found and_rect10:=(and_rect1 ((or ((R Xx) Xy)) ((R Xy) Xx))):(((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))->((forall (Xx0:a), ((R Xx0) Xx0))->((or ((R Xx) Xy)) ((R Xy) Xx))))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Instantiate: x3:=(fun (x6:a)=> ((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))->((forall (Xx0:a), ((R Xx0) Xx0))->((or ((R Xx) x6)) ((R x6) Xx))))):(a->Prop)
% Found (fun (x6:(x3 Xx))=> and_rect10) as proof of ((x3 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x6:(x3 Xx))=> and_rect10) as proof of ((x3 Xx)->((x3 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect20 (fun (x6:(x3 Xx))=> and_rect10)) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect2 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x3 Xx))=> and_rect10)) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x3 Xx))=> and_rect10)) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x5:(forall (Xx0:a), ((R Xx0) Xx0)))=> (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x3 Xx))=> and_rect10))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x4:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (x5:(forall (Xx0:a), ((R Xx0) Xx0)))=> (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x3 Xx))=> and_rect10))) as proof of ((forall (Xx0:a), ((R Xx0) Xx0))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x4:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (x5:(forall (Xx0:a), ((R Xx0) Xx0)))=> (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x3 Xx))=> and_rect10))) as proof of ((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))->((forall (Xx0:a), ((R Xx0) Xx0))->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect10 (fun (x4:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (x5:(forall (Xx0:a), ((R Xx0) Xx0)))=> (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x3 Xx))=> and_rect10)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect1 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x4:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (x5:(forall (Xx0:a), ((R Xx0) Xx0)))=> (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x3 Xx))=> (and_rect1 ((or ((R Xx) Xy)) ((R Xy) Xx))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x4:((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->((forall (Xx:a), ((R Xx) Xx))->P)))=> (((((and_rect (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx))) P) x4) x1)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x4:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (x5:(forall (Xx0:a), ((R Xx0) Xx0)))=> (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x3 Xx))=> ((fun (P:Type) (x4:((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->((forall (Xx:a), ((R Xx) Xx))->P)))=> (((((and_rect (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx))) P) x4) x1)) ((or ((R Xx) Xy)) ((R Xy) Xx))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x3 Xx)) (x3 Xy)))=> (((fun (P:Type) (x4:((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->((forall (Xx:a), ((R Xx) Xx))->P)))=> (((((and_rect (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx))) P) x4) x1)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x4:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (x5:(forall (Xx0:a), ((R Xx0) Xx0)))=> (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x3 Xx))=> ((fun (P:Type) (x4:((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->((forall (Xx:a), ((R Xx) Xx))->P)))=> (((((and_rect (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx))) P) x4) x1)) ((or ((R Xx) Xy)) ((R Xy) Xx)))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x3 Xx)) (x3 Xy)))=> (((fun (P:Type) (x4:((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->((forall (Xx:a), ((R Xx) Xx))->P)))=> (((((and_rect (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx))) P) x4) x1)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x4:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (x5:(forall (Xx0:a), ((R Xx0) Xx0)))=> (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x6:(x3 Xx))=> ((fun (P:Type) (x4:((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->((forall (Xx:a), ((R Xx) Xx))->P)))=> (((((and_rect (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx))) P) x4) x1)) ((or ((R Xx) Xy)) ((R Xy) Xx)))))))) as proof of (((and (x3 Xx)) (x3 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found x01:(Xy Xx)
% Instantiate: x6:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x6 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x6 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x6 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x6 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x6 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x1:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x1 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x4:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x4 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x4 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x4 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x4 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x4 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x4 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x4 Xx)))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx)
% Found (eq_sym0000 ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x1 Xx))
% Found (eq_sym0000 ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x1 Xx))
% Found ((fun (x2:(((eq a) Xx) Xx))=> ((eq_sym000 x2) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x1 Xx))
% Found ((fun (x2:(((eq a) Xx) Xx))=> (((eq_sym00 Xx) x2) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x1 Xx))
% Found ((fun (x2:(((eq a) Xx) Xx))=> ((((eq_sym0 Xx) Xx) x2) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x1 Xx))
% Found ((fun (x2:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x2) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x1 Xx))
% Found ((fun (x2:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x2) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (Xx:a)=> ((fun (x2:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x2) Xy)) ((eq_ref a) Xx))) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a)=> ((fun (x2:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x2) Xy)) ((eq_ref a) Xx))) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x6 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x6 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x6:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x6 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x6 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x6 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x6:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x6 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x6 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x6 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x6 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x6 Xx)))
% Found eq_ref000:=(eq_ref00 (fun (x7:a)=> (x5 Xy))):((x5 Xy)->(x5 Xy))
% Found (eq_ref00 (fun (x7:a)=> (x5 Xy))) as proof of ((x5 Xy)->((R Xy) Xx))
% Found ((eq_ref0 Xx) (fun (x7:a)=> (x5 Xy))) as proof of ((x5 Xy)->((R Xy) Xx))
% Found (((eq_ref a) Xx) (fun (x7:a)=> (x5 Xy))) as proof of ((x5 Xy)->((R Xy) Xx))
% Found (((eq_ref a) Xx) (fun (x7:a)=> (x5 Xy))) as proof of ((x5 Xy)->((R Xy) Xx))
% Found (fun (x6:(x5 Xx))=> (((eq_ref a) Xx) (fun (x7:a)=> (x5 Xy)))) as proof of ((x5 Xy)->((R Xy) Xx))
% Found (fun (x6:(x5 Xx))=> (((eq_ref a) Xx) (fun (x7:a)=> (x5 Xy)))) as proof of ((x5 Xx)->((x5 Xy)->((R Xy) Xx)))
% Found (and_rect20 (fun (x6:(x5 Xx))=> (((eq_ref a) Xx) (fun (x7:a)=> (x5 Xy))))) as proof of ((R Xy) Xx)
% Found ((and_rect2 ((R Xy) Xx)) (fun (x6:(x5 Xx))=> (((eq_ref a) Xx) (fun (x7:a)=> (x5 Xy))))) as proof of ((R Xy) Xx)
% Found (((fun (P:Type) (x6:((x5 Xx)->((x5 Xy)->P)))=> (((((and_rect (x5 Xx)) (x5 Xy)) P) x6) x00)) ((R Xy) Xx)) (fun (x6:(x5 Xx))=> (((eq_ref a) Xx) (fun (x7:a)=> (x5 Xy))))) as proof of ((R Xy) Xx)
% Found (((fun (P:Type) (x6:((x5 Xx)->((x5 Xy)->P)))=> (((((and_rect (x5 Xx)) (x5 Xy)) P) x6) x00)) ((R Xy) Xx)) (fun (x6:(x5 Xx))=> (((eq_ref a) Xx) (fun (x7:a)=> (x5 Xy))))) as proof of ((R Xy) Xx)
% Found (or_intror00 (((fun (P:Type) (x6:((x5 Xx)->((x5 Xy)->P)))=> (((((and_rect (x5 Xx)) (x5 Xy)) P) x6) x00)) ((R Xy) Xx)) (fun (x6:(x5 Xx))=> (((eq_ref a) Xx) (fun (x7:a)=> (x5 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((or_intror0 ((R Xy) Xx)) (((fun (P:Type) (x6:((x5 Xx)->((x5 Xy)->P)))=> (((((and_rect (x5 Xx)) (x5 Xy)) P) x6) x00)) ((R Xy) Xx)) (fun (x6:(x5 Xx))=> (((eq_ref a) Xx) (fun (x7:a)=> (x5 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((or_intror ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x6:((x5 Xx)->((x5 Xy)->P)))=> (((((and_rect (x5 Xx)) (x5 Xy)) P) x6) x00)) ((R Xy) Xx)) (fun (x6:(x5 Xx))=> (((eq_ref a) Xx) (fun (x7:a)=> (x5 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x5 Xx)) (x5 Xy)))=> (((or_intror ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x6:((x5 Xx)->((x5 Xy)->P)))=> (((((and_rect (x5 Xx)) (x5 Xy)) P) x6) x00)) ((R Xy) Xx)) (fun (x6:(x5 Xx))=> (((eq_ref a) Xx) (fun (x7:a)=> (x5 Xy))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x5 Xx)) (x5 Xy)))=> (((or_intror ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x6:((x5 Xx)->((x5 Xy)->P)))=> (((((and_rect (x5 Xx)) (x5 Xy)) P) x6) x00)) ((R Xy) Xx)) (fun (x6:(x5 Xx))=> (((eq_ref a) Xx) (fun (x7:a)=> (x5 Xy))))))) as proof of (((and (x5 Xx)) (x5 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found eq_ref000:=(eq_ref00 x5):((x5 Xy)->(x5 Xy))
% Found (eq_ref00 x5) as proof of ((x5 Xy)->((R Xx) Xy))
% Found ((eq_ref0 Xy) x5) as proof of ((x5 Xy)->((R Xx) Xy))
% Found (((eq_ref a) Xy) x5) as proof of ((x5 Xy)->((R Xx) Xy))
% Found (((eq_ref a) Xy) x5) as proof of ((x5 Xy)->((R Xx) Xy))
% Found (fun (x6:(x5 Xx))=> (((eq_ref a) Xy) x5)) as proof of ((x5 Xy)->((R Xx) Xy))
% Found (fun (x6:(x5 Xx))=> (((eq_ref a) Xy) x5)) as proof of ((x5 Xx)->((x5 Xy)->((R Xx) Xy)))
% Found (and_rect20 (fun (x6:(x5 Xx))=> (((eq_ref a) Xy) x5))) as proof of ((R Xx) Xy)
% Found ((and_rect2 ((R Xx) Xy)) (fun (x6:(x5 Xx))=> (((eq_ref a) Xy) x5))) as proof of ((R Xx) Xy)
% Found (((fun (P:Type) (x6:((x5 Xx)->((x5 Xy)->P)))=> (((((and_rect (x5 Xx)) (x5 Xy)) P) x6) x00)) ((R Xx) Xy)) (fun (x6:(x5 Xx))=> (((eq_ref a) Xy) x5))) as proof of ((R Xx) Xy)
% Found (((fun (P:Type) (x6:((x5 Xx)->((x5 Xy)->P)))=> (((((and_rect (x5 Xx)) (x5 Xy)) P) x6) x00)) ((R Xx) Xy)) (fun (x6:(x5 Xx))=> (((eq_ref a) Xy) x5))) as proof of ((R Xx) Xy)
% Found (or_introl00 (((fun (P:Type) (x6:((x5 Xx)->((x5 Xy)->P)))=> (((((and_rect (x5 Xx)) (x5 Xy)) P) x6) x00)) ((R Xx) Xy)) (fun (x6:(x5 Xx))=> (((eq_ref a) Xy) x5)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((or_introl0 ((R Xy) Xx)) (((fun (P:Type) (x6:((x5 Xx)->((x5 Xy)->P)))=> (((((and_rect (x5 Xx)) (x5 Xy)) P) x6) x00)) ((R Xx) Xy)) (fun (x6:(x5 Xx))=> (((eq_ref a) Xy) x5)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((or_introl ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x6:((x5 Xx)->((x5 Xy)->P)))=> (((((and_rect (x5 Xx)) (x5 Xy)) P) x6) x00)) ((R Xx) Xy)) (fun (x6:(x5 Xx))=> (((eq_ref a) Xy) x5)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x5 Xx)) (x5 Xy)))=> (((or_introl ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x6:((x5 Xx)->((x5 Xy)->P)))=> (((((and_rect (x5 Xx)) (x5 Xy)) P) x6) x00)) ((R Xx) Xy)) (fun (x6:(x5 Xx))=> (((eq_ref a) Xy) x5))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x5 Xx)) (x5 Xy)))=> (((or_introl ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x6:((x5 Xx)->((x5 Xy)->P)))=> (((((and_rect (x5 Xx)) (x5 Xy)) P) x6) x00)) ((R Xx) Xy)) (fun (x6:(x5 Xx))=> (((eq_ref a) Xy) x5))))) as proof of (((and (x5 Xx)) (x5 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found eq_ref000:=(eq_ref00 (fun (x7:Prop)=> (x5 Xy))):((x5 Xy)->(x5 Xy))
% Found (eq_ref00 (fun (x7:Prop)=> (x5 Xy))) as proof of ((x5 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found ((eq_ref0 ((R Xx) Xy)) (fun (x7:Prop)=> (x5 Xy))) as proof of ((x5 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found (((eq_ref Prop) ((R Xx) Xy)) (fun (x7:Prop)=> (x5 Xy))) as proof of ((x5 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found (((eq_ref Prop) ((R Xx) Xy)) (fun (x7:Prop)=> (x5 Xy))) as proof of ((x5 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found (fun (x6:(x5 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x7:Prop)=> (x5 Xy)))) as proof of ((x5 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found (fun (x6:(x5 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x7:Prop)=> (x5 Xy)))) as proof of ((x5 Xx)->((x5 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy))))
% Found (and_rect20 (fun (x6:(x5 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x7:Prop)=> (x5 Xy))))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found ((and_rect2 ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x6:(x5 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x7:Prop)=> (x5 Xy))))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found (((fun (P:Type) (x6:((x5 Xx)->((x5 Xy)->P)))=> (((((and_rect (x5 Xx)) (x5 Xy)) P) x6) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x6:(x5 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x7:Prop)=> (x5 Xy))))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found (((fun (P:Type) (x6:((x5 Xx)->((x5 Xy)->P)))=> (((((and_rect (x5 Xx)) (x5 Xy)) P) x6) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x6:(x5 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x7:Prop)=> (x5 Xy))))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found (or_comm_i00 (((fun (P:Type) (x6:((x5 Xx)->((x5 Xy)->P)))=> (((((and_rect (x5 Xx)) (x5 Xy)) P) x6) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x6:(x5 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x7:Prop)=> (x5 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((or_comm_i0 ((R Xx) Xy)) (((fun (P:Type) (x6:((x5 Xx)->((x5 Xy)->P)))=> (((((and_rect (x5 Xx)) (x5 Xy)) P) x6) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x6:(x5 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x7:Prop)=> (x5 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((or_comm_i ((R Xy) Xx)) ((R Xx) Xy)) (((fun (P:Type) (x6:((x5 Xx)->((x5 Xy)->P)))=> (((((and_rect (x5 Xx)) (x5 Xy)) P) x6) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x6:(x5 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x7:Prop)=> (x5 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x5 Xx)) (x5 Xy)))=> (((or_comm_i ((R Xy) Xx)) ((R Xx) Xy)) (((fun (P:Type) (x6:((x5 Xx)->((x5 Xy)->P)))=> (((((and_rect (x5 Xx)) (x5 Xy)) P) x6) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x6:(x5 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x7:Prop)=> (x5 Xy))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x5 Xx)) (x5 Xy)))=> (((or_comm_i ((R Xy) Xx)) ((R Xx) Xy)) (((fun (P:Type) (x6:((x5 Xx)->((x5 Xy)->P)))=> (((((and_rect (x5 Xx)) (x5 Xy)) P) x6) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x6:(x5 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x7:Prop)=> (x5 Xy))))))) as proof of (((and (x5 Xx)) (x5 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x3 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x3 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x3 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x3:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x3 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x3 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x3 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x3 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x4:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x4 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x4 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x4 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x4 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x4 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x4 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x4 Xx)))
% Found or_intror00:=(or_intror0 (x4 Xy)):((x4 Xy)->((or ((R Xx) Xy)) (x4 Xy)))
% Found (or_intror0 (x4 Xy)) as proof of ((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_intror ((R Xx) Xy)) (x4 Xy)) as proof of ((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_intror ((R Xx) Xy)) (x4 Xy)) as proof of ((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x5:(x4 Xx))=> ((or_intror ((R Xx) Xy)) (x4 Xy))) as proof of ((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x5:(x4 Xx))=> ((or_intror ((R Xx) Xy)) (x4 Xy))) as proof of ((x4 Xx)->((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect00 (fun (x5:(x4 Xx))=> ((or_intror ((R Xx) Xy)) (x4 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect0 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:(x4 Xx))=> ((or_intror ((R Xx) Xy)) (x4 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:(x4 Xx))=> ((or_intror ((R Xx) Xy)) (x4 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x4 Xx)) (x4 Xy)))=> (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:(x4 Xx))=> ((or_intror ((R Xx) Xy)) (x4 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x4 Xx)) (x4 Xy)))=> (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:(x4 Xx))=> ((or_intror ((R Xx) Xy)) (x4 Xy))))) as proof of (((and (x4 Xx)) (x4 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found or_intror00:=(or_intror0 (x1 Xy)):((x1 Xy)->((or ((R Xy) Xx)) (x1 Xy)))
% Found (or_intror0 (x1 Xy)) as proof of ((x1 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found ((or_intror ((R Xy) Xx)) (x1 Xy)) as proof of ((x1 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found ((or_intror ((R Xy) Xx)) (x1 Xy)) as proof of ((x1 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found (fun (x6:(x1 Xx))=> ((or_intror ((R Xy) Xx)) (x1 Xy))) as proof of ((x1 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found (fun (x6:(x1 Xx))=> ((or_intror ((R Xy) Xx)) (x1 Xy))) as proof of ((x1 Xx)->((x1 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy))))
% Found (and_rect20 (fun (x6:(x1 Xx))=> ((or_intror ((R Xy) Xx)) (x1 Xy)))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found ((and_rect2 ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x6:(x1 Xx))=> ((or_intror ((R Xy) Xx)) (x1 Xy)))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x6:(x1 Xx))=> ((or_intror ((R Xy) Xx)) (x1 Xy)))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x6:(x1 Xx))=> ((or_intror ((R Xy) Xx)) (x1 Xy)))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found (or_comm_i00 (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x6:(x1 Xx))=> ((or_intror ((R Xy) Xx)) (x1 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((or_comm_i0 ((R Xx) Xy)) (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x6:(x1 Xx))=> ((or_intror ((R Xy) Xx)) (x1 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((or_comm_i ((R Xy) Xx)) ((R Xx) Xy)) (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x6:(x1 Xx))=> ((or_intror ((R Xy) Xx)) (x1 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x1 Xx)) (x1 Xy)))=> (((or_comm_i ((R Xy) Xx)) ((R Xx) Xy)) (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x6:(x1 Xx))=> ((or_intror ((R Xy) Xx)) (x1 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x1 Xx)) (x1 Xy)))=> (((or_comm_i ((R Xy) Xx)) ((R Xx) Xy)) (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x6:(x1 Xx))=> ((or_intror ((R Xy) Xx)) (x1 Xy)))))) as proof of (((and (x1 Xx)) (x1 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found eq_ref000:=(eq_ref00 x1):((x1 Xy)->(x1 Xy))
% Found (eq_ref00 x1) as proof of ((x1 Xy)->((R Xx) Xy))
% Found ((eq_ref0 Xy) x1) as proof of ((x1 Xy)->((R Xx) Xy))
% Found (((eq_ref a) Xy) x1) as proof of ((x1 Xy)->((R Xx) Xy))
% Found (((eq_ref a) Xy) x1) as proof of ((x1 Xy)->((R Xx) Xy))
% Found (fun (x6:(x1 Xx))=> (((eq_ref a) Xy) x1)) as proof of ((x1 Xy)->((R Xx) Xy))
% Found (fun (x6:(x1 Xx))=> (((eq_ref a) Xy) x1)) as proof of ((x1 Xx)->((x1 Xy)->((R Xx) Xy)))
% Found (and_rect20 (fun (x6:(x1 Xx))=> (((eq_ref a) Xy) x1))) as proof of ((R Xx) Xy)
% Found ((and_rect2 ((R Xx) Xy)) (fun (x6:(x1 Xx))=> (((eq_ref a) Xy) x1))) as proof of ((R Xx) Xy)
% Found (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((R Xx) Xy)) (fun (x6:(x1 Xx))=> (((eq_ref a) Xy) x1))) as proof of ((R Xx) Xy)
% Found (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((R Xx) Xy)) (fun (x6:(x1 Xx))=> (((eq_ref a) Xy) x1))) as proof of ((R Xx) Xy)
% Found (or_introl00 (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((R Xx) Xy)) (fun (x6:(x1 Xx))=> (((eq_ref a) Xy) x1)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((or_introl0 ((R Xy) Xx)) (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((R Xx) Xy)) (fun (x6:(x1 Xx))=> (((eq_ref a) Xy) x1)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((or_introl ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((R Xx) Xy)) (fun (x6:(x1 Xx))=> (((eq_ref a) Xy) x1)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x1 Xx)) (x1 Xy)))=> (((or_introl ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((R Xx) Xy)) (fun (x6:(x1 Xx))=> (((eq_ref a) Xy) x1))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x1 Xx)) (x1 Xy)))=> (((or_introl ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((R Xx) Xy)) (fun (x6:(x1 Xx))=> (((eq_ref a) Xy) x1))))) as proof of (((and (x1 Xx)) (x1 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found x6:(x1 Xx)
% Instantiate: x1:=(R Xy):(a->Prop)
% Found (fun (x7:(x1 Xy))=> x6) as proof of ((R Xy) Xx)
% Found (fun (x6:(x1 Xx)) (x7:(x1 Xy))=> x6) as proof of ((x1 Xy)->((R Xy) Xx))
% Found (fun (x6:(x1 Xx)) (x7:(x1 Xy))=> x6) as proof of ((x1 Xx)->((x1 Xy)->((R Xy) Xx)))
% Found (and_rect20 (fun (x6:(x1 Xx)) (x7:(x1 Xy))=> x6)) as proof of ((R Xy) Xx)
% Found ((and_rect2 ((R Xy) Xx)) (fun (x6:(x1 Xx)) (x7:(x1 Xy))=> x6)) as proof of ((R Xy) Xx)
% Found (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((R Xy) Xx)) (fun (x6:(x1 Xx)) (x7:(x1 Xy))=> x6)) as proof of ((R Xy) Xx)
% Found (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((R Xy) Xx)) (fun (x6:(x1 Xx)) (x7:(x1 Xy))=> x6)) as proof of ((R Xy) Xx)
% Found (or_intror00 (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((R Xy) Xx)) (fun (x6:(x1 Xx)) (x7:(x1 Xy))=> x6))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((or_intror0 ((R Xy) Xx)) (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((R Xy) Xx)) (fun (x6:(x1 Xx)) (x7:(x1 Xy))=> x6))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((or_intror ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((R Xy) Xx)) (fun (x6:(x1 Xx)) (x7:(x1 Xy))=> x6))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x1 Xx)) (x1 Xy)))=> (((or_intror ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x6:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x6) x00)) ((R Xy) Xx)) (fun (x6:(x1 Xx)) (x7:(x1 Xy))=> x6)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found or_introl00:=(or_introl0 ((R Xx) Xy)):((x3 Xy)->((or (x3 Xy)) ((R Xx) Xy)))
% Found (or_introl0 ((R Xx) Xy)) as proof of ((x3 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found ((or_introl (x3 Xy)) ((R Xx) Xy)) as proof of ((x3 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found ((or_introl (x3 Xy)) ((R Xx) Xy)) as proof of ((x3 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found (fun (x6:(x3 Xx))=> ((or_introl (x3 Xy)) ((R Xx) Xy))) as proof of ((x3 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found (fun (x6:(x3 Xx))=> ((or_introl (x3 Xy)) ((R Xx) Xy))) as proof of ((x3 Xx)->((x3 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy))))
% Found (and_rect20 (fun (x6:(x3 Xx))=> ((or_introl (x3 Xy)) ((R Xx) Xy)))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found ((and_rect2 ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x6:(x3 Xx))=> ((or_introl (x3 Xy)) ((R Xx) Xy)))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x6:(x3 Xx))=> ((or_introl (x3 Xy)) ((R Xx) Xy)))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x6:(x3 Xx))=> ((or_introl (x3 Xy)) ((R Xx) Xy)))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found (or_comm_i00 (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x6:(x3 Xx))=> ((or_introl (x3 Xy)) ((R Xx) Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((or_comm_i0 ((R Xx) Xy)) (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x6:(x3 Xx))=> ((or_introl (x3 Xy)) ((R Xx) Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((or_comm_i ((R Xy) Xx)) ((R Xx) Xy)) (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x6:(x3 Xx))=> ((or_introl (x3 Xy)) ((R Xx) Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x3 Xx)) (x3 Xy)))=> (((or_comm_i ((R Xy) Xx)) ((R Xx) Xy)) (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x6:(x3 Xx))=> ((or_introl (x3 Xy)) ((R Xx) Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x3 Xx)) (x3 Xy)))=> (((or_comm_i ((R Xy) Xx)) ((R Xx) Xy)) (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x6:(x3 Xx))=> ((or_introl (x3 Xy)) ((R Xx) Xy)))))) as proof of (((and (x3 Xx)) (x3 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found x7:(x3 Xy)
% Instantiate: x3:=(R Xx):(a->Prop)
% Found (fun (x7:(x3 Xy))=> x7) as proof of ((R Xx) Xy)
% Found (fun (x6:(x3 Xx)) (x7:(x3 Xy))=> x7) as proof of ((x3 Xy)->((R Xx) Xy))
% Found (fun (x6:(x3 Xx)) (x7:(x3 Xy))=> x7) as proof of ((x3 Xx)->((x3 Xy)->((R Xx) Xy)))
% Found (and_rect20 (fun (x6:(x3 Xx)) (x7:(x3 Xy))=> x7)) as proof of ((R Xx) Xy)
% Found ((and_rect2 ((R Xx) Xy)) (fun (x6:(x3 Xx)) (x7:(x3 Xy))=> x7)) as proof of ((R Xx) Xy)
% Found (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((R Xx) Xy)) (fun (x6:(x3 Xx)) (x7:(x3 Xy))=> x7)) as proof of ((R Xx) Xy)
% Found (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((R Xx) Xy)) (fun (x6:(x3 Xx)) (x7:(x3 Xy))=> x7)) as proof of ((R Xx) Xy)
% Found (or_introl00 (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((R Xx) Xy)) (fun (x6:(x3 Xx)) (x7:(x3 Xy))=> x7))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((or_introl0 ((R Xy) Xx)) (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((R Xx) Xy)) (fun (x6:(x3 Xx)) (x7:(x3 Xy))=> x7))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((or_introl ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((R Xx) Xy)) (fun (x6:(x3 Xx)) (x7:(x3 Xy))=> x7))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x3 Xx)) (x3 Xy)))=> (((or_introl ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((R Xx) Xy)) (fun (x6:(x3 Xx)) (x7:(x3 Xy))=> x7)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x3 Xx)) (x3 Xy)))=> (((or_introl ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((R Xx) Xy)) (fun (x6:(x3 Xx)) (x7:(x3 Xy))=> x7)))) as proof of (((and (x3 Xx)) (x3 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found eq_ref000:=(eq_ref00 (fun (x7:a)=> (x3 Xy))):((x3 Xy)->(x3 Xy))
% Found (eq_ref00 (fun (x7:a)=> (x3 Xy))) as proof of ((x3 Xy)->((R Xy) Xx))
% Found ((eq_ref0 Xx) (fun (x7:a)=> (x3 Xy))) as proof of ((x3 Xy)->((R Xy) Xx))
% Found (((eq_ref a) Xx) (fun (x7:a)=> (x3 Xy))) as proof of ((x3 Xy)->((R Xy) Xx))
% Found (((eq_ref a) Xx) (fun (x7:a)=> (x3 Xy))) as proof of ((x3 Xy)->((R Xy) Xx))
% Found (fun (x6:(x3 Xx))=> (((eq_ref a) Xx) (fun (x7:a)=> (x3 Xy)))) as proof of ((x3 Xy)->((R Xy) Xx))
% Found (fun (x6:(x3 Xx))=> (((eq_ref a) Xx) (fun (x7:a)=> (x3 Xy)))) as proof of ((x3 Xx)->((x3 Xy)->((R Xy) Xx)))
% Found (and_rect20 (fun (x6:(x3 Xx))=> (((eq_ref a) Xx) (fun (x7:a)=> (x3 Xy))))) as proof of ((R Xy) Xx)
% Found ((and_rect2 ((R Xy) Xx)) (fun (x6:(x3 Xx))=> (((eq_ref a) Xx) (fun (x7:a)=> (x3 Xy))))) as proof of ((R Xy) Xx)
% Found (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((R Xy) Xx)) (fun (x6:(x3 Xx))=> (((eq_ref a) Xx) (fun (x7:a)=> (x3 Xy))))) as proof of ((R Xy) Xx)
% Found (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((R Xy) Xx)) (fun (x6:(x3 Xx))=> (((eq_ref a) Xx) (fun (x7:a)=> (x3 Xy))))) as proof of ((R Xy) Xx)
% Found (or_intror00 (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((R Xy) Xx)) (fun (x6:(x3 Xx))=> (((eq_ref a) Xx) (fun (x7:a)=> (x3 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((or_intror0 ((R Xy) Xx)) (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((R Xy) Xx)) (fun (x6:(x3 Xx))=> (((eq_ref a) Xx) (fun (x7:a)=> (x3 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((or_intror ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((R Xy) Xx)) (fun (x6:(x3 Xx))=> (((eq_ref a) Xx) (fun (x7:a)=> (x3 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x3 Xx)) (x3 Xy)))=> (((or_intror ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((R Xy) Xx)) (fun (x6:(x3 Xx))=> (((eq_ref a) Xx) (fun (x7:a)=> (x3 Xy))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x3 Xx)) (x3 Xy)))=> (((or_intror ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x6:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x6) x00)) ((R Xy) Xx)) (fun (x6:(x3 Xx))=> (((eq_ref a) Xx) (fun (x7:a)=> (x3 Xy))))))) as proof of (((and (x3 Xx)) (x3 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x1:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x1 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x1:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x1 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found or_introl00:=(or_introl0 ((R Xy) Xx)):((x1 Xy)->((or (x1 Xy)) ((R Xy) Xx)))
% Found (or_introl0 ((R Xy) Xx)) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_introl (x1 Xy)) ((R Xy) Xx)) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_introl (x1 Xy)) ((R Xy) Xx)) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x5:(x1 Xx))=> ((or_introl (x1 Xy)) ((R Xy) Xx))) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x5:(x1 Xx))=> ((or_introl (x1 Xy)) ((R Xy) Xx))) as proof of ((x1 Xx)->((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect00 (fun (x5:(x1 Xx))=> ((or_introl (x1 Xy)) ((R Xy) Xx)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect0 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:(x1 Xx))=> ((or_introl (x1 Xy)) ((R Xy) Xx)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:(x1 Xx))=> ((or_introl (x1 Xy)) ((R Xy) Xx)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x1 Xx)) (x1 Xy)))=> (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:(x1 Xx))=> ((or_introl (x1 Xy)) ((R Xy) Xx))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x1 Xx)) (x1 Xy)))=> (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:(x1 Xx))=> ((or_introl (x1 Xy)) ((R Xy) Xx))))) as proof of (((and (x1 Xx)) (x1 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x8:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x8 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx)
% Found (eq_sym0000 ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x3 Xx))
% Found (eq_sym0000 ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x3 Xx))
% Found ((fun (x4:(((eq a) Xx) Xx))=> ((eq_sym000 x4) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x3 Xx))
% Found ((fun (x4:(((eq a) Xx) Xx))=> (((eq_sym00 Xx) x4) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x3 Xx))
% Found ((fun (x4:(((eq a) Xx) Xx))=> ((((eq_sym0 Xx) Xx) x4) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x3 Xx))
% Found ((fun (x4:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x4) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x3 Xx))
% Found ((fun (x4:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x4) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x3 Xx))
% Found (fun (Xx:a)=> ((fun (x4:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x4) Xy)) ((eq_ref a) Xx))) as proof of ((Xy Xx)->(x3 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x3 Xx)->(Xy Xx))))) (Xx:a)=> ((fun (x4:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x4) Xy)) ((eq_ref a) Xx))) as proof of (forall (Xx:a), ((Xy Xx)->(x3 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x1:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x1 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x4:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x4 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x4 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x4 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x4 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x4 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x4 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x4 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x6 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x6 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x6:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x6 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x6 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x6 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x8:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x8 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx)
% Found (eq_sym0000 ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x1 Xx))
% Found (eq_sym0000 ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x1 Xx))
% Found ((fun (x4:(((eq a) Xx) Xx))=> ((eq_sym000 x4) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x1 Xx))
% Found ((fun (x4:(((eq a) Xx) Xx))=> (((eq_sym00 Xx) x4) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x1 Xx))
% Found ((fun (x4:(((eq a) Xx) Xx))=> ((((eq_sym0 Xx) Xx) x4) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x1 Xx))
% Found ((fun (x4:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x4) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x1 Xx))
% Found ((fun (x4:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x4) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (Xx:a)=> ((fun (x4:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x4) Xy)) ((eq_ref a) Xx))) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a)=> ((fun (x4:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x4) Xy)) ((eq_ref a) Xx))) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x8:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x8 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x3 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x3 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x3 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x3:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x3 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x3 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x3 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x3 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x6 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x6 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x6:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x6 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x6 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x6 Xx)))
% Found or_intror00:=(or_intror0 (x6 Xy)):((x6 Xy)->((or ((R Xx) Xy)) (x6 Xy)))
% Found (or_intror0 (x6 Xy)) as proof of ((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_intror ((R Xx) Xy)) (x6 Xy)) as proof of ((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_intror ((R Xx) Xy)) (x6 Xy)) as proof of ((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x7:(x6 Xx))=> ((or_intror ((R Xx) Xy)) (x6 Xy))) as proof of ((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x7:(x6 Xx))=> ((or_intror ((R Xx) Xy)) (x6 Xy))) as proof of ((x6 Xx)->((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect10 (fun (x7:(x6 Xx))=> ((or_intror ((R Xx) Xy)) (x6 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect1 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x6 Xx))=> ((or_intror ((R Xx) Xy)) (x6 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x6 Xx))=> ((or_intror ((R Xx) Xy)) (x6 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x6 Xx)) (x6 Xy)))=> (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x6 Xx))=> ((or_intror ((R Xx) Xy)) (x6 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x6 Xx)) (x6 Xy)))=> (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x6 Xx))=> ((or_intror ((R Xx) Xy)) (x6 Xy))))) as proof of (((and (x6 Xx)) (x6 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x4 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x4 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x4 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x8:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x8 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x4:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x4 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x4 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x4 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x4 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x6 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x6 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x6:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x6 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x6 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x6 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x8:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x8 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x8:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x8 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x8:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x8 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x1:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x1 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found or_introl00:=(or_introl0 ((R Xy) Xx)):((x1 Xy)->((or (x1 Xy)) ((R Xy) Xx)))
% Found (or_introl0 ((R Xy) Xx)) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_introl (x1 Xy)) ((R Xy) Xx)) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_introl (x1 Xy)) ((R Xy) Xx)) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x7:(x1 Xx))=> ((or_introl (x1 Xy)) ((R Xy) Xx))) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x7:(x1 Xx))=> ((or_introl (x1 Xy)) ((R Xy) Xx))) as proof of ((x1 Xx)->((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect10 (fun (x7:(x1 Xx))=> ((or_introl (x1 Xy)) ((R Xy) Xx)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect1 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> ((or_introl (x1 Xy)) ((R Xy) Xx)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x7:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> ((or_introl (x1 Xy)) ((R Xy) Xx)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x1 Xx)) (x1 Xy)))=> (((fun (P:Type) (x7:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> ((or_introl (x1 Xy)) ((R Xy) Xx))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x1 Xx)) (x1 Xy)))=> (((fun (P:Type) (x7:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> ((or_introl (x1 Xy)) ((R Xy) Xx))))) as proof of (((and (x1 Xx)) (x1 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found eq_ref000:=(eq_ref00 (fun (x8:Prop)=> (x4 Xy))):((x4 Xy)->(x4 Xy))
% Found (eq_ref00 (fun (x8:Prop)=> (x4 Xy))) as proof of ((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((eq_ref0 ((R Xy) Xx)) (fun (x8:Prop)=> (x4 Xy))) as proof of ((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (((eq_ref Prop) ((R Xy) Xx)) (fun (x8:Prop)=> (x4 Xy))) as proof of ((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (((eq_ref Prop) ((R Xy) Xx)) (fun (x8:Prop)=> (x4 Xy))) as proof of ((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x7:(x4 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x8:Prop)=> (x4 Xy)))) as proof of ((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x7:(x4 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x8:Prop)=> (x4 Xy)))) as proof of ((x4 Xx)->((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect10 (fun (x7:(x4 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x8:Prop)=> (x4 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect1 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x4 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x8:Prop)=> (x4 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x4 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x8:Prop)=> (x4 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x4 Xx)) (x4 Xy)))=> (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x4 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x8:Prop)=> (x4 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x4 Xx)) (x4 Xy)))=> (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x4 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x8:Prop)=> (x4 Xy)))))) as proof of (((and (x4 Xx)) (x4 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x5 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x5 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x5 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x5 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x5 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x5 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x5 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x1:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x1 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x5:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x5 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x5 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x5 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x5 Xx)))
% Found eq_ref000:=(eq_ref00 (fun (x8:Prop)=> (x6 Xy))):((x6 Xy)->(x6 Xy))
% Found (eq_ref00 (fun (x8:Prop)=> (x6 Xy))) as proof of ((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((eq_ref0 ((R Xy) Xx)) (fun (x8:Prop)=> (x6 Xy))) as proof of ((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (((eq_ref Prop) ((R Xy) Xx)) (fun (x8:Prop)=> (x6 Xy))) as proof of ((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (((eq_ref Prop) ((R Xy) Xx)) (fun (x8:Prop)=> (x6 Xy))) as proof of ((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x7:(x6 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x8:Prop)=> (x6 Xy)))) as proof of ((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x7:(x6 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x8:Prop)=> (x6 Xy)))) as proof of ((x6 Xx)->((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect10 (fun (x7:(x6 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x8:Prop)=> (x6 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect1 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x6 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x8:Prop)=> (x6 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x6 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x8:Prop)=> (x6 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x6 Xx)) (x6 Xy)))=> (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x6 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x8:Prop)=> (x6 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x6 Xx)) (x6 Xy)))=> (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x6 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x8:Prop)=> (x6 Xy)))))) as proof of (((and (x6 Xx)) (x6 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x6 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x6 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x6 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x6 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x6:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x6 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x6 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x6 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x6:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x6 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x6 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x6 Xx)))
% Found or_intror00:=(or_intror0 (x6 Xy)):((x6 Xy)->((or ((R Xx) Xy)) (x6 Xy)))
% Found (or_intror0 (x6 Xy)) as proof of ((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_intror ((R Xx) Xy)) (x6 Xy)) as proof of ((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_intror ((R Xx) Xy)) (x6 Xy)) as proof of ((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x7:(x6 Xx))=> ((or_intror ((R Xx) Xy)) (x6 Xy))) as proof of ((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x7:(x6 Xx))=> ((or_intror ((R Xx) Xy)) (x6 Xy))) as proof of ((x6 Xx)->((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect10 (fun (x7:(x6 Xx))=> ((or_intror ((R Xx) Xy)) (x6 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect1 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x6 Xx))=> ((or_intror ((R Xx) Xy)) (x6 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x6 Xx))=> ((or_intror ((R Xx) Xy)) (x6 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x6 Xx)) (x6 Xy)))=> (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x6 Xx))=> ((or_intror ((R Xx) Xy)) (x6 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x6 Xx)) (x6 Xy)))=> (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x6 Xx))=> ((or_intror ((R Xx) Xy)) (x6 Xy))))) as proof of (((and (x6 Xx)) (x6 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x6 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x6 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x6:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x6 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x6 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x6 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x1:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x1 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found eq_ref000:=(eq_ref00 (fun (x8:Prop)=> (x3 Xy))):((x3 Xy)->(x3 Xy))
% Found (eq_ref00 (fun (x8:Prop)=> (x3 Xy))) as proof of ((x3 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((eq_ref0 ((R Xy) Xx)) (fun (x8:Prop)=> (x3 Xy))) as proof of ((x3 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (((eq_ref Prop) ((R Xy) Xx)) (fun (x8:Prop)=> (x3 Xy))) as proof of ((x3 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (((eq_ref Prop) ((R Xy) Xx)) (fun (x8:Prop)=> (x3 Xy))) as proof of ((x3 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x7:(x3 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x8:Prop)=> (x3 Xy)))) as proof of ((x3 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x7:(x3 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x8:Prop)=> (x3 Xy)))) as proof of ((x3 Xx)->((x3 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect10 (fun (x7:(x3 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x8:Prop)=> (x3 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect1 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x3 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x8:Prop)=> (x3 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x7:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x3 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x8:Prop)=> (x3 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x3 Xx)) (x3 Xy)))=> (((fun (P:Type) (x7:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x3 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x8:Prop)=> (x3 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x3 Xx)) (x3 Xy)))=> (((fun (P:Type) (x7:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x3 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x8:Prop)=> (x3 Xy)))))) as proof of (((and (x3 Xx)) (x3 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found x01:(Xy Xx)
% Instantiate: x3:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x3 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x3 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x3 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x3 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x3 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x3 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x3 Xx)))
% Found ex_ind00:=(ex_ind0 ((or ((R Xx) Xy)) ((R Xy) Xx))):((forall (x:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((x Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x U)) (x V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x Xx0))))))->((or ((R Xx) Xy)) ((R Xy) Xx))))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Instantiate: x1:=(fun (x5:a)=> (forall (x50:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((x50 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x50 U)) (x50 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x50 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x50 Xx0))))))->((or ((R Xx) x5)) ((R x5) Xx))))):(a->Prop)
% Found (fun (x5:(x1 Xx))=> ex_ind00) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x5:(x1 Xx))=> ex_ind00) as proof of ((x1 Xx)->((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect00 (fun (x5:(x1 Xx))=> ex_ind00)) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect0 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:(x1 Xx))=> ex_ind00)) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:(x1 Xx))=> ex_ind00)) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x4:((and ((and (forall (Xx0:(a->Prop)), ((x3 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x3 U)) (x3 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x3 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x3 Xx0)))))))=> (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:(x1 Xx))=> ex_ind00))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x3:((a->Prop)->Prop)) (x4:((and ((and (forall (Xx0:(a->Prop)), ((x3 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x3 U)) (x3 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x3 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x3 Xx0)))))))=> (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:(x1 Xx))=> ex_ind00))) as proof of (((and ((and (forall (Xx0:(a->Prop)), ((x3 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x3 U)) (x3 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x3 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x3 Xx0))))))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x3:((a->Prop)->Prop)) (x4:((and ((and (forall (Xx0:(a->Prop)), ((x3 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x3 U)) (x3 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x3 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x3 Xx0)))))))=> (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:(x1 Xx))=> ex_ind00))) as proof of (forall (x:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((x Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x U)) (x V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x Xx0))))))->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (ex_ind00 (fun (x3:((a->Prop)->Prop)) (x4:((and ((and (forall (Xx0:(a->Prop)), ((x3 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x3 U)) (x3 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x3 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x3 Xx0)))))))=> (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:(x1 Xx))=> ex_ind00)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((ex_ind0 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x3:((a->Prop)->Prop)) (x4:((and ((and (forall (Xx0:(a->Prop)), ((x3 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x3 U)) (x3 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x3 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x3 Xx0)))))))=> (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:(x1 Xx))=> (ex_ind0 ((or ((R Xx) Xy)) ((R Xy) Xx))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Prop) (x3:(forall (x:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((x Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x U)) (x V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((x Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(x Xx))))))->P)))=> (((((ex_ind ((a->Prop)->Prop)) (fun (M:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((M Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (M U)) (M V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((M Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(M Xx)))))))) P) x3) x2)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x3:((a->Prop)->Prop)) (x4:((and ((and (forall (Xx0:(a->Prop)), ((x3 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x3 U)) (x3 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x3 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x3 Xx0)))))))=> (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:(x1 Xx))=> ((fun (P:Prop) (x3:(forall (x:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((x Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x U)) (x V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((x Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(x Xx))))))->P)))=> (((((ex_ind ((a->Prop)->Prop)) (fun (M:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((M Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (M U)) (M V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((M Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(M Xx)))))))) P) x3) x2)) ((or ((R Xx) Xy)) ((R Xy) Xx))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x1:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x1 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x1:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x1 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x4 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x4 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x4 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x4:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x4 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x4 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x4 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x4 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x4:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x4 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x4 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x4 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x4 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x4 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x4 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x4 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x1:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x1 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found or_intror00:=(or_intror0 (x4 Xy)):((x4 Xy)->((or ((R Xx) Xy)) (x4 Xy)))
% Found (or_intror0 (x4 Xy)) as proof of ((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_intror ((R Xx) Xy)) (x4 Xy)) as proof of ((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_intror ((R Xx) Xy)) (x4 Xy)) as proof of ((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x7:(x4 Xx))=> ((or_intror ((R Xx) Xy)) (x4 Xy))) as proof of ((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x7:(x4 Xx))=> ((or_intror ((R Xx) Xy)) (x4 Xy))) as proof of ((x4 Xx)->((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect10 (fun (x7:(x4 Xx))=> ((or_intror ((R Xx) Xy)) (x4 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect1 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x4 Xx))=> ((or_intror ((R Xx) Xy)) (x4 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x4 Xx))=> ((or_intror ((R Xx) Xy)) (x4 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x4 Xx)) (x4 Xy)))=> (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x4 Xx))=> ((or_intror ((R Xx) Xy)) (x4 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x4 Xx)) (x4 Xy)))=> (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x4 Xx))=> ((or_intror ((R Xx) Xy)) (x4 Xy))))) as proof of (((and (x4 Xx)) (x4 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x3 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x3 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x3 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x3:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x3 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x3 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x3 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x3 Xx)))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx)
% Found (eq_sym0000 ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x5 Xx))
% Found (eq_sym0000 ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x5 Xx))
% Found ((fun (x6:(((eq a) Xx) Xx))=> ((eq_sym000 x6) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x5 Xx))
% Found ((fun (x6:(((eq a) Xx) Xx))=> (((eq_sym00 Xx) x6) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x5 Xx))
% Found ((fun (x6:(((eq a) Xx) Xx))=> ((((eq_sym0 Xx) Xx) x6) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x5 Xx))
% Found ((fun (x6:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x6) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x5 Xx))
% Found ((fun (x6:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x6) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x5 Xx))
% Found (fun (Xx:a)=> ((fun (x6:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x6) Xy)) ((eq_ref a) Xx))) as proof of ((Xy Xx)->(x5 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x5 Xx)->(Xy Xx))))) (Xx:a)=> ((fun (x6:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x6) Xy)) ((eq_ref a) Xx))) as proof of (forall (Xx:a), ((Xy Xx)->(x5 Xx)))
% Found x5:(x4 Xx)
% Instantiate: x4:=(R Xy):(a->Prop)
% Found (fun (x6:(x4 Xy))=> x5) as proof of ((R Xy) Xx)
% Found (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x5) as proof of ((x4 Xy)->((R Xy) Xx))
% Found (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x5) as proof of ((x4 Xx)->((x4 Xy)->((R Xy) Xx)))
% Found (and_rect00 (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x5)) as proof of ((R Xy) Xx)
% Found ((and_rect0 ((R Xy) Xx)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x5)) as proof of ((R Xy) Xx)
% Found (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((R Xy) Xx)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x5)) as proof of ((R Xy) Xx)
% Found (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((R Xy) Xx)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x5)) as proof of ((R Xy) Xx)
% Found (or_intror00 (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((R Xy) Xx)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x5))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((or_intror0 ((R Xy) Xx)) (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((R Xy) Xx)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x5))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((or_intror ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((R Xy) Xx)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x5))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x4 Xx)) (x4 Xy)))=> (((or_intror ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((R Xy) Xx)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x5)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found x6:(x4 Xy)
% Instantiate: x4:=(R Xx):(a->Prop)
% Found (fun (x6:(x4 Xy))=> x6) as proof of ((R Xx) Xy)
% Found (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x6) as proof of ((x4 Xy)->((R Xx) Xy))
% Found (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x6) as proof of ((x4 Xx)->((x4 Xy)->((R Xx) Xy)))
% Found (and_rect00 (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x6)) as proof of ((R Xx) Xy)
% Found ((and_rect0 ((R Xx) Xy)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x6)) as proof of ((R Xx) Xy)
% Found (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((R Xx) Xy)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x6)) as proof of ((R Xx) Xy)
% Found (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((R Xx) Xy)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x6)) as proof of ((R Xx) Xy)
% Found (or_introl00 (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((R Xx) Xy)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x6))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((or_introl0 ((R Xy) Xx)) (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((R Xx) Xy)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x6))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((or_introl ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((R Xx) Xy)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x6))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x4 Xx)) (x4 Xy)))=> (((or_introl ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((R Xx) Xy)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x6)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x4 Xx)) (x4 Xy)))=> (((or_introl ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((R Xx) Xy)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x6)))) as proof of (((and (x4 Xx)) (x4 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found or_introl00:=(or_introl0 ((R Xx) Xy)):((x4 Xy)->((or (x4 Xy)) ((R Xx) Xy)))
% Found (or_introl0 ((R Xx) Xy)) as proof of ((x4 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found ((or_introl (x4 Xy)) ((R Xx) Xy)) as proof of ((x4 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found ((or_introl (x4 Xy)) ((R Xx) Xy)) as proof of ((x4 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found (fun (x5:(x4 Xx))=> ((or_introl (x4 Xy)) ((R Xx) Xy))) as proof of ((x4 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found (fun (x5:(x4 Xx))=> ((or_introl (x4 Xy)) ((R Xx) Xy))) as proof of ((x4 Xx)->((x4 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy))))
% Found (and_rect00 (fun (x5:(x4 Xx))=> ((or_introl (x4 Xy)) ((R Xx) Xy)))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found ((and_rect0 ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x5:(x4 Xx))=> ((or_introl (x4 Xy)) ((R Xx) Xy)))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x5:(x4 Xx))=> ((or_introl (x4 Xy)) ((R Xx) Xy)))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x5:(x4 Xx))=> ((or_introl (x4 Xy)) ((R Xx) Xy)))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found (or_comm_i00 (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x5:(x4 Xx))=> ((or_introl (x4 Xy)) ((R Xx) Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((or_comm_i0 ((R Xx) Xy)) (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x5:(x4 Xx))=> ((or_introl (x4 Xy)) ((R Xx) Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((or_comm_i ((R Xy) Xx)) ((R Xx) Xy)) (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x5:(x4 Xx))=> ((or_introl (x4 Xy)) ((R Xx) Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x4 Xx)) (x4 Xy)))=> (((or_comm_i ((R Xy) Xx)) ((R Xx) Xy)) (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x5:(x4 Xx))=> ((or_introl (x4 Xy)) ((R Xx) Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x4 Xx)) (x4 Xy)))=> (((or_comm_i ((R Xy) Xx)) ((R Xx) Xy)) (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x5:(x4 Xx))=> ((or_introl (x4 Xy)) ((R Xx) Xy)))))) as proof of (((and (x4 Xx)) (x4 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found or_intror00:=(or_intror0 (x1 Xy)):((x1 Xy)->((or ((R Xx) Xy)) (x1 Xy)))
% Found (or_intror0 (x1 Xy)) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_intror ((R Xx) Xy)) (x1 Xy)) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_intror ((R Xx) Xy)) (x1 Xy)) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x7:(x1 Xx))=> ((or_intror ((R Xx) Xy)) (x1 Xy))) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x7:(x1 Xx))=> ((or_intror ((R Xx) Xy)) (x1 Xy))) as proof of ((x1 Xx)->((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect10 (fun (x7:(x1 Xx))=> ((or_intror ((R Xx) Xy)) (x1 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect1 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> ((or_intror ((R Xx) Xy)) (x1 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x7:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> ((or_intror ((R Xx) Xy)) (x1 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x1 Xx)) (x1 Xy)))=> (((fun (P:Type) (x7:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> ((or_intror ((R Xx) Xy)) (x1 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x1 Xx)) (x1 Xy)))=> (((fun (P:Type) (x7:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> ((or_intror ((R Xx) Xy)) (x1 Xy))))) as proof of (((and (x1 Xx)) (x1 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found or_intror00:=(or_intror0 (x1 Xy)):((x1 Xy)->((or ((R Xx) Xy)) (x1 Xy)))
% Found (or_intror0 (x1 Xy)) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_intror ((R Xx) Xy)) (x1 Xy)) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_intror ((R Xx) Xy)) (x1 Xy)) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x7:(x1 Xx))=> ((or_intror ((R Xx) Xy)) (x1 Xy))) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x7:(x1 Xx))=> ((or_intror ((R Xx) Xy)) (x1 Xy))) as proof of ((x1 Xx)->((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect10 (fun (x7:(x1 Xx))=> ((or_intror ((R Xx) Xy)) (x1 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect1 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> ((or_intror ((R Xx) Xy)) (x1 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x7:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> ((or_intror ((R Xx) Xy)) (x1 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x1 Xx)) (x1 Xy)))=> (((fun (P:Type) (x7:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> ((or_intror ((R Xx) Xy)) (x1 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x1 Xx)) (x1 Xy)))=> (((fun (P:Type) (x7:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> ((or_intror ((R Xx) Xy)) (x1 Xy))))) as proof of (((and (x1 Xx)) (x1 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx)
% Found (eq_sym0000 ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x1 Xx))
% Found (eq_sym0000 ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x1 Xx))
% Found ((fun (x6:(((eq a) Xx) Xx))=> ((eq_sym000 x6) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x1 Xx))
% Found ((fun (x6:(((eq a) Xx) Xx))=> (((eq_sym00 Xx) x6) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x1 Xx))
% Found ((fun (x6:(((eq a) Xx) Xx))=> ((((eq_sym0 Xx) Xx) x6) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x1 Xx))
% Found ((fun (x6:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x6) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x1 Xx))
% Found ((fun (x6:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x6) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (Xx:a)=> ((fun (x6:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x6) Xy)) ((eq_ref a) Xx))) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a)=> ((fun (x6:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x6) Xy)) ((eq_ref a) Xx))) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx)
% Found (eq_sym0000 ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x3 Xx))
% Found (eq_sym0000 ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x3 Xx))
% Found ((fun (x6:(((eq a) Xx) Xx))=> ((eq_sym000 x6) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x3 Xx))
% Found ((fun (x6:(((eq a) Xx) Xx))=> (((eq_sym00 Xx) x6) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x3 Xx))
% Found ((fun (x6:(((eq a) Xx) Xx))=> ((((eq_sym0 Xx) Xx) x6) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x3 Xx))
% Found ((fun (x6:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x6) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x3 Xx))
% Found ((fun (x6:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x6) Xy)) ((eq_ref a) Xx)) as proof of ((Xy Xx)->(x3 Xx))
% Found (fun (Xx:a)=> ((fun (x6:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x6) Xy)) ((eq_ref a) Xx))) as proof of ((Xy Xx)->(x3 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x3 Xx)->(Xy Xx))))) (Xx:a)=> ((fun (x6:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x6) Xy)) ((eq_ref a) Xx))) as proof of (forall (Xx:a), ((Xy Xx)->(x3 Xx)))
% Found eq_ref000:=(eq_ref00 x1):((x1 Xy)->(x1 Xy))
% Found (eq_ref00 x1) as proof of ((x1 Xy)->((R Xx) Xy))
% Found ((eq_ref0 Xy) x1) as proof of ((x1 Xy)->((R Xx) Xy))
% Found (((eq_ref a) Xy) x1) as proof of ((x1 Xy)->((R Xx) Xy))
% Found (((eq_ref a) Xy) x1) as proof of ((x1 Xy)->((R Xx) Xy))
% Found (fun (x5:(x1 Xx))=> (((eq_ref a) Xy) x1)) as proof of ((x1 Xy)->((R Xx) Xy))
% Found (fun (x5:(x1 Xx))=> (((eq_ref a) Xy) x1)) as proof of ((x1 Xx)->((x1 Xy)->((R Xx) Xy)))
% Found (and_rect00 (fun (x5:(x1 Xx))=> (((eq_ref a) Xy) x1))) as proof of ((R Xx) Xy)
% Found ((and_rect0 ((R Xx) Xy)) (fun (x5:(x1 Xx))=> (((eq_ref a) Xy) x1))) as proof of ((R Xx) Xy)
% Found (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((R Xx) Xy)) (fun (x5:(x1 Xx))=> (((eq_ref a) Xy) x1))) as proof of ((R Xx) Xy)
% Found (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((R Xx) Xy)) (fun (x5:(x1 Xx))=> (((eq_ref a) Xy) x1))) as proof of ((R Xx) Xy)
% Found (or_introl00 (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((R Xx) Xy)) (fun (x5:(x1 Xx))=> (((eq_ref a) Xy) x1)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((or_introl0 ((R Xy) Xx)) (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((R Xx) Xy)) (fun (x5:(x1 Xx))=> (((eq_ref a) Xy) x1)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((or_introl ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((R Xx) Xy)) (fun (x5:(x1 Xx))=> (((eq_ref a) Xy) x1)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x1 Xx)) (x1 Xy)))=> (((or_introl ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((R Xx) Xy)) (fun (x5:(x1 Xx))=> (((eq_ref a) Xy) x1))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x1 Xx)) (x1 Xy)))=> (((or_introl ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((R Xx) Xy)) (fun (x5:(x1 Xx))=> (((eq_ref a) Xy) x1))))) as proof of (((and (x1 Xx)) (x1 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found eq_ref000:=(eq_ref00 (fun (x6:a)=> (x1 Xy))):((x1 Xy)->(x1 Xy))
% Found (eq_ref00 (fun (x6:a)=> (x1 Xy))) as proof of ((x1 Xy)->((R Xy) Xx))
% Found ((eq_ref0 Xx) (fun (x6:a)=> (x1 Xy))) as proof of ((x1 Xy)->((R Xy) Xx))
% Found (((eq_ref a) Xx) (fun (x6:a)=> (x1 Xy))) as proof of ((x1 Xy)->((R Xy) Xx))
% Found (((eq_ref a) Xx) (fun (x6:a)=> (x1 Xy))) as proof of ((x1 Xy)->((R Xy) Xx))
% Found (fun (x5:(x1 Xx))=> (((eq_ref a) Xx) (fun (x6:a)=> (x1 Xy)))) as proof of ((x1 Xy)->((R Xy) Xx))
% Found (fun (x5:(x1 Xx))=> (((eq_ref a) Xx) (fun (x6:a)=> (x1 Xy)))) as proof of ((x1 Xx)->((x1 Xy)->((R Xy) Xx)))
% Found (and_rect00 (fun (x5:(x1 Xx))=> (((eq_ref a) Xx) (fun (x6:a)=> (x1 Xy))))) as proof of ((R Xy) Xx)
% Found ((and_rect0 ((R Xy) Xx)) (fun (x5:(x1 Xx))=> (((eq_ref a) Xx) (fun (x6:a)=> (x1 Xy))))) as proof of ((R Xy) Xx)
% Found (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((R Xy) Xx)) (fun (x5:(x1 Xx))=> (((eq_ref a) Xx) (fun (x6:a)=> (x1 Xy))))) as proof of ((R Xy) Xx)
% Found (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((R Xy) Xx)) (fun (x5:(x1 Xx))=> (((eq_ref a) Xx) (fun (x6:a)=> (x1 Xy))))) as proof of ((R Xy) Xx)
% Found (or_intror00 (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((R Xy) Xx)) (fun (x5:(x1 Xx))=> (((eq_ref a) Xx) (fun (x6:a)=> (x1 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((or_intror0 ((R Xy) Xx)) (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((R Xy) Xx)) (fun (x5:(x1 Xx))=> (((eq_ref a) Xx) (fun (x6:a)=> (x1 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((or_intror ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((R Xy) Xx)) (fun (x5:(x1 Xx))=> (((eq_ref a) Xx) (fun (x6:a)=> (x1 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x1 Xx)) (x1 Xy)))=> (((or_intror ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((R Xy) Xx)) (fun (x5:(x1 Xx))=> (((eq_ref a) Xx) (fun (x6:a)=> (x1 Xy))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x1 Xx)) (x1 Xy)))=> (((or_intror ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((R Xy) Xx)) (fun (x5:(x1 Xx))=> (((eq_ref a) Xx) (fun (x6:a)=> (x1 Xy))))))) as proof of (((and (x1 Xx)) (x1 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found or_intror00:=(or_intror0 (x1 Xy)):((x1 Xy)->((or ((R Xy) Xx)) (x1 Xy)))
% Found (or_intror0 (x1 Xy)) as proof of ((x1 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found ((or_intror ((R Xy) Xx)) (x1 Xy)) as proof of ((x1 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found ((or_intror ((R Xy) Xx)) (x1 Xy)) as proof of ((x1 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found (fun (x5:(x1 Xx))=> ((or_intror ((R Xy) Xx)) (x1 Xy))) as proof of ((x1 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found (fun (x5:(x1 Xx))=> ((or_intror ((R Xy) Xx)) (x1 Xy))) as proof of ((x1 Xx)->((x1 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy))))
% Found (and_rect00 (fun (x5:(x1 Xx))=> ((or_intror ((R Xy) Xx)) (x1 Xy)))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found ((and_rect0 ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x5:(x1 Xx))=> ((or_intror ((R Xy) Xx)) (x1 Xy)))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x5:(x1 Xx))=> ((or_intror ((R Xy) Xx)) (x1 Xy)))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x5:(x1 Xx))=> ((or_intror ((R Xy) Xx)) (x1 Xy)))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found (or_comm_i00 (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x5:(x1 Xx))=> ((or_intror ((R Xy) Xx)) (x1 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((or_comm_i0 ((R Xx) Xy)) (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x5:(x1 Xx))=> ((or_intror ((R Xy) Xx)) (x1 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((or_comm_i ((R Xy) Xx)) ((R Xx) Xy)) (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x5:(x1 Xx))=> ((or_intror ((R Xy) Xx)) (x1 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x1 Xx)) (x1 Xy)))=> (((or_comm_i ((R Xy) Xx)) ((R Xx) Xy)) (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x5:(x1 Xx))=> ((or_intror ((R Xy) Xx)) (x1 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x1 Xx)) (x1 Xy)))=> (((or_comm_i ((R Xy) Xx)) ((R Xx) Xy)) (((fun (P:Type) (x5:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x5) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x5:(x1 Xx))=> ((or_intror ((R Xy) Xx)) (x1 Xy)))))) as proof of (((and (x1 Xx)) (x1 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq ((a->Prop)->Prop)) a0) (fun (x:(a->Prop))=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x2:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->Prop)->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x10 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x10 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x10:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x10 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x10 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x10 Xx)))
% Found eq_ref00:=(eq_ref0 (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(x1 Xx)))))):(((eq Prop) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(x1 Xx)))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(x1 Xx))))))
% Found (eq_ref0 (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(x1 Xx)))))) as proof of (((eq Prop) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(x1 Xx)))))) b)
% Found ((eq_ref Prop) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(x1 Xx)))))) as proof of (((eq Prop) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(x1 Xx)))))) b)
% Found ((eq_ref Prop) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(x1 Xx)))))) as proof of (((eq Prop) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(x1 Xx)))))) b)
% Found ((eq_ref Prop) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(x1 Xx)))))) as proof of (((eq Prop) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(x1 Xx)))))) b)
% Found x01:(Xy Xx)
% Instantiate: x3:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x3 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x3 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x3 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x3 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x3 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x3 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x3 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x6:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x6 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x6 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x6 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x6 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x6 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x8:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x8 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found or_introl00:=(or_introl0 ((R Xy) Xx)):((x8 Xy)->((or (x8 Xy)) ((R Xy) Xx)))
% Found (or_introl0 ((R Xy) Xx)) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_introl (x8 Xy)) ((R Xy) Xx)) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_introl (x8 Xy)) ((R Xy) Xx)) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x8 Xx))=> ((or_introl (x8 Xy)) ((R Xy) Xx))) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x8 Xx))=> ((or_introl (x8 Xy)) ((R Xy) Xx))) as proof of ((x8 Xx)->((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect20 (fun (x9:(x8 Xx))=> ((or_introl (x8 Xy)) ((R Xy) Xx)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect2 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x8 Xx))=> ((or_introl (x8 Xy)) ((R Xy) Xx)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x8 Xx))=> ((or_introl (x8 Xy)) ((R Xy) Xx)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x8 Xx)) (x8 Xy)))=> (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x8 Xx))=> ((or_introl (x8 Xy)) ((R Xy) Xx))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x8 Xx)) (x8 Xy)))=> (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x8 Xx))=> ((or_introl (x8 Xy)) ((R Xy) Xx))))) as proof of (((and (x8 Xx)) (x8 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) (fun (x:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (x Xx)) (x Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(x Xx))))))))
% Found (eta_expansion_dep00 (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) b)
% Found ((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx)))))))) b)
% Found x01:(Xy Xx)
% Instantiate: x10:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x10 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x10 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x10 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x10 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x10 Xx)))
% Found and_rect00:=(and_rect0 ((or ((R Xx) Xy)) ((R Xy) Xx))):((((and (forall (Xx0:(a->Prop)), ((x3 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x3 U)) (x3 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))->((forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x3 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x3 Xx0)))))->((or ((R Xx) Xy)) ((R Xy) Xx))))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Instantiate: x1:=(fun (x7:a)=> (((and (forall (Xx0:(a->Prop)), ((x3 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x3 U)) (x3 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))->((forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x3 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x3 Xx0)))))->((or ((R Xx) x7)) ((R x7) Xx))))):(a->Prop)
% Found (fun (x7:(x1 Xx))=> and_rect00) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x7:(x1 Xx))=> and_rect00) as proof of ((x1 Xx)->((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect10 (fun (x7:(x1 Xx))=> and_rect00)) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect1 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> and_rect00)) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x7:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> and_rect00)) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x6:(forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x3 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x3 Xx0))))))=> (((fun (P:Type) (x7:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> and_rect00))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x5:((and (forall (Xx0:(a->Prop)), ((x3 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x3 U)) (x3 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (x6:(forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x3 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x3 Xx0))))))=> (((fun (P:Type) (x7:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> and_rect00))) as proof of ((forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x3 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x3 Xx0)))))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x5:((and (forall (Xx0:(a->Prop)), ((x3 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x3 U)) (x3 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (x6:(forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x3 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x3 Xx0))))))=> (((fun (P:Type) (x7:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> and_rect00))) as proof of (((and (forall (Xx0:(a->Prop)), ((x3 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x3 U)) (x3 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))->((forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x3 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x3 Xx0)))))->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect00 (fun (x5:((and (forall (Xx0:(a->Prop)), ((x3 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x3 U)) (x3 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (x6:(forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x3 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x3 Xx0))))))=> (((fun (P:Type) (x7:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> and_rect00)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect0 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:((and (forall (Xx0:(a->Prop)), ((x3 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x3 U)) (x3 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (x6:(forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x3 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x3 Xx0))))))=> (((fun (P:Type) (x7:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> (and_rect0 ((or ((R Xx) Xy)) ((R Xy) Xx))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x5:(((and (forall (Xx:(a->Prop)), ((x3 Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x3 U)) (x3 V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))->((forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((x3 Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(x3 Xx)))))->P)))=> (((((and_rect ((and (forall (Xx:(a->Prop)), ((x3 Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x3 U)) (x3 V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((x3 Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(x3 Xx)))))) P) x5) x4)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:((and (forall (Xx0:(a->Prop)), ((x3 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x3 U)) (x3 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (x6:(forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x3 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x3 Xx0))))))=> (((fun (P:Type) (x7:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> ((fun (P:Type) (x5:(((and (forall (Xx:(a->Prop)), ((x3 Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x3 U)) (x3 V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))->((forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((x3 Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(x3 Xx)))))->P)))=> (((((and_rect ((and (forall (Xx:(a->Prop)), ((x3 Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x3 U)) (x3 V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((x3 Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(x3 Xx)))))) P) x5) x4)) ((or ((R Xx) Xy)) ((R Xy) Xx))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x1 Xx)) (x1 Xy)))=> (((fun (P:Type) (x5:(((and (forall (Xx:(a->Prop)), ((x3 Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x3 U)) (x3 V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))->((forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((x3 Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(x3 Xx)))))->P)))=> (((((and_rect ((and (forall (Xx:(a->Prop)), ((x3 Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x3 U)) (x3 V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((x3 Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(x3 Xx)))))) P) x5) x4)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:((and (forall (Xx0:(a->Prop)), ((x3 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x3 U)) (x3 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (x6:(forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x3 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x3 Xx0))))))=> (((fun (P:Type) (x7:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> ((fun (P:Type) (x5:(((and (forall (Xx:(a->Prop)), ((x3 Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x3 U)) (x3 V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))->((forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((x3 Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(x3 Xx)))))->P)))=> (((((and_rect ((and (forall (Xx:(a->Prop)), ((x3 Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x3 U)) (x3 V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((x3 Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(x3 Xx)))))) P) x5) x4)) ((or ((R Xx) Xy)) ((R Xy) Xx)))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x1 Xx)) (x1 Xy)))=> (((fun (P:Type) (x5:(((and (forall (Xx:(a->Prop)), ((x3 Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x3 U)) (x3 V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))->((forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((x3 Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(x3 Xx)))))->P)))=> (((((and_rect ((and (forall (Xx:(a->Prop)), ((x3 Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x3 U)) (x3 V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((x3 Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(x3 Xx)))))) P) x5) x4)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:((and (forall (Xx0:(a->Prop)), ((x3 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x3 U)) (x3 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (x6:(forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x3 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x3 Xx0))))))=> (((fun (P:Type) (x7:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> ((fun (P:Type) (x5:(((and (forall (Xx:(a->Prop)), ((x3 Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x3 U)) (x3 V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))->((forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((x3 Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(x3 Xx)))))->P)))=> (((((and_rect ((and (forall (Xx:(a->Prop)), ((x3 Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x3 U)) (x3 V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((x3 Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(x3 Xx)))))) P) x5) x4)) ((or ((R Xx) Xy)) ((R Xy) Xx)))))))) as proof of (((and (x1 Xx)) (x1 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found and_rect00:=(and_rect0 ((or ((R Xx) Xy)) ((R Xy) Xx))):((((and (forall (Xx0:(a->Prop)), ((x2 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x2 U)) (x2 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))->((forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x2 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x2 Xx0)))))->((or ((R Xx) Xy)) ((R Xy) Xx))))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Instantiate: x4:=(fun (x7:a)=> (((and (forall (Xx0:(a->Prop)), ((x2 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x2 U)) (x2 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))->((forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x2 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x2 Xx0)))))->((or ((R Xx) x7)) ((R x7) Xx))))):(a->Prop)
% Found (fun (x7:(x4 Xx))=> and_rect00) as proof of ((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x7:(x4 Xx))=> and_rect00) as proof of ((x4 Xx)->((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect10 (fun (x7:(x4 Xx))=> and_rect00)) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect1 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x4 Xx))=> and_rect00)) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x4 Xx))=> and_rect00)) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x6:(forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x2 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x2 Xx0))))))=> (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x4 Xx))=> and_rect00))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x5:((and (forall (Xx0:(a->Prop)), ((x2 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x2 U)) (x2 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (x6:(forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x2 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x2 Xx0))))))=> (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x4 Xx))=> and_rect00))) as proof of ((forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x2 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x2 Xx0)))))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x5:((and (forall (Xx0:(a->Prop)), ((x2 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x2 U)) (x2 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (x6:(forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x2 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x2 Xx0))))))=> (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x4 Xx))=> and_rect00))) as proof of (((and (forall (Xx0:(a->Prop)), ((x2 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x2 U)) (x2 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))->((forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x2 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x2 Xx0)))))->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect00 (fun (x5:((and (forall (Xx0:(a->Prop)), ((x2 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x2 U)) (x2 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (x6:(forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x2 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x2 Xx0))))))=> (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x4 Xx))=> and_rect00)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect0 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:((and (forall (Xx0:(a->Prop)), ((x2 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x2 U)) (x2 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (x6:(forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x2 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x2 Xx0))))))=> (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x4 Xx))=> (and_rect0 ((or ((R Xx) Xy)) ((R Xy) Xx))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x5:(((and (forall (Xx:(a->Prop)), ((x2 Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x2 U)) (x2 V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))->((forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((x2 Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(x2 Xx)))))->P)))=> (((((and_rect ((and (forall (Xx:(a->Prop)), ((x2 Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x2 U)) (x2 V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((x2 Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(x2 Xx)))))) P) x5) x3)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:((and (forall (Xx0:(a->Prop)), ((x2 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x2 U)) (x2 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (x6:(forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x2 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x2 Xx0))))))=> (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x4 Xx))=> ((fun (P:Type) (x5:(((and (forall (Xx:(a->Prop)), ((x2 Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x2 U)) (x2 V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))->((forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((x2 Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(x2 Xx)))))->P)))=> (((((and_rect ((and (forall (Xx:(a->Prop)), ((x2 Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x2 U)) (x2 V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((x2 Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(x2 Xx)))))) P) x5) x3)) ((or ((R Xx) Xy)) ((R Xy) Xx))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x4 Xx)) (x4 Xy)))=> (((fun (P:Type) (x5:(((and (forall (Xx:(a->Prop)), ((x2 Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x2 U)) (x2 V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))->((forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((x2 Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(x2 Xx)))))->P)))=> (((((and_rect ((and (forall (Xx:(a->Prop)), ((x2 Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x2 U)) (x2 V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((x2 Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(x2 Xx)))))) P) x5) x3)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:((and (forall (Xx0:(a->Prop)), ((x2 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x2 U)) (x2 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (x6:(forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x2 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x2 Xx0))))))=> (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x4 Xx))=> ((fun (P:Type) (x5:(((and (forall (Xx:(a->Prop)), ((x2 Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x2 U)) (x2 V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))->((forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((x2 Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(x2 Xx)))))->P)))=> (((((and_rect ((and (forall (Xx:(a->Prop)), ((x2 Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x2 U)) (x2 V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((x2 Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(x2 Xx)))))) P) x5) x3)) ((or ((R Xx) Xy)) ((R Xy) Xx)))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x4 Xx)) (x4 Xy)))=> (((fun (P:Type) (x5:(((and (forall (Xx:(a->Prop)), ((x2 Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x2 U)) (x2 V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))->((forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((x2 Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(x2 Xx)))))->P)))=> (((((and_rect ((and (forall (Xx:(a->Prop)), ((x2 Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x2 U)) (x2 V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((x2 Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(x2 Xx)))))) P) x5) x3)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:((and (forall (Xx0:(a->Prop)), ((x2 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x2 U)) (x2 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (x6:(forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x2 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x2 Xx0))))))=> (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x4 Xx))=> ((fun (P:Type) (x5:(((and (forall (Xx:(a->Prop)), ((x2 Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x2 U)) (x2 V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))->((forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((x2 Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(x2 Xx)))))->P)))=> (((((and_rect ((and (forall (Xx:(a->Prop)), ((x2 Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x2 U)) (x2 V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((x2 Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(x2 Xx)))))) P) x5) x3)) ((or ((R Xx) Xy)) ((R Xy) Xx)))))))) as proof of (((and (x4 Xx)) (x4 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found x01:(Xy Xx)
% Instantiate: x10:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x10 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x10 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x10 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x10 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x10 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x10 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x10 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x10:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x10 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x10 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x10 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x1:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x1 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found or_introl00:=(or_introl0 ((R Xy) Xx)):((x1 Xy)->((or (x1 Xy)) ((R Xy) Xx)))
% Found (or_introl0 ((R Xy) Xx)) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_introl (x1 Xy)) ((R Xy) Xx)) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_introl (x1 Xy)) ((R Xy) Xx)) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x1 Xx))=> ((or_introl (x1 Xy)) ((R Xy) Xx))) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x1 Xx))=> ((or_introl (x1 Xy)) ((R Xy) Xx))) as proof of ((x1 Xx)->((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect20 (fun (x9:(x1 Xx))=> ((or_introl (x1 Xy)) ((R Xy) Xx)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect2 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x1 Xx))=> ((or_introl (x1 Xy)) ((R Xy) Xx)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x9:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x1 Xx))=> ((or_introl (x1 Xy)) ((R Xy) Xx)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x1 Xx)) (x1 Xy)))=> (((fun (P:Type) (x9:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x1 Xx))=> ((or_introl (x1 Xy)) ((R Xy) Xx))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x1 Xx)) (x1 Xy)))=> (((fun (P:Type) (x9:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x1 Xx))=> ((or_introl (x1 Xy)) ((R Xy) Xx))))) as proof of (((and (x1 Xx)) (x1 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found or_intror00:=(or_intror0 (x4 Xy)):((x4 Xy)->((or ((R Xx) Xy)) (x4 Xy)))
% Found (or_intror0 (x4 Xy)) as proof of ((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_intror ((R Xx) Xy)) (x4 Xy)) as proof of ((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_intror ((R Xx) Xy)) (x4 Xy)) as proof of ((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x4 Xx))=> ((or_intror ((R Xx) Xy)) (x4 Xy))) as proof of ((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x4 Xx))=> ((or_intror ((R Xx) Xy)) (x4 Xy))) as proof of ((x4 Xx)->((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect20 (fun (x9:(x4 Xx))=> ((or_intror ((R Xx) Xy)) (x4 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect2 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x4 Xx))=> ((or_intror ((R Xx) Xy)) (x4 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x4 Xx))=> ((or_intror ((R Xx) Xy)) (x4 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x4 Xx)) (x4 Xy)))=> (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x4 Xx))=> ((or_intror ((R Xx) Xy)) (x4 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x4 Xx)) (x4 Xy)))=> (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x4 Xx))=> ((or_intror ((R Xx) Xy)) (x4 Xy))))) as proof of (((and (x4 Xx)) (x4 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found or_introl00:=(or_introl0 ((R Xy) Xx)):((x6 Xy)->((or (x6 Xy)) ((R Xy) Xx)))
% Found (or_introl0 ((R Xy) Xx)) as proof of ((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_introl (x6 Xy)) ((R Xy) Xx)) as proof of ((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_introl (x6 Xy)) ((R Xy) Xx)) as proof of ((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x6 Xx))=> ((or_introl (x6 Xy)) ((R Xy) Xx))) as proof of ((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x6 Xx))=> ((or_introl (x6 Xy)) ((R Xy) Xx))) as proof of ((x6 Xx)->((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect20 (fun (x9:(x6 Xx))=> ((or_introl (x6 Xy)) ((R Xy) Xx)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect2 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x6 Xx))=> ((or_introl (x6 Xy)) ((R Xy) Xx)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x6 Xx))=> ((or_introl (x6 Xy)) ((R Xy) Xx)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x6 Xx)) (x6 Xy)))=> (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x6 Xx))=> ((or_introl (x6 Xy)) ((R Xy) Xx))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x6 Xx)) (x6 Xy)))=> (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x6 Xx))=> ((or_introl (x6 Xy)) ((R Xy) Xx))))) as proof of (((and (x6 Xx)) (x6 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found eq_ref000:=(eq_ref00 (fun (x10:Prop)=> (x8 Xy))):((x8 Xy)->(x8 Xy))
% Found (eq_ref00 (fun (x10:Prop)=> (x8 Xy))) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((eq_ref0 ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy))) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy))) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy))) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy)))) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy)))) as proof of ((x8 Xx)->((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect20 (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect2 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x8 Xx)) (x8 Xy)))=> (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x8 Xx)) (x8 Xy)))=> (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy)))))) as proof of (((and (x8 Xx)) (x8 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x5 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x5 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x5 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x5 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x5 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x5 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x5 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x5:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x5 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x5 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x5 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x5 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x8:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x8 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((R Xx) Xy)) ((R Xy) Xx))))) (forall (Xy:(a->Prop)), (((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((S Xx)->(Xy Xx))))->(forall (Xx:a), ((Xy Xx)->(S Xx))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq ((a->Prop)->Prop)) a0) (fun (x:(a->Prop))=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x4:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->Prop)->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x6 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x6 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x6:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x6 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x6 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x6 Xx)))
% Found eq_ref000:=(eq_ref00 (fun (x10:Prop)=> (x8 Xy))):((x8 Xy)->(x8 Xy))
% Found (eq_ref00 (fun (x10:Prop)=> (x8 Xy))) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((eq_ref0 ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy))) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy))) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy))) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy)))) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy)))) as proof of ((x8 Xx)->((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect20 (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect2 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x8 Xx)) (x8 Xy)))=> (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x8 Xx)) (x8 Xy)))=> (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy)))))) as proof of (((and (x8 Xx)) (x8 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found x01:(Xy Xx)
% Instantiate: x6:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x6 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x6 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x6 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x6 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x6 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x8:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x8 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x8:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x8 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x10:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x10 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x10 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x10 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x10 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x10 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x10 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x10 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x10:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x10 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x10 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x10 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x10 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x10 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x10:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x10 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x10 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x10 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x10 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x10 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x10 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x10 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x10:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x10 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x10 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x10 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x10:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x10 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x10 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x10 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x10 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x10 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x10:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x10 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x10 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x10 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x10 Xx)))
% Found and_rect00:=(and_rect0 ((or ((R Xx) Xy)) ((R Xy) Xx))):((((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))->((forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0)))->((or ((R Xx) Xy)) ((R Xy) Xx))))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Instantiate: x4:=(fun (x7:a)=> (((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))->((forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0)))->((or ((R Xx) x7)) ((R x7) Xx))))):(a->Prop)
% Found (fun (x7:(x4 Xx))=> and_rect00) as proof of ((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x7:(x4 Xx))=> and_rect00) as proof of ((x4 Xx)->((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect10 (fun (x7:(x4 Xx))=> and_rect00)) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect1 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x4 Xx))=> and_rect00)) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x4 Xx))=> and_rect00)) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x6:(forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x4 Xx))=> and_rect00))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x5:((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))) (x6:(forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x4 Xx))=> and_rect00))) as proof of ((forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0)))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x5:((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))) (x6:(forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x4 Xx))=> and_rect00))) as proof of (((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))->((forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0)))->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect00 (fun (x5:((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))) (x6:(forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x4 Xx))=> and_rect00)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect0 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))) (x6:(forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x4 Xx))=> (and_rect0 ((or ((R Xx) Xy)) ((R Xy) Xx))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x5:(((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))->P)))=> (((((and_rect ((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) P) x5) x0)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))) (x6:(forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x4 Xx))=> ((fun (P:Type) (x5:(((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))->P)))=> (((((and_rect ((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) P) x5) x0)) ((or ((R Xx) Xy)) ((R Xy) Xx))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x4 Xx)) (x4 Xy)))=> (((fun (P:Type) (x5:(((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))->P)))=> (((((and_rect ((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) P) x5) x0)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))) (x6:(forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x4 Xx))=> ((fun (P:Type) (x5:(((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))->P)))=> (((((and_rect ((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) P) x5) x0)) ((or ((R Xx) Xy)) ((R Xy) Xx)))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x4 Xx)) (x4 Xy)))=> (((fun (P:Type) (x5:(((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))->P)))=> (((((and_rect ((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) P) x5) x0)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))) (x6:(forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x4 Xx))=> ((fun (P:Type) (x5:(((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))->P)))=> (((((and_rect ((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) P) x5) x0)) ((or ((R Xx) Xy)) ((R Xy) Xx)))))))) as proof of (((and (x4 Xx)) (x4 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found eq_ref000:=(eq_ref00 (fun (x10:Prop)=> (x3 Xy))):((x3 Xy)->(x3 Xy))
% Found (eq_ref00 (fun (x10:Prop)=> (x3 Xy))) as proof of ((x3 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((eq_ref0 ((R Xy) Xx)) (fun (x10:Prop)=> (x3 Xy))) as proof of ((x3 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x3 Xy))) as proof of ((x3 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x3 Xy))) as proof of ((x3 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x3 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x3 Xy)))) as proof of ((x3 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x3 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x3 Xy)))) as proof of ((x3 Xx)->((x3 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect20 (fun (x9:(x3 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x3 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect2 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x3 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x3 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x9:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x3 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x3 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x3 Xx)) (x3 Xy)))=> (((fun (P:Type) (x9:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x3 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x3 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x3 Xx)) (x3 Xy)))=> (((fun (P:Type) (x9:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x3 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x3 Xy)))))) as proof of (((and (x3 Xx)) (x3 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found or_introl00:=(or_introl0 ((R Xy) Xx)):((x6 Xy)->((or (x6 Xy)) ((R Xy) Xx)))
% Found (or_introl0 ((R Xy) Xx)) as proof of ((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_introl (x6 Xy)) ((R Xy) Xx)) as proof of ((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_introl (x6 Xy)) ((R Xy) Xx)) as proof of ((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x6 Xx))=> ((or_introl (x6 Xy)) ((R Xy) Xx))) as proof of ((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x6 Xx))=> ((or_introl (x6 Xy)) ((R Xy) Xx))) as proof of ((x6 Xx)->((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect20 (fun (x9:(x6 Xx))=> ((or_introl (x6 Xy)) ((R Xy) Xx)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect2 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x6 Xx))=> ((or_introl (x6 Xy)) ((R Xy) Xx)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x6 Xx))=> ((or_introl (x6 Xy)) ((R Xy) Xx)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x6 Xx)) (x6 Xy)))=> (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x6 Xx))=> ((or_introl (x6 Xy)) ((R Xy) Xx))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x6 Xx)) (x6 Xy)))=> (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x6 Xx))=> ((or_introl (x6 Xy)) ((R Xy) Xx))))) as proof of (((and (x6 Xx)) (x6 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x1:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x1 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x3:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x3 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x3 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x3 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x3 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x3 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x3 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x3 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x3 Xx)))
% 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_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x1:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x1 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x1:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x1 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x1 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x1 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x1 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x4 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x4 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x4 Xx)))
% Found or_intror00:=(or_intror0 (x4 Xy)):((x4 Xy)->((or ((R Xx) Xy)) (x4 Xy)))
% Found (or_intror0 (x4 Xy)) as proof of ((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_intror ((R Xx) Xy)) (x4 Xy)) as proof of ((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_intror ((R Xx) Xy)) (x4 Xy)) as proof of ((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x4 Xx))=> ((or_intror ((R Xx) Xy)) (x4 Xy))) as proof of ((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x4 Xx))=> ((or_intror ((R Xx) Xy)) (x4 Xy))) as proof of ((x4 Xx)->((x4 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect20 (fun (x9:(x4 Xx))=> ((or_intror ((R Xx) Xy)) (x4 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect2 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x4 Xx))=> ((or_intror ((R Xx) Xy)) (x4 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x4 Xx))=> ((or_intror ((R Xx) Xy)) (x4 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x4 Xx)) (x4 Xy)))=> (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x4 Xx))=> ((or_intror ((R Xx) Xy)) (x4 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x4 Xx)) (x4 Xy)))=> (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x4 Xx))=> ((or_intror ((R Xx) Xy)) (x4 Xy))))) as proof of (((and (x4 Xx)) (x4 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found or_intror00:=(or_intror0 (x8 Xy)):((x8 Xy)->((or ((R Xx) Xy)) (x8 Xy)))
% Found (or_intror0 (x8 Xy)) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_intror ((R Xx) Xy)) (x8 Xy)) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_intror ((R Xx) Xy)) (x8 Xy)) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x8 Xx))=> ((or_intror ((R Xx) Xy)) (x8 Xy))) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x8 Xx))=> ((or_intror ((R Xx) Xy)) (x8 Xy))) as proof of ((x8 Xx)->((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect20 (fun (x9:(x8 Xx))=> ((or_intror ((R Xx) Xy)) (x8 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect2 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x8 Xx))=> ((or_intror ((R Xx) Xy)) (x8 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x8 Xx))=> ((or_intror ((R Xx) Xy)) (x8 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x8 Xx)) (x8 Xy)))=> (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x8 Xx))=> ((or_intror ((R Xx) Xy)) (x8 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x8 Xx)) (x8 Xy)))=> (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x8 Xx))=> ((or_intror ((R Xx) Xy)) (x8 Xy))))) as proof of (((and (x8 Xx)) (x8 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ex_ind00:=(ex_ind0 ((or ((R Xx) Xy)) ((R Xy) Xx))):((forall (x:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((x Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x U)) (x V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x Xx0))))))->((or ((R Xx) Xy)) ((R Xy) Xx))))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Instantiate: x3:=(fun (x7:a)=> (forall (x70:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((x70 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x70 U)) (x70 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x70 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x70 Xx0))))))->((or ((R Xx) x7)) ((R x7) Xx))))):(a->Prop)
% Found (fun (x7:(x3 Xx))=> ex_ind00) as proof of ((x3 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x7:(x3 Xx))=> ex_ind00) as proof of ((x3 Xx)->((x3 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect10 (fun (x7:(x3 Xx))=> ex_ind00)) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect1 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x3 Xx))=> ex_ind00)) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x7:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x3 Xx))=> ex_ind00)) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x6:((and ((and (forall (Xx0:(a->Prop)), ((x5 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x5 U)) (x5 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x5 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x5 Xx0)))))))=> (((fun (P:Type) (x7:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x3 Xx))=> ex_ind00))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x5:((a->Prop)->Prop)) (x6:((and ((and (forall (Xx0:(a->Prop)), ((x5 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x5 U)) (x5 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x5 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x5 Xx0)))))))=> (((fun (P:Type) (x7:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x3 Xx))=> ex_ind00))) as proof of (((and ((and (forall (Xx0:(a->Prop)), ((x5 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x5 U)) (x5 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x5 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x5 Xx0))))))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x5:((a->Prop)->Prop)) (x6:((and ((and (forall (Xx0:(a->Prop)), ((x5 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x5 U)) (x5 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x5 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x5 Xx0)))))))=> (((fun (P:Type) (x7:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x3 Xx))=> ex_ind00))) as proof of (forall (x:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((x Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x U)) (x V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x Xx0))))))->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (ex_ind00 (fun (x5:((a->Prop)->Prop)) (x6:((and ((and (forall (Xx0:(a->Prop)), ((x5 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x5 U)) (x5 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x5 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x5 Xx0)))))))=> (((fun (P:Type) (x7:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x3 Xx))=> ex_ind00)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((ex_ind0 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:((a->Prop)->Prop)) (x6:((and ((and (forall (Xx0:(a->Prop)), ((x5 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x5 U)) (x5 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x5 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x5 Xx0)))))))=> (((fun (P:Type) (x7:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x3 Xx))=> (ex_ind0 ((or ((R Xx) Xy)) ((R Xy) Xx))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Prop) (x5:(forall (x:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((x Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x U)) (x V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((x Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(x Xx))))))->P)))=> (((((ex_ind ((a->Prop)->Prop)) (fun (M:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((M Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (M U)) (M V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((M Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(M Xx)))))))) P) x5) x4)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:((a->Prop)->Prop)) (x6:((and ((and (forall (Xx0:(a->Prop)), ((x5 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x5 U)) (x5 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xy0:((a->Prop)->Prop)), (((and ((and (forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(X2 Xx0)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy0 U)) (Xy0 V))->((or (forall (Xx0:a), ((U Xx0)->(V Xx0)))) (forall (Xx0:a), ((V Xx0)->(U Xx0)))))))) (forall (Xx0:(a->Prop)), ((x5 Xx0)->(Xy0 Xx0))))->(forall (Xx0:(a->Prop)), ((Xy0 Xx0)->(x5 Xx0)))))))=> (((fun (P:Type) (x7:((x3 Xx)->((x3 Xy)->P)))=> (((((and_rect (x3 Xx)) (x3 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x3 Xx))=> ((fun (P:Prop) (x5:(forall (x:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((x Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (x U)) (x V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((x Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(x Xx))))))->P)))=> (((((ex_ind ((a->Prop)->Prop)) (fun (M:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((M Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (M U)) (M V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xy:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((Xy Xx)->(X2 Xx)))) (forall (U:(a->Prop)) (V:(a->Prop)), (((and (Xy U)) (Xy V))->((or (forall (Xx:a), ((U Xx)->(V Xx)))) (forall (Xx:a), ((V Xx)->(U Xx)))))))) (forall (Xx:(a->Prop)), ((M Xx)->(Xy Xx))))->(forall (Xx:(a->Prop)), ((Xy Xx)->(M Xx)))))))) P) x5) x4)) ((or ((R Xx) Xy)) ((R Xy) Xx))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found x01:(Xy Xx)
% Instantiate: x4:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x4 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x4 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x4 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x4 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x4 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x4 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x4 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x4 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x4:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x4 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x4 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x4 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x4 Xx)))
% Found or_intror00:=(or_intror0 (x6 Xy)):((x6 Xy)->((or ((R Xx) Xy)) (x6 Xy)))
% Found (or_intror0 (x6 Xy)) as proof of ((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_intror ((R Xx) Xy)) (x6 Xy)) as proof of ((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_intror ((R Xx) Xy)) (x6 Xy)) as proof of ((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x6 Xx))=> ((or_intror ((R Xx) Xy)) (x6 Xy))) as proof of ((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x6 Xx))=> ((or_intror ((R Xx) Xy)) (x6 Xy))) as proof of ((x6 Xx)->((x6 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect20 (fun (x9:(x6 Xx))=> ((or_intror ((R Xx) Xy)) (x6 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect2 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x6 Xx))=> ((or_intror ((R Xx) Xy)) (x6 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x6 Xx))=> ((or_intror ((R Xx) Xy)) (x6 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x6 Xx)) (x6 Xy)))=> (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x6 Xx))=> ((or_intror ((R Xx) Xy)) (x6 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x6 Xx)) (x6 Xy)))=> (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x6 Xx))=> ((or_intror ((R Xx) Xy)) (x6 Xy))))) as proof of (((and (x6 Xx)) (x6 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found or_introl00:=(or_introl0 ((R Xy) Xx)):((x8 Xy)->((or (x8 Xy)) ((R Xy) Xx)))
% Found (or_introl0 ((R Xy) Xx)) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_introl (x8 Xy)) ((R Xy) Xx)) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_introl (x8 Xy)) ((R Xy) Xx)) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x8 Xx))=> ((or_introl (x8 Xy)) ((R Xy) Xx))) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x8 Xx))=> ((or_introl (x8 Xy)) ((R Xy) Xx))) as proof of ((x8 Xx)->((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect20 (fun (x9:(x8 Xx))=> ((or_introl (x8 Xy)) ((R Xy) Xx)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect2 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x8 Xx))=> ((or_introl (x8 Xy)) ((R Xy) Xx)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x8 Xx))=> ((or_introl (x8 Xy)) ((R Xy) Xx)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x8 Xx)) (x8 Xy)))=> (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x8 Xx))=> ((or_introl (x8 Xy)) ((R Xy) Xx))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x8 Xx)) (x8 Xy)))=> (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x8 Xx))=> ((or_introl (x8 Xy)) ((R Xy) Xx))))) as proof of (((and (x8 Xx)) (x8 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found or_intror00:=(or_intror0 (x8 Xy)):((x8 Xy)->((or ((R Xx) Xy)) (x8 Xy)))
% Found (or_intror0 (x8 Xy)) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_intror ((R Xx) Xy)) (x8 Xy)) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((or_intror ((R Xx) Xy)) (x8 Xy)) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x8 Xx))=> ((or_intror ((R Xx) Xy)) (x8 Xy))) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x8 Xx))=> ((or_intror ((R Xx) Xy)) (x8 Xy))) as proof of ((x8 Xx)->((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect20 (fun (x9:(x8 Xx))=> ((or_intror ((R Xx) Xy)) (x8 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect2 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x8 Xx))=> ((or_intror ((R Xx) Xy)) (x8 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x8 Xx))=> ((or_intror ((R Xx) Xy)) (x8 Xy)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x8 Xx)) (x8 Xy)))=> (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x8 Xx))=> ((or_intror ((R Xx) Xy)) (x8 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x8 Xx)) (x8 Xy)))=> (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x8 Xx))=> ((or_intror ((R Xx) Xy)) (x8 Xy))))) as proof of (((and (x8 Xx)) (x8 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x8:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x8 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found eq_ref000:=(eq_ref00 (fun (x10:Prop)=> (x8 Xy))):((x8 Xy)->(x8 Xy))
% Found (eq_ref00 (fun (x10:Prop)=> (x8 Xy))) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((eq_ref0 ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy))) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy))) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy))) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy)))) as proof of ((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy)))) as proof of ((x8 Xx)->((x8 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect20 (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect2 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x8 Xx)) (x8 Xy)))=> (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x8 Xx)) (x8 Xy)))=> (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x8 Xy)))))) as proof of (((and (x8 Xx)) (x8 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found x01:(Xy Xx)
% Instantiate: x8:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x8 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x8:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x8 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found x8:(x6 Xy)
% Instantiate: x6:=(R Xx):(a->Prop)
% Found (fun (x8:(x6 Xy))=> x8) as proof of ((R Xx) Xy)
% Found (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x8) as proof of ((x6 Xy)->((R Xx) Xy))
% Found (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x8) as proof of ((x6 Xx)->((x6 Xy)->((R Xx) Xy)))
% Found (and_rect10 (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x8)) as proof of ((R Xx) Xy)
% Found ((and_rect1 ((R Xx) Xy)) (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x8)) as proof of ((R Xx) Xy)
% Found (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((R Xx) Xy)) (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x8)) as proof of ((R Xx) Xy)
% Found (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((R Xx) Xy)) (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x8)) as proof of ((R Xx) Xy)
% Found (or_introl00 (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((R Xx) Xy)) (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x8))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((or_introl0 ((R Xy) Xx)) (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((R Xx) Xy)) (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x8))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((or_introl ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((R Xx) Xy)) (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x8))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x6 Xx)) (x6 Xy)))=> (((or_introl ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((R Xx) Xy)) (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x8)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x6 Xx)) (x6 Xy)))=> (((or_introl ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((R Xx) Xy)) (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x8)))) as proof of (((and (x6 Xx)) (x6 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found x7:(x6 Xx)
% Instantiate: x6:=(R Xy):(a->Prop)
% Found (fun (x8:(x6 Xy))=> x7) as proof of ((R Xy) Xx)
% Found (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x7) as proof of ((x6 Xy)->((R Xy) Xx))
% Found (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x7) as proof of ((x6 Xx)->((x6 Xy)->((R Xy) Xx)))
% Found (and_rect10 (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x7)) as proof of ((R Xy) Xx)
% Found ((and_rect1 ((R Xy) Xx)) (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x7)) as proof of ((R Xy) Xx)
% Found (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((R Xy) Xx)) (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x7)) as proof of ((R Xy) Xx)
% Found (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((R Xy) Xx)) (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x7)) as proof of ((R Xy) Xx)
% Found (or_intror00 (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((R Xy) Xx)) (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x7))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((or_intror0 ((R Xy) Xx)) (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((R Xy) Xx)) (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x7))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((or_intror ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((R Xy) Xx)) (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x7))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x6 Xx)) (x6 Xy)))=> (((or_intror ((R Xx) Xy)) ((R Xy) Xx)) (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((R Xy) Xx)) (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x7)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x8:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x8 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found x01:(Xy Xx)
% Instantiate: x8:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x8 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found eq_ref000:=(eq_ref00 (fun (x8:Prop)=> (x6 Xy))):((x6 Xy)->(x6 Xy))
% Found (eq_ref00 (fun (x8:Prop)=> (x6 Xy))) as proof of ((x6 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found ((eq_ref0 ((R Xx) Xy)) (fun (x8:Prop)=> (x6 Xy))) as proof of ((x6 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found (((eq_ref Prop) ((R Xx) Xy)) (fun (x8:Prop)=> (x6 Xy))) as proof of ((x6 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found (((eq_ref Prop) ((R Xx) Xy)) (fun (x8:Prop)=> (x6 Xy))) as proof of ((x6 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found (fun (x7:(x6 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x8:Prop)=> (x6 Xy)))) as proof of ((x6 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy)))
% Found (fun (x7:(x6 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x8:Prop)=> (x6 Xy)))) as proof of ((x6 Xx)->((x6 Xy)->((or ((R Xy) Xx)) ((R Xx) Xy))))
% Found (and_rect10 (fun (x7:(x6 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x8:Prop)=> (x6 Xy))))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found ((and_rect1 ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x7:(x6 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x8:Prop)=> (x6 Xy))))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x7:(x6 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x8:Prop)=> (x6 Xy))))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x7:(x6 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x8:Prop)=> (x6 Xy))))) as proof of ((or ((R Xy) Xx)) ((R Xx) Xy))
% Found (or_comm_i00 (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x7:(x6 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x8:Prop)=> (x6 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((or_comm_i0 ((R Xx) Xy)) (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x7:(x6 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x8:Prop)=> (x6 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((or_comm_i ((R Xy) Xx)) ((R Xx) Xy)) (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x7:(x6 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x8:Prop)=> (x6 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x6 Xx)) (x6 Xy)))=> (((or_comm_i ((R Xy) Xx)) ((R Xx) Xy)) (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x7:(x6 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x8:Prop)=> (x6 Xy))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x6 Xx)) (x6 Xy)))=> (((or_comm_i ((R Xy) Xx)) ((R Xx) Xy)) (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((or ((R Xy) Xx)) ((R Xx) Xy))) (fun (x7:(x6 Xx))=> (((eq_ref Prop) ((R Xx) Xy)) (fun (x8:Prop)=> (x6 Xy))))))) as proof of (((and (x6 Xx)) (x6 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found x01:(Xy Xx)
% Instantiate: x8:=Xy:(a->Prop)
% Found (fun (x01:(Xy Xx))=> x01) as proof of (x8 Xx)
% Found (fun (Xx:a) (x01:(Xy Xx))=> x01) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a) (x01:(Xy Xx))=> x01) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found eq_ref000:=(eq_ref00 Xy):((Xy Xx)->(Xy Xx))
% Found (eq_ref00 Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found ((eq_ref0 Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) Xy) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of ((Xy Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy0:a), (((and (Xy Xx)) (Xy Xy0))->((or ((R Xx) Xy0)) ((R Xy0) Xx))))) (forall (Xx:a), ((x8 Xx)->(Xy Xx))))) (Xx:a)=> (((eq_ref a) Xx) Xy)) as proof of (forall (Xx:a), ((Xy Xx)->(x8 Xx)))
% Found and_rect00:=(and_rect0 ((or ((R Xx) Xy)) ((R Xy) Xx))):((((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))->((forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0)))->((or ((R Xx) Xy)) ((R Xy) Xx))))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Instantiate: x1:=(fun (x7:a)=> (((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))->((forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0)))->((or ((R Xx) x7)) ((R x7) Xx))))):(a->Prop)
% Found (fun (x7:(x1 Xx))=> and_rect00) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x7:(x1 Xx))=> and_rect00) as proof of ((x1 Xx)->((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect10 (fun (x7:(x1 Xx))=> and_rect00)) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect1 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> and_rect00)) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x7:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> and_rect00)) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x6:(forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x7:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> and_rect00))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x5:((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))) (x6:(forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x7:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> and_rect00))) as proof of ((forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0)))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x5:((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))) (x6:(forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x7:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> and_rect00))) as proof of (((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))->((forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0)))->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect00 (fun (x5:((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))) (x6:(forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x7:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> and_rect00)))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect0 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))) (x6:(forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x7:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> (and_rect0 ((or ((R Xx) Xy)) ((R Xy) Xx))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x5:(((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))->P)))=> (((((and_rect ((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) P) x5) x0)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))) (x6:(forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x7:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> ((fun (P:Type) (x5:(((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))->P)))=> (((((and_rect ((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) P) x5) x0)) ((or ((R Xx) Xy)) ((R Xy) Xx))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x1 Xx)) (x1 Xy)))=> (((fun (P:Type) (x5:(((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))->P)))=> (((((and_rect ((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) P) x5) x0)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))) (x6:(forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x7:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> ((fun (P:Type) (x5:(((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))->P)))=> (((((and_rect ((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) P) x5) x0)) ((or ((R Xx) Xy)) ((R Xy) Xx)))))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x1 Xx)) (x1 Xy)))=> (((fun (P:Type) (x5:(((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))->P)))=> (((((and_rect ((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) P) x5) x0)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x5:((and (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xz))->((R Xx0) Xz)))) (forall (Xx0:a), ((R Xx0) Xx0)))) (x6:(forall (Xx0:a) (Xy0:a), (((and ((R Xx0) Xy0)) ((R Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x7:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x7) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x7:(x1 Xx))=> ((fun (P:Type) (x5:(((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))->P)))=> (((((and_rect ((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xx:a), ((R Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) P) x5) x0)) ((or ((R Xx) Xy)) ((R Xy) Xx)))))))) as proof of (((and (x1 Xx)) (x1 Xy))->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found eq_ref000:=(eq_ref00 (fun (x10:Prop)=> (x1 Xy))):((x1 Xy)->(x1 Xy))
% Found (eq_ref00 (fun (x10:Prop)=> (x1 Xy))) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found ((eq_ref0 ((R Xy) Xx)) (fun (x10:Prop)=> (x1 Xy))) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x1 Xy))) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x1 Xy))) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x1 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x1 Xy)))) as proof of ((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx)))
% Found (fun (x9:(x1 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x1 Xy)))) as proof of ((x1 Xx)->((x1 Xy)->((or ((R Xx) Xy)) ((R Xy) Xx))))
% Found (and_rect20 (fun (x9:(x1 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x1 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found ((and_rect2 ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x1 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x1 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (((fun (P:Type) (x9:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x1 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x1 Xy))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (x00:((and (x1 Xx)) (x1 Xy)))=> (((fun (P:Type) (x9:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 Xy)) P) x9) x00)) ((or ((R Xx) Xy)) ((R Xy) Xx))) (fun (x9:(x1 Xx))=> (((eq_ref Prop) ((R Xy) Xx)) (fun (x10:Prop)=> (x1 Xy)))))) as proof of ((or ((R Xx) Xy)) ((R Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x1 Xx)) (x1 Xy)))=> (((fun (P:Type) (x9:((x1 Xx)->((x1 Xy)->P)))=> (((((and_rect (x1 Xx)) (x1 X
% EOF
%------------------------------------------------------------------------------