%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV281^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n114.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 13:33:59 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  : SEV281^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n114.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 08:43:21 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 0x14b0560>, <kernel.Type object at 0x14b0ea8>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0xeae128>, <kernel.DependentProduct object at 0x14b08c0>) of role type named cR
% Using role type
% Declaring cR:(a->(a->Prop))
% FOF formula (((and ((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx))))))))) of role conjecture named cTHM548_pme
% Conjecture to prove = (((and ((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx))))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(((and ((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))))']
% Parameter a:Type.
% Parameter cR:(a->(a->Prop)).
% Trying to prove (((and ((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(x0 Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(x0 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))) (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(x0 Xx)))
% Found x000:(T Xx)
% Instantiate: x0:=T:(a->Prop)
% Found (fun (x000:(T Xx))=> x000) as proof of (x0 Xx)
% Found (fun (Xx:a) (x000:(T Xx))=> x000) as proof of ((T Xx)->(x0 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))) (Xx:a) (x000:(T Xx))=> x000) as proof of (forall (Xx:a), ((T Xx)->(x0 Xx)))
% Found x01:(T Xx)
% Instantiate: x2:=T:(a->Prop)
% Found (fun (x01:(T Xx))=> x01) as proof of (x2 Xx)
% Found (fun (Xx:a) (x01:(T Xx))=> x01) as proof of ((T Xx)->(x2 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))) (Xx:a) (x01:(T Xx))=> x01) as proof of (forall (Xx:a), ((T Xx)->(x2 Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(x2 Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(x2 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x2 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x2 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(x2 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))) (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(x2 Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(x0 Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(x0 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))) (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(x0 Xx)))
% Found x000:(T Xx)
% Instantiate: x0:=T:(a->Prop)
% Found (fun (x000:(T Xx))=> x000) as proof of (x0 Xx)
% Found (fun (Xx:a) (x000:(T Xx))=> x000) as proof of ((T Xx)->(x0 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))) (Xx:a) (x000:(T Xx))=> x000) as proof of (forall (Xx:a), ((T Xx)->(x0 Xx)))
% Found eq_ref000:=(eq_ref00 (fun (x2:Prop)=> (x0 Xy))):((x0 Xy)->(x0 Xy))
% Found (eq_ref00 (fun (x2:Prop)=> (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((eq_ref0 ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (((eq_ref Prop) ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (((eq_ref Prop) ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy)))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy)))) as proof of ((x0 Xx)->((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect00 (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect0 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x1:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x1) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P:Type) (x1:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x1) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P:Type) (x1:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x1) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy)))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(x4 Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(x4 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x4 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x4 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(x4 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))) (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(x4 Xx)))
% Found x01:(T Xx)
% Instantiate: x4:=T:(a->Prop)
% Found (fun (x01:(T Xx))=> x01) as proof of (x4 Xx)
% Found (fun (Xx:a) (x01:(T Xx))=> x01) as proof of ((T Xx)->(x4 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))) (Xx:a) (x01:(T Xx))=> x01) as proof of (forall (Xx:a), ((T Xx)->(x4 Xx)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))) (fun (x:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (x Xx)) (x Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x Xx))))))))
% Found (eta_expansion_dep00 (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))) b)
% Found ((eta_expansion_dep0 (fun (x1:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x1:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x1:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x1:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))) b)
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(x0 Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(x0 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))) (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(x0 Xx)))
% Found x000:(T Xx)
% Instantiate: x0:=T:(a->Prop)
% Found (fun (x000:(T Xx))=> x000) as proof of (x0 Xx)
% Found (fun (Xx:a) (x000:(T Xx))=> x000) as proof of ((T Xx)->(x0 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))) (Xx:a) (x000:(T Xx))=> x000) as proof of (forall (Xx:a), ((T Xx)->(x0 Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(x2 Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(x2 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x2 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x2 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(x2 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))) (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(x2 Xx)))
% Found x01:(T Xx)
% Instantiate: x2:=T:(a->Prop)
% Found (fun (x01:(T Xx))=> x01) as proof of (x2 Xx)
% Found (fun (Xx:a) (x01:(T Xx))=> x01) as proof of ((T Xx)->(x2 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))) (Xx:a) (x01:(T Xx))=> x01) as proof of (forall (Xx:a), ((T Xx)->(x2 Xx)))
% Found or_intror00:=(or_intror0 (x2 Xy)):((x2 Xy)->((or ((cR Xx) Xy)) (x2 Xy)))
% Found (or_intror0 (x2 Xy)) as proof of ((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_intror ((cR Xx) Xy)) (x2 Xy)) as proof of ((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_intror ((cR Xx) Xy)) (x2 Xy)) as proof of ((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x3:(x2 Xx))=> ((or_intror ((cR Xx) Xy)) (x2 Xy))) as proof of ((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x3:(x2 Xx))=> ((or_intror ((cR Xx) Xy)) (x2 Xy))) as proof of ((x2 Xx)->((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect10 (fun (x3:(x2 Xx))=> ((or_intror ((cR Xx) Xy)) (x2 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect1 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x2 Xx))=> ((or_intror ((cR Xx) Xy)) (x2 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x3:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x2 Xx))=> ((or_intror ((cR Xx) Xy)) (x2 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x2 Xx)) (x2 Xy)))=> (((fun (P:Type) (x3:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x2 Xx))=> ((or_intror ((cR Xx) Xy)) (x2 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x2 Xx)) (x2 Xy)))=> (((fun (P:Type) (x3:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x2 Xx))=> ((or_intror ((cR Xx) Xy)) (x2 Xy))))) as proof of (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found eta_expansion000:=(eta_expansion00 (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))) (fun (x:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (x Xx)) (x Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x Xx))))))))
% Found (eta_expansion00 (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))) b)
% Found ((eta_expansion0 Prop) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))) b)
% Found eq_ref000:=(eq_ref00 (fun (x4:Prop)=> (x0 Xy))):((x0 Xy)->(x0 Xy))
% Found (eq_ref00 (fun (x4:Prop)=> (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((eq_ref0 ((cR Xy) Xx)) (fun (x4:Prop)=> (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (((eq_ref Prop) ((cR Xy) Xx)) (fun (x4:Prop)=> (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (((eq_ref Prop) ((cR Xy) Xx)) (fun (x4:Prop)=> (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x3:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x4:Prop)=> (x0 Xy)))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x3:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x4:Prop)=> (x0 Xy)))) as proof of ((x0 Xx)->((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect10 (fun (x3:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x4:Prop)=> (x0 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect1 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x4:Prop)=> (x0 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x4:Prop)=> (x0 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x4:Prop)=> (x0 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x4:Prop)=> (x0 Xy)))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(x6 Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(x6 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x6 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x6 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x6 Xx)->(T Xx))))) (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(x6 Xx)))
% Found x01:(T Xx)
% Instantiate: x6:=T:(a->Prop)
% Found (fun (x01:(T Xx))=> x01) as proof of (x6 Xx)
% Found (fun (Xx:a) (x01:(T Xx))=> x01) as proof of ((T Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x6 Xx)->(T Xx))))) (Xx:a) (x01:(T Xx))=> x01) as proof of (forall (Xx:a), ((T Xx)->(x6 Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(x0 Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(x0 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))) (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(x0 Xx)))
% Found x000:(T Xx)
% Instantiate: x0:=T:(a->Prop)
% Found (fun (x000:(T Xx))=> x000) as proof of (x0 Xx)
% Found (fun (Xx:a) (x000:(T Xx))=> x000) as proof of ((T Xx)->(x0 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))) (Xx:a) (x000:(T Xx))=> x000) as proof of (forall (Xx:a), ((T Xx)->(x0 Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(x2 Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(x2 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x2 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x2 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(x2 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))) (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(x2 Xx)))
% Found x01:(T Xx)
% Instantiate: x2:=T:(a->Prop)
% Found (fun (x01:(T Xx))=> x01) as proof of (x2 Xx)
% Found (fun (Xx:a) (x01:(T Xx))=> x01) as proof of ((T Xx)->(x2 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))) (Xx:a) (x01:(T Xx))=> x01) as proof of (forall (Xx:a), ((T Xx)->(x2 Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(x4 Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(x4 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x4 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x4 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(x4 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))) (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(x4 Xx)))
% Found x01:(T Xx)
% Instantiate: x4:=T:(a->Prop)
% Found (fun (x01:(T Xx))=> x01) as proof of (x4 Xx)
% Found (fun (Xx:a) (x01:(T Xx))=> x01) as proof of ((T Xx)->(x4 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))) (Xx:a) (x01:(T Xx))=> x01) as proof of (forall (Xx:a), ((T Xx)->(x4 Xx)))
% Found eq_ref000:=(eq_ref00 (fun (x2:a)=> (x0 Xy))):((x0 Xy)->(x0 Xy))
% Found (eq_ref00 (fun (x2:a)=> (x0 Xy))) as proof of ((x0 Xy)->((cR Xy) Xx))
% Found ((eq_ref0 Xx) (fun (x2:a)=> (x0 Xy))) as proof of ((x0 Xy)->((cR Xy) Xx))
% Found (((eq_ref a) Xx) (fun (x2:a)=> (x0 Xy))) as proof of ((x0 Xy)->((cR Xy) Xx))
% Found (((eq_ref a) Xx) (fun (x2:a)=> (x0 Xy))) as proof of ((x0 Xy)->((cR Xy) Xx))
% Found (fun (x1:(x0 Xx))=> (((eq_ref a) Xx) (fun (x2:a)=> (x0 Xy)))) as proof of ((x0 Xy)->((cR Xy) Xx))
% Found (fun (x1:(x0 Xx))=> (((eq_ref a) Xx) (fun (x2:a)=> (x0 Xy)))) as proof of ((x0 Xx)->((x0 Xy)->((cR Xy) Xx)))
% Found (and_rect00 (fun (x1:(x0 Xx))=> (((eq_ref a) Xx) (fun (x2:a)=> (x0 Xy))))) as proof of ((cR Xy) Xx)
% Found ((and_rect0 ((cR Xy) Xx)) (fun (x1:(x0 Xx))=> (((eq_ref a) Xx) (fun (x2:a)=> (x0 Xy))))) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x1:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x1) x00)) ((cR Xy) Xx)) (fun (x1:(x0 Xx))=> (((eq_ref a) Xx) (fun (x2:a)=> (x0 Xy))))) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x1:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x1) x00)) ((cR Xy) Xx)) (fun (x1:(x0 Xx))=> (((eq_ref a) Xx) (fun (x2:a)=> (x0 Xy))))) as proof of ((cR Xy) Xx)
% Found (or_intror00 (((fun (P:Type) (x1:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x1) x00)) ((cR Xy) Xx)) (fun (x1:(x0 Xx))=> (((eq_ref a) Xx) (fun (x2:a)=> (x0 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_intror0 ((cR Xy) Xx)) (((fun (P:Type) (x1:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x1) x00)) ((cR Xy) Xx)) (fun (x1:(x0 Xx))=> (((eq_ref a) Xx) (fun (x2:a)=> (x0 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x1:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x1) x00)) ((cR Xy) Xx)) (fun (x1:(x0 Xx))=> (((eq_ref a) Xx) (fun (x2:a)=> (x0 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x1:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x1) x00)) ((cR Xy) Xx)) (fun (x1:(x0 Xx))=> (((eq_ref a) Xx) (fun (x2:a)=> (x0 Xy))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x1:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x1) x00)) ((cR Xy) Xx)) (fun (x1:(x0 Xx))=> (((eq_ref a) Xx) (fun (x2:a)=> (x0 Xy))))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found x2:(x0 Xy)
% Instantiate: x0:=(cR Xx):(a->Prop)
% Found (fun (x2:(x0 Xy))=> x2) as proof of ((cR Xx) Xy)
% Found (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2) as proof of ((x0 Xy)->((cR Xx) Xy))
% Found (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2) as proof of ((x0 Xx)->((x0 Xy)->((cR Xx) Xy)))
% Found (and_rect00 (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2)) as proof of ((cR Xx) Xy)
% Found ((and_rect0 ((cR Xx) Xy)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2)) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x1:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x1) x00)) ((cR Xx) Xy)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2)) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x1:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x1) x00)) ((cR Xx) Xy)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2)) as proof of ((cR Xx) Xy)
% Found (or_introl00 (((fun (P:Type) (x1:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x1) x00)) ((cR Xx) Xy)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_introl0 ((cR Xy) Xx)) (((fun (P:Type) (x1:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x1) x00)) ((cR Xx) Xy)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x1:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x1) x00)) ((cR Xx) Xy)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x1:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x1) x00)) ((cR Xx) Xy)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x1:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x1) x00)) ((cR Xx) Xy)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2)))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found or_introl00:=(or_introl0 ((cR Xx) Xy)):((x0 Xy)->((or (x0 Xy)) ((cR Xx) Xy)))
% Found (or_introl0 ((cR Xx) Xy)) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((or_introl (x0 Xy)) ((cR Xx) Xy)) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((or_introl (x0 Xy)) ((cR Xx) Xy)) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x1:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy))) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x1:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy))) as proof of ((x0 Xx)->((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy))))
% Found (and_rect00 (fun (x1:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found ((and_rect0 ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x1:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x1:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x1) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x1:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x1:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x1) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x1:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (or_comm_i00 (((fun (P:Type) (x1:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x1) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x1:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_comm_i0 ((cR Xx) Xy)) (((fun (P:Type) (x1:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x1) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x1:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x1:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x1) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x1:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x1:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x1) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x1:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x1:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x1) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x1:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))) (fun (x:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (x Xx)) (x Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x Xx))))))))
% Found (eta_expansion_dep00 (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))) b)
% Found or_introl00:=(or_introl0 ((cR Xy) Xx)):((x4 Xy)->((or (x4 Xy)) ((cR Xy) Xx)))
% Found (or_introl0 ((cR Xy) Xx)) as proof of ((x4 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_introl (x4 Xy)) ((cR Xy) Xx)) as proof of ((x4 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_introl (x4 Xy)) ((cR Xy) Xx)) as proof of ((x4 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x5:(x4 Xx))=> ((or_introl (x4 Xy)) ((cR Xy) Xx))) as proof of ((x4 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x5:(x4 Xx))=> ((or_introl (x4 Xy)) ((cR Xy) Xx))) as proof of ((x4 Xx)->((x4 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect20 (fun (x5:(x4 Xx))=> ((or_introl (x4 Xy)) ((cR Xy) Xx)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect2 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x4 Xx))=> ((or_introl (x4 Xy)) ((cR Xy) Xx)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x4 Xx))=> ((or_introl (x4 Xy)) ((cR Xy) Xx)))) as proof of ((or ((cR Xx) Xy)) ((cR 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 ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x4 Xx))=> ((or_introl (x4 Xy)) ((cR Xy) Xx))))) as proof of ((or ((cR Xx) Xy)) ((cR 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 ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x4 Xx))=> ((or_introl (x4 Xy)) ((cR Xy) Xx))))) as proof of (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found or_intror00:=(or_intror0 (x0 Xy)):((x0 Xy)->((or ((cR Xx) Xy)) (x0 Xy)))
% Found (or_intror0 (x0 Xy)) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_intror ((cR Xx) Xy)) (x0 Xy)) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_intror ((cR Xx) Xy)) (x0 Xy)) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x5:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x5:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy))) as proof of ((x0 Xx)->((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect20 (fun (x5:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect2 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found eq_ref000:=(eq_ref00 (fun (x6:Prop)=> (x2 Xy))):((x2 Xy)->(x2 Xy))
% Found (eq_ref00 (fun (x6:Prop)=> (x2 Xy))) as proof of ((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((eq_ref0 ((cR Xy) Xx)) (fun (x6:Prop)=> (x2 Xy))) as proof of ((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (((eq_ref Prop) ((cR Xy) Xx)) (fun (x6:Prop)=> (x2 Xy))) as proof of ((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (((eq_ref Prop) ((cR Xy) Xx)) (fun (x6:Prop)=> (x2 Xy))) as proof of ((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x5:(x2 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x6:Prop)=> (x2 Xy)))) as proof of ((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x5:(x2 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x6:Prop)=> (x2 Xy)))) as proof of ((x2 Xx)->((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect20 (fun (x5:(x2 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x6:Prop)=> (x2 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect2 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x2 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x6:Prop)=> (x2 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x2 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x6:Prop)=> (x2 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x2 Xx)) (x2 Xy)))=> (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x2 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x6:Prop)=> (x2 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x2 Xx)) (x2 Xy)))=> (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x2 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x6:Prop)=> (x2 Xy)))))) as proof of (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found and_rect00:=(and_rect0 ((or ((cR Xx) Xy)) ((cR Xy) Xx))):((((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))->((forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0)))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Instantiate: x0:=(fun (x3:a)=> (((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))->((forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0)))->((or ((cR Xx) x3)) ((cR x3) Xx))))):(a->Prop)
% Found (fun (x3:(x0 Xx))=> and_rect00) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x3:(x0 Xx))=> and_rect00) as proof of ((x0 Xx)->((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect10 (fun (x3:(x0 Xx))=> and_rect00)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect1 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> and_rect00)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> and_rect00)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x2:(forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> and_rect00))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x1:((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))) (x2:(forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> and_rect00))) as proof of ((forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0)))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x1:((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))) (x2:(forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> and_rect00))) as proof of (((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))->((forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0)))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect00 (fun (x1:((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))) (x2:(forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> and_rect00)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect0 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x1:((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))) (x2:(forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> (and_rect0 ((or ((cR Xx) Xy)) ((cR Xy) Xx))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x1:(((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))->P)))=> (((((and_rect ((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))) P) x1) x)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x1:((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))) (x2:(forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> ((fun (P:Type) (x1:(((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))->P)))=> (((((and_rect ((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))) P) x1) x)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P:Type) (x1:(((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))->P)))=> (((((and_rect ((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))) P) x1) x)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x1:((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))) (x2:(forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> ((fun (P:Type) (x1:(((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))->P)))=> (((((and_rect ((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))) P) x1) x)) ((or ((cR Xx) Xy)) ((cR Xy) Xx)))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P:Type) (x1:(((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))->P)))=> (((((and_rect ((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))) P) x1) x)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x1:((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))) (x2:(forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> ((fun (P:Type) (x1:(((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))->P)))=> (((((and_rect ((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))) P) x1) x)) ((or ((cR Xx) Xy)) ((cR Xy) Xx)))))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found eq_ref000:=(eq_ref00 (fun (x4:a)=> (x2 Xy))):((x2 Xy)->(x2 Xy))
% Found (eq_ref00 (fun (x4:a)=> (x2 Xy))) as proof of ((x2 Xy)->((cR Xy) Xx))
% Found ((eq_ref0 Xx) (fun (x4:a)=> (x2 Xy))) as proof of ((x2 Xy)->((cR Xy) Xx))
% Found (((eq_ref a) Xx) (fun (x4:a)=> (x2 Xy))) as proof of ((x2 Xy)->((cR Xy) Xx))
% Found (((eq_ref a) Xx) (fun (x4:a)=> (x2 Xy))) as proof of ((x2 Xy)->((cR Xy) Xx))
% Found (fun (x3:(x2 Xx))=> (((eq_ref a) Xx) (fun (x4:a)=> (x2 Xy)))) as proof of ((x2 Xy)->((cR Xy) Xx))
% Found (fun (x3:(x2 Xx))=> (((eq_ref a) Xx) (fun (x4:a)=> (x2 Xy)))) as proof of ((x2 Xx)->((x2 Xy)->((cR Xy) Xx)))
% Found (and_rect10 (fun (x3:(x2 Xx))=> (((eq_ref a) Xx) (fun (x4:a)=> (x2 Xy))))) as proof of ((cR Xy) Xx)
% Found ((and_rect1 ((cR Xy) Xx)) (fun (x3:(x2 Xx))=> (((eq_ref a) Xx) (fun (x4:a)=> (x2 Xy))))) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x3:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x3) x00)) ((cR Xy) Xx)) (fun (x3:(x2 Xx))=> (((eq_ref a) Xx) (fun (x4:a)=> (x2 Xy))))) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x3:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x3) x00)) ((cR Xy) Xx)) (fun (x3:(x2 Xx))=> (((eq_ref a) Xx) (fun (x4:a)=> (x2 Xy))))) as proof of ((cR Xy) Xx)
% Found (or_intror00 (((fun (P:Type) (x3:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x3) x00)) ((cR Xy) Xx)) (fun (x3:(x2 Xx))=> (((eq_ref a) Xx) (fun (x4:a)=> (x2 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_intror0 ((cR Xy) Xx)) (((fun (P:Type) (x3:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x3) x00)) ((cR Xy) Xx)) (fun (x3:(x2 Xx))=> (((eq_ref a) Xx) (fun (x4:a)=> (x2 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x3:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x3) x00)) ((cR Xy) Xx)) (fun (x3:(x2 Xx))=> (((eq_ref a) Xx) (fun (x4:a)=> (x2 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x2 Xx)) (x2 Xy)))=> (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x3:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x3) x00)) ((cR Xy) Xx)) (fun (x3:(x2 Xx))=> (((eq_ref a) Xx) (fun (x4:a)=> (x2 Xy))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x2 Xx)) (x2 Xy)))=> (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x3:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x3) x00)) ((cR Xy) Xx)) (fun (x3:(x2 Xx))=> (((eq_ref a) Xx) (fun (x4:a)=> (x2 Xy))))))) as proof of (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found x4:(x2 Xy)
% Instantiate: x2:=(cR Xx):(a->Prop)
% Found (fun (x4:(x2 Xy))=> x4) as proof of ((cR Xx) Xy)
% Found (fun (x3:(x2 Xx)) (x4:(x2 Xy))=> x4) as proof of ((x2 Xy)->((cR Xx) Xy))
% Found (fun (x3:(x2 Xx)) (x4:(x2 Xy))=> x4) as proof of ((x2 Xx)->((x2 Xy)->((cR Xx) Xy)))
% Found (and_rect10 (fun (x3:(x2 Xx)) (x4:(x2 Xy))=> x4)) as proof of ((cR Xx) Xy)
% Found ((and_rect1 ((cR Xx) Xy)) (fun (x3:(x2 Xx)) (x4:(x2 Xy))=> x4)) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x3:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x3) x00)) ((cR Xx) Xy)) (fun (x3:(x2 Xx)) (x4:(x2 Xy))=> x4)) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x3:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x3) x00)) ((cR Xx) Xy)) (fun (x3:(x2 Xx)) (x4:(x2 Xy))=> x4)) as proof of ((cR Xx) Xy)
% Found (or_introl00 (((fun (P:Type) (x3:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x3) x00)) ((cR Xx) Xy)) (fun (x3:(x2 Xx)) (x4:(x2 Xy))=> x4))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_introl0 ((cR Xy) Xx)) (((fun (P:Type) (x3:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x3) x00)) ((cR Xx) Xy)) (fun (x3:(x2 Xx)) (x4:(x2 Xy))=> x4))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x3:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x3) x00)) ((cR Xx) Xy)) (fun (x3:(x2 Xx)) (x4:(x2 Xy))=> x4))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x2 Xx)) (x2 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x3:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x3) x00)) ((cR Xx) Xy)) (fun (x3:(x2 Xx)) (x4:(x2 Xy))=> x4)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x2 Xx)) (x2 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x3:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x3) x00)) ((cR Xx) Xy)) (fun (x3:(x2 Xx)) (x4:(x2 Xy))=> x4)))) as proof of (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found or_introl00:=(or_introl0 ((cR Xx) Xy)):((x2 Xy)->((or (x2 Xy)) ((cR Xx) Xy)))
% Found (or_introl0 ((cR Xx) Xy)) as proof of ((x2 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((or_introl (x2 Xy)) ((cR Xx) Xy)) as proof of ((x2 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((or_introl (x2 Xy)) ((cR Xx) Xy)) as proof of ((x2 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x3:(x2 Xx))=> ((or_introl (x2 Xy)) ((cR Xx) Xy))) as proof of ((x2 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x3:(x2 Xx))=> ((or_introl (x2 Xy)) ((cR Xx) Xy))) as proof of ((x2 Xx)->((x2 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy))))
% Found (and_rect10 (fun (x3:(x2 Xx))=> ((or_introl (x2 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found ((and_rect1 ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x3:(x2 Xx))=> ((or_introl (x2 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x3:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x3) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x3:(x2 Xx))=> ((or_introl (x2 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x3:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x3) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x3:(x2 Xx))=> ((or_introl (x2 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (or_comm_i00 (((fun (P:Type) (x3:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x3) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x3:(x2 Xx))=> ((or_introl (x2 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_comm_i0 ((cR Xx) Xy)) (((fun (P:Type) (x3:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x3) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x3:(x2 Xx))=> ((or_introl (x2 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x3:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x3) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x3:(x2 Xx))=> ((or_introl (x2 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x2 Xx)) (x2 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x3:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x3) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x3:(x2 Xx))=> ((or_introl (x2 Xy)) ((cR Xx) Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x2 Xx)) (x2 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x3:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x3) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x3:(x2 Xx))=> ((or_introl (x2 Xy)) ((cR Xx) Xy)))))) as proof of (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(x8 Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x8 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x8 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x8 Xx)->(T Xx))))) (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(x8 Xx)))
% Found x01:(T Xx)
% Instantiate: x8:=T:(a->Prop)
% Found (fun (x01:(T Xx))=> x01) as proof of (x8 Xx)
% Found (fun (Xx:a) (x01:(T Xx))=> x01) as proof of ((T Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x8 Xx)->(T Xx))))) (Xx:a) (x01:(T Xx))=> x01) as proof of (forall (Xx:a), ((T Xx)->(x8 Xx)))
% Found eta_expansion000:=(eta_expansion00 (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))) (fun (x:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (x Xx)) (x Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x Xx))))))))
% Found (eta_expansion00 (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))) b)
% Found ((eta_expansion0 Prop) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))) b)
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(x0 Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(x0 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))) (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(x0 Xx)))
% Found x000:(T Xx)
% Instantiate: x0:=T:(a->Prop)
% Found (fun (x000:(T Xx))=> x000) as proof of (x0 Xx)
% Found (fun (Xx:a) (x000:(T Xx))=> x000) as proof of ((T Xx)->(x0 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))) (Xx:a) (x000:(T Xx))=> x000) as proof of (forall (Xx:a), ((T Xx)->(x0 Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(x2 Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(x2 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x2 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x2 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(x2 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))) (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(x2 Xx)))
% Found x01:(T Xx)
% Instantiate: x2:=T:(a->Prop)
% Found (fun (x01:(T Xx))=> x01) as proof of (x2 Xx)
% Found (fun (Xx:a) (x01:(T Xx))=> x01) as proof of ((T Xx)->(x2 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))) (Xx:a) (x01:(T Xx))=> x01) as proof of (forall (Xx:a), ((T Xx)->(x2 Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(x4 Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(x4 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x4 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x4 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(x4 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))) (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(x4 Xx)))
% Found eq_ref000:=(eq_ref00 x0):((x0 Xy)->(x0 Xy))
% Found (eq_ref00 x0) as proof of ((x0 Xy)->((cR Xx) Xy))
% Found ((eq_ref0 Xy) x0) as proof of ((x0 Xy)->((cR Xx) Xy))
% Found (((eq_ref a) Xy) x0) as proof of ((x0 Xy)->((cR Xx) Xy))
% Found (((eq_ref a) Xy) x0) as proof of ((x0 Xy)->((cR Xx) Xy))
% Found (fun (x3:(x0 Xx))=> (((eq_ref a) Xy) x0)) as proof of ((x0 Xy)->((cR Xx) Xy))
% Found (fun (x3:(x0 Xx))=> (((eq_ref a) Xy) x0)) as proof of ((x0 Xx)->((x0 Xy)->((cR Xx) Xy)))
% Found (and_rect10 (fun (x3:(x0 Xx))=> (((eq_ref a) Xy) x0))) as proof of ((cR Xx) Xy)
% Found ((and_rect1 ((cR Xx) Xy)) (fun (x3:(x0 Xx))=> (((eq_ref a) Xy) x0))) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((cR Xx) Xy)) (fun (x3:(x0 Xx))=> (((eq_ref a) Xy) x0))) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((cR Xx) Xy)) (fun (x3:(x0 Xx))=> (((eq_ref a) Xy) x0))) as proof of ((cR Xx) Xy)
% Found (or_introl00 (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((cR Xx) Xy)) (fun (x3:(x0 Xx))=> (((eq_ref a) Xy) x0)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_introl0 ((cR Xy) Xx)) (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((cR Xx) Xy)) (fun (x3:(x0 Xx))=> (((eq_ref a) Xy) x0)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((cR Xx) Xy)) (fun (x3:(x0 Xx))=> (((eq_ref a) Xy) x0)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((cR Xx) Xy)) (fun (x3:(x0 Xx))=> (((eq_ref a) Xy) x0))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((cR Xx) Xy)) (fun (x3:(x0 Xx))=> (((eq_ref a) Xy) x0))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found eq_ref000:=(eq_ref00 (fun (x4:a)=> (x0 Xy))):((x0 Xy)->(x0 Xy))
% Found (eq_ref00 (fun (x4:a)=> (x0 Xy))) as proof of ((x0 Xy)->((cR Xy) Xx))
% Found ((eq_ref0 Xx) (fun (x4:a)=> (x0 Xy))) as proof of ((x0 Xy)->((cR Xy) Xx))
% Found (((eq_ref a) Xx) (fun (x4:a)=> (x0 Xy))) as proof of ((x0 Xy)->((cR Xy) Xx))
% Found (((eq_ref a) Xx) (fun (x4:a)=> (x0 Xy))) as proof of ((x0 Xy)->((cR Xy) Xx))
% Found (fun (x3:(x0 Xx))=> (((eq_ref a) Xx) (fun (x4:a)=> (x0 Xy)))) as proof of ((x0 Xy)->((cR Xy) Xx))
% Found (fun (x3:(x0 Xx))=> (((eq_ref a) Xx) (fun (x4:a)=> (x0 Xy)))) as proof of ((x0 Xx)->((x0 Xy)->((cR Xy) Xx)))
% Found (and_rect10 (fun (x3:(x0 Xx))=> (((eq_ref a) Xx) (fun (x4:a)=> (x0 Xy))))) as proof of ((cR Xy) Xx)
% Found ((and_rect1 ((cR Xy) Xx)) (fun (x3:(x0 Xx))=> (((eq_ref a) Xx) (fun (x4:a)=> (x0 Xy))))) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((cR Xy) Xx)) (fun (x3:(x0 Xx))=> (((eq_ref a) Xx) (fun (x4:a)=> (x0 Xy))))) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((cR Xy) Xx)) (fun (x3:(x0 Xx))=> (((eq_ref a) Xx) (fun (x4:a)=> (x0 Xy))))) as proof of ((cR Xy) Xx)
% Found (or_intror00 (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((cR Xy) Xx)) (fun (x3:(x0 Xx))=> (((eq_ref a) Xx) (fun (x4:a)=> (x0 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_intror0 ((cR Xy) Xx)) (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((cR Xy) Xx)) (fun (x3:(x0 Xx))=> (((eq_ref a) Xx) (fun (x4:a)=> (x0 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((cR Xy) Xx)) (fun (x3:(x0 Xx))=> (((eq_ref a) Xx) (fun (x4:a)=> (x0 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((cR Xy) Xx)) (fun (x3:(x0 Xx))=> (((eq_ref a) Xx) (fun (x4:a)=> (x0 Xy))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((cR Xy) Xx)) (fun (x3:(x0 Xx))=> (((eq_ref a) Xx) (fun (x4:a)=> (x0 Xy))))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found eq_ref000:=(eq_ref00 (fun (x4:Prop)=> (x0 Xy))):((x0 Xy)->(x0 Xy))
% Found (eq_ref00 (fun (x4:Prop)=> (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((eq_ref0 ((cR Xx) Xy)) (fun (x4:Prop)=> (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (((eq_ref Prop) ((cR Xx) Xy)) (fun (x4:Prop)=> (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (((eq_ref Prop) ((cR Xx) Xy)) (fun (x4:Prop)=> (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x3:(x0 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x4:Prop)=> (x0 Xy)))) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x3:(x0 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x4:Prop)=> (x0 Xy)))) as proof of ((x0 Xx)->((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy))))
% Found (and_rect10 (fun (x3:(x0 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x4:Prop)=> (x0 Xy))))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found ((and_rect1 ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x3:(x0 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x4:Prop)=> (x0 Xy))))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x3:(x0 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x4:Prop)=> (x0 Xy))))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x3:(x0 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x4:Prop)=> (x0 Xy))))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (or_comm_i00 (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x3:(x0 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x4:Prop)=> (x0 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_comm_i0 ((cR Xx) Xy)) (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x3:(x0 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x4:Prop)=> (x0 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x3:(x0 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x4:Prop)=> (x0 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x3:(x0 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x4:Prop)=> (x0 Xy))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x3:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x3) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x3:(x0 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x4:Prop)=> (x0 Xy))))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found x01:(T Xx)
% Instantiate: x4:=T:(a->Prop)
% Found (fun (x01:(T Xx))=> x01) as proof of (x4 Xx)
% Found (fun (Xx:a) (x01:(T Xx))=> x01) as proof of ((T Xx)->(x4 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))) (Xx:a) (x01:(T Xx))=> x01) as proof of (forall (Xx:a), ((T Xx)->(x4 Xx)))
% Found x01:(T Xx)
% Instantiate: x6:=T:(a->Prop)
% Found (fun (x01:(T Xx))=> x01) as proof of (x6 Xx)
% Found (fun (Xx:a) (x01:(T Xx))=> x01) as proof of ((T Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x6 Xx)->(T Xx))))) (Xx:a) (x01:(T Xx))=> x01) as proof of (forall (Xx:a), ((T Xx)->(x6 Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(x6 Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(x6 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x6 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x6 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x6 Xx)->(T Xx))))) (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(x6 Xx)))
% Found or_intror00:=(or_intror0 (x6 Xy)):((x6 Xy)->((or ((cR Xx) Xy)) (x6 Xy)))
% Found (or_intror0 (x6 Xy)) as proof of ((x6 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_intror ((cR Xx) Xy)) (x6 Xy)) as proof of ((x6 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_intror ((cR Xx) Xy)) (x6 Xy)) as proof of ((x6 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x7:(x6 Xx))=> ((or_intror ((cR Xx) Xy)) (x6 Xy))) as proof of ((x6 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x7:(x6 Xx))=> ((or_intror ((cR Xx) Xy)) (x6 Xy))) as proof of ((x6 Xx)->((x6 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect30 (fun (x7:(x6 Xx))=> ((or_intror ((cR Xx) Xy)) (x6 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect3 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x6 Xx))=> ((or_intror ((cR Xx) Xy)) (x6 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x6 Xx))=> ((or_intror ((cR Xx) Xy)) (x6 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR 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 ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x6 Xx))=> ((or_intror ((cR Xx) Xy)) (x6 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR 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 ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x6 Xx))=> ((or_intror ((cR Xx) Xy)) (x6 Xy))))) as proof of (((and (x6 Xx)) (x6 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found and_rect10:=(and_rect1 ((or ((cR Xx) Xy)) ((cR Xy) Xx))):((((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))->((forall (Xx0:a), ((cR Xx0) Xx0))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Instantiate: x0:=(fun (x5:a)=> (((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))->((forall (Xx0:a), ((cR Xx0) Xx0))->((or ((cR Xx) x5)) ((cR x5) Xx))))):(a->Prop)
% Found (fun (x5:(x0 Xx))=> and_rect10) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x5:(x0 Xx))=> and_rect10) as proof of ((x0 Xx)->((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect20 (fun (x5:(x0 Xx))=> and_rect10)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect2 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x0 Xx))=> and_rect10)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x0 Xx))=> and_rect10)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x0 Xx))=> and_rect10))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x0 Xx))=> and_rect10))) as proof of ((forall (Xx0:a), ((cR Xx0) Xx0))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x0 Xx))=> and_rect10))) as proof of (((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))->((forall (Xx0:a), ((cR Xx0) Xx0))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect10 (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x0 Xx))=> and_rect10)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect1 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x0 Xx))=> (and_rect1 ((or ((cR Xx) Xy)) ((cR Xy) Xx))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))->((forall (Xx:a), ((cR Xx) Xx))->P)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx))) P) x3) x1)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x0 Xx))=> ((fun (P:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))->((forall (Xx:a), ((cR Xx) Xx))->P)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx))) P) x3) x1)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))->((forall (Xx:a), ((cR Xx) Xx))->P)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx))) P) x3) x1)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x0 Xx))=> ((fun (P:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))->((forall (Xx:a), ((cR Xx) Xx))->P)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx))) P) x3) x1)) ((or ((cR Xx) Xy)) ((cR Xy) Xx)))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))->((forall (Xx:a), ((cR Xx) Xx))->P)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx))) P) x3) x1)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x0 Xx))=> ((fun (P:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))->((forall (Xx:a), ((cR Xx) Xx))->P)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx))) P) x3) x1)) ((or ((cR Xx) Xy)) ((cR Xy) Xx)))))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found and_rect10:=(and_rect1 ((or ((cR Xx) Xy)) ((cR Xy) Xx))):((((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))->((forall (Xx0:a), ((cR Xx0) Xx0))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Instantiate: x2:=(fun (x5:a)=> (((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))->((forall (Xx0:a), ((cR Xx0) Xx0))->((or ((cR Xx) x5)) ((cR x5) Xx))))):(a->Prop)
% Found (fun (x5:(x2 Xx))=> and_rect10) as proof of ((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x5:(x2 Xx))=> and_rect10) as proof of ((x2 Xx)->((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect20 (fun (x5:(x2 Xx))=> and_rect10)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect2 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x2 Xx))=> and_rect10)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x2 Xx))=> and_rect10)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x2 Xx))=> and_rect10))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x2 Xx))=> and_rect10))) as proof of ((forall (Xx0:a), ((cR Xx0) Xx0))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x2 Xx))=> and_rect10))) as proof of (((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))->((forall (Xx0:a), ((cR Xx0) Xx0))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect10 (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x2 Xx))=> and_rect10)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect1 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x2 Xx))=> (and_rect1 ((or ((cR Xx) Xy)) ((cR Xy) Xx))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))->((forall (Xx:a), ((cR Xx) Xx))->P)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx))) P) x3) x0)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x2 Xx))=> ((fun (P:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))->((forall (Xx:a), ((cR Xx) Xx))->P)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx))) P) x3) x0)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x2 Xx)) (x2 Xy)))=> (((fun (P:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))->((forall (Xx:a), ((cR Xx) Xx))->P)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx))) P) x3) x0)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x2 Xx))=> ((fun (P:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))->((forall (Xx:a), ((cR Xx) Xx))->P)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx))) P) x3) x0)) ((or ((cR Xx) Xy)) ((cR Xy) Xx)))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x2 Xx)) (x2 Xy)))=> (((fun (P:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))->((forall (Xx:a), ((cR Xx) Xx))->P)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx))) P) x3) x0)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:(x2 Xx))=> ((fun (P:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))->((forall (Xx:a), ((cR Xx) Xx))->P)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx))) P) x3) x0)) ((or ((cR Xx) Xy)) ((cR Xy) Xx)))))))) as proof of (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found or_intror00:=(or_intror0 (x0 Xy)):((x0 Xy)->((or ((cR Xx) Xy)) (x0 Xy)))
% Found (or_intror0 (x0 Xy)) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_intror ((cR Xx) Xy)) (x0 Xy)) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_intror ((cR Xx) Xy)) (x0 Xy)) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x7:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x7:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy))) as proof of ((x0 Xx)->((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect30 (fun (x7:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect3 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found or_intror00:=(or_intror0 (x2 Xy)):((x2 Xy)->((or ((cR Xx) Xy)) (x2 Xy)))
% Found (or_intror0 (x2 Xy)) as proof of ((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_intror ((cR Xx) Xy)) (x2 Xy)) as proof of ((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_intror ((cR Xx) Xy)) (x2 Xy)) as proof of ((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x7:(x2 Xx))=> ((or_intror ((cR Xx) Xy)) (x2 Xy))) as proof of ((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x7:(x2 Xx))=> ((or_intror ((cR Xx) Xy)) (x2 Xy))) as proof of ((x2 Xx)->((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect30 (fun (x7:(x2 Xx))=> ((or_intror ((cR Xx) Xy)) (x2 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect3 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x2 Xx))=> ((or_intror ((cR Xx) Xy)) (x2 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x2 Xx))=> ((or_intror ((cR Xx) Xy)) (x2 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x2 Xx)) (x2 Xy)))=> (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x2 Xx))=> ((or_intror ((cR Xx) Xy)) (x2 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x2 Xx)) (x2 Xy)))=> (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x2 Xx))=> ((or_intror ((cR Xx) Xy)) (x2 Xy))))) as proof of (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found or_intror00:=(or_intror0 (x4 Xy)):((x4 Xy)->((or ((cR Xx) Xy)) (x4 Xy)))
% Found (or_intror0 (x4 Xy)) as proof of ((x4 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_intror ((cR Xx) Xy)) (x4 Xy)) as proof of ((x4 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_intror ((cR Xx) Xy)) (x4 Xy)) as proof of ((x4 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x7:(x4 Xx))=> ((or_intror ((cR Xx) Xy)) (x4 Xy))) as proof of ((x4 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x7:(x4 Xx))=> ((or_intror ((cR Xx) Xy)) (x4 Xy))) as proof of ((x4 Xx)->((x4 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect30 (fun (x7:(x4 Xx))=> ((or_intror ((cR Xx) Xy)) (x4 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect3 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x4 Xx))=> ((or_intror ((cR Xx) Xy)) (x4 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x4 Xx))=> ((or_intror ((cR Xx) Xy)) (x4 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR 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 ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x4 Xx))=> ((or_intror ((cR Xx) Xy)) (x4 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR 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 ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x4 Xx))=> ((or_intror ((cR Xx) Xy)) (x4 Xy))))) as proof of (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR 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 ((T Xx)->(x0 Xx))
% Found (eq_sym0000 ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x1:(((eq a) Xx) Xx))=> ((eq_sym000 x1) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x1:(((eq a) Xx) Xx))=> (((eq_sym00 Xx) x1) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x1:(((eq a) Xx) Xx))=> ((((eq_sym0 Xx) Xx) x1) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x1:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x1) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x1:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x1) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found (fun (Xx:a)=> ((fun (x1:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x1) T)) ((eq_ref a) Xx))) as proof of ((T Xx)->(x0 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))) (Xx:a)=> ((fun (x1:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x1) T)) ((eq_ref a) Xx))) as proof of (forall (Xx:a), ((T Xx)->(x0 Xx)))
% Found x5:(x4 Xx)
% Instantiate: x4:=(cR Xy):(a->Prop)
% Found (fun (x6:(x4 Xy))=> x5) as proof of ((cR Xy) Xx)
% Found (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x5) as proof of ((x4 Xy)->((cR Xy) Xx))
% Found (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x5) as proof of ((x4 Xx)->((x4 Xy)->((cR Xy) Xx)))
% Found (and_rect20 (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x5)) as proof of ((cR Xy) Xx)
% Found ((and_rect2 ((cR Xy) Xx)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x5)) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((cR Xy) Xx)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x5)) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((cR Xy) Xx)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x5)) as proof of ((cR Xy) Xx)
% Found (or_intror00 (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((cR Xy) Xx)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x5))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_intror0 ((cR Xy) Xx)) (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((cR Xy) Xx)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x5))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((cR Xy) Xx)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x5))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x4 Xx)) (x4 Xy)))=> (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((cR Xy) Xx)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x5)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found x6:(x4 Xy)
% Instantiate: x4:=(cR Xx):(a->Prop)
% Found (fun (x6:(x4 Xy))=> x6) as proof of ((cR Xx) Xy)
% Found (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x6) as proof of ((x4 Xy)->((cR Xx) Xy))
% Found (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x6) as proof of ((x4 Xx)->((x4 Xy)->((cR Xx) Xy)))
% Found (and_rect20 (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x6)) as proof of ((cR Xx) Xy)
% Found ((and_rect2 ((cR Xx) Xy)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x6)) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((cR Xx) Xy)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x6)) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((cR Xx) Xy)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x6)) as proof of ((cR Xx) Xy)
% Found (or_introl00 (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((cR Xx) Xy)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x6))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_introl0 ((cR Xy) Xx)) (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((cR Xx) Xy)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x6))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((cR Xx) Xy)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x6))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x4 Xx)) (x4 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((cR Xx) Xy)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x6)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x4 Xx)) (x4 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((cR Xx) Xy)) (fun (x5:(x4 Xx)) (x6:(x4 Xy))=> x6)))) as proof of (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found or_introl00:=(or_introl0 ((cR Xx) Xy)):((x4 Xy)->((or (x4 Xy)) ((cR Xx) Xy)))
% Found (or_introl0 ((cR Xx) Xy)) as proof of ((x4 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((or_introl (x4 Xy)) ((cR Xx) Xy)) as proof of ((x4 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((or_introl (x4 Xy)) ((cR Xx) Xy)) as proof of ((x4 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x5:(x4 Xx))=> ((or_introl (x4 Xy)) ((cR Xx) Xy))) as proof of ((x4 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x5:(x4 Xx))=> ((or_introl (x4 Xy)) ((cR Xx) Xy))) as proof of ((x4 Xx)->((x4 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy))))
% Found (and_rect20 (fun (x5:(x4 Xx))=> ((or_introl (x4 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found ((and_rect2 ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x5:(x4 Xx))=> ((or_introl (x4 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x5:(x4 Xx))=> ((or_introl (x4 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x5:(x4 Xx))=> ((or_introl (x4 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR 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 ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x5:(x4 Xx))=> ((or_introl (x4 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_comm_i0 ((cR Xx) Xy)) (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x5:(x4 Xx))=> ((or_introl (x4 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x5:(x4 Xx))=> ((or_introl (x4 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x4 Xx)) (x4 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x5:(x4 Xx))=> ((or_introl (x4 Xy)) ((cR Xx) Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x4 Xx)) (x4 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x5:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x5) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x5:(x4 Xx))=> ((or_introl (x4 Xy)) ((cR Xx) Xy)))))) as proof of (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))) (fun (x:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (x Xx)) (x Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x Xx))))))))
% Found (eta_expansion_dep00 (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))) b)
% Found ((eta_expansion_dep0 (fun (x9:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x9:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x9:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x9:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and (forall (Xx:a) (Xy:a), (((and (S Xx)) (S Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx)))))))) b)
% Found eq_ref000:=(eq_ref00 x0):((x0 Xy)->(x0 Xy))
% Found (eq_ref00 x0) as proof of ((x0 Xy)->((cR Xx) Xy))
% Found ((eq_ref0 Xy) x0) as proof of ((x0 Xy)->((cR Xx) Xy))
% Found (((eq_ref a) Xy) x0) as proof of ((x0 Xy)->((cR Xx) Xy))
% Found (((eq_ref a) Xy) x0) as proof of ((x0 Xy)->((cR Xx) Xy))
% Found (fun (x5:(x0 Xx))=> (((eq_ref a) Xy) x0)) as proof of ((x0 Xy)->((cR Xx) Xy))
% Found (fun (x5:(x0 Xx))=> (((eq_ref a) Xy) x0)) as proof of ((x0 Xx)->((x0 Xy)->((cR Xx) Xy)))
% Found (and_rect20 (fun (x5:(x0 Xx))=> (((eq_ref a) Xy) x0))) as proof of ((cR Xx) Xy)
% Found ((and_rect2 ((cR Xx) Xy)) (fun (x5:(x0 Xx))=> (((eq_ref a) Xy) x0))) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((cR Xx) Xy)) (fun (x5:(x0 Xx))=> (((eq_ref a) Xy) x0))) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((cR Xx) Xy)) (fun (x5:(x0 Xx))=> (((eq_ref a) Xy) x0))) as proof of ((cR Xx) Xy)
% Found (or_introl00 (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((cR Xx) Xy)) (fun (x5:(x0 Xx))=> (((eq_ref a) Xy) x0)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_introl0 ((cR Xy) Xx)) (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((cR Xx) Xy)) (fun (x5:(x0 Xx))=> (((eq_ref a) Xy) x0)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((cR Xx) Xy)) (fun (x5:(x0 Xx))=> (((eq_ref a) Xy) x0)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((cR Xx) Xy)) (fun (x5:(x0 Xx))=> (((eq_ref a) Xy) x0))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((cR Xx) Xy)) (fun (x5:(x0 Xx))=> (((eq_ref a) Xy) x0))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found x5:(x0 Xx)
% Instantiate: x0:=(cR Xy):(a->Prop)
% Found (fun (x6:(x0 Xy))=> x5) as proof of ((cR Xy) Xx)
% Found (fun (x5:(x0 Xx)) (x6:(x0 Xy))=> x5) as proof of ((x0 Xy)->((cR Xy) Xx))
% Found (fun (x5:(x0 Xx)) (x6:(x0 Xy))=> x5) as proof of ((x0 Xx)->((x0 Xy)->((cR Xy) Xx)))
% Found (and_rect20 (fun (x5:(x0 Xx)) (x6:(x0 Xy))=> x5)) as proof of ((cR Xy) Xx)
% Found ((and_rect2 ((cR Xy) Xx)) (fun (x5:(x0 Xx)) (x6:(x0 Xy))=> x5)) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((cR Xy) Xx)) (fun (x5:(x0 Xx)) (x6:(x0 Xy))=> x5)) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((cR Xy) Xx)) (fun (x5:(x0 Xx)) (x6:(x0 Xy))=> x5)) as proof of ((cR Xy) Xx)
% Found (or_intror00 (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((cR Xy) Xx)) (fun (x5:(x0 Xx)) (x6:(x0 Xy))=> x5))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_intror0 ((cR Xy) Xx)) (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((cR Xy) Xx)) (fun (x5:(x0 Xx)) (x6:(x0 Xy))=> x5))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((cR Xy) Xx)) (fun (x5:(x0 Xx)) (x6:(x0 Xy))=> x5))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((cR Xy) Xx)) (fun (x5:(x0 Xx)) (x6:(x0 Xy))=> x5)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found or_introl00:=(or_introl0 ((cR Xx) Xy)):((x0 Xy)->((or (x0 Xy)) ((cR Xx) Xy)))
% Found (or_introl0 ((cR Xx) Xy)) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((or_introl (x0 Xy)) ((cR Xx) Xy)) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((or_introl (x0 Xy)) ((cR Xx) Xy)) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x5:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy))) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x5:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy))) as proof of ((x0 Xx)->((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy))))
% Found (and_rect20 (fun (x5:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found ((and_rect2 ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x5:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x5:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x5:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (or_comm_i00 (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x5:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_comm_i0 ((cR Xx) Xy)) (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x5:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x5:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x5:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x5:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x5) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x5:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found x6:(x2 Xy)
% Instantiate: x2:=(cR Xx):(a->Prop)
% Found (fun (x6:(x2 Xy))=> x6) as proof of ((cR Xx) Xy)
% Found (fun (x5:(x2 Xx)) (x6:(x2 Xy))=> x6) as proof of ((x2 Xy)->((cR Xx) Xy))
% Found (fun (x5:(x2 Xx)) (x6:(x2 Xy))=> x6) as proof of ((x2 Xx)->((x2 Xy)->((cR Xx) Xy)))
% Found (and_rect20 (fun (x5:(x2 Xx)) (x6:(x2 Xy))=> x6)) as proof of ((cR Xx) Xy)
% Found ((and_rect2 ((cR Xx) Xy)) (fun (x5:(x2 Xx)) (x6:(x2 Xy))=> x6)) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((cR Xx) Xy)) (fun (x5:(x2 Xx)) (x6:(x2 Xy))=> x6)) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((cR Xx) Xy)) (fun (x5:(x2 Xx)) (x6:(x2 Xy))=> x6)) as proof of ((cR Xx) Xy)
% Found (or_introl00 (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((cR Xx) Xy)) (fun (x5:(x2 Xx)) (x6:(x2 Xy))=> x6))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_introl0 ((cR Xy) Xx)) (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((cR Xx) Xy)) (fun (x5:(x2 Xx)) (x6:(x2 Xy))=> x6))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((cR Xx) Xy)) (fun (x5:(x2 Xx)) (x6:(x2 Xy))=> x6))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x2 Xx)) (x2 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((cR Xx) Xy)) (fun (x5:(x2 Xx)) (x6:(x2 Xy))=> x6)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x2 Xx)) (x2 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((cR Xx) Xy)) (fun (x5:(x2 Xx)) (x6:(x2 Xy))=> x6)))) as proof of (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found x5:(x2 Xx)
% Instantiate: x2:=(cR Xy):(a->Prop)
% Found (fun (x6:(x2 Xy))=> x5) as proof of ((cR Xy) Xx)
% Found (fun (x5:(x2 Xx)) (x6:(x2 Xy))=> x5) as proof of ((x2 Xy)->((cR Xy) Xx))
% Found (fun (x5:(x2 Xx)) (x6:(x2 Xy))=> x5) as proof of ((x2 Xx)->((x2 Xy)->((cR Xy) Xx)))
% Found (and_rect20 (fun (x5:(x2 Xx)) (x6:(x2 Xy))=> x5)) as proof of ((cR Xy) Xx)
% Found ((and_rect2 ((cR Xy) Xx)) (fun (x5:(x2 Xx)) (x6:(x2 Xy))=> x5)) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((cR Xy) Xx)) (fun (x5:(x2 Xx)) (x6:(x2 Xy))=> x5)) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((cR Xy) Xx)) (fun (x5:(x2 Xx)) (x6:(x2 Xy))=> x5)) as proof of ((cR Xy) Xx)
% Found (or_intror00 (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((cR Xy) Xx)) (fun (x5:(x2 Xx)) (x6:(x2 Xy))=> x5))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_intror0 ((cR Xy) Xx)) (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((cR Xy) Xx)) (fun (x5:(x2 Xx)) (x6:(x2 Xy))=> x5))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((cR Xy) Xx)) (fun (x5:(x2 Xx)) (x6:(x2 Xy))=> x5))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x2 Xx)) (x2 Xy)))=> (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((cR Xy) Xx)) (fun (x5:(x2 Xx)) (x6:(x2 Xy))=> x5)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found or_intror00:=(or_intror0 (x2 Xy)):((x2 Xy)->((or ((cR Xy) Xx)) (x2 Xy)))
% Found (or_intror0 (x2 Xy)) as proof of ((x2 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((or_intror ((cR Xy) Xx)) (x2 Xy)) as proof of ((x2 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((or_intror ((cR Xy) Xx)) (x2 Xy)) as proof of ((x2 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x5:(x2 Xx))=> ((or_intror ((cR Xy) Xx)) (x2 Xy))) as proof of ((x2 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x5:(x2 Xx))=> ((or_intror ((cR Xy) Xx)) (x2 Xy))) as proof of ((x2 Xx)->((x2 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy))))
% Found (and_rect20 (fun (x5:(x2 Xx))=> ((or_intror ((cR Xy) Xx)) (x2 Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found ((and_rect2 ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x5:(x2 Xx))=> ((or_intror ((cR Xy) Xx)) (x2 Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x5:(x2 Xx))=> ((or_intror ((cR Xy) Xx)) (x2 Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x5:(x2 Xx))=> ((or_intror ((cR Xy) Xx)) (x2 Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (or_comm_i00 (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x5:(x2 Xx))=> ((or_intror ((cR Xy) Xx)) (x2 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_comm_i0 ((cR Xx) Xy)) (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x5:(x2 Xx))=> ((or_intror ((cR Xy) Xx)) (x2 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x5:(x2 Xx))=> ((or_intror ((cR Xy) Xx)) (x2 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x2 Xx)) (x2 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x5:(x2 Xx))=> ((or_intror ((cR Xy) Xx)) (x2 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x2 Xx)) (x2 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x5:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x5) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x5:(x2 Xx))=> ((or_intror ((cR Xy) Xx)) (x2 Xy)))))) as proof of (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found and_rect20:=(and_rect2 ((or ((cR Xx) Xy)) ((cR Xy) Xx))):((((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz)))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Instantiate: x0:=(fun (x7:a)=> (((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz)))->((or ((cR Xx) x7)) ((cR x7) Xx))))):(a->Prop)
% Found (fun (x7:(x0 Xx))=> and_rect20) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x7:(x0 Xx))=> and_rect20) as proof of ((x0 Xx)->((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect30 (fun (x7:(x0 Xx))=> and_rect20)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect3 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x0 Xx))=> and_rect20)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x0 Xx))=> and_rect20)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x6:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))=> (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x0 Xx))=> and_rect20))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x5:((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (x6:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))=> (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x0 Xx))=> and_rect20))) as proof of ((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz)))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x5:((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (x6:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))=> (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x0 Xx))=> and_rect20))) as proof of (((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz)))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect20 (fun (x5:((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (x6:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))=> (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x0 Xx))=> and_rect20)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect2 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (x6:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))=> (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x0 Xx))=> (and_rect2 ((or ((cR Xx) Xy)) ((cR Xy) Xx))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x5:(((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))->P)))=> (((((and_rect ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))) P) x5) x3)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (x6:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))=> (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x0 Xx))=> ((fun (P:Type) (x5:(((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))->P)))=> (((((and_rect ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))) P) x5) x3)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P:Type) (x5:(((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))->P)))=> (((((and_rect ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))) P) x5) x3)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (x6:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))=> (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x0 Xx))=> ((fun (P:Type) (x5:(((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))->P)))=> (((((and_rect ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))) P) x5) x3)) ((or ((cR Xx) Xy)) ((cR Xy) Xx)))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P:Type) (x5:(((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))->P)))=> (((((and_rect ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))) P) x5) x3)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (x6:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))=> (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x0 Xx))=> ((fun (P:Type) (x5:(((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))->P)))=> (((((and_rect ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))) P) x5) x3)) ((or ((cR Xx) Xy)) ((cR Xy) Xx)))))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found and_rect20:=(and_rect2 ((or ((cR Xx) Xy)) ((cR Xy) Xx))):((((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz)))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Instantiate: x2:=(fun (x7:a)=> (((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz)))->((or ((cR Xx) x7)) ((cR x7) Xx))))):(a->Prop)
% Found (fun (x7:(x2 Xx))=> and_rect20) as proof of ((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x7:(x2 Xx))=> and_rect20) as proof of ((x2 Xx)->((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect30 (fun (x7:(x2 Xx))=> and_rect20)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect3 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x2 Xx))=> and_rect20)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x2 Xx))=> and_rect20)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x6:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))=> (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x2 Xx))=> and_rect20))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x5:((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (x6:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))=> (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x2 Xx))=> and_rect20))) as proof of ((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz)))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x5:((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (x6:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))=> (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x2 Xx))=> and_rect20))) as proof of (((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz)))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect20 (fun (x5:((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (x6:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))=> (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x2 Xx))=> and_rect20)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect2 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (x6:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))=> (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x2 Xx))=> (and_rect2 ((or ((cR Xx) Xy)) ((cR Xy) Xx))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x5:(((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))->P)))=> (((((and_rect ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))) P) x5) x3)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (x6:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))=> (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x2 Xx))=> ((fun (P:Type) (x5:(((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))->P)))=> (((((and_rect ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))) P) x5) x3)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x2 Xx)) (x2 Xy)))=> (((fun (P:Type) (x5:(((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))->P)))=> (((((and_rect ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))) P) x5) x3)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (x6:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))=> (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x2 Xx))=> ((fun (P:Type) (x5:(((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))->P)))=> (((((and_rect ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))) P) x5) x3)) ((or ((cR Xx) Xy)) ((cR Xy) Xx)))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x2 Xx)) (x2 Xy)))=> (((fun (P:Type) (x5:(((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))->P)))=> (((((and_rect ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))) P) x5) x3)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (x6:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))=> (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x2 Xx))=> ((fun (P:Type) (x5:(((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))->P)))=> (((((and_rect ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))) P) x5) x3)) ((or ((cR Xx) Xy)) ((cR Xy) Xx)))))))) as proof of (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found or_introl00:=(or_introl0 ((cR Xy) Xx)):((x8 Xy)->((or (x8 Xy)) ((cR Xy) Xx)))
% Found (or_introl0 ((cR Xy) Xx)) as proof of ((x8 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_introl (x8 Xy)) ((cR Xy) Xx)) as proof of ((x8 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_introl (x8 Xy)) ((cR Xy) Xx)) as proof of ((x8 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x9:(x8 Xx))=> ((or_introl (x8 Xy)) ((cR Xy) Xx))) as proof of ((x8 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x9:(x8 Xx))=> ((or_introl (x8 Xy)) ((cR Xy) Xx))) as proof of ((x8 Xx)->((x8 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect30 (fun (x9:(x8 Xx))=> ((or_introl (x8 Xy)) ((cR Xy) Xx)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect3 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x8 Xx))=> ((or_introl (x8 Xy)) ((cR Xy) Xx)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x8 Xx))=> ((or_introl (x8 Xy)) ((cR Xy) Xx)))) as proof of ((or ((cR Xx) Xy)) ((cR 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 ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x8 Xx))=> ((or_introl (x8 Xy)) ((cR Xy) Xx))))) as proof of ((or ((cR Xx) Xy)) ((cR 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 ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x8 Xx))=> ((or_introl (x8 Xy)) ((cR Xy) Xx))))) as proof of (((and (x8 Xx)) (x8 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found and_rect20:=(and_rect2 ((or ((cR Xx) Xy)) ((cR Xy) Xx))):((((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz)))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Instantiate: x4:=(fun (x7:a)=> (((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz)))->((or ((cR Xx) x7)) ((cR x7) Xx))))):(a->Prop)
% Found (fun (x7:(x4 Xx))=> and_rect20) as proof of ((x4 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x7:(x4 Xx))=> and_rect20) as proof of ((x4 Xx)->((x4 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect30 (fun (x7:(x4 Xx))=> and_rect20)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect3 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x4 Xx))=> and_rect20)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x4 Xx))=> and_rect20)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x6:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))=> (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x4 Xx))=> and_rect20))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x5:((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (x6:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))=> (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x4 Xx))=> and_rect20))) as proof of ((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz)))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x5:((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (x6:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))=> (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x4 Xx))=> and_rect20))) as proof of (((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz)))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect20 (fun (x5:((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (x6:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))=> (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x4 Xx))=> and_rect20)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect2 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (x6:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))=> (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x4 Xx))=> (and_rect2 ((or ((cR Xx) Xy)) ((cR Xy) Xx))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x5:(((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))->P)))=> (((((and_rect ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))) P) x5) x2)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (x6:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))=> (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x4 Xx))=> ((fun (P:Type) (x5:(((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))->P)))=> (((((and_rect ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))) P) x5) x2)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x4 Xx)) (x4 Xy)))=> (((fun (P:Type) (x5:(((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))->P)))=> (((((and_rect ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))) P) x5) x2)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (x6:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))=> (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x4 Xx))=> ((fun (P:Type) (x5:(((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))->P)))=> (((((and_rect ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))) P) x5) x2)) ((or ((cR Xx) Xy)) ((cR Xy) Xx)))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x4 Xx)) (x4 Xy)))=> (((fun (P:Type) (x5:(((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))->P)))=> (((((and_rect ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))) P) x5) x2)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x5:((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (x6:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))=> (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(x4 Xx))=> ((fun (P:Type) (x5:(((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))->P)))=> (((((and_rect ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))) P) x5) x2)) ((or ((cR Xx) Xy)) ((cR Xy) Xx)))))))) as proof of (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR 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 ((T Xx)->(x2 Xx))
% Found (eq_sym0000 ((eq_ref a) Xx)) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x3:(((eq a) Xx) Xx))=> ((eq_sym000 x3) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x3:(((eq a) Xx) Xx))=> (((eq_sym00 Xx) x3) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x3:(((eq a) Xx) Xx))=> ((((eq_sym0 Xx) Xx) x3) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x3:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x3) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x3:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x3) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x2 Xx))
% Found (fun (Xx:a)=> ((fun (x3:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x3) T)) ((eq_ref a) Xx))) as proof of ((T Xx)->(x2 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))) (Xx:a)=> ((fun (x3:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x3) T)) ((eq_ref a) Xx))) as proof of (forall (Xx:a), ((T Xx)->(x2 Xx)))
% Found or_intror00:=(or_intror0 (x0 Xy)):((x0 Xy)->((or ((cR Xx) Xy)) (x0 Xy)))
% Found (or_intror0 (x0 Xy)) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_intror ((cR Xx) Xy)) (x0 Xy)) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_intror ((cR Xx) Xy)) (x0 Xy)) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x9:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x9:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy))) as proof of ((x0 Xx)->((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect30 (fun (x9:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect3 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found or_introl00:=(or_introl0 ((cR Xy) Xx)):((x2 Xy)->((or (x2 Xy)) ((cR Xy) Xx)))
% Found (or_introl0 ((cR Xy) Xx)) as proof of ((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_introl (x2 Xy)) ((cR Xy) Xx)) as proof of ((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_introl (x2 Xy)) ((cR Xy) Xx)) as proof of ((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x9:(x2 Xx))=> ((or_introl (x2 Xy)) ((cR Xy) Xx))) as proof of ((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x9:(x2 Xx))=> ((or_introl (x2 Xy)) ((cR Xy) Xx))) as proof of ((x2 Xx)->((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect30 (fun (x9:(x2 Xx))=> ((or_introl (x2 Xy)) ((cR Xy) Xx)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect3 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x2 Xx))=> ((or_introl (x2 Xy)) ((cR Xy) Xx)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x2 Xx))=> ((or_introl (x2 Xy)) ((cR Xy) Xx)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x2 Xx)) (x2 Xy)))=> (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x2 Xx))=> ((or_introl (x2 Xy)) ((cR Xy) Xx))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x2 Xx)) (x2 Xy)))=> (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x2 Xx))=> ((or_introl (x2 Xy)) ((cR Xy) Xx))))) as proof of (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found eq_ref000:=(eq_ref00 (fun (x10:Prop)=> (x4 Xy))):((x4 Xy)->(x4 Xy))
% Found (eq_ref00 (fun (x10:Prop)=> (x4 Xy))) as proof of ((x4 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((eq_ref0 ((cR Xy) Xx)) (fun (x10:Prop)=> (x4 Xy))) as proof of ((x4 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (((eq_ref Prop) ((cR Xy) Xx)) (fun (x10:Prop)=> (x4 Xy))) as proof of ((x4 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (((eq_ref Prop) ((cR Xy) Xx)) (fun (x10:Prop)=> (x4 Xy))) as proof of ((x4 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x9:(x4 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x10:Prop)=> (x4 Xy)))) as proof of ((x4 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x9:(x4 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x10:Prop)=> (x4 Xy)))) as proof of ((x4 Xx)->((x4 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect30 (fun (x9:(x4 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x10:Prop)=> (x4 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect3 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x4 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x10:Prop)=> (x4 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x4 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x10:Prop)=> (x4 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR 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 ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x4 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x10:Prop)=> (x4 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR 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 ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x4 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x10:Prop)=> (x4 Xy)))))) as proof of (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found or_introl00:=(or_introl0 ((cR Xy) Xx)):((x6 Xy)->((or (x6 Xy)) ((cR Xy) Xx)))
% Found (or_introl0 ((cR Xy) Xx)) as proof of ((x6 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_introl (x6 Xy)) ((cR Xy) Xx)) as proof of ((x6 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_introl (x6 Xy)) ((cR Xy) Xx)) as proof of ((x6 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x9:(x6 Xx))=> ((or_introl (x6 Xy)) ((cR Xy) Xx))) as proof of ((x6 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x9:(x6 Xx))=> ((or_introl (x6 Xy)) ((cR Xy) Xx))) as proof of ((x6 Xx)->((x6 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect30 (fun (x9:(x6 Xx))=> ((or_introl (x6 Xy)) ((cR Xy) Xx)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect3 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x6 Xx))=> ((or_introl (x6 Xy)) ((cR Xy) Xx)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x6 Xx))=> ((or_introl (x6 Xy)) ((cR Xy) Xx)))) as proof of ((or ((cR Xx) Xy)) ((cR 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 ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x6 Xx))=> ((or_introl (x6 Xy)) ((cR Xy) Xx))))) as proof of ((or ((cR Xx) Xy)) ((cR 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 ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x6 Xx))=> ((or_introl (x6 Xy)) ((cR Xy) Xx))))) as proof of (((and (x6 Xx)) (x6 Xy))->((or ((cR Xx) Xy)) ((cR 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 ((T Xx)->(x0 Xx))
% Found (eq_sym0000 ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x3:(((eq a) Xx) Xx))=> ((eq_sym000 x3) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x3:(((eq a) Xx) Xx))=> (((eq_sym00 Xx) x3) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x3:(((eq a) Xx) Xx))=> ((((eq_sym0 Xx) Xx) x3) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x3:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x3) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x3:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x3) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found (fun (Xx:a)=> ((fun (x3:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x3) T)) ((eq_ref a) Xx))) as proof of ((T Xx)->(x0 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))) (Xx:a)=> ((fun (x3:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x3) T)) ((eq_ref a) Xx))) as proof of (forall (Xx:a), ((T Xx)->(x0 Xx)))
% Found or_introl00:=(or_introl0 ((cR Xx) Xy)):((x6 Xy)->((or (x6 Xy)) ((cR Xx) Xy)))
% Found (or_introl0 ((cR Xx) Xy)) as proof of ((x6 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((or_introl (x6 Xy)) ((cR Xx) Xy)) as proof of ((x6 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((or_introl (x6 Xy)) ((cR Xx) Xy)) as proof of ((x6 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x7:(x6 Xx))=> ((or_introl (x6 Xy)) ((cR Xx) Xy))) as proof of ((x6 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x7:(x6 Xx))=> ((or_introl (x6 Xy)) ((cR Xx) Xy))) as proof of ((x6 Xx)->((x6 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy))))
% Found (and_rect30 (fun (x7:(x6 Xx))=> ((or_introl (x6 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found ((and_rect3 ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x6 Xx))=> ((or_introl (x6 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x6 Xx))=> ((or_introl (x6 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x6 Xx))=> ((or_introl (x6 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR 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 ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x6 Xx))=> ((or_introl (x6 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_comm_i0 ((cR Xx) Xy)) (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x6 Xx))=> ((or_introl (x6 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x6 Xx))=> ((or_introl (x6 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x6 Xx)) (x6 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x6 Xx))=> ((or_introl (x6 Xy)) ((cR Xx) Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x6 Xx)) (x6 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x6 Xx))=> ((or_introl (x6 Xy)) ((cR Xx) Xy)))))) as proof of (((and (x6 Xx)) (x6 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found x7:(x6 Xx)
% Instantiate: x6:=(cR Xy):(a->Prop)
% Found (fun (x8:(x6 Xy))=> x7) as proof of ((cR Xy) Xx)
% Found (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x7) as proof of ((x6 Xy)->((cR Xy) Xx))
% Found (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x7) as proof of ((x6 Xx)->((x6 Xy)->((cR Xy) Xx)))
% Found (and_rect30 (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x7)) as proof of ((cR Xy) Xx)
% Found ((and_rect3 ((cR Xy) Xx)) (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x7)) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((cR Xy) Xx)) (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x7)) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((cR Xy) Xx)) (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x7)) as proof of ((cR Xy) Xx)
% Found (or_intror00 (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((cR Xy) Xx)) (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x7))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_intror0 ((cR Xy) Xx)) (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((cR Xy) Xx)) (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x7))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((cR Xy) Xx)) (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x7))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x6 Xx)) (x6 Xy)))=> (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((cR Xy) Xx)) (fun (x7:(x6 Xx)) (x8:(x6 Xy))=> x7)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found eq_ref000:=(eq_ref00 x6):((x6 Xy)->(x6 Xy))
% Found (eq_ref00 x6) as proof of ((x6 Xy)->((cR Xx) Xy))
% Found ((eq_ref0 Xy) x6) as proof of ((x6 Xy)->((cR Xx) Xy))
% Found (((eq_ref a) Xy) x6) as proof of ((x6 Xy)->((cR Xx) Xy))
% Found (((eq_ref a) Xy) x6) as proof of ((x6 Xy)->((cR Xx) Xy))
% Found (fun (x7:(x6 Xx))=> (((eq_ref a) Xy) x6)) as proof of ((x6 Xy)->((cR Xx) Xy))
% Found (fun (x7:(x6 Xx))=> (((eq_ref a) Xy) x6)) as proof of ((x6 Xx)->((x6 Xy)->((cR Xx) Xy)))
% Found (and_rect30 (fun (x7:(x6 Xx))=> (((eq_ref a) Xy) x6))) as proof of ((cR Xx) Xy)
% Found ((and_rect3 ((cR Xx) Xy)) (fun (x7:(x6 Xx))=> (((eq_ref a) Xy) x6))) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((cR Xx) Xy)) (fun (x7:(x6 Xx))=> (((eq_ref a) Xy) x6))) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((cR Xx) Xy)) (fun (x7:(x6 Xx))=> (((eq_ref a) Xy) x6))) as proof of ((cR Xx) Xy)
% Found (or_introl00 (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((cR Xx) Xy)) (fun (x7:(x6 Xx))=> (((eq_ref a) Xy) x6)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_introl0 ((cR Xy) Xx)) (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((cR Xx) Xy)) (fun (x7:(x6 Xx))=> (((eq_ref a) Xy) x6)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((cR Xx) Xy)) (fun (x7:(x6 Xx))=> (((eq_ref a) Xy) x6)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x6 Xx)) (x6 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((cR Xx) Xy)) (fun (x7:(x6 Xx))=> (((eq_ref a) Xy) x6))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x6 Xx)) (x6 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x7:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x7) x00)) ((cR Xx) Xy)) (fun (x7:(x6 Xx))=> (((eq_ref a) Xy) x6))))) as proof of (((and (x6 Xx)) (x6 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found eq_ref000:=(eq_ref00 x0):((x0 Xy)->(x0 Xy))
% Found (eq_ref00 x0) as proof of ((x0 Xy)->((cR Xx) Xy))
% Found ((eq_ref0 Xy) x0) as proof of ((x0 Xy)->((cR Xx) Xy))
% Found (((eq_ref a) Xy) x0) as proof of ((x0 Xy)->((cR Xx) Xy))
% Found (((eq_ref a) Xy) x0) as proof of ((x0 Xy)->((cR Xx) Xy))
% Found (fun (x7:(x0 Xx))=> (((eq_ref a) Xy) x0)) as proof of ((x0 Xy)->((cR Xx) Xy))
% Found (fun (x7:(x0 Xx))=> (((eq_ref a) Xy) x0)) as proof of ((x0 Xx)->((x0 Xy)->((cR Xx) Xy)))
% Found (and_rect30 (fun (x7:(x0 Xx))=> (((eq_ref a) Xy) x0))) as proof of ((cR Xx) Xy)
% Found ((and_rect3 ((cR Xx) Xy)) (fun (x7:(x0 Xx))=> (((eq_ref a) Xy) x0))) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((cR Xx) Xy)) (fun (x7:(x0 Xx))=> (((eq_ref a) Xy) x0))) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((cR Xx) Xy)) (fun (x7:(x0 Xx))=> (((eq_ref a) Xy) x0))) as proof of ((cR Xx) Xy)
% Found (or_introl00 (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((cR Xx) Xy)) (fun (x7:(x0 Xx))=> (((eq_ref a) Xy) x0)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_introl0 ((cR Xy) Xx)) (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((cR Xx) Xy)) (fun (x7:(x0 Xx))=> (((eq_ref a) Xy) x0)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((cR Xx) Xy)) (fun (x7:(x0 Xx))=> (((eq_ref a) Xy) x0)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((cR Xx) Xy)) (fun (x7:(x0 Xx))=> (((eq_ref a) Xy) x0))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((cR Xx) Xy)) (fun (x7:(x0 Xx))=> (((eq_ref a) Xy) x0))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found or_introl00:=(or_introl0 ((cR Xx) Xy)):((x0 Xy)->((or (x0 Xy)) ((cR Xx) Xy)))
% Found (or_introl0 ((cR Xx) Xy)) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((or_introl (x0 Xy)) ((cR Xx) Xy)) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((or_introl (x0 Xy)) ((cR Xx) Xy)) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x7:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy))) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x7:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy))) as proof of ((x0 Xx)->((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy))))
% Found (and_rect30 (fun (x7:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found ((and_rect3 ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (or_comm_i00 (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_comm_i0 ((cR Xx) Xy)) (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found eq_ref000:=(eq_ref00 (fun (x8:a)=> (x0 Xy))):((x0 Xy)->(x0 Xy))
% Found (eq_ref00 (fun (x8:a)=> (x0 Xy))) as proof of ((x0 Xy)->((cR Xy) Xx))
% Found ((eq_ref0 Xx) (fun (x8:a)=> (x0 Xy))) as proof of ((x0 Xy)->((cR Xy) Xx))
% Found (((eq_ref a) Xx) (fun (x8:a)=> (x0 Xy))) as proof of ((x0 Xy)->((cR Xy) Xx))
% Found (((eq_ref a) Xx) (fun (x8:a)=> (x0 Xy))) as proof of ((x0 Xy)->((cR Xy) Xx))
% Found (fun (x7:(x0 Xx))=> (((eq_ref a) Xx) (fun (x8:a)=> (x0 Xy)))) as proof of ((x0 Xy)->((cR Xy) Xx))
% Found (fun (x7:(x0 Xx))=> (((eq_ref a) Xx) (fun (x8:a)=> (x0 Xy)))) as proof of ((x0 Xx)->((x0 Xy)->((cR Xy) Xx)))
% Found (and_rect30 (fun (x7:(x0 Xx))=> (((eq_ref a) Xx) (fun (x8:a)=> (x0 Xy))))) as proof of ((cR Xy) Xx)
% Found ((and_rect3 ((cR Xy) Xx)) (fun (x7:(x0 Xx))=> (((eq_ref a) Xx) (fun (x8:a)=> (x0 Xy))))) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((cR Xy) Xx)) (fun (x7:(x0 Xx))=> (((eq_ref a) Xx) (fun (x8:a)=> (x0 Xy))))) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((cR Xy) Xx)) (fun (x7:(x0 Xx))=> (((eq_ref a) Xx) (fun (x8:a)=> (x0 Xy))))) as proof of ((cR Xy) Xx)
% Found (or_intror00 (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((cR Xy) Xx)) (fun (x7:(x0 Xx))=> (((eq_ref a) Xx) (fun (x8:a)=> (x0 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_intror0 ((cR Xy) Xx)) (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((cR Xy) Xx)) (fun (x7:(x0 Xx))=> (((eq_ref a) Xx) (fun (x8:a)=> (x0 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((cR Xy) Xx)) (fun (x7:(x0 Xx))=> (((eq_ref a) Xx) (fun (x8:a)=> (x0 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((cR Xy) Xx)) (fun (x7:(x0 Xx))=> (((eq_ref a) Xx) (fun (x8:a)=> (x0 Xy))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x7:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x7) x00)) ((cR Xy) Xx)) (fun (x7:(x0 Xx))=> (((eq_ref a) Xx) (fun (x8:a)=> (x0 Xy))))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found or_intror00:=(or_intror0 (x2 Xy)):((x2 Xy)->((or ((cR Xy) Xx)) (x2 Xy)))
% Found (or_intror0 (x2 Xy)) as proof of ((x2 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((or_intror ((cR Xy) Xx)) (x2 Xy)) as proof of ((x2 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((or_intror ((cR Xy) Xx)) (x2 Xy)) as proof of ((x2 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x7:(x2 Xx))=> ((or_intror ((cR Xy) Xx)) (x2 Xy))) as proof of ((x2 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x7:(x2 Xx))=> ((or_intror ((cR Xy) Xx)) (x2 Xy))) as proof of ((x2 Xx)->((x2 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy))))
% Found (and_rect30 (fun (x7:(x2 Xx))=> ((or_intror ((cR Xy) Xx)) (x2 Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found ((and_rect3 ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x2 Xx))=> ((or_intror ((cR Xy) Xx)) (x2 Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x2 Xx))=> ((or_intror ((cR Xy) Xx)) (x2 Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x2 Xx))=> ((or_intror ((cR Xy) Xx)) (x2 Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (or_comm_i00 (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x2 Xx))=> ((or_intror ((cR Xy) Xx)) (x2 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_comm_i0 ((cR Xx) Xy)) (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x2 Xx))=> ((or_intror ((cR Xy) Xx)) (x2 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x2 Xx))=> ((or_intror ((cR Xy) Xx)) (x2 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x2 Xx)) (x2 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x2 Xx))=> ((or_intror ((cR Xy) Xx)) (x2 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x2 Xx)) (x2 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x2 Xx))=> ((or_intror ((cR Xy) Xx)) (x2 Xy)))))) as proof of (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found x7:(x2 Xx)
% Instantiate: x2:=(cR Xy):(a->Prop)
% Found (fun (x8:(x2 Xy))=> x7) as proof of ((cR Xy) Xx)
% Found (fun (x7:(x2 Xx)) (x8:(x2 Xy))=> x7) as proof of ((x2 Xy)->((cR Xy) Xx))
% Found (fun (x7:(x2 Xx)) (x8:(x2 Xy))=> x7) as proof of ((x2 Xx)->((x2 Xy)->((cR Xy) Xx)))
% Found (and_rect30 (fun (x7:(x2 Xx)) (x8:(x2 Xy))=> x7)) as proof of ((cR Xy) Xx)
% Found ((and_rect3 ((cR Xy) Xx)) (fun (x7:(x2 Xx)) (x8:(x2 Xy))=> x7)) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((cR Xy) Xx)) (fun (x7:(x2 Xx)) (x8:(x2 Xy))=> x7)) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((cR Xy) Xx)) (fun (x7:(x2 Xx)) (x8:(x2 Xy))=> x7)) as proof of ((cR Xy) Xx)
% Found (or_intror00 (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((cR Xy) Xx)) (fun (x7:(x2 Xx)) (x8:(x2 Xy))=> x7))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_intror0 ((cR Xy) Xx)) (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((cR Xy) Xx)) (fun (x7:(x2 Xx)) (x8:(x2 Xy))=> x7))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((cR Xy) Xx)) (fun (x7:(x2 Xx)) (x8:(x2 Xy))=> x7))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x2 Xx)) (x2 Xy)))=> (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((cR Xy) Xx)) (fun (x7:(x2 Xx)) (x8:(x2 Xy))=> x7)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found x8:(x2 Xy)
% Instantiate: x2:=(cR Xx):(a->Prop)
% Found (fun (x8:(x2 Xy))=> x8) as proof of ((cR Xx) Xy)
% Found (fun (x7:(x2 Xx)) (x8:(x2 Xy))=> x8) as proof of ((x2 Xy)->((cR Xx) Xy))
% Found (fun (x7:(x2 Xx)) (x8:(x2 Xy))=> x8) as proof of ((x2 Xx)->((x2 Xy)->((cR Xx) Xy)))
% Found (and_rect30 (fun (x7:(x2 Xx)) (x8:(x2 Xy))=> x8)) as proof of ((cR Xx) Xy)
% Found ((and_rect3 ((cR Xx) Xy)) (fun (x7:(x2 Xx)) (x8:(x2 Xy))=> x8)) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((cR Xx) Xy)) (fun (x7:(x2 Xx)) (x8:(x2 Xy))=> x8)) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((cR Xx) Xy)) (fun (x7:(x2 Xx)) (x8:(x2 Xy))=> x8)) as proof of ((cR Xx) Xy)
% Found (or_introl00 (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((cR Xx) Xy)) (fun (x7:(x2 Xx)) (x8:(x2 Xy))=> x8))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_introl0 ((cR Xy) Xx)) (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((cR Xx) Xy)) (fun (x7:(x2 Xx)) (x8:(x2 Xy))=> x8))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((cR Xx) Xy)) (fun (x7:(x2 Xx)) (x8:(x2 Xy))=> x8))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x2 Xx)) (x2 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((cR Xx) Xy)) (fun (x7:(x2 Xx)) (x8:(x2 Xy))=> x8)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x2 Xx)) (x2 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x7:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x7) x00)) ((cR Xx) Xy)) (fun (x7:(x2 Xx)) (x8:(x2 Xy))=> x8)))) as proof of (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found or_introl00:=(or_introl0 ((cR Xx) Xy)):((x4 Xy)->((or (x4 Xy)) ((cR Xx) Xy)))
% Found (or_introl0 ((cR Xx) Xy)) as proof of ((x4 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((or_introl (x4 Xy)) ((cR Xx) Xy)) as proof of ((x4 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((or_introl (x4 Xy)) ((cR Xx) Xy)) as proof of ((x4 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x7:(x4 Xx))=> ((or_introl (x4 Xy)) ((cR Xx) Xy))) as proof of ((x4 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x7:(x4 Xx))=> ((or_introl (x4 Xy)) ((cR Xx) Xy))) as proof of ((x4 Xx)->((x4 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy))))
% Found (and_rect30 (fun (x7:(x4 Xx))=> ((or_introl (x4 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found ((and_rect3 ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x4 Xx))=> ((or_introl (x4 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x4 Xx))=> ((or_introl (x4 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x4 Xx))=> ((or_introl (x4 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (or_comm_i00 (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x4 Xx))=> ((or_introl (x4 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_comm_i0 ((cR Xx) Xy)) (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x4 Xx))=> ((or_introl (x4 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x4 Xx))=> ((or_introl (x4 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x4 Xx)) (x4 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x4 Xx))=> ((or_introl (x4 Xy)) ((cR Xx) Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x4 Xx)) (x4 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x7:(x4 Xx))=> ((or_introl (x4 Xy)) ((cR Xx) Xy)))))) as proof of (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found x8:(x4 Xy)
% Instantiate: x4:=(cR Xx):(a->Prop)
% Found (fun (x8:(x4 Xy))=> x8) as proof of ((cR Xx) Xy)
% Found (fun (x7:(x4 Xx)) (x8:(x4 Xy))=> x8) as proof of ((x4 Xy)->((cR Xx) Xy))
% Found (fun (x7:(x4 Xx)) (x8:(x4 Xy))=> x8) as proof of ((x4 Xx)->((x4 Xy)->((cR Xx) Xy)))
% Found (and_rect30 (fun (x7:(x4 Xx)) (x8:(x4 Xy))=> x8)) as proof of ((cR Xx) Xy)
% Found ((and_rect3 ((cR Xx) Xy)) (fun (x7:(x4 Xx)) (x8:(x4 Xy))=> x8)) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((cR Xx) Xy)) (fun (x7:(x4 Xx)) (x8:(x4 Xy))=> x8)) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((cR Xx) Xy)) (fun (x7:(x4 Xx)) (x8:(x4 Xy))=> x8)) as proof of ((cR Xx) Xy)
% Found (or_introl00 (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((cR Xx) Xy)) (fun (x7:(x4 Xx)) (x8:(x4 Xy))=> x8))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_introl0 ((cR Xy) Xx)) (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((cR Xx) Xy)) (fun (x7:(x4 Xx)) (x8:(x4 Xy))=> x8))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((cR Xx) Xy)) (fun (x7:(x4 Xx)) (x8:(x4 Xy))=> x8))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x4 Xx)) (x4 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((cR Xx) Xy)) (fun (x7:(x4 Xx)) (x8:(x4 Xy))=> x8)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x4 Xx)) (x4 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((cR Xx) Xy)) (fun (x7:(x4 Xx)) (x8:(x4 Xy))=> x8)))) as proof of (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found eq_ref000:=(eq_ref00 (fun (x8:a)=> (x4 Xy))):((x4 Xy)->(x4 Xy))
% Found (eq_ref00 (fun (x8:a)=> (x4 Xy))) as proof of ((x4 Xy)->((cR Xy) Xx))
% Found ((eq_ref0 Xx) (fun (x8:a)=> (x4 Xy))) as proof of ((x4 Xy)->((cR Xy) Xx))
% Found (((eq_ref a) Xx) (fun (x8:a)=> (x4 Xy))) as proof of ((x4 Xy)->((cR Xy) Xx))
% Found (((eq_ref a) Xx) (fun (x8:a)=> (x4 Xy))) as proof of ((x4 Xy)->((cR Xy) Xx))
% Found (fun (x7:(x4 Xx))=> (((eq_ref a) Xx) (fun (x8:a)=> (x4 Xy)))) as proof of ((x4 Xy)->((cR Xy) Xx))
% Found (fun (x7:(x4 Xx))=> (((eq_ref a) Xx) (fun (x8:a)=> (x4 Xy)))) as proof of ((x4 Xx)->((x4 Xy)->((cR Xy) Xx)))
% Found (and_rect30 (fun (x7:(x4 Xx))=> (((eq_ref a) Xx) (fun (x8:a)=> (x4 Xy))))) as proof of ((cR Xy) Xx)
% Found ((and_rect3 ((cR Xy) Xx)) (fun (x7:(x4 Xx))=> (((eq_ref a) Xx) (fun (x8:a)=> (x4 Xy))))) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((cR Xy) Xx)) (fun (x7:(x4 Xx))=> (((eq_ref a) Xx) (fun (x8:a)=> (x4 Xy))))) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((cR Xy) Xx)) (fun (x7:(x4 Xx))=> (((eq_ref a) Xx) (fun (x8:a)=> (x4 Xy))))) as proof of ((cR Xy) Xx)
% Found (or_intror00 (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((cR Xy) Xx)) (fun (x7:(x4 Xx))=> (((eq_ref a) Xx) (fun (x8:a)=> (x4 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_intror0 ((cR Xy) Xx)) (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((cR Xy) Xx)) (fun (x7:(x4 Xx))=> (((eq_ref a) Xx) (fun (x8:a)=> (x4 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((cR Xy) Xx)) (fun (x7:(x4 Xx))=> (((eq_ref a) Xx) (fun (x8:a)=> (x4 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x4 Xx)) (x4 Xy)))=> (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((cR Xy) Xx)) (fun (x7:(x4 Xx))=> (((eq_ref a) Xx) (fun (x8:a)=> (x4 Xy))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x4 Xx)) (x4 Xy)))=> (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x7:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x7) x00)) ((cR Xy) Xx)) (fun (x7:(x4 Xx))=> (((eq_ref a) Xx) (fun (x8:a)=> (x4 Xy))))))) as proof of (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR 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 ((T Xx)->(x4 Xx))
% Found (eq_sym0000 ((eq_ref a) Xx)) as proof of ((T Xx)->(x4 Xx))
% Found ((fun (x5:(((eq a) Xx) Xx))=> ((eq_sym000 x5) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x4 Xx))
% Found ((fun (x5:(((eq a) Xx) Xx))=> (((eq_sym00 Xx) x5) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x4 Xx))
% Found ((fun (x5:(((eq a) Xx) Xx))=> ((((eq_sym0 Xx) Xx) x5) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x4 Xx))
% Found ((fun (x5:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x5) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x4 Xx))
% Found ((fun (x5:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x5) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x4 Xx))
% Found (fun (Xx:a)=> ((fun (x5:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x5) T)) ((eq_ref a) Xx))) as proof of ((T Xx)->(x4 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))) (Xx:a)=> ((fun (x5:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x5) T)) ((eq_ref a) Xx))) as proof of (forall (Xx:a), ((T Xx)->(x4 Xx)))
% Found ex_ind00:=(ex_ind0 ((or ((cR Xx) Xy)) ((cR Xy) Xx))):((forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Instantiate: x0:=(fun (x9:a)=> (forall (x90:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x90 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x90 Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))->((or ((cR Xx) x9)) ((cR x9) Xx))))):(a->Prop)
% Found (fun (x9:(x0 Xx))=> ex_ind00) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x9:(x0 Xx))=> ex_ind00) as proof of ((x0 Xx)->((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect30 (fun (x9:(x0 Xx))=> ex_ind00)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect3 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x0 Xx))=> ex_ind00)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x0 Xx))=> ex_ind00)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x0 Xx))=> ex_ind00))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x7:(a->(a->Prop))) (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x0 Xx))=> ex_ind00))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x7:(a->(a->Prop))) (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x0 Xx))=> ex_ind00))) as proof of (forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (ex_ind00 (fun (x7:(a->(a->Prop))) (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x0 Xx))=> ex_ind00)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((ex_ind0 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(a->(a->Prop))) (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x0 Xx))=> (ex_ind0 ((or ((cR Xx) Xy)) ((cR Xy) Xx))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Prop) (x7:(forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x Xy) Xx))))->(((eq a) Xy) Xz))))))))->P)))=> (((((ex_ind (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz)))))))))) P) x7) x5)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(a->(a->Prop))) (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x0 Xx))=> ((fun (P:Prop) (x7:(forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x Xy) Xx))))->(((eq a) Xy) Xz))))))))->P)))=> (((((ex_ind (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz)))))))))) P) x7) x5)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P:Prop) (x7:(forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x Xy) Xx))))->(((eq a) Xy) Xz))))))))->P)))=> (((((ex_ind (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz)))))))))) P) x7) x5)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(a->(a->Prop))) (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x0 Xx))=> ((fun (P:Prop) (x7:(forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x Xy) Xx))))->(((eq a) Xy) Xz))))))))->P)))=> (((((ex_ind (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz)))))))))) P) x7) x5)) ((or ((cR Xx) Xy)) ((cR Xy) Xx)))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P:Prop) (x7:(forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x Xy) Xx))))->(((eq a) Xy) Xz))))))))->P)))=> (((((ex_ind (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz)))))))))) P) x7) x5)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(a->(a->Prop))) (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x0 Xx))=> ((fun (P:Prop) (x7:(forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x Xy) Xx))))->(((eq a) Xy) Xz))))))))->P)))=> (((((ex_ind (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz)))))))))) P) x7) x5)) ((or ((cR Xx) Xy)) ((cR Xy) Xx)))))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ex_ind00:=(ex_ind0 ((or ((cR Xx) Xy)) ((cR Xy) Xx))):((forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Instantiate: x2:=(fun (x9:a)=> (forall (x90:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x90 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x90 Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))->((or ((cR Xx) x9)) ((cR x9) Xx))))):(a->Prop)
% Found (fun (x9:(x2 Xx))=> ex_ind00) as proof of ((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x9:(x2 Xx))=> ex_ind00) as proof of ((x2 Xx)->((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect30 (fun (x9:(x2 Xx))=> ex_ind00)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect3 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x2 Xx))=> ex_ind00)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x2 Xx))=> ex_ind00)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x2 Xx))=> ex_ind00))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x7:(a->(a->Prop))) (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x2 Xx))=> ex_ind00))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x7:(a->(a->Prop))) (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x2 Xx))=> ex_ind00))) as proof of (forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (ex_ind00 (fun (x7:(a->(a->Prop))) (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x2 Xx))=> ex_ind00)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((ex_ind0 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(a->(a->Prop))) (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x2 Xx))=> (ex_ind0 ((or ((cR Xx) Xy)) ((cR Xy) Xx))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Prop) (x7:(forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x Xy) Xx))))->(((eq a) Xy) Xz))))))))->P)))=> (((((ex_ind (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz)))))))))) P) x7) x5)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(a->(a->Prop))) (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x2 Xx))=> ((fun (P:Prop) (x7:(forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x Xy) Xx))))->(((eq a) Xy) Xz))))))))->P)))=> (((((ex_ind (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz)))))))))) P) x7) x5)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x2 Xx)) (x2 Xy)))=> (((fun (P:Prop) (x7:(forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x Xy) Xx))))->(((eq a) Xy) Xz))))))))->P)))=> (((((ex_ind (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz)))))))))) P) x7) x5)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(a->(a->Prop))) (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x2 Xx))=> ((fun (P:Prop) (x7:(forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x Xy) Xx))))->(((eq a) Xy) Xz))))))))->P)))=> (((((ex_ind (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz)))))))))) P) x7) x5)) ((or ((cR Xx) Xy)) ((cR Xy) Xx)))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x2 Xx)) (x2 Xy)))=> (((fun (P:Prop) (x7:(forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x Xy) Xx))))->(((eq a) Xy) Xz))))))))->P)))=> (((((ex_ind (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz)))))))))) P) x7) x5)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(a->(a->Prop))) (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x2 Xx))=> ((fun (P:Prop) (x7:(forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x Xy) Xx))))->(((eq a) Xy) Xz))))))))->P)))=> (((((ex_ind (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz)))))))))) P) x7) x5)) ((or ((cR Xx) Xy)) ((cR Xy) Xx)))))))) as proof of (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ex_ind00:=(ex_ind0 ((or ((cR Xx) Xy)) ((cR Xy) Xx))):((forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Instantiate: x4:=(fun (x9:a)=> (forall (x90:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x90 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x90 Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))->((or ((cR Xx) x9)) ((cR x9) Xx))))):(a->Prop)
% Found (fun (x9:(x4 Xx))=> ex_ind00) as proof of ((x4 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x9:(x4 Xx))=> ex_ind00) as proof of ((x4 Xx)->((x4 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect30 (fun (x9:(x4 Xx))=> ex_ind00)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect3 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x4 Xx))=> ex_ind00)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x4 Xx))=> ex_ind00)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x4 Xx))=> ex_ind00))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x7:(a->(a->Prop))) (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x4 Xx))=> ex_ind00))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x7:(a->(a->Prop))) (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x4 Xx))=> ex_ind00))) as proof of (forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (ex_ind00 (fun (x7:(a->(a->Prop))) (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x4 Xx))=> ex_ind00)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((ex_ind0 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(a->(a->Prop))) (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x4 Xx))=> (ex_ind0 ((or ((cR Xx) Xy)) ((cR Xy) Xx))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Prop) (x7:(forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x Xy) Xx))))->(((eq a) Xy) Xz))))))))->P)))=> (((((ex_ind (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz)))))))))) P) x7) x5)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(a->(a->Prop))) (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x4 Xx))=> ((fun (P:Prop) (x7:(forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x Xy) Xx))))->(((eq a) Xy) Xz))))))))->P)))=> (((((ex_ind (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz)))))))))) P) x7) x5)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x4 Xx)) (x4 Xy)))=> (((fun (P:Prop) (x7:(forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x Xy) Xx))))->(((eq a) Xy) Xz))))))))->P)))=> (((((ex_ind (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz)))))))))) P) x7) x5)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(a->(a->Prop))) (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x4 Xx))=> ((fun (P:Prop) (x7:(forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x Xy) Xx))))->(((eq a) Xy) Xz))))))))->P)))=> (((((ex_ind (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz)))))))))) P) x7) x5)) ((or ((cR Xx) Xy)) ((cR Xy) Xx)))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x4 Xx)) (x4 Xy)))=> (((fun (P:Prop) (x7:(forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x Xy) Xx))))->(((eq a) Xy) Xz))))))))->P)))=> (((((ex_ind (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz)))))))))) P) x7) x5)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(a->(a->Prop))) (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x4 Xx))=> ((fun (P:Prop) (x7:(forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x Xy) Xx))))->(((eq a) Xy) Xz))))))))->P)))=> (((((ex_ind (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz)))))))))) P) x7) x5)) ((or ((cR Xx) Xy)) ((cR Xy) Xx)))))))) as proof of (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ex_ind00:=(ex_ind0 ((or ((cR Xx) Xy)) ((cR Xy) Xx))):((forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Instantiate: x6:=(fun (x9:a)=> (forall (x90:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x90 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x90 Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))->((or ((cR Xx) x9)) ((cR x9) Xx))))):(a->Prop)
% Found (fun (x9:(x6 Xx))=> ex_ind00) as proof of ((x6 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x9:(x6 Xx))=> ex_ind00) as proof of ((x6 Xx)->((x6 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect30 (fun (x9:(x6 Xx))=> ex_ind00)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect3 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x6 Xx))=> ex_ind00)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x6 Xx))=> ex_ind00)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x6 Xx))=> ex_ind00))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x7:(a->(a->Prop))) (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x6 Xx))=> ex_ind00))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x7:(a->(a->Prop))) (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x6 Xx))=> ex_ind00))) as proof of (forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (ex_ind00 (fun (x7:(a->(a->Prop))) (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x6 Xx))=> ex_ind00)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((ex_ind0 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(a->(a->Prop))) (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x6 Xx))=> (ex_ind0 ((or ((cR Xx) Xy)) ((cR Xy) Xx))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P:Prop) (x7:(forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x Xy) Xx))))->(((eq a) Xy) Xz))))))))->P)))=> (((((ex_ind (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz)))))))))) P) x7) x4)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(a->(a->Prop))) (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x6 Xx))=> ((fun (P:Prop) (x7:(forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x Xy) Xx))))->(((eq a) Xy) Xz))))))))->P)))=> (((((ex_ind (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz)))))))))) P) x7) x4)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x6 Xx)) (x6 Xy)))=> (((fun (P:Prop) (x7:(forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x Xy) Xx))))->(((eq a) Xy) Xz))))))))->P)))=> (((((ex_ind (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz)))))))))) P) x7) x4)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(a->(a->Prop))) (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x6 Xx))=> ((fun (P:Prop) (x7:(forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x Xy) Xx))))->(((eq a) Xy) Xz))))))))->P)))=> (((((ex_ind (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz)))))))))) P) x7) x4)) ((or ((cR Xx) Xy)) ((cR Xy) Xx)))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x6 Xx)) (x6 Xy)))=> (((fun (P:Prop) (x7:(forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x Xy) Xx))))->(((eq a) Xy) Xz))))))))->P)))=> (((((ex_ind (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz)))))))))) P) x7) x4)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x7:(a->(a->Prop))) (x8:(forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((x7 Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((x7 Xy0) Xx0))))->(((eq a) Xy0) Xz)))))))))=> (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x9:(x6 Xx))=> ((fun (P:Prop) (x7:(forall (x:(a->(a->Prop))), ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x Xy) Xx))))->(((eq a) Xy) Xz))))))))->P)))=> (((((ex_ind (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz)))))))))) P) x7) x4)) ((or ((cR Xx) Xy)) ((cR Xy) Xx)))))))) as proof of (((and (x6 Xx)) (x6 Xy))->((or ((cR Xx) Xy)) ((cR 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 ((T Xx)->(x0 Xx))
% Found (eq_sym0000 ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x5:(((eq a) Xx) Xx))=> ((eq_sym000 x5) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x5:(((eq a) Xx) Xx))=> (((eq_sym00 Xx) x5) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x5:(((eq a) Xx) Xx))=> ((((eq_sym0 Xx) Xx) x5) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x5:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x5) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x5:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x5) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found (fun (Xx:a)=> ((fun (x5:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x5) T)) ((eq_ref a) Xx))) as proof of ((T Xx)->(x0 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))) (Xx:a)=> ((fun (x5:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x5) T)) ((eq_ref a) Xx))) as proof of (forall (Xx:a), ((T Xx)->(x0 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 ((T Xx)->(x2 Xx))
% Found (eq_sym0000 ((eq_ref a) Xx)) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x5:(((eq a) Xx) Xx))=> ((eq_sym000 x5) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x5:(((eq a) Xx) Xx))=> (((eq_sym00 Xx) x5) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x5:(((eq a) Xx) Xx))=> ((((eq_sym0 Xx) Xx) x5) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x5:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x5) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x5:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x5) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x2 Xx))
% Found (fun (Xx:a)=> ((fun (x5:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x5) T)) ((eq_ref a) Xx))) as proof of ((T Xx)->(x2 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))) (Xx:a)=> ((fun (x5:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x5) T)) ((eq_ref a) Xx))) as proof of (forall (Xx:a), ((T Xx)->(x2 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx))))))))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq ((a->Prop)->Prop)) a0) (fun (x:(a->Prop))=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion (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_ref00:=(eq_ref0 (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))):(((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))
% Found (eq_ref0 (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) b)
% Found eq_ref000:=(eq_ref00 (fun (x10:a)=> (x8 Xy))):((x8 Xy)->(x8 Xy))
% Found (eq_ref00 (fun (x10:a)=> (x8 Xy))) as proof of ((x8 Xy)->((cR Xy) Xx))
% Found ((eq_ref0 Xx) (fun (x10:a)=> (x8 Xy))) as proof of ((x8 Xy)->((cR Xy) Xx))
% Found (((eq_ref a) Xx) (fun (x10:a)=> (x8 Xy))) as proof of ((x8 Xy)->((cR Xy) Xx))
% Found (((eq_ref a) Xx) (fun (x10:a)=> (x8 Xy))) as proof of ((x8 Xy)->((cR Xy) Xx))
% Found (fun (x9:(x8 Xx))=> (((eq_ref a) Xx) (fun (x10:a)=> (x8 Xy)))) as proof of ((x8 Xy)->((cR Xy) Xx))
% Found (fun (x9:(x8 Xx))=> (((eq_ref a) Xx) (fun (x10:a)=> (x8 Xy)))) as proof of ((x8 Xx)->((x8 Xy)->((cR Xy) Xx)))
% Found (and_rect30 (fun (x9:(x8 Xx))=> (((eq_ref a) Xx) (fun (x10:a)=> (x8 Xy))))) as proof of ((cR Xy) Xx)
% Found ((and_rect3 ((cR Xy) Xx)) (fun (x9:(x8 Xx))=> (((eq_ref a) Xx) (fun (x10:a)=> (x8 Xy))))) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x8 Xx))=> (((eq_ref a) Xx) (fun (x10:a)=> (x8 Xy))))) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x8 Xx))=> (((eq_ref a) Xx) (fun (x10:a)=> (x8 Xy))))) as proof of ((cR Xy) Xx)
% Found (or_intror00 (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x8 Xx))=> (((eq_ref a) Xx) (fun (x10:a)=> (x8 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_intror0 ((cR Xy) Xx)) (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x8 Xx))=> (((eq_ref a) Xx) (fun (x10:a)=> (x8 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x8 Xx))=> (((eq_ref a) Xx) (fun (x10:a)=> (x8 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x8 Xx)) (x8 Xy)))=> (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x8 Xx))=> (((eq_ref a) Xx) (fun (x10:a)=> (x8 Xy))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x8 Xx)) (x8 Xy)))=> (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x8 Xx))=> (((eq_ref a) Xx) (fun (x10:a)=> (x8 Xy))))))) as proof of (((and (x8 Xx)) (x8 Xy))->((or ((cR Xx) Xy)) ((cR 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 ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((eq_ref0 ((cR Xx) Xy)) (fun (x10:Prop)=> (x8 Xy))) as proof of ((x8 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (((eq_ref Prop) ((cR Xx) Xy)) (fun (x10:Prop)=> (x8 Xy))) as proof of ((x8 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (((eq_ref Prop) ((cR Xx) Xy)) (fun (x10:Prop)=> (x8 Xy))) as proof of ((x8 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x10:Prop)=> (x8 Xy)))) as proof of ((x8 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x10:Prop)=> (x8 Xy)))) as proof of ((x8 Xx)->((x8 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy))))
% Found (and_rect30 (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x10:Prop)=> (x8 Xy))))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found ((and_rect3 ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x10:Prop)=> (x8 Xy))))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x10:Prop)=> (x8 Xy))))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x10:Prop)=> (x8 Xy))))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (or_comm_i00 (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x10:Prop)=> (x8 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_comm_i0 ((cR Xx) Xy)) (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x10:Prop)=> (x8 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x10:Prop)=> (x8 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x8 Xx)) (x8 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x10:Prop)=> (x8 Xy))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x8 Xx)) (x8 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x8 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x10:Prop)=> (x8 Xy))))))) as proof of (((and (x8 Xx)) (x8 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found eq_ref000:=(eq_ref00 x8):((x8 Xy)->(x8 Xy))
% Found (eq_ref00 x8) as proof of ((x8 Xy)->((cR Xx) Xy))
% Found ((eq_ref0 Xy) x8) as proof of ((x8 Xy)->((cR Xx) Xy))
% Found (((eq_ref a) Xy) x8) as proof of ((x8 Xy)->((cR Xx) Xy))
% Found (((eq_ref a) Xy) x8) as proof of ((x8 Xy)->((cR Xx) Xy))
% Found (fun (x9:(x8 Xx))=> (((eq_ref a) Xy) x8)) as proof of ((x8 Xy)->((cR Xx) Xy))
% Found (fun (x9:(x8 Xx))=> (((eq_ref a) Xy) x8)) as proof of ((x8 Xx)->((x8 Xy)->((cR Xx) Xy)))
% Found (and_rect30 (fun (x9:(x8 Xx))=> (((eq_ref a) Xy) x8))) as proof of ((cR Xx) Xy)
% Found ((and_rect3 ((cR Xx) Xy)) (fun (x9:(x8 Xx))=> (((eq_ref a) Xy) x8))) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x8 Xx))=> (((eq_ref a) Xy) x8))) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x8 Xx))=> (((eq_ref a) Xy) x8))) as proof of ((cR Xx) Xy)
% Found (or_introl00 (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x8 Xx))=> (((eq_ref a) Xy) x8)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_introl0 ((cR Xy) Xx)) (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x8 Xx))=> (((eq_ref a) Xy) x8)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x8 Xx))=> (((eq_ref a) Xy) x8)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x8 Xx)) (x8 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x8 Xx))=> (((eq_ref a) Xy) x8))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x8 Xx)) (x8 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x9:((x8 Xx)->((x8 Xy)->P)))=> (((((and_rect (x8 Xx)) (x8 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x8 Xx))=> (((eq_ref a) Xy) x8))))) as proof of (((and (x8 Xx)) (x8 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found or_introl00:=(or_introl0 ((cR Xx) Xy)):((x0 Xy)->((or (x0 Xy)) ((cR Xx) Xy)))
% Found (or_introl0 ((cR Xx) Xy)) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((or_introl (x0 Xy)) ((cR Xx) Xy)) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((or_introl (x0 Xy)) ((cR Xx) Xy)) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x9:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy))) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x9:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy))) as proof of ((x0 Xx)->((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy))))
% Found (and_rect30 (fun (x9:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found ((and_rect3 ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (or_comm_i00 (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_comm_i0 ((cR Xx) Xy)) (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found x9:(x0 Xx)
% Instantiate: x0:=(cR Xy):(a->Prop)
% Found (fun (x10:(x0 Xy))=> x9) as proof of ((cR Xy) Xx)
% Found (fun (x9:(x0 Xx)) (x10:(x0 Xy))=> x9) as proof of ((x0 Xy)->((cR Xy) Xx))
% Found (fun (x9:(x0 Xx)) (x10:(x0 Xy))=> x9) as proof of ((x0 Xx)->((x0 Xy)->((cR Xy) Xx)))
% Found (and_rect30 (fun (x9:(x0 Xx)) (x10:(x0 Xy))=> x9)) as proof of ((cR Xy) Xx)
% Found ((and_rect3 ((cR Xy) Xx)) (fun (x9:(x0 Xx)) (x10:(x0 Xy))=> x9)) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x0 Xx)) (x10:(x0 Xy))=> x9)) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x0 Xx)) (x10:(x0 Xy))=> x9)) as proof of ((cR Xy) Xx)
% Found (or_intror00 (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x0 Xx)) (x10:(x0 Xy))=> x9))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_intror0 ((cR Xy) Xx)) (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x0 Xx)) (x10:(x0 Xy))=> x9))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x0 Xx)) (x10:(x0 Xy))=> x9))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x0 Xx)) (x10:(x0 Xy))=> x9)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found eq_ref000:=(eq_ref00 x0):((x0 Xy)->(x0 Xy))
% Found (eq_ref00 x0) as proof of ((x0 Xy)->((cR Xx) Xy))
% Found ((eq_ref0 Xy) x0) as proof of ((x0 Xy)->((cR Xx) Xy))
% Found (((eq_ref a) Xy) x0) as proof of ((x0 Xy)->((cR Xx) Xy))
% Found (((eq_ref a) Xy) x0) as proof of ((x0 Xy)->((cR Xx) Xy))
% Found (fun (x9:(x0 Xx))=> (((eq_ref a) Xy) x0)) as proof of ((x0 Xy)->((cR Xx) Xy))
% Found (fun (x9:(x0 Xx))=> (((eq_ref a) Xy) x0)) as proof of ((x0 Xx)->((x0 Xy)->((cR Xx) Xy)))
% Found (and_rect30 (fun (x9:(x0 Xx))=> (((eq_ref a) Xy) x0))) as proof of ((cR Xx) Xy)
% Found ((and_rect3 ((cR Xx) Xy)) (fun (x9:(x0 Xx))=> (((eq_ref a) Xy) x0))) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x0 Xx))=> (((eq_ref a) Xy) x0))) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x0 Xx))=> (((eq_ref a) Xy) x0))) as proof of ((cR Xx) Xy)
% Found (or_introl00 (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x0 Xx))=> (((eq_ref a) Xy) x0)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_introl0 ((cR Xy) Xx)) (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x0 Xx))=> (((eq_ref a) Xy) x0)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x0 Xx))=> (((eq_ref a) Xy) x0)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x0 Xx))=> (((eq_ref a) Xy) x0))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x9:((x0 Xx)->((x0 Xy)->P)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x0 Xx))=> (((eq_ref a) Xy) x0))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found x9:(x2 Xx)
% Instantiate: x2:=(cR Xy):(a->Prop)
% Found (fun (x10:(x2 Xy))=> x9) as proof of ((cR Xy) Xx)
% Found (fun (x9:(x2 Xx)) (x10:(x2 Xy))=> x9) as proof of ((x2 Xy)->((cR Xy) Xx))
% Found (fun (x9:(x2 Xx)) (x10:(x2 Xy))=> x9) as proof of ((x2 Xx)->((x2 Xy)->((cR Xy) Xx)))
% Found (and_rect30 (fun (x9:(x2 Xx)) (x10:(x2 Xy))=> x9)) as proof of ((cR Xy) Xx)
% Found ((and_rect3 ((cR Xy) Xx)) (fun (x9:(x2 Xx)) (x10:(x2 Xy))=> x9)) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x2 Xx)) (x10:(x2 Xy))=> x9)) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x2 Xx)) (x10:(x2 Xy))=> x9)) as proof of ((cR Xy) Xx)
% Found (or_intror00 (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x2 Xx)) (x10:(x2 Xy))=> x9))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_intror0 ((cR Xy) Xx)) (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x2 Xx)) (x10:(x2 Xy))=> x9))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x2 Xx)) (x10:(x2 Xy))=> x9))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x2 Xx)) (x2 Xy)))=> (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x2 Xx)) (x10:(x2 Xy))=> x9)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found x10:(x2 Xy)
% Instantiate: x2:=(cR Xx):(a->Prop)
% Found (fun (x10:(x2 Xy))=> x10) as proof of ((cR Xx) Xy)
% Found (fun (x9:(x2 Xx)) (x10:(x2 Xy))=> x10) as proof of ((x2 Xy)->((cR Xx) Xy))
% Found (fun (x9:(x2 Xx)) (x10:(x2 Xy))=> x10) as proof of ((x2 Xx)->((x2 Xy)->((cR Xx) Xy)))
% Found (and_rect30 (fun (x9:(x2 Xx)) (x10:(x2 Xy))=> x10)) as proof of ((cR Xx) Xy)
% Found ((and_rect3 ((cR Xx) Xy)) (fun (x9:(x2 Xx)) (x10:(x2 Xy))=> x10)) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x2 Xx)) (x10:(x2 Xy))=> x10)) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x2 Xx)) (x10:(x2 Xy))=> x10)) as proof of ((cR Xx) Xy)
% Found (or_introl00 (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x2 Xx)) (x10:(x2 Xy))=> x10))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_introl0 ((cR Xy) Xx)) (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x2 Xx)) (x10:(x2 Xy))=> x10))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x2 Xx)) (x10:(x2 Xy))=> x10))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x2 Xx)) (x2 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x2 Xx)) (x10:(x2 Xy))=> x10)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x2 Xx)) (x2 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x2 Xx)) (x10:(x2 Xy))=> x10)))) as proof of (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found or_introl00:=(or_introl0 ((cR Xx) Xy)):((x2 Xy)->((or (x2 Xy)) ((cR Xx) Xy)))
% Found (or_introl0 ((cR Xx) Xy)) as proof of ((x2 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((or_introl (x2 Xy)) ((cR Xx) Xy)) as proof of ((x2 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((or_introl (x2 Xy)) ((cR Xx) Xy)) as proof of ((x2 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x9:(x2 Xx))=> ((or_introl (x2 Xy)) ((cR Xx) Xy))) as proof of ((x2 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x9:(x2 Xx))=> ((or_introl (x2 Xy)) ((cR Xx) Xy))) as proof of ((x2 Xx)->((x2 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy))))
% Found (and_rect30 (fun (x9:(x2 Xx))=> ((or_introl (x2 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found ((and_rect3 ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x2 Xx))=> ((or_introl (x2 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x2 Xx))=> ((or_introl (x2 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x2 Xx))=> ((or_introl (x2 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (or_comm_i00 (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x2 Xx))=> ((or_introl (x2 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_comm_i0 ((cR Xx) Xy)) (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x2 Xx))=> ((or_introl (x2 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x2 Xx))=> ((or_introl (x2 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x2 Xx)) (x2 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x2 Xx))=> ((or_introl (x2 Xy)) ((cR Xx) Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x2 Xx)) (x2 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x9:((x2 Xx)->((x2 Xy)->P)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x2 Xx))=> ((or_introl (x2 Xy)) ((cR Xx) Xy)))))) as proof of (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found eq_ref000:=(eq_ref00 x4):((x4 Xy)->(x4 Xy))
% Found (eq_ref00 x4) as proof of ((x4 Xy)->((cR Xx) Xy))
% Found ((eq_ref0 Xy) x4) as proof of ((x4 Xy)->((cR Xx) Xy))
% Found (((eq_ref a) Xy) x4) as proof of ((x4 Xy)->((cR Xx) Xy))
% Found (((eq_ref a) Xy) x4) as proof of ((x4 Xy)->((cR Xx) Xy))
% Found (fun (x9:(x4 Xx))=> (((eq_ref a) Xy) x4)) as proof of ((x4 Xy)->((cR Xx) Xy))
% Found (fun (x9:(x4 Xx))=> (((eq_ref a) Xy) x4)) as proof of ((x4 Xx)->((x4 Xy)->((cR Xx) Xy)))
% Found (and_rect30 (fun (x9:(x4 Xx))=> (((eq_ref a) Xy) x4))) as proof of ((cR Xx) Xy)
% Found ((and_rect3 ((cR Xx) Xy)) (fun (x9:(x4 Xx))=> (((eq_ref a) Xy) x4))) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x4 Xx))=> (((eq_ref a) Xy) x4))) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x4 Xx))=> (((eq_ref a) Xy) x4))) as proof of ((cR Xx) Xy)
% Found (or_introl00 (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x4 Xx))=> (((eq_ref a) Xy) x4)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_introl0 ((cR Xy) Xx)) (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x4 Xx))=> (((eq_ref a) Xy) x4)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x4 Xx))=> (((eq_ref a) Xy) x4)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x4 Xx)) (x4 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x4 Xx))=> (((eq_ref a) Xy) x4))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x4 Xx)) (x4 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x4 Xx))=> (((eq_ref a) Xy) x4))))) as proof of (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found or_intror00:=(or_intror0 (x4 Xy)):((x4 Xy)->((or ((cR Xy) Xx)) (x4 Xy)))
% Found (or_intror0 (x4 Xy)) as proof of ((x4 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((or_intror ((cR Xy) Xx)) (x4 Xy)) as proof of ((x4 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((or_intror ((cR Xy) Xx)) (x4 Xy)) as proof of ((x4 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x9:(x4 Xx))=> ((or_intror ((cR Xy) Xx)) (x4 Xy))) as proof of ((x4 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x9:(x4 Xx))=> ((or_intror ((cR Xy) Xx)) (x4 Xy))) as proof of ((x4 Xx)->((x4 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy))))
% Found (and_rect30 (fun (x9:(x4 Xx))=> ((or_intror ((cR Xy) Xx)) (x4 Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found ((and_rect3 ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x4 Xx))=> ((or_intror ((cR Xy) Xx)) (x4 Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x4 Xx))=> ((or_intror ((cR Xy) Xx)) (x4 Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x4 Xx))=> ((or_intror ((cR Xy) Xx)) (x4 Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (or_comm_i00 (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x4 Xx))=> ((or_intror ((cR Xy) Xx)) (x4 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_comm_i0 ((cR Xx) Xy)) (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x4 Xx))=> ((or_intror ((cR Xy) Xx)) (x4 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x4 Xx))=> ((or_intror ((cR Xy) Xx)) (x4 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x4 Xx)) (x4 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x4 Xx))=> ((or_intror ((cR Xy) Xx)) (x4 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x4 Xx)) (x4 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x4 Xx))=> ((or_intror ((cR Xy) Xx)) (x4 Xy)))))) as proof of (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found x9:(x4 Xx)
% Instantiate: x4:=(cR Xy):(a->Prop)
% Found (fun (x10:(x4 Xy))=> x9) as proof of ((cR Xy) Xx)
% Found (fun (x9:(x4 Xx)) (x10:(x4 Xy))=> x9) as proof of ((x4 Xy)->((cR Xy) Xx))
% Found (fun (x9:(x4 Xx)) (x10:(x4 Xy))=> x9) as proof of ((x4 Xx)->((x4 Xy)->((cR Xy) Xx)))
% Found (and_rect30 (fun (x9:(x4 Xx)) (x10:(x4 Xy))=> x9)) as proof of ((cR Xy) Xx)
% Found ((and_rect3 ((cR Xy) Xx)) (fun (x9:(x4 Xx)) (x10:(x4 Xy))=> x9)) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x4 Xx)) (x10:(x4 Xy))=> x9)) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x4 Xx)) (x10:(x4 Xy))=> x9)) as proof of ((cR Xy) Xx)
% Found (or_intror00 (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x4 Xx)) (x10:(x4 Xy))=> x9))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_intror0 ((cR Xy) Xx)) (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x4 Xx)) (x10:(x4 Xy))=> x9))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x4 Xx)) (x10:(x4 Xy))=> x9))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x4 Xx)) (x4 Xy)))=> (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x9:((x4 Xx)->((x4 Xy)->P)))=> (((((and_rect (x4 Xx)) (x4 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x4 Xx)) (x10:(x4 Xy))=> x9)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found eq_ref000:=(eq_ref00 (fun (x10:Prop)=> (x6 Xy))):((x6 Xy)->(x6 Xy))
% Found (eq_ref00 (fun (x10:Prop)=> (x6 Xy))) as proof of ((x6 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((eq_ref0 ((cR Xx) Xy)) (fun (x10:Prop)=> (x6 Xy))) as proof of ((x6 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (((eq_ref Prop) ((cR Xx) Xy)) (fun (x10:Prop)=> (x6 Xy))) as proof of ((x6 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (((eq_ref Prop) ((cR Xx) Xy)) (fun (x10:Prop)=> (x6 Xy))) as proof of ((x6 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x9:(x6 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x10:Prop)=> (x6 Xy)))) as proof of ((x6 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x9:(x6 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x10:Prop)=> (x6 Xy)))) as proof of ((x6 Xx)->((x6 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy))))
% Found (and_rect30 (fun (x9:(x6 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x10:Prop)=> (x6 Xy))))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found ((and_rect3 ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x6 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x10:Prop)=> (x6 Xy))))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x6 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x10:Prop)=> (x6 Xy))))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x6 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x10:Prop)=> (x6 Xy))))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (or_comm_i00 (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x6 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x10:Prop)=> (x6 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_comm_i0 ((cR Xx) Xy)) (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x6 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x10:Prop)=> (x6 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x6 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x10:Prop)=> (x6 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x6 Xx)) (x6 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x6 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x10:Prop)=> (x6 Xy))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x6 Xx)) (x6 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x9:(x6 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x10:Prop)=> (x6 Xy))))))) as proof of (((and (x6 Xx)) (x6 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found eq_ref000:=(eq_ref00 x6):((x6 Xy)->(x6 Xy))
% Found (eq_ref00 x6) as proof of ((x6 Xy)->((cR Xx) Xy))
% Found ((eq_ref0 Xy) x6) as proof of ((x6 Xy)->((cR Xx) Xy))
% Found (((eq_ref a) Xy) x6) as proof of ((x6 Xy)->((cR Xx) Xy))
% Found (((eq_ref a) Xy) x6) as proof of ((x6 Xy)->((cR Xx) Xy))
% Found (fun (x9:(x6 Xx))=> (((eq_ref a) Xy) x6)) as proof of ((x6 Xy)->((cR Xx) Xy))
% Found (fun (x9:(x6 Xx))=> (((eq_ref a) Xy) x6)) as proof of ((x6 Xx)->((x6 Xy)->((cR Xx) Xy)))
% Found (and_rect30 (fun (x9:(x6 Xx))=> (((eq_ref a) Xy) x6))) as proof of ((cR Xx) Xy)
% Found ((and_rect3 ((cR Xx) Xy)) (fun (x9:(x6 Xx))=> (((eq_ref a) Xy) x6))) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x6 Xx))=> (((eq_ref a) Xy) x6))) as proof of ((cR Xx) Xy)
% Found (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x6 Xx))=> (((eq_ref a) Xy) x6))) as proof of ((cR Xx) Xy)
% Found (or_introl00 (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x6 Xx))=> (((eq_ref a) Xy) x6)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_introl0 ((cR Xy) Xx)) (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x6 Xx))=> (((eq_ref a) Xy) x6)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x6 Xx))=> (((eq_ref a) Xy) x6)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x6 Xx)) (x6 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x6 Xx))=> (((eq_ref a) Xy) x6))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x6 Xx)) (x6 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((cR Xx) Xy)) (fun (x9:(x6 Xx))=> (((eq_ref a) Xy) x6))))) as proof of (((and (x6 Xx)) (x6 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found x9:(x6 Xx)
% Instantiate: x6:=(cR Xy):(a->Prop)
% Found (fun (x10:(x6 Xy))=> x9) as proof of ((cR Xy) Xx)
% Found (fun (x9:(x6 Xx)) (x10:(x6 Xy))=> x9) as proof of ((x6 Xy)->((cR Xy) Xx))
% Found (fun (x9:(x6 Xx)) (x10:(x6 Xy))=> x9) as proof of ((x6 Xx)->((x6 Xy)->((cR Xy) Xx)))
% Found (and_rect30 (fun (x9:(x6 Xx)) (x10:(x6 Xy))=> x9)) as proof of ((cR Xy) Xx)
% Found ((and_rect3 ((cR Xy) Xx)) (fun (x9:(x6 Xx)) (x10:(x6 Xy))=> x9)) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x6 Xx)) (x10:(x6 Xy))=> x9)) as proof of ((cR Xy) Xx)
% Found (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x6 Xx)) (x10:(x6 Xy))=> x9)) as proof of ((cR Xy) Xx)
% Found (or_intror00 (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x6 Xx)) (x10:(x6 Xy))=> x9))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_intror0 ((cR Xy) Xx)) (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x6 Xx)) (x10:(x6 Xy))=> x9))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x6 Xx)) (x10:(x6 Xy))=> x9))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x6 Xx)) (x6 Xy)))=> (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P:Type) (x9:((x6 Xx)->((x6 Xy)->P)))=> (((((and_rect (x6 Xx)) (x6 Xy)) P) x9) x00)) ((cR Xy) Xx)) (fun (x9:(x6 Xx)) (x10:(x6 Xy))=> x9)))) as proof of ((or ((cR Xx) Xy)) ((cR 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx))))))))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq ((a->Prop)->Prop)) a0) (fun (x:(a->Prop))=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion (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_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 ((T Xx)->(x6 Xx))
% Found (eq_sym0000 ((eq_ref a) Xx)) as proof of ((T Xx)->(x6 Xx))
% Found ((fun (x7:(((eq a) Xx) Xx))=> ((eq_sym000 x7) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x6 Xx))
% Found ((fun (x7:(((eq a) Xx) Xx))=> (((eq_sym00 Xx) x7) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x6 Xx))
% Found ((fun (x7:(((eq a) Xx) Xx))=> ((((eq_sym0 Xx) Xx) x7) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x6 Xx))
% Found ((fun (x7:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x7) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x6 Xx))
% Found ((fun (x7:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x7) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x6 Xx))
% Found (fun (Xx:a)=> ((fun (x7:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x7) T)) ((eq_ref a) Xx))) as proof of ((T Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x6 Xx)->(T Xx))))) (Xx:a)=> ((fun (x7:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x7) T)) ((eq_ref a) Xx))) as proof of (forall (Xx:a), ((T Xx)->(x6 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 x30:=(x3 Xx):((cR Xx) Xx)
% Found (x3 Xx) as proof of ((cR Xx) Xx)
% Found (x3 Xx) as proof of ((cR Xx) Xx)
% Found x30:=(x3 Xx):((cR Xx) Xx)
% Found (x3 Xx) as proof of ((cR Xx) Xx)
% Found (x3 Xx) as proof of ((cR Xx) Xx)
% Found ((conj10 (x3 Xx)) (x3 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found (((conj1 ((cR Xx) Xx)) (x3 Xx)) (x3 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found (x1000 ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx))) as proof of ((T Xx)->(x4 Xx))
% Found (x1000 ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx))) as proof of ((T Xx)->(x4 Xx))
% Found ((fun (x5:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((x100 x5) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx))) as proof of ((T Xx)->(x4 Xx))
% Found ((fun (x5:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> (((x10 Xx) x5) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx))) as proof of ((T Xx)->(x4 Xx))
% Found ((fun (x5:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x1 Xx) Xx) x5) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx))) as proof of ((T Xx)->(x4 Xx))
% Found ((fun (x5:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x1 Xx) Xx) x5) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx))) as proof of ((T Xx)->(x4 Xx))
% Found (fun (Xx:a)=> ((fun (x5:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x1 Xx) Xx) x5) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx)))) as proof of ((T Xx)->(x4 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))) (Xx:a)=> ((fun (x5:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x1 Xx) Xx) x5) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(x4 Xx)))
% Found eq_ref00:=(eq_ref0 (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))):(((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx))))))
% Found (eq_ref0 (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))) b)
% 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 ((T Xx)->(x0 Xx))
% Found (eq_sym0000 ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x7:(((eq a) Xx) Xx))=> ((eq_sym000 x7) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x7:(((eq a) Xx) Xx))=> (((eq_sym00 Xx) x7) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x7:(((eq a) Xx) Xx))=> ((((eq_sym0 Xx) Xx) x7) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x7:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x7) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x7:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x7) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found (fun (Xx:a)=> ((fun (x7:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x7) T)) ((eq_ref a) Xx))) as proof of ((T Xx)->(x0 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))) (Xx:a)=> ((fun (x7:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x7) T)) ((eq_ref a) Xx))) as proof of (forall (Xx:a), ((T Xx)->(x0 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 ((T Xx)->(x2 Xx))
% Found (eq_sym0000 ((eq_ref a) Xx)) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x7:(((eq a) Xx) Xx))=> ((eq_sym000 x7) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x7:(((eq a) Xx) Xx))=> (((eq_sym00 Xx) x7) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x7:(((eq a) Xx) Xx))=> ((((eq_sym0 Xx) Xx) x7) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x7:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x7) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x7:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x7) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x2 Xx))
% Found (fun (Xx:a)=> ((fun (x7:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x7) T)) ((eq_ref a) Xx))) as proof of ((T Xx)->(x2 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))) (Xx:a)=> ((fun (x7:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x7) T)) ((eq_ref a) Xx))) as proof of (forall (Xx:a), ((T Xx)->(x2 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 ((T Xx)->(x4 Xx))
% Found (eq_sym0000 ((eq_ref a) Xx)) as proof of ((T Xx)->(x4 Xx))
% Found ((fun (x7:(((eq a) Xx) Xx))=> ((eq_sym000 x7) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x4 Xx))
% Found ((fun (x7:(((eq a) Xx) Xx))=> (((eq_sym00 Xx) x7) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x4 Xx))
% Found ((fun (x7:(((eq a) Xx) Xx))=> ((((eq_sym0 Xx) Xx) x7) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x4 Xx))
% Found ((fun (x7:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x7) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x4 Xx))
% Found ((fun (x7:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x7) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x4 Xx))
% Found (fun (Xx:a)=> ((fun (x7:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x7) T)) ((eq_ref a) Xx))) as proof of ((T Xx)->(x4 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))) (Xx:a)=> ((fun (x7:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x7) T)) ((eq_ref a) Xx))) as proof of (forall (Xx:a), ((T Xx)->(x4 Xx)))
% Found x40:=(x4 Xx):((cR Xx) Xx)
% Found (x4 Xx) as proof of ((cR Xx) Xx)
% Found (x4 Xx) as proof of ((cR Xx) Xx)
% Found x40:=(x4 Xx):((cR Xx) Xx)
% Found (x4 Xx) as proof of ((cR Xx) Xx)
% Found (x4 Xx) as proof of ((cR Xx) Xx)
% Found ((conj10 (x4 Xx)) (x4 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found (((conj1 ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found (x2000 ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of ((T Xx)->(x0 Xx))
% Found (x2000 ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x5:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((x200 x5) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x5:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> (((x20 Xx) x5) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x5:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x2 Xx) Xx) x5) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x5:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x2 Xx) Xx) x5) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of ((T Xx)->(x0 Xx))
% Found (fun (Xx:a)=> ((fun (x5:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x2 Xx) Xx) x5) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)))) as proof of ((T Xx)->(x0 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))) (Xx:a)=> ((fun (x5:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x2 Xx) Xx) x5) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(x0 Xx)))
% Found x40:=(x4 Xx):((cR Xx) Xx)
% Found (x4 Xx) as proof of ((cR Xx) Xx)
% Found (x4 Xx) as proof of ((cR Xx) Xx)
% Found x40:=(x4 Xx):((cR Xx) Xx)
% Found (x4 Xx) as proof of ((cR Xx) Xx)
% Found (x4 Xx) as proof of ((cR Xx) Xx)
% Found ((conj10 (x4 Xx)) (x4 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found (((conj1 ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found (x1000 ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of ((T Xx)->(x2 Xx))
% Found (x1000 ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x5:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((x100 x5) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x5:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> (((x10 Xx) x5) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x5:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x1 Xx) Xx) x5) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x5:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x1 Xx) Xx) x5) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of ((T Xx)->(x2 Xx))
% Found (fun (Xx:a)=> ((fun (x5:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x1 Xx) Xx) x5) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)))) as proof of ((T Xx)->(x2 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))) (Xx:a)=> ((fun (x5:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x1 Xx) Xx) x5) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(x2 Xx)))
% Found eq_ref00:=(eq_ref0 (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))):(((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))
% Found (eq_ref0 (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) b)
% 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx))))))))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq ((a->Prop)->Prop)) a0) (fun (x:(a->Prop))=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) b)
% Found (((eta_expansion (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_sym:=(fun (T:Type) (a:T) (b:T) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq T) x) a))) ((eq_ref T) a))):(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a)))
% Instantiate: b:=(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a))):Prop
% Found eq_sym as proof of b
% Found eq_ref000:=(eq_ref00 (fun (x2:Prop)=> (x0 Xy))):((x0 Xy)->(x0 Xy))
% Found (eq_ref00 (fun (x2:Prop)=> (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((eq_ref0 ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (((eq_ref Prop) ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (((eq_ref Prop) ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy)))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy)))) as proof of ((x0 Xx)->((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect00 (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect0 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy)))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found eq_ref000:=(eq_ref00 (fun (x2:Prop)=> (x0 Xy))):((x0 Xy)->(x0 Xy))
% Found (eq_ref00 (fun (x2:Prop)=> (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((eq_ref0 ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (((eq_ref Prop) ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (((eq_ref Prop) ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy)))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy)))) as proof of ((x0 Xx)->((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect00 (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect0 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x2:Prop)=> (x0 Xy)))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found eq_sym:=(fun (T:Type) (a:T) (b:T) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq T) x) a))) ((eq_ref T) a))):(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a)))
% Instantiate: b:=(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a))):Prop
% Found eq_sym as proof of b
% Found eq_ref00:=(eq_ref0 (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))):(((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx))))))
% Found (eq_ref0 (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))) b)
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(x0 Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(x0 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))) (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(x0 Xx)))
% Found x000:(T Xx)
% Instantiate: x0:=T:(a->Prop)
% Found (fun (x000:(T Xx))=> x000) as proof of (x0 Xx)
% Found (fun (Xx:a) (x000:(T Xx))=> x000) as proof of ((T Xx)->(x0 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))) (Xx:a) (x000:(T Xx))=> x000) as proof of (forall (Xx:a), ((T Xx)->(x0 Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(x0 Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(x0 Xx))
% Found (fun (x1:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))) (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(x0 Xx)))
% Found x2:(T Xx)
% Instantiate: x0:=T:(a->Prop)
% Found (fun (x2:(T Xx))=> x2) as proof of (x0 Xx)
% Found (fun (Xx:a) (x2:(T Xx))=> x2) as proof of ((T Xx)->(x0 Xx))
% Found (fun (x1:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))) (Xx:a) (x2:(T Xx))=> x2) as proof of (forall (Xx:a), ((T Xx)->(x0 Xx)))
% Found eq_ref00:=(eq_ref0 (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))):(((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))
% Found (eq_ref0 (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) b)
% Found eq_ref00:=(eq_ref0 (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))):(((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx))))))
% Found (eq_ref0 (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))) b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))))
% Found (((eq_trans000 ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))))
% Found ((((eq_trans00 ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))))
% Found (((((eq_trans0 (f x0)) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))))
% Found ((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))))
% Found (((eq_trans000 ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))))
% Found ((((eq_trans00 ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))))
% Found (((((eq_trans0 (f x0)) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))))
% Found ((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))))
% Found eq_sym:=(fun (T:Type) (a:T) (b:T) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq T) x) a))) ((eq_ref T) a))):(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a)))
% Instantiate: b:=(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a))):Prop
% Found eq_sym as proof of b
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% Found (eq_ref00 P0) as proof of (P1 (f x0))
% Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% Found (eq_ref00 P0) as proof of (P1 (f x0))
% Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% 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 ((T Xx)->(x8 Xx))
% Found (eq_sym0000 ((eq_ref a) Xx)) as proof of ((T Xx)->(x8 Xx))
% Found ((fun (x9:(((eq a) Xx) Xx))=> ((eq_sym000 x9) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x8 Xx))
% Found ((fun (x9:(((eq a) Xx) Xx))=> (((eq_sym00 Xx) x9) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x8 Xx))
% Found ((fun (x9:(((eq a) Xx) Xx))=> ((((eq_sym0 Xx) Xx) x9) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x8 Xx))
% Found ((fun (x9:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x9) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x8 Xx))
% Found ((fun (x9:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x9) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x8 Xx))
% Found (fun (Xx:a)=> ((fun (x9:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x9) T)) ((eq_ref a) Xx))) as proof of ((T Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x8 Xx)->(T Xx))))) (Xx:a)=> ((fun (x9:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x9) T)) ((eq_ref a) Xx))) as proof of (forall (Xx:a), ((T Xx)->(x8 Xx)))
% Found x30:=(x3 Xx):((cR Xx) Xx)
% Found (x3 Xx) as proof of ((cR Xx) Xx)
% Found (x3 Xx) as proof of ((cR Xx) Xx)
% Found x30:=(x3 Xx):((cR Xx) Xx)
% Found (x3 Xx) as proof of ((cR Xx) Xx)
% Found (x3 Xx) as proof of ((cR Xx) Xx)
% Found ((conj10 (x3 Xx)) (x3 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found (((conj1 ((cR Xx) Xx)) (x3 Xx)) (x3 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found (x1000 ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx))) as proof of ((T Xx)->(x6 Xx))
% Found (x1000 ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx))) as proof of ((T Xx)->(x6 Xx))
% Found ((fun (x7:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((x100 x7) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx))) as proof of ((T Xx)->(x6 Xx))
% Found ((fun (x7:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> (((x10 Xx) x7) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx))) as proof of ((T Xx)->(x6 Xx))
% Found ((fun (x7:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x1 Xx) Xx) x7) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx))) as proof of ((T Xx)->(x6 Xx))
% Found ((fun (x7:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x1 Xx) Xx) x7) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx))) as proof of ((T Xx)->(x6 Xx))
% Found (fun (Xx:a)=> ((fun (x7:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x1 Xx) Xx) x7) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx)))) as proof of ((T Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x6 Xx)->(T Xx))))) (Xx:a)=> ((fun (x7:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x1 Xx) Xx) x7) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(x6 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx))))))))
% 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_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 ((T Xx)->(x0 Xx))
% Found (eq_sym0000 ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x9:(((eq a) Xx) Xx))=> ((eq_sym000 x9) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x9:(((eq a) Xx) Xx))=> (((eq_sym00 Xx) x9) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x9:(((eq a) Xx) Xx))=> ((((eq_sym0 Xx) Xx) x9) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x9:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x9) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x9:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x9) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x0 Xx))
% Found (fun (Xx:a)=> ((fun (x9:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x9) T)) ((eq_ref a) Xx))) as proof of ((T Xx)->(x0 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))) (Xx:a)=> ((fun (x9:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x9) T)) ((eq_ref a) Xx))) as proof of (forall (Xx:a), ((T Xx)->(x0 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 ((T Xx)->(x2 Xx))
% Found (eq_sym0000 ((eq_ref a) Xx)) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x9:(((eq a) Xx) Xx))=> ((eq_sym000 x9) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x9:(((eq a) Xx) Xx))=> (((eq_sym00 Xx) x9) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x9:(((eq a) Xx) Xx))=> ((((eq_sym0 Xx) Xx) x9) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x9:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x9) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x9:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x9) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x2 Xx))
% Found (fun (Xx:a)=> ((fun (x9:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x9) T)) ((eq_ref a) Xx))) as proof of ((T Xx)->(x2 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))) (Xx:a)=> ((fun (x9:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x9) T)) ((eq_ref a) Xx))) as proof of (forall (Xx:a), ((T Xx)->(x2 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 ((T Xx)->(x4 Xx))
% Found (eq_sym0000 ((eq_ref a) Xx)) as proof of ((T Xx)->(x4 Xx))
% Found ((fun (x9:(((eq a) Xx) Xx))=> ((eq_sym000 x9) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x4 Xx))
% Found ((fun (x9:(((eq a) Xx) Xx))=> (((eq_sym00 Xx) x9) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x4 Xx))
% Found ((fun (x9:(((eq a) Xx) Xx))=> ((((eq_sym0 Xx) Xx) x9) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x4 Xx))
% Found ((fun (x9:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x9) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x4 Xx))
% Found ((fun (x9:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x9) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x4 Xx))
% Found (fun (Xx:a)=> ((fun (x9:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x9) T)) ((eq_ref a) Xx))) as proof of ((T Xx)->(x4 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))) (Xx:a)=> ((fun (x9:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x9) T)) ((eq_ref a) Xx))) as proof of (forall (Xx:a), ((T 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 ((T Xx)->(x6 Xx))
% Found (eq_sym0000 ((eq_ref a) Xx)) as proof of ((T Xx)->(x6 Xx))
% Found ((fun (x9:(((eq a) Xx) Xx))=> ((eq_sym000 x9) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x6 Xx))
% Found ((fun (x9:(((eq a) Xx) Xx))=> (((eq_sym00 Xx) x9) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x6 Xx))
% Found ((fun (x9:(((eq a) Xx) Xx))=> ((((eq_sym0 Xx) Xx) x9) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x6 Xx))
% Found ((fun (x9:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x9) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x6 Xx))
% Found ((fun (x9:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x9) T)) ((eq_ref a) Xx)) as proof of ((T Xx)->(x6 Xx))
% Found (fun (Xx:a)=> ((fun (x9:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x9) T)) ((eq_ref a) Xx))) as proof of ((T Xx)->(x6 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x6 Xx)->(T Xx))))) (Xx:a)=> ((fun (x9:(((eq a) Xx) Xx))=> (((((eq_sym a) Xx) Xx) x9) T)) ((eq_ref a) Xx))) as proof of (forall (Xx:a), ((T Xx)->(x6 Xx)))
% Found x40:=(x4 Xx):((cR Xx) Xx)
% Found (x4 Xx) as proof of ((cR Xx) Xx)
% Found (x4 Xx) as proof of ((cR Xx) Xx)
% Found x40:=(x4 Xx):((cR Xx) Xx)
% Found (x4 Xx) as proof of ((cR Xx) Xx)
% Found (x4 Xx) as proof of ((cR Xx) Xx)
% Found ((conj10 (x4 Xx)) (x4 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found (((conj1 ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found (x2000 ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of ((T Xx)->(x0 Xx))
% Found (x2000 ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x7:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((x200 x7) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x7:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> (((x20 Xx) x7) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x7:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x2 Xx) Xx) x7) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x7:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x2 Xx) Xx) x7) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of ((T Xx)->(x0 Xx))
% Found (fun (Xx:a)=> ((fun (x7:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x2 Xx) Xx) x7) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)))) as proof of ((T Xx)->(x0 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))) (Xx:a)=> ((fun (x7:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x2 Xx) Xx) x7) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(x0 Xx)))
% Found x40:=(x4 Xx):((cR Xx) Xx)
% Found (x4 Xx) as proof of ((cR Xx) Xx)
% Found (x4 Xx) as proof of ((cR Xx) Xx)
% Found x40:=(x4 Xx):((cR Xx) Xx)
% Found (x4 Xx) as proof of ((cR Xx) Xx)
% Found (x4 Xx) as proof of ((cR Xx) Xx)
% Found ((conj10 (x4 Xx)) (x4 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found (((conj1 ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found (x1000 ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of ((T Xx)->(x2 Xx))
% Found (x1000 ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x7:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((x100 x7) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x7:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> (((x10 Xx) x7) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x7:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x1 Xx) Xx) x7) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x7:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x1 Xx) Xx) x7) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of ((T Xx)->(x2 Xx))
% Found (fun (Xx:a)=> ((fun (x7:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x1 Xx) Xx) x7) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)))) as proof of ((T Xx)->(x2 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))) (Xx:a)=> ((fun (x7:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x1 Xx) Xx) x7) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(x2 Xx)))
% Found x30:=(x3 Xx):((cR Xx) Xx)
% Found (x3 Xx) as proof of ((cR Xx) Xx)
% Found (x3 Xx) as proof of ((cR Xx) Xx)
% Found x30:=(x3 Xx):((cR Xx) Xx)
% Found (x3 Xx) as proof of ((cR Xx) Xx)
% Found (x3 Xx) as proof of ((cR Xx) Xx)
% Found ((conj10 (x3 Xx)) (x3 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found (((conj1 ((cR Xx) Xx)) (x3 Xx)) (x3 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found (x1000 ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx))) as proof of ((T Xx)->(x4 Xx))
% Found (x1000 ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx))) as proof of ((T Xx)->(x4 Xx))
% Found ((fun (x7:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((x100 x7) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx))) as proof of ((T Xx)->(x4 Xx))
% Found ((fun (x7:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> (((x10 Xx) x7) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx))) as proof of ((T Xx)->(x4 Xx))
% Found ((fun (x7:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x1 Xx) Xx) x7) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx))) as proof of ((T Xx)->(x4 Xx))
% Found ((fun (x7:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x1 Xx) Xx) x7) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx))) as proof of ((T Xx)->(x4 Xx))
% Found (fun (Xx:a)=> ((fun (x7:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x1 Xx) Xx) x7) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx)))) as proof of ((T Xx)->(x4 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))) (Xx:a)=> ((fun (x7:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x1 Xx) Xx) x7) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(x4 Xx)))
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b
% Found x40:=(x4 Xx):((cR Xx) Xx)
% Found (x4 Xx) as proof of ((cR Xx) Xx)
% Found (x4 Xx) as proof of ((cR Xx) Xx)
% Found x40:=(x4 Xx):((cR Xx) Xx)
% Found (x4 Xx) as proof of ((cR Xx) Xx)
% Found (x4 Xx) as proof of ((cR Xx) Xx)
% Found ((conj10 (x4 Xx)) (x4 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found (((conj1 ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found (fun (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of ((forall (Xx0:a), ((cR Xx0) Xx0))->((and ((cR Xx) Xx)) ((cR Xx) Xx)))
% Found (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of (((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))->((forall (Xx0:a), ((cR Xx0) Xx0))->((and ((cR Xx) Xx)) ((cR Xx) Xx))))
% Found (and_rect10 (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)))) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found ((and_rect1 ((and ((cR Xx) Xx)) ((cR Xx) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)))) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found (((fun (P0:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))->((forall (Xx:a), ((cR Xx) Xx))->P0)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx))) P0) x3) x1)) ((and ((cR Xx) Xx)) ((cR Xx) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)))) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found (((fun (P0:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))->((forall (Xx:a), ((cR Xx) Xx))->P0)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx))) P0) x3) x1)) ((and ((cR Xx) Xx)) ((cR Xx) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)))) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found (x2000 (((fun (P0:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))->((forall (Xx:a), ((cR Xx) Xx))->P0)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx))) P0) x3) x1)) ((and ((cR Xx) Xx)) ((cR Xx) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))))) as proof of ((T Xx)->(x0 Xx))
% Found (x2000 (((fun (P0:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))->((forall (Xx:a), ((cR Xx) Xx))->P0)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx))) P0) x3) x1)) ((and ((cR Xx) Xx)) ((cR Xx) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))))) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x3:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((x200 x3) T)) (((fun (P0:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))->((forall (Xx0:a), ((cR Xx0) Xx0))->P0)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0))) P0) x3) x1)) ((and ((cR Xx) Xx)) ((cR Xx) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))))) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x3:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> (((x20 Xx) x3) T)) (((fun (P0:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))->((forall (Xx0:a), ((cR Xx0) Xx0))->P0)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0))) P0) x3) x1)) ((and ((cR Xx) Xx)) ((cR Xx) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))))) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x3:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x2 Xx) Xx) x3) T)) (((fun (P0:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))->((forall (Xx0:a), ((cR Xx0) Xx0))->P0)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0))) P0) x3) x1)) ((and ((cR Xx) Xx)) ((cR Xx) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))))) as proof of ((T Xx)->(x0 Xx))
% Found ((fun (x3:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x2 Xx) Xx) x3) T)) (((fun (P0:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))->((forall (Xx0:a), ((cR Xx0) Xx0))->P0)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0))) P0) x3) x1)) ((and ((cR Xx) Xx)) ((cR Xx) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))))) as proof of ((T Xx)->(x0 Xx))
% Found (fun (Xx:a)=> ((fun (x3:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x2 Xx) Xx) x3) T)) (((fun (P0:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))->((forall (Xx0:a), ((cR Xx0) Xx0))->P0)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0))) P0) x3) x1)) ((and ((cR Xx) Xx)) ((cR Xx) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)))))) as proof of ((T Xx)->(x0 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))) (Xx:a)=> ((fun (x3:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x2 Xx) Xx) x3) T)) (((fun (P0:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))->((forall (Xx0:a), ((cR Xx0) Xx0))->P0)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0))) P0) x3) x1)) ((and ((cR Xx) Xx)) ((cR Xx) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)))))) as proof of (forall (Xx:a), ((T Xx)->(x0 Xx)))
% Found x40:=(x4 Xx):((cR Xx) Xx)
% Found (x4 Xx) as proof of ((cR Xx) Xx)
% Found (x4 Xx) as proof of ((cR Xx) Xx)
% Found x40:=(x4 Xx):((cR Xx) Xx)
% Found (x4 Xx) as proof of ((cR Xx) Xx)
% Found (x4 Xx) as proof of ((cR Xx) Xx)
% Found ((conj10 (x4 Xx)) (x4 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found (((conj1 ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found (fun (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of ((forall (Xx0:a), ((cR Xx0) Xx0))->((and ((cR Xx) Xx)) ((cR Xx) Xx)))
% Found (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))) as proof of (((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))->((forall (Xx0:a), ((cR Xx0) Xx0))->((and ((cR Xx) Xx)) ((cR Xx) Xx))))
% Found (and_rect10 (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)))) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found ((and_rect1 ((and ((cR Xx) Xx)) ((cR Xx) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)))) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found (((fun (P0:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))->((forall (Xx:a), ((cR Xx) Xx))->P0)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx))) P0) x3) x0)) ((and ((cR Xx) Xx)) ((cR Xx) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)))) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found (((fun (P0:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))->((forall (Xx:a), ((cR Xx) Xx))->P0)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx))) P0) x3) x0)) ((and ((cR Xx) Xx)) ((cR Xx) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)))) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found (x1000 (((fun (P0:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))->((forall (Xx:a), ((cR Xx) Xx))->P0)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx))) P0) x3) x0)) ((and ((cR Xx) Xx)) ((cR Xx) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))))) as proof of ((T Xx)->(x2 Xx))
% Found (x1000 (((fun (P0:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))->((forall (Xx:a), ((cR Xx) Xx))->P0)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx))) P0) x3) x0)) ((and ((cR Xx) Xx)) ((cR Xx) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))))) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x3:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((x100 x3) T)) (((fun (P0:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))->((forall (Xx0:a), ((cR Xx0) Xx0))->P0)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0))) P0) x3) x0)) ((and ((cR Xx) Xx)) ((cR Xx) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))))) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x3:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> (((x10 Xx) x3) T)) (((fun (P0:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))->((forall (Xx0:a), ((cR Xx0) Xx0))->P0)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0))) P0) x3) x0)) ((and ((cR Xx) Xx)) ((cR Xx) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))))) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x3:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x1 Xx) Xx) x3) T)) (((fun (P0:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))->((forall (Xx0:a), ((cR Xx0) Xx0))->P0)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0))) P0) x3) x0)) ((and ((cR Xx) Xx)) ((cR Xx) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))))) as proof of ((T Xx)->(x2 Xx))
% Found ((fun (x3:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x1 Xx) Xx) x3) T)) (((fun (P0:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))->((forall (Xx0:a), ((cR Xx0) Xx0))->P0)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0))) P0) x3) x0)) ((and ((cR Xx) Xx)) ((cR Xx) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx))))) as proof of ((T Xx)->(x2 Xx))
% Found (fun (Xx:a)=> ((fun (x3:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x1 Xx) Xx) x3) T)) (((fun (P0:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))->((forall (Xx0:a), ((cR Xx0) Xx0))->P0)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0))) P0) x3) x0)) ((and ((cR Xx) Xx)) ((cR Xx) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)))))) as proof of ((T Xx)->(x2 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))) (Xx:a)=> ((fun (x3:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x1 Xx) Xx) x3) T)) (((fun (P0:Type) (x3:(((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))->((forall (Xx0:a), ((cR Xx0) Xx0))->P0)))=> (((((and_rect ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0))) P0) x3) x0)) ((and ((cR Xx) Xx)) ((cR Xx) Xx))) (fun (x3:((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx0:a), ((X Xx0)->((W Xy) Xx0))))->(((eq a) Xy) Xz))))))))))) (forall (Xx0:a) (Xy:a) (Xz:a), (((and ((cR Xx0) Xy)) ((cR Xy) Xz))->((cR Xx0) Xz))))) (x4:(forall (Xx0:a), ((cR Xx0) Xx0)))=> ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x4 Xx)) (x4 Xx)))))) as proof of (forall (Xx:a), ((T Xx)->(x2 Xx)))
% Found or_intror00:=(or_intror0 (x2 Xy)):((x2 Xy)->((or ((cR Xx) Xy)) (x2 Xy)))
% Found (or_intror0 (x2 Xy)) as proof of ((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_intror ((cR Xx) Xy)) (x2 Xy)) as proof of ((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_intror ((cR Xx) Xy)) (x2 Xy)) as proof of ((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x3:(x2 Xx))=> ((or_intror ((cR Xx) Xy)) (x2 Xy))) as proof of ((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x3:(x2 Xx))=> ((or_intror ((cR Xx) Xy)) (x2 Xy))) as proof of ((x2 Xx)->((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect10 (fun (x3:(x2 Xx))=> ((or_intror ((cR Xx) Xy)) (x2 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect1 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x2 Xx))=> ((or_intror ((cR Xx) Xy)) (x2 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P0:Type) (x3:((x2 Xx)->((x2 Xy)->P0)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x2 Xx))=> ((or_intror ((cR Xx) Xy)) (x2 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x2 Xx)) (x2 Xy)))=> (((fun (P0:Type) (x3:((x2 Xx)->((x2 Xy)->P0)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x2 Xx))=> ((or_intror ((cR Xx) Xy)) (x2 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x2 Xx)) (x2 Xy)))=> (((fun (P0:Type) (x3:((x2 Xx)->((x2 Xy)->P0)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x2 Xx))=> ((or_intror ((cR Xx) Xy)) (x2 Xy))))) as proof of (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(x2 Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(x2 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x2 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x2 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(x2 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))) (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(x2 Xx)))
% Found x01:(T Xx)
% Instantiate: x2:=T:(a->Prop)
% Found (fun (x01:(T Xx))=> x01) as proof of (x2 Xx)
% Found (fun (Xx:a) (x01:(T Xx))=> x01) as proof of ((T Xx)->(x2 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))) (Xx:a) (x01:(T Xx))=> x01) as proof of (forall (Xx:a), ((T Xx)->(x2 Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(x2 Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(x2 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x2 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x2 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(x2 Xx))
% Found (fun (x3:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))) (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(x2 Xx)))
% Found x4:(T Xx)
% Instantiate: x2:=T:(a->Prop)
% Found (fun (x4:(T Xx))=> x4) as proof of (x2 Xx)
% Found (fun (Xx:a) (x4:(T Xx))=> x4) as proof of ((T Xx)->(x2 Xx))
% Found (fun (x3:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))) (Xx:a) (x4:(T Xx))=> x4) as proof of (forall (Xx:a), ((T Xx)->(x2 Xx)))
% Found eq_ref000:=(eq_ref00 (fun (x4:Prop)=> (x2 Xy))):((x2 Xy)->(x2 Xy))
% Found (eq_ref00 (fun (x4:Prop)=> (x2 Xy))) as proof of ((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((eq_ref0 ((cR Xy) Xx)) (fun (x4:Prop)=> (x2 Xy))) as proof of ((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (((eq_ref Prop) ((cR Xy) Xx)) (fun (x4:Prop)=> (x2 Xy))) as proof of ((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (((eq_ref Prop) ((cR Xy) Xx)) (fun (x4:Prop)=> (x2 Xy))) as proof of ((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x3:(x2 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x4:Prop)=> (x2 Xy)))) as proof of ((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x3:(x2 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x4:Prop)=> (x2 Xy)))) as proof of ((x2 Xx)->((x2 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect10 (fun (x3:(x2 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x4:Prop)=> (x2 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect1 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x2 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x4:Prop)=> (x2 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P0:Type) (x3:((x2 Xx)->((x2 Xy)->P0)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x2 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x4:Prop)=> (x2 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x2 Xx)) (x2 Xy)))=> (((fun (P0:Type) (x3:((x2 Xx)->((x2 Xy)->P0)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x2 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x4:Prop)=> (x2 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x2 Xx)) (x2 Xy)))=> (((fun (P0:Type) (x3:((x2 Xx)->((x2 Xy)->P0)))=> (((((and_rect (x2 Xx)) (x2 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x2 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x4:Prop)=> (x2 Xy)))))) as proof of (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))))
% Found (((eq_trans000 ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))))
% Found ((((eq_trans00 ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx))))))) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))))) as proof of (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))))
% Found (((((eq_trans0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx))))))) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))))) as proof of (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))))
% Found ((((((eq_trans Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx))))))) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))))) as proof of (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))))
% Found (((eq_trans000 ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))))
% Found ((((eq_trans00 ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx))))))) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))))) as proof of (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))))
% Found (((((eq_trans0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx))))))) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))))) as proof of (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))))
% Found ((((((eq_trans Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx))))))) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))))) as proof of (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and (x2 Xx)) (x2 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))))
% Found x1:(x0 Xx)
% Instantiate: x0:=(cR Xy):(a->Prop)
% Found (fun (x2:(x0 Xy))=> x1) as proof of ((cR Xy) Xx)
% Found (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x1) as proof of ((x0 Xy)->((cR Xy) Xx))
% Found (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x1) as proof of ((x0 Xx)->((x0 Xy)->((cR Xy) Xx)))
% Found (and_rect00 (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x1)) as proof of ((cR Xy) Xx)
% Found ((and_rect0 ((cR Xy) Xx)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x1)) as proof of ((cR Xy) Xx)
% Found (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((cR Xy) Xx)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x1)) as proof of ((cR Xy) Xx)
% Found (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((cR Xy) Xx)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x1)) as proof of ((cR Xy) Xx)
% Found (or_intror00 (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((cR Xy) Xx)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x1))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_intror0 ((cR Xy) Xx)) (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((cR Xy) Xx)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x1))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((cR Xy) Xx)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x1))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((cR Xy) Xx)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x1)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found x2:(x0 Xy)
% Instantiate: x0:=(cR Xx):(a->Prop)
% Found (fun (x2:(x0 Xy))=> x2) as proof of ((cR Xx) Xy)
% Found (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2) as proof of ((x0 Xy)->((cR Xx) Xy))
% Found (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2) as proof of ((x0 Xx)->((x0 Xy)->((cR Xx) Xy)))
% Found (and_rect00 (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2)) as proof of ((cR Xx) Xy)
% Found ((and_rect0 ((cR Xx) Xy)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2)) as proof of ((cR Xx) Xy)
% Found (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((cR Xx) Xy)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2)) as proof of ((cR Xx) Xy)
% Found (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((cR Xx) Xy)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2)) as proof of ((cR Xx) Xy)
% Found (or_introl00 (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((cR Xx) Xy)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_introl0 ((cR Xy) Xx)) (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((cR Xx) Xy)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((cR Xx) Xy)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((cR Xx) Xy)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((cR Xx) Xy)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2)))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found or_introl00:=(or_introl0 ((cR Xx) Xy)):((x0 Xy)->((or (x0 Xy)) ((cR Xx) Xy)))
% Found (or_introl0 ((cR Xx) Xy)) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((or_introl (x0 Xy)) ((cR Xx) Xy)) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((or_introl (x0 Xy)) ((cR Xx) Xy)) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x1:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy))) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x1:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy))) as proof of ((x0 Xx)->((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy))))
% Found (and_rect00 (fun (x1:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found ((and_rect0 ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x1:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x1:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x1:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (or_comm_i00 (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x1:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_comm_i0 ((cR Xx) Xy)) (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x1:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x1:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x1:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x1:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xx) Xy)))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found or_intror00:=(or_intror0 (x0 Xy)):((x0 Xy)->((or ((cR Xx) Xy)) (x0 Xy)))
% Found (or_intror0 (x0 Xy)) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_intror ((cR Xx) Xy)) (x0 Xy)) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_intror ((cR Xx) Xy)) (x0 Xy)) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x3:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x3:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy))) as proof of ((x0 Xx)->((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect10 (fun (x3:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect1 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found eq_ref00:=(eq_ref0 (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x6 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x6 Xx)))))):(((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x6 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x6 Xx)))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x6 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x6 Xx))))))
% Found (eq_ref0 (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x6 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x6 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x6 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x6 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x6 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x6 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x6 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x6 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x6 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x6 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x6 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x6 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x6 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x6 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x6 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x6 Xx)))))) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x2))->(P0 (f x2)))
% Found (eq_ref00 P0) as proof of (P1 (f x2))
% Found ((eq_ref0 (f x2)) P0) as proof of (P1 (f x2))
% Found (((eq_ref Prop) (f x2)) P0) as proof of (P1 (f x2))
% Found (((eq_ref Prop) (f x2)) P0) as proof of (P1 (f x2))
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x2))->(P0 (f x2)))
% Found (eq_ref00 P0) as proof of (P1 (f x2))
% Found ((eq_ref0 (f x2)) P0) as proof of (P1 (f x2))
% Found (((eq_ref Prop) (f x2)) P0) as proof of (P1 (f x2))
% Found (((eq_ref Prop) (f x2)) P0) as proof of (P1 (f x2))
% Found eq_ref000:=(eq_ref00 (fun (x2:a)=> (x0 Xy))):((x0 Xy)->(x0 Xy))
% Found (eq_ref00 (fun (x2:a)=> (x0 Xy))) as proof of ((x0 Xy)->((cR Xy) Xx))
% Found ((eq_ref0 Xx) (fun (x2:a)=> (x0 Xy))) as proof of ((x0 Xy)->((cR Xy) Xx))
% Found (((eq_ref a) Xx) (fun (x2:a)=> (x0 Xy))) as proof of ((x0 Xy)->((cR Xy) Xx))
% Found (((eq_ref a) Xx) (fun (x2:a)=> (x0 Xy))) as proof of ((x0 Xy)->((cR Xy) Xx))
% Found (fun (x1:(x0 Xx))=> (((eq_ref a) Xx) (fun (x2:a)=> (x0 Xy)))) as proof of ((x0 Xy)->((cR Xy) Xx))
% Found (fun (x1:(x0 Xx))=> (((eq_ref a) Xx) (fun (x2:a)=> (x0 Xy)))) as proof of ((x0 Xx)->((x0 Xy)->((cR Xy) Xx)))
% Found (and_rect00 (fun (x1:(x0 Xx))=> (((eq_ref a) Xx) (fun (x2:a)=> (x0 Xy))))) as proof of ((cR Xy) Xx)
% Found ((and_rect0 ((cR Xy) Xx)) (fun (x1:(x0 Xx))=> (((eq_ref a) Xx) (fun (x2:a)=> (x0 Xy))))) as proof of ((cR Xy) Xx)
% Found (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((cR Xy) Xx)) (fun (x1:(x0 Xx))=> (((eq_ref a) Xx) (fun (x2:a)=> (x0 Xy))))) as proof of ((cR Xy) Xx)
% Found (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((cR Xy) Xx)) (fun (x1:(x0 Xx))=> (((eq_ref a) Xx) (fun (x2:a)=> (x0 Xy))))) as proof of ((cR Xy) Xx)
% Found (or_intror00 (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((cR Xy) Xx)) (fun (x1:(x0 Xx))=> (((eq_ref a) Xx) (fun (x2:a)=> (x0 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_intror0 ((cR Xy) Xx)) (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((cR Xy) Xx)) (fun (x1:(x0 Xx))=> (((eq_ref a) Xx) (fun (x2:a)=> (x0 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((cR Xy) Xx)) (fun (x1:(x0 Xx))=> (((eq_ref a) Xx) (fun (x2:a)=> (x0 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((cR Xy) Xx)) (fun (x1:(x0 Xx))=> (((eq_ref a) Xx) (fun (x2:a)=> (x0 Xy))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_intror ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((cR Xy) Xx)) (fun (x1:(x0 Xx))=> (((eq_ref a) Xx) (fun (x2:a)=> (x0 Xy))))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found x2:(x0 Xy)
% Instantiate: x0:=(cR Xx):(a->Prop)
% Found (fun (x2:(x0 Xy))=> x2) as proof of ((cR Xx) Xy)
% Found (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2) as proof of ((x0 Xy)->((cR Xx) Xy))
% Found (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2) as proof of ((x0 Xx)->((x0 Xy)->((cR Xx) Xy)))
% Found (and_rect00 (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2)) as proof of ((cR Xx) Xy)
% Found ((and_rect0 ((cR Xx) Xy)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2)) as proof of ((cR Xx) Xy)
% Found (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((cR Xx) Xy)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2)) as proof of ((cR Xx) Xy)
% Found (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((cR Xx) Xy)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2)) as proof of ((cR Xx) Xy)
% Found (or_introl00 (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((cR Xx) Xy)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_introl0 ((cR Xy) Xx)) (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((cR Xx) Xy)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((cR Xx) Xy)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((cR Xx) Xy)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_introl ((cR Xx) Xy)) ((cR Xy) Xx)) (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((cR Xx) Xy)) (fun (x1:(x0 Xx)) (x2:(x0 Xy))=> x2)))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found eq_ref000:=(eq_ref00 (fun (x2:Prop)=> (x0 Xy))):((x0 Xy)->(x0 Xy))
% Found (eq_ref00 (fun (x2:Prop)=> (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found ((eq_ref0 ((cR Xx) Xy)) (fun (x2:Prop)=> (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (((eq_ref Prop) ((cR Xx) Xy)) (fun (x2:Prop)=> (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (((eq_ref Prop) ((cR Xx) Xy)) (fun (x2:Prop)=> (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x2:Prop)=> (x0 Xy)))) as proof of ((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy)))
% Found (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x2:Prop)=> (x0 Xy)))) as proof of ((x0 Xx)->((x0 Xy)->((or ((cR Xy) Xx)) ((cR Xx) Xy))))
% Found (and_rect00 (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x2:Prop)=> (x0 Xy))))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found ((and_rect0 ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x2:Prop)=> (x0 Xy))))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x2:Prop)=> (x0 Xy))))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x2:Prop)=> (x0 Xy))))) as proof of ((or ((cR Xy) Xx)) ((cR Xx) Xy))
% Found (or_comm_i00 (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x2:Prop)=> (x0 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((or_comm_i0 ((cR Xx) Xy)) (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x2:Prop)=> (x0 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x2:Prop)=> (x0 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x2:Prop)=> (x0 Xy))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((or_comm_i ((cR Xy) Xx)) ((cR Xx) Xy)) (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((or ((cR Xy) Xx)) ((cR Xx) Xy))) (fun (x1:(x0 Xx))=> (((eq_ref Prop) ((cR Xx) Xy)) (fun (x2:Prop)=> (x0 Xy))))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found x000:(T Xx)
% Instantiate: x0:=T:(a->Prop)
% Found (fun (x000:(T Xx))=> x000) as proof of (x0 Xx)
% Found (fun (Xx:a) (x000:(T Xx))=> x000) as proof of ((T Xx)->(x0 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))) (Xx:a) (x000:(T Xx))=> x000) as proof of (forall (Xx:a), ((T Xx)->(x0 Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(x0 Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(x0 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))) (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(x0 Xx)))
% Found or_introl00:=(or_introl0 ((cR Xy) Xx)):((x0 Xy)->((or (x0 Xy)) ((cR Xy) Xx)))
% Found (or_introl0 ((cR Xy) Xx)) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_introl (x0 Xy)) ((cR Xy) Xx)) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_introl (x0 Xy)) ((cR Xy) Xx)) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x3:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xy) Xx))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x3:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xy) Xx))) as proof of ((x0 Xx)->((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect10 (fun (x3:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xy) Xx)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect1 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xy) Xx)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xy) Xx)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xy) Xx))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xy) Xx))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(x2 Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(x2 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x2 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x2 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(x2 Xx))
% Found (fun (x3:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))) (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(x2 Xx)))
% Found x4:(T Xx)
% Instantiate: x2:=T:(a->Prop)
% Found (fun (x4:(T Xx))=> x4) as proof of (x2 Xx)
% Found (fun (Xx:a) (x4:(T Xx))=> x4) as proof of ((T Xx)->(x2 Xx))
% Found (fun (x3:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))) (Xx:a) (x4:(T Xx))=> x4) as proof of (forall (Xx:a), ((T Xx)->(x2 Xx)))
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b
% Found eq_ref00:=(eq_ref0 (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))):(((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))
% Found (eq_ref0 (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))) b)
% Found eq_ref00:=(eq_ref0 (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))):(((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx))))))
% Found (eq_ref0 (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x2 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x2 Xx)))))) b)
% Found eq_ref00:=(eq_ref0 (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))):(((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx))))))
% Found (eq_ref0 (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))) as proof of (((eq Prop) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))) b)
% Found eq_ref00:=(eq_ref0 ((cR Xy) Xx)):(((eq Prop) ((cR Xy) Xx)) ((cR Xy) Xx))
% Found (eq_ref0 ((cR Xy) Xx)) as proof of (((eq Prop) ((cR Xy) Xx)) b)
% Found ((eq_ref Prop) ((cR Xy) Xx)) as proof of (((eq Prop) ((cR Xy) Xx)) b)
% Found ((eq_ref Prop) ((cR Xy) Xx)) as proof of (((eq Prop) ((cR Xy) Xx)) b)
% Found ((eq_ref Prop) ((cR Xy) Xx)) as proof of (((eq Prop) ((cR Xy) Xx)) b)
% Found or_introl00:=(or_introl0 ((cR Xy) Xx)):((x0 Xy)->((or (x0 Xy)) ((cR Xy) Xx)))
% Found (or_introl0 ((cR Xy) Xx)) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_introl (x0 Xy)) ((cR Xy) Xx)) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_introl (x0 Xy)) ((cR Xy) Xx)) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x2:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xy) Xx))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x2:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xy) Xx))) as proof of ((x0 Xx)->((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect10 (fun (x2:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xy) Xx)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect1 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x2:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xy) Xx)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P0:Type) (x2:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x2) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x2:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xy) Xx)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P0:Type) (x2:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x2) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x2:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xy) Xx))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P0:Type) (x2:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x2) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x2:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xy) Xx))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found or_introl00:=(or_introl0 ((cR Xy) Xx)):((x0 Xy)->((or (x0 Xy)) ((cR Xy) Xx)))
% Found (or_introl0 ((cR Xy) Xx)) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_introl (x0 Xy)) ((cR Xy) Xx)) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_introl (x0 Xy)) ((cR Xy) Xx)) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x3:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xy) Xx))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x3:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xy) Xx))) as proof of ((x0 Xx)->((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect10 (fun (x3:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xy) Xx)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect1 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xy) Xx)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xy) Xx)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xy) Xx))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xy) Xx))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found or_intror00:=(or_intror0 (x0 Xy)):((x0 Xy)->((or ((cR Xx) Xy)) (x0 Xy)))
% Found (or_intror0 (x0 Xy)) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_intror ((cR Xx) Xy)) (x0 Xy)) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_intror ((cR Xx) Xy)) (x0 Xy)) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x1:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x1:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy))) as proof of ((x0 Xx)->((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect00 (fun (x1:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect0 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x1:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x1:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x1:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P0:Type) (x1:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x1) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x1:(x0 Xx))=> ((or_intror ((cR Xx) Xy)) (x0 Xy))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found x000:(T Xx)
% Instantiate: x0:=T:(a->Prop)
% Found (fun (x000:(T Xx))=> x000) as proof of (x0 Xx)
% Found (fun (Xx:a) (x000:(T Xx))=> x000) as proof of ((T Xx)->(x0 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))) (Xx:a) (x000:(T Xx))=> x000) as proof of (forall (Xx:a), ((T Xx)->(x0 Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(x0 Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(x0 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))) (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(x0 Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(x0 Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(x0 Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(x0 Xx))
% Found (fun (x3:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))) (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(x0 Xx)))
% Found x4:(T Xx)
% Instantiate: x0:=T:(a->Prop)
% Found (fun (x4:(T Xx))=> x4) as proof of (x0 Xx)
% Found (fun (Xx:a) (x4:(T Xx))=> x4) as proof of ((T Xx)->(x0 Xx))
% Found (fun (x3:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))) (Xx:a) (x4:(T Xx))=> x4) as proof of (forall (Xx:a), ((T Xx)->(x0 Xx)))
% Found or_introl00:=(or_introl0 ((cR Xy) Xx)):((x0 Xy)->((or (x0 Xy)) ((cR Xy) Xx)))
% Found (or_introl0 ((cR Xy) Xx)) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_introl (x0 Xy)) ((cR Xy) Xx)) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((or_introl (x0 Xy)) ((cR Xy) Xx)) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x2:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xy) Xx))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x2:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xy) Xx))) as proof of ((x0 Xx)->((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect00 (fun (x2:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xy) Xx)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect0 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x2:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xy) Xx)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P0:Type) (x2:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x2) x1)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x2:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xy) Xx)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x1:((and (x0 Xx)) (x0 Xy)))=> (((fun (P0:Type) (x2:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x2) x1)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x2:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xy) Xx))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x1:((and (x0 Xx)) (x0 Xy)))=> (((fun (P0:Type) (x2:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x2) x1)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x2:(x0 Xx))=> ((or_introl (x0 Xy)) ((cR Xy) Xx))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found eq_ref000:=(eq_ref00 (fun (x4:Prop)=> (x0 Xy))):((x0 Xy)->(x0 Xy))
% Found (eq_ref00 (fun (x4:Prop)=> (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found ((eq_ref0 ((cR Xy) Xx)) (fun (x4:Prop)=> (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (((eq_ref Prop) ((cR Xy) Xx)) (fun (x4:Prop)=> (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (((eq_ref Prop) ((cR Xy) Xx)) (fun (x4:Prop)=> (x0 Xy))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x3:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x4:Prop)=> (x0 Xy)))) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x3:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x4:Prop)=> (x0 Xy)))) as proof of ((x0 Xx)->((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect10 (fun (x3:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x4:Prop)=> (x0 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect1 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x4:Prop)=> (x0 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x4:Prop)=> (x0 Xy))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x4:Prop)=> (x0 Xy)))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> (((eq_ref Prop) ((cR Xy) Xx)) (fun (x4:Prop)=> (x0 Xy)))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T 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 ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((S Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(S Xx))))))))
% 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 iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b
% Found and_rect00:=(and_rect0 ((or ((cR Xx) Xy)) ((cR Xy) Xx))):((((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))->((forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0)))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Instantiate: x0:=(fun (x3:a)=> (((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))->((forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0)))->((or ((cR Xx) x3)) ((cR x3) Xx))))):(a->Prop)
% Found (fun (x3:(x0 Xx))=> and_rect00) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x3:(x0 Xx))=> and_rect00) as proof of ((x0 Xx)->((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect10 (fun (x3:(x0 Xx))=> and_rect00)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect1 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> and_rect00)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> and_rect00)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x2:(forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> and_rect00))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x1:((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))) (x2:(forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> and_rect00))) as proof of ((forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0)))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x1:((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))) (x2:(forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> and_rect00))) as proof of (((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))->((forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0)))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect00 (fun (x1:((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))) (x2:(forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> and_rect00)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect0 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x1:((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))) (x2:(forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> (and_rect0 ((or ((cR Xx) Xy)) ((cR Xy) Xx))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P0:Type) (x1:(((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))->P0)))=> (((((and_rect ((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))) P0) x1) x)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x1:((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))) (x2:(forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> ((fun (P0:Type) (x1:(((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))->P0)))=> (((((and_rect ((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))) P0) x1) x)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P0:Type) (x1:(((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))->P0)))=> (((((and_rect ((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))) P0) x1) x)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x1:((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))) (x2:(forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> ((fun (P0:Type) (x1:(((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))->P0)))=> (((((and_rect ((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))) P0) x1) x)) ((or ((cR Xx) Xy)) ((cR Xy) Xx)))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P0:Type) (x1:(((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))->P0)))=> (((((and_rect ((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))) P0) x1) x)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x1:((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))) (x2:(forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> ((fun (P0:Type) (x1:(((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))->P0)))=> (((((and_rect ((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))) P0) x1) x)) ((or ((cR Xx) Xy)) ((cR Xy) Xx)))))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found x30:=(x3 Xx):((cR Xx) Xx)
% Found (x3 Xx) as proof of ((cR Xx) Xx)
% Found (x3 Xx) as proof of ((cR Xx) Xx)
% Found x30:=(x3 Xx):((cR Xx) Xx)
% Found (x3 Xx) as proof of ((cR Xx) Xx)
% Found (x3 Xx) as proof of ((cR Xx) Xx)
% Found ((conj10 (x3 Xx)) (x3 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found (((conj1 ((cR Xx) Xx)) (x3 Xx)) (x3 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx)) as proof of ((and ((cR Xx) Xx)) ((cR Xx) Xx))
% Found (x1000 ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx))) as proof of ((T Xx)->(x8 Xx))
% Found (x1000 ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx))) as proof of ((T Xx)->(x8 Xx))
% Found ((fun (x9:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((x100 x9) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx))) as proof of ((T Xx)->(x8 Xx))
% Found ((fun (x9:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> (((x10 Xx) x9) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx))) as proof of ((T Xx)->(x8 Xx))
% Found ((fun (x9:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x1 Xx) Xx) x9) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx))) as proof of ((T Xx)->(x8 Xx))
% Found ((fun (x9:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x1 Xx) Xx) x9) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx))) as proof of ((T Xx)->(x8 Xx))
% Found (fun (Xx:a)=> ((fun (x9:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x1 Xx) Xx) x9) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx)))) as proof of ((T Xx)->(x8 Xx))
% Found (fun (x00:((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x8 Xx)->(T Xx))))) (Xx:a)=> ((fun (x9:((and ((cR Xx) Xx)) ((cR Xx) Xx)))=> ((((x1 Xx) Xx) x9) T)) ((((conj ((cR Xx) Xx)) ((cR Xx) Xx)) (x3 Xx)) (x3 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(x8 Xx)))
% Found and_rect00:=(and_rect0 ((or ((cR Xx) Xy)) ((cR Xy) Xx))):((((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))->((forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0)))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Instantiate: x0:=(fun (x3:a)=> (((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))->((forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0)))->((or ((cR Xx) x3)) ((cR x3) Xx))))):(a->Prop)
% Found (fun (x3:(x0 Xx))=> and_rect00) as proof of ((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x3:(x0 Xx))=> and_rect00) as proof of ((x0 Xx)->((x0 Xy)->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect10 (fun (x3:(x0 Xx))=> and_rect00)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect1 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> and_rect00)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> and_rect00)) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x2:(forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> and_rect00))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x1:((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))) (x2:(forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> and_rect00))) as proof of ((forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0)))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found (fun (x1:((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))) (x2:(forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> and_rect00))) as proof of (((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))->((forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0)))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))
% Found (and_rect00 (fun (x1:((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))) (x2:(forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> and_rect00)))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found ((and_rect0 ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x1:((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))) (x2:(forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> (and_rect0 ((or ((cR Xx) Xy)) ((cR Xy) Xx))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (((fun (P0:Type) (x1:(((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))->P0)))=> (((((and_rect ((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))) P0) x1) x)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x1:((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))) (x2:(forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> ((fun (P0:Type) (x1:(((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))->P0)))=> (((((and_rect ((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))) P0) x1) x)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P0:Type) (x1:(((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))->P0)))=> (((((and_rect ((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))) P0) x1) x)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x1:((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))) (x2:(forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> ((fun (P0:Type) (x1:(((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))->P0)))=> (((((and_rect ((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))) P0) x1) x)) ((or ((cR Xx) Xy)) ((cR Xy) Xx)))))))) as proof of ((or ((cR Xx) Xy)) ((cR Xy) Xx))
% Found (fun (Xy:a) (x00:((and (x0 Xx)) (x0 Xy)))=> (((fun (P0:Type) (x1:(((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))->P0)))=> (((((and_rect ((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))) P0) x1) x)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x1:((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx0:a), ((X Xx0)->((W Xz) Xx0))))) (forall (Xy0:a), (((and (X Xy0)) (forall (Xx0:a), ((X Xx0)->((W Xy0) Xx0))))->(((eq a) Xy0) Xz))))))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xz))->((cR Xx0) Xz))))) (forall (Xx0:a), ((cR Xx0) Xx0)))) (x2:(forall (Xx0:a) (Xy0:a), (((and ((cR Xx0) Xy0)) ((cR Xy0) Xx0))->(((eq a) Xx0) Xy0))))=> (((fun (P0:Type) (x3:((x0 Xx)->((x0 Xy)->P0)))=> (((((and_rect (x0 Xx)) (x0 Xy)) P0) x3) x00)) ((or ((cR Xx) Xy)) ((cR Xy) Xx))) (fun (x3:(x0 Xx))=> ((fun (P0:Type) (x1:(((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))->((forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))->P0)))=> (((((and_rect ((and ((and ((ex (a->(a->Prop))) (fun (W:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((W Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((W Xy) Xx))))->(((eq a) Xy) Xz))))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))) (forall (Xx:a), ((cR Xx) Xx)))) (forall (Xx:a) (Xy:a), (((and ((cR Xx) Xy)) ((cR Xy) Xx))->(((eq a) Xx) Xy)))) P0) x1) x)) ((or ((cR Xx) Xy)) ((cR Xy) Xx)))))))) as proof of (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and (x0 Xx)) (x0 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x0 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) b)
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x4))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x4)) ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))))
% Found (((eq_trans000 ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx))))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x4)) ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))))
% Found ((((eq_trans00 ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx))))))) ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx))))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))))) as proof of (((eq Prop) (f x4)) ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))))
% Found (((((eq_trans0 (f x4)) ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx))))))) ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx))))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))))) as proof of (((eq Prop) (f x4)) ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))))
% Found ((((((eq_trans Prop) (f x4)) ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx))))))) ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx))))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))))) as proof of (((eq Prop) (f x4)) ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) b)
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x4))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x4)) ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))))
% Found (((eq_trans000 ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx))))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x4)) ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))))
% Found ((((eq_trans00 ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx))))))) ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx))))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (Xx:a), ((x4 Xx)->(T Xx))))->(forall (Xx:a), ((T Xx)->(x4 Xx)))))))) as proof of (((eq Prop) (f x4)) ((and (forall (Xx:a) (Xy:a), (((and (x4 Xx)) (x4 Xy))->((or ((cR Xx) Xy)) ((cR Xy) Xx))))) (forall (T:(a->Prop)), (((and (forall (Xx:a) (Xy:a), (((and (T Xx)) (T Xy))
% EOF
%------------------------------------------------------------------------------