%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEU989^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n105.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 13:33:28 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  : SEU989^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n105.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 11:48:11 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 0x1de97e8>, <kernel.Type object at 0x1de9b90>) of role type named b_type
% Using role type
% Declaring b:Type
% FOF formula (<kernel.Constant object at 0x1fbd3b0>, <kernel.Type object at 0x1de9878>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (((ex (b->(a->Prop))) (fun (S:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((S Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((S Xx) Xy1)) ((S Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((S Xx1) Xy)) ((S Xx2) Xy))->(((eq b) Xx1) Xx2))))))->((ex (a->(b->Prop))) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2))))))) of role conjecture named cTHM554_pme
% Conjecture to prove = (((ex (b->(a->Prop))) (fun (S:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((S Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((S Xx) Xy1)) ((S Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((S Xx1) Xy)) ((S Xx2) Xy))->(((eq b) Xx1) Xx2))))))->((ex (a->(b->Prop))) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2))))))):Prop
% Parameter b_DUMMY:b.
% Parameter a_DUMMY:a.
% We need to prove ['(((ex (b->(a->Prop))) (fun (S:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((S Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((S Xx) Xy1)) ((S Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((S Xx1) Xy)) ((S Xx2) Xy))->(((eq b) Xx1) Xx2))))))->((ex (a->(b->Prop))) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))))']
% Parameter b:Type.
% Parameter a:Type.
% Trying to prove (((ex (b->(a->Prop))) (fun (S:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((S Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((S Xx) Xy1)) ((S Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((S Xx1) Xy)) ((S Xx2) Xy))->(((eq b) Xx1) Xx2))))))->((ex (a->(b->Prop))) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))):(((eq ((a->(b->Prop))->Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) (fun (x:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x Xx) Xy1)) ((x Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found (eta_expansion_dep00 (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found ((eta_expansion_dep0 (fun (x1:(a->(b->Prop)))=> Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found (((eta_expansion_dep (a->(b->Prop))) (fun (x1:(a->(b->Prop)))=> Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found (((eta_expansion_dep (a->(b->Prop))) (fun (x1:(a->(b->Prop)))=> Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found (((eta_expansion_dep (a->(b->Prop))) (fun (x1:(a->(b->Prop)))=> Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found eq_ref000:=(eq_ref00 (x0 Xx)):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy2)))
% Found (and_rect00 (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found ((and_rect0 (P Xy2)) (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x2:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x2) x00)) (P Xy2)) (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found (fun (x1:(P Xy1))=> (((fun (P0:Type) (x2:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x2) x00)) (P Xy2)) (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))))) as proof of (P Xy2)
% Found eq_ref00:=(eq_ref0 (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))):(((eq ((a->(b->Prop))->Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found (eq_ref0 (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found ((eq_ref ((a->(b->Prop))->Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found ((eq_ref ((a->(b->Prop))->Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found ((eq_ref ((a->(b->Prop))->Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found eq_ref00:=(eq_ref0 (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))):(((eq ((a->(b->Prop))->Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found (eq_ref0 (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found ((eq_ref ((a->(b->Prop))->Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found ((eq_ref ((a->(b->Prop))->Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found ((eq_ref ((a->(b->Prop))->Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found eq_ref000:=(eq_ref00 (x2 Xx)):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(P Xy2)))
% Found (and_rect00 (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))) as proof of (P Xy2)
% Found ((and_rect0 (P Xy2)) (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x4:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x4) x00)) (P Xy2)) (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))) as proof of (P Xy2)
% Found (fun (x3:(P Xy1))=> (((fun (P0:Type) (x4:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x4) x00)) (P Xy2)) (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))))) as proof of (P Xy2)
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq b) Xy1) Xy2)->(((eq b) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((eq_trans000 c) x1)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((eq_trans00 Xy1) c) x1)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((((eq_trans0 Xy1) Xy1) c) x1)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x1)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x1)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x1:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x1)) Xy2)) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x1:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x1)) Xy2)) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(((eq b) Xy1) Xy2)))
% Found (and_rect00 (fun (x1:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x1)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect0 (((eq b) Xy1) Xy2)) (fun (x1:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x1)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy1) Xy2)) (fun (x1:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x1)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy1) Xy2)) (fun (x1:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x1)) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy1) Xy2)) (fun (x1:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x1)) Xy2)))) as proof of (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found x4:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))
% Instantiate: x6:=(fun (x10:a) (x90:b)=> ((x0 x90) x10)):(a->(b->Prop))
% Found x4 as proof of (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x6 Xx) Xy))))
% Found eq_ref000:=(eq_ref00 (x0 Xx)):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy2)))
% Found (and_rect00 (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found ((and_rect0 (P Xy2)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy2)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found (fun (x3:(P Xy1))=> (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy2)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))))) as proof of (P Xy2)
% Found x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))
% Instantiate: x0:=(fun (x10:a) (x90:b)=> ((x1 x90) x10)):(a->(b->Prop))
% Found x5 as proof of (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy))))
% Found x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))
% Instantiate: x2:=(fun (x10:a) (x90:b)=> ((x0 x90) x10)):(a->(b->Prop))
% Found x5 as proof of (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy))))
% Found x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))
% Instantiate: x4:=(fun (x10:a) (x90:b)=> ((x0 x90) x10)):(a->(b->Prop))
% Found x5 as proof of (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x4 Xx) Xy))))
% Found eta_expansion000:=(eta_expansion00 (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))):(((eq ((a->(b->Prop))->Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) (fun (x:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x Xx) Xy1)) ((x Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found (eta_expansion00 (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found ((eta_expansion0 Prop) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found (((eta_expansion (a->(b->Prop))) Prop) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found (((eta_expansion (a->(b->Prop))) Prop) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found (((eta_expansion (a->(b->Prop))) Prop) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq b) Xy1) Xy2)->(((eq b) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((eq_trans000 c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((eq_trans00 Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((((eq_trans0 Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(((eq b) Xy1) Xy2)))
% Found (and_rect00 (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect0 (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_ref000:=(eq_ref00 (x4 Xx)):(((x4 Xx) Xy2)->((x4 Xx) Xy2))
% Found (eq_ref00 (x4 Xx)) as proof of (((x4 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x4 Xx)) as proof of (((x4 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x4 Xx)) as proof of (((x4 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x4 Xx)) as proof of (((x4 Xx) Xy2)->(P Xy2))
% Found (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx))) as proof of (((x4 Xx) Xy2)->(P Xy2))
% Found (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx))) as proof of (((x4 Xx) Xy1)->(((x4 Xx) Xy2)->(P Xy2)))
% Found (and_rect10 (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx)))) as proof of (P Xy2)
% Found ((and_rect1 (P Xy2)) (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x6:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx)))) as proof of (P Xy2)
% Found (fun (x5:(P Xy1))=> (((fun (P0:Type) (x6:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx))))) as proof of (P Xy2)
% Found eq_ref000:=(eq_ref00 (fun (x3:b)=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x3:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy1)))
% Found (and_rect00 (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect0 (P Xy1)) (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x2:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x2) x00)) (P Xy1)) (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x1:(P Xy2))=> (((fun (P0:Type) (x2:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x2) x00)) (P Xy1)) (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2)))))) as proof of (P Xy1)
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq b) Xy1) Xy2)->(((eq b) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((eq_trans000 c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((eq_trans00 Xy1) c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((((eq_trans0 Xy1) Xy1) c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(((eq b) Xy1) Xy2)))
% Found (and_rect00 (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect0 (((eq b) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_ref000:=(eq_ref00 (x0 Xx)):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy2)))
% Found (and_rect10 (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found ((and_rect1 (P Xy2)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found (fun (x5:(P Xy1))=> (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))))) as proof of (P Xy2)
% Found eq_ref000:=(eq_ref00 (x2 Xx)):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(P Xy2)))
% Found (and_rect10 (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))) as proof of (P Xy2)
% Found ((and_rect1 (P Xy2)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x6:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))) as proof of (P Xy2)
% Found (fun (x5:(P Xy1))=> (((fun (P0:Type) (x6:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))))) as proof of (P Xy2)
% Found x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))
% Instantiate: x0:=(fun (x10:a) (x90:b)=> ((x1 x90) x10)):(a->(b->Prop))
% Found x5 as proof of (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy))))
% Found x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))
% Instantiate: x2:=(fun (x10:a) (x90:b)=> ((x0 x90) x10)):(a->(b->Prop))
% Found x5 as proof of (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy))))
% Found x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))
% Instantiate: x4:=(fun (x10:a) (x90:b)=> ((x0 x90) x10)):(a->(b->Prop))
% Found x5 as proof of (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x4 Xx) Xy))))
% Found eq_ref000:=(eq_ref00 (x0 Xx)):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy2)))
% Found (and_rect00 (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found ((and_rect0 (P Xy2)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy2)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found (fun (x3:(P Xy1))=> (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy2)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))))) as proof of (P Xy2)
% Found eq_ref000:=(eq_ref00 (x0 Xx)):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy2)))
% Found (and_rect00 (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found ((and_rect0 (P Xy2)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy2)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found (fun (x3:((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x2 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy20))->(((eq a) Xy1) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x2 Xx1) Xy)) ((x2 Xx2) Xy))->(((eq b) Xx1) Xx2)))))=> (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy2)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))))) as proof of (P Xy2)
% Found (fun (x2:(b->(a->Prop))) (x3:((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x2 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy20))->(((eq a) Xy1) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x2 Xx1) Xy)) ((x2 Xx2) Xy))->(((eq b) Xx1) Xx2)))))=> (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy2)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))))) as proof of (((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x2 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy20))->(((eq a) Xy1) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x2 Xx1) Xy)) ((x2 Xx2) Xy))->(((eq b) Xx1) Xx2))))->(P Xy2))
% Found (fun (x2:(b->(a->Prop))) (x3:((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x2 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy20))->(((eq a) Xy1) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x2 Xx1) Xy)) ((x2 Xx2) Xy))->(((eq b) Xx1) Xx2)))))=> (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy2)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))))) as proof of (forall (x:(b->(a->Prop))), (((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x Xx) Xy1)) ((x Xx) Xy20))->(((eq a) Xy1) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x Xx1) Xy)) ((x Xx2) Xy))->(((eq b) Xx1) Xx2))))->(P Xy2)))
% Found (ex_ind00 (fun (x2:(b->(a->Prop))) (x3:((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x2 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy20))->(((eq a) Xy1) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x2 Xx1) Xy)) ((x2 Xx2) Xy))->(((eq b) Xx1) Xx2)))))=> (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy2)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))))) as proof of (P Xy2)
% Found ((ex_ind0 (P Xy2)) (fun (x2:(b->(a->Prop))) (x3:((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x2 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy20))->(((eq a) Xy1) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x2 Xx1) Xy)) ((x2 Xx2) Xy))->(((eq b) Xx1) Xx2)))))=> (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy2)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))))) as proof of (P Xy2)
% Found (((fun (P0:Prop) (x2:(forall (x:(b->(a->Prop))), (((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x Xx) Xy1)) ((x Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x Xx1) Xy)) ((x Xx2) Xy))->(((eq b) Xx1) Xx2))))->P0)))=> (((((ex_ind (b->(a->Prop))) (fun (S:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((S Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((S Xx) Xy1)) ((S Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((S Xx1) Xy)) ((S Xx2) Xy))->(((eq b) Xx1) Xx2)))))) P0) x2) x)) (P Xy2)) (fun (x2:(b->(a->Prop))) (x3:((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x2 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy20))->(((eq a) Xy1) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x2 Xx1) Xy)) ((x2 Xx2) Xy))->(((eq b) Xx1) Xx2)))))=> (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy2)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))))) as proof of (P Xy2)
% Found (fun (x1:(P Xy1))=> (((fun (P0:Prop) (x2:(forall (x:(b->(a->Prop))), (((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x Xx) Xy1)) ((x Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x Xx1) Xy)) ((x Xx2) Xy))->(((eq b) Xx1) Xx2))))->P0)))=> (((((ex_ind (b->(a->Prop))) (fun (S:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((S Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((S Xx) Xy1)) ((S Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((S Xx1) Xy)) ((S Xx2) Xy))->(((eq b) Xx1) Xx2)))))) P0) x2) x)) (P Xy2)) (fun (x2:(b->(a->Prop))) (x3:((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x2 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy20))->(((eq a) Xy1) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x2 Xx1) Xy)) ((x2 Xx2) Xy))->(((eq b) Xx1) Xx2)))))=> (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy2)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))))))) as proof of (P Xy2)
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq b) Xy1) Xy2)->(((eq b) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x4 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((eq_trans000 c) x5)) Xy2) as proof of (((x4 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((eq_trans00 Xy1) c) x5)) Xy2) as proof of (((x4 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((((eq_trans0 Xy1) Xy1) c) x5)) Xy2) as proof of (((x4 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2) as proof of (((x4 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2) as proof of (((x4 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x5:((x4 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2)) as proof of (((x4 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x5:((x4 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2)) as proof of (((x4 Xx) Xy1)->(((x4 Xx) Xy2)->(((eq b) Xy1) Xy2)))
% Found (and_rect10 (fun (x5:((x4 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect1 (((eq b) Xy1) Xy2)) (fun (x5:((x4 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x5:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x4 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2)))=> (((fun (P:Type) (x5:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x4 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2)))=> (((fun (P:Type) (x5:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x4 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2)))) as proof of (((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_ref000:=(eq_ref00 (fun (x5:b)=> ((x2 Xx) Xy2))):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (fun (x5:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(P Xy1)))
% Found (and_rect00 (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect0 (P Xy1)) (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x4:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x4) x00)) (P Xy1)) (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x3:(P Xy2))=> (((fun (P0:Type) (x4:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x4) x00)) (P Xy1)) (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2)))))) as proof of (P Xy1)
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq b) Xy1) Xy2)->(((eq b) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((eq_trans000 c) x5)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((eq_trans00 Xy1) c) x5)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((((eq_trans0 Xy1) Xy1) c) x5)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x5:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2)) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x5:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2)) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(((eq b) Xy1) Xy2)))
% Found (and_rect10 (fun (x5:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect1 (((eq b) Xy1) Xy2)) (fun (x5:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2)))) as proof of (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq b) Xy1) Xy2)->(((eq b) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((eq_trans000 c) x5)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((eq_trans00 Xy1) c) x5)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((((eq_trans0 Xy1) Xy1) c) x5)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x5:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2)) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x5:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2)) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(((eq b) Xy1) Xy2)))
% Found (and_rect10 (fun (x5:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect1 (((eq b) Xy1) Xy2)) (fun (x5:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2)))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))
% Instantiate: x0:=(fun (x10:a) (x90:b)=> ((x1 x90) x10)):(a->(b->Prop))
% Found x5 as proof of (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy))))
% Found x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))
% Instantiate: x2:=(fun (x10:a) (x90:b)=> ((x0 x90) x10)):(a->(b->Prop))
% Found x5 as proof of (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy))))
% Found eq_ref000:=(eq_ref00 (x6 Xx)):(((x6 Xx) Xy2)->((x6 Xx) Xy2))
% Found (eq_ref00 (x6 Xx)) as proof of (((x6 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x6 Xx)) as proof of (((x6 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x6 Xx)) as proof of (((x6 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x6 Xx)) as proof of (((x6 Xx) Xy2)->(P Xy2))
% Found (fun (x8:((x6 Xx) Xy1))=> (((eq_ref b) Xy2) (x6 Xx))) as proof of (((x6 Xx) Xy2)->(P Xy2))
% Found (fun (x8:((x6 Xx) Xy1))=> (((eq_ref b) Xy2) (x6 Xx))) as proof of (((x6 Xx) Xy1)->(((x6 Xx) Xy2)->(P Xy2)))
% Found (and_rect20 (fun (x8:((x6 Xx) Xy1))=> (((eq_ref b) Xy2) (x6 Xx)))) as proof of (P Xy2)
% Found ((and_rect2 (P Xy2)) (fun (x8:((x6 Xx) Xy1))=> (((eq_ref b) Xy2) (x6 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x8:(((x6 Xx) Xy1)->(((x6 Xx) Xy2)->P0)))=> (((((and_rect ((x6 Xx) Xy1)) ((x6 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x6 Xx) Xy1))=> (((eq_ref b) Xy2) (x6 Xx)))) as proof of (P Xy2)
% Found (fun (x7:(P Xy1))=> (((fun (P0:Type) (x8:(((x6 Xx) Xy1)->(((x6 Xx) Xy2)->P0)))=> (((((and_rect ((x6 Xx) Xy1)) ((x6 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x6 Xx) Xy1))=> (((eq_ref b) Xy2) (x6 Xx))))) as proof of (P Xy2)
% Found eq_ref000:=(eq_ref00 (fun (x5:b)=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x5:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy1)))
% Found (and_rect00 (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect0 (P Xy1)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy1)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x3:(P Xy2))=> (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy1)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2)))))) as proof of (P Xy1)
% Found eq_ref000:=(eq_ref00 P):((P Xy1)->(P Xy1))
% Found (eq_ref00 P) as proof of (P0 Xy1)
% Found ((eq_ref0 Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref b) Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref b) Xy1) P) as proof of (P0 Xy1)
% Found eq_ref000:=(eq_ref00 (x0 Xx)):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy2)))
% Found (and_rect20 (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found ((and_rect2 (P Xy2)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x8:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found (fun (x7:(P Xy1))=> (((fun (P0:Type) (x8:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))))) as proof of (P Xy2)
% Found eq_ref000:=(eq_ref00 (x2 Xx)):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(P Xy2)))
% Found (and_rect20 (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))) as proof of (P Xy2)
% Found ((and_rect2 (P Xy2)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x8:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))) as proof of (P Xy2)
% Found (fun (x7:(P Xy1))=> (((fun (P0:Type) (x8:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))))) as proof of (P Xy2)
% Found eq_ref000:=(eq_ref00 (x4 Xx)):(((x4 Xx) Xy2)->((x4 Xx) Xy2))
% Found (eq_ref00 (x4 Xx)) as proof of (((x4 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x4 Xx)) as proof of (((x4 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x4 Xx)) as proof of (((x4 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x4 Xx)) as proof of (((x4 Xx) Xy2)->(P Xy2))
% Found (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx))) as proof of (((x4 Xx) Xy2)->(P Xy2))
% Found (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx))) as proof of (((x4 Xx) Xy1)->(((x4 Xx) Xy2)->(P Xy2)))
% Found (and_rect20 (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx)))) as proof of (P Xy2)
% Found ((and_rect2 (P Xy2)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x8:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx)))) as proof of (P Xy2)
% Found (fun (x7:(P Xy1))=> (((fun (P0:Type) (x8:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx))))) as proof of (P Xy2)
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq b) Xy1) Xy2)->(((eq b) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((eq_trans000 c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((eq_trans00 Xy1) c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((((eq_trans0 Xy1) Xy1) c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(((eq b) Xy1) Xy2)))
% Found (and_rect00 (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect0 (((eq b) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x2:((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))))=> (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x1:(b->(a->Prop))) (x2:((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))))=> (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))->(((eq b) Xy1) Xy2))
% Found (fun (x1:(b->(a->Prop))) (x2:((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))))=> (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)))) as proof of (forall (x:(b->(a->Prop))), (((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x Xx) Xy10)) ((x Xx) Xy20))->(((eq a) Xy10) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x Xx1) Xy)) ((x Xx2) Xy))->(((eq b) Xx1) Xx2))))->(((eq b) Xy1) Xy2)))
% Found (ex_ind00 (fun (x1:(b->(a->Prop))) (x2:((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))))=> (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))))) as proof of (((eq b) Xy1) Xy2)
% Found ((ex_ind0 (((eq b) Xy1) Xy2)) (fun (x1:(b->(a->Prop))) (x2:((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))))=> (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Prop) (x1:(forall (x:(b->(a->Prop))), (((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x Xx) Xy1)) ((x Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x Xx1) Xy)) ((x Xx2) Xy))->(((eq b) Xx1) Xx2))))->P)))=> (((((ex_ind (b->(a->Prop))) (fun (S:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((S Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((S Xx) Xy1)) ((S Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((S Xx1) Xy)) ((S Xx2) Xy))->(((eq b) Xx1) Xx2)))))) P) x1) x)) (((eq b) Xy1) Xy2)) (fun (x1:(b->(a->Prop))) (x2:((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))))=> (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Prop) (x1:(forall (x:(b->(a->Prop))), (((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x Xx) Xy1)) ((x Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x Xx1) Xy)) ((x Xx2) Xy))->(((eq b) Xx1) Xx2))))->P)))=> (((((ex_ind (b->(a->Prop))) (fun (S:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((S Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((S Xx) Xy1)) ((S Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((S Xx1) Xy)) ((S Xx2) Xy))->(((eq b) Xx1) Xx2)))))) P) x1) x)) (((eq b) Xy1) Xy2)) (fun (x1:(b->(a->Prop))) (x2:((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))))=> (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Prop) (x1:(forall (x:(b->(a->Prop))), (((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x Xx) Xy1)) ((x Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x Xx1) Xy)) ((x Xx2) Xy))->(((eq b) Xx1) Xx2))))->P)))=> (((((ex_ind (b->(a->Prop))) (fun (S:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((S Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((S Xx) Xy1)) ((S Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((S Xx1) Xy)) ((S Xx2) Xy))->(((eq b) Xx1) Xx2)))))) P) x1) x)) (((eq b) Xy1) Xy2)) (fun (x1:(b->(a->Prop))) (x2:((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))))=> (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)))))) as proof of (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_ref00:=(eq_ref0 Xy1):(((eq b) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found eq_ref000:=(eq_ref00 (x0 Xx)):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy2)))
% Found (and_rect10 (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found ((and_rect1 (P Xy2)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found (fun (x5:(P Xy1))=> (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))))) as proof of (P Xy2)
% Found eq_ref000:=(eq_ref00 (x2 Xx)):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(P Xy2)))
% Found (and_rect10 (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))) as proof of (P Xy2)
% Found ((and_rect1 (P Xy2)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x6:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))) as proof of (P Xy2)
% Found (fun (x5:(P Xy1))=> (((fun (P0:Type) (x6:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))))) as proof of (P Xy2)
% Found eq_ref000:=(eq_ref00 (x0 Xx)):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy2)))
% Found (and_rect10 (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found ((and_rect1 (P Xy2)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found (fun (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))))) as proof of (P Xy2)
% Found (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy20))->(((eq a) Xy1) Xy20))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))))) as proof of ((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))->(P Xy2))
% Found (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy20))->(((eq a) Xy1) Xy20))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))))) as proof of (((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy20))->(((eq a) Xy1) Xy20))))->((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))->(P Xy2)))
% Found (and_rect00 (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy20))->(((eq a) Xy1) Xy20))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))))) as proof of (P Xy2)
% Found ((and_rect0 (P Xy2)) (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy20))->(((eq a) Xy1) Xy20))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x4:(((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy2))->(((eq a) Xy1) Xy2))))->((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))->P0)))=> (((((and_rect ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))) P0) x4) x2)) (P Xy2)) (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy20))->(((eq a) Xy1) Xy20))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))))) as proof of (P Xy2)
% Found (fun (x3:(P Xy1))=> (((fun (P0:Type) (x4:(((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy2))->(((eq a) Xy1) Xy2))))->((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))->P0)))=> (((((and_rect ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))) P0) x4) x2)) (P Xy2)) (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy20))->(((eq a) Xy1) Xy20))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))))))) as proof of (P Xy2)
% Found eq_ref000:=(eq_ref00 (x2 Xx)):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(P Xy2)))
% Found (and_rect10 (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))) as proof of (P Xy2)
% Found ((and_rect1 (P Xy2)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x6:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))) as proof of (P Xy2)
% Found (fun (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))))) as proof of (P Xy2)
% Found (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy20))->(((eq a) Xy1) Xy20))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))))) as proof of ((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2)))->(P Xy2))
% Found (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy20))->(((eq a) Xy1) Xy20))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))))) as proof of (((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy20))->(((eq a) Xy1) Xy20))))->((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2)))->(P Xy2)))
% Found (and_rect00 (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy20))->(((eq a) Xy1) Xy20))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))))) as proof of (P Xy2)
% Found ((and_rect0 (P Xy2)) (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy20))->(((eq a) Xy1) Xy20))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x4:(((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2))))->((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2)))->P0)))=> (((((and_rect ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2)))) P0) x4) x1)) (P Xy2)) (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy20))->(((eq a) Xy1) Xy20))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))))) as proof of (P Xy2)
% Found (fun (x3:(P Xy1))=> (((fun (P0:Type) (x4:(((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2))))->((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2)))->P0)))=> (((((and_rect ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2)))) P0) x4) x1)) (P Xy2)) (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy20))->(((eq a) Xy1) Xy20))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))))))) as proof of (P Xy2)
% Found x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))
% Instantiate: x0:=(fun (x10:a) (x90:b)=> ((x1 x90) x10)):(a->(b->Prop))
% Found x5 as proof of (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy))))
% Found eq_ref000:=(eq_ref00 P):((P Xy1)->(P Xy1))
% Found (eq_ref00 P) as proof of (P0 Xy1)
% Found ((eq_ref0 Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref b) Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref b) Xy1) P) as proof of (P0 Xy1)
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq b) Xy1) Xy2)->(((eq b) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x6 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((eq_trans000 c) x7)) Xy2) as proof of (((x6 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((eq_trans00 Xy1) c) x7)) Xy2) as proof of (((x6 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((((eq_trans0 Xy1) Xy1) c) x7)) Xy2) as proof of (((x6 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2) as proof of (((x6 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2) as proof of (((x6 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x7:((x6 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)) as proof of (((x6 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x7:((x6 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)) as proof of (((x6 Xx) Xy1)->(((x6 Xx) Xy2)->(((eq b) Xy1) Xy2)))
% Found (and_rect20 (fun (x7:((x6 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect2 (((eq b) Xy1) Xy2)) (fun (x7:((x6 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x7:(((x6 Xx) Xy1)->(((x6 Xx) Xy2)->P)))=> (((((and_rect ((x6 Xx) Xy1)) ((x6 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x6 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x6 Xx) Xy1)) ((x6 Xx) Xy2)))=> (((fun (P:Type) (x7:(((x6 Xx) Xy1)->(((x6 Xx) Xy2)->P)))=> (((((and_rect ((x6 Xx) Xy1)) ((x6 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x6 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x6 Xx) Xy1)) ((x6 Xx) Xy2)))=> (((fun (P:Type) (x7:(((x6 Xx) Xy1)->(((x6 Xx) Xy2)->P)))=> (((((and_rect ((x6 Xx) Xy1)) ((x6 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x6 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)))) as proof of (((and ((x6 Xx) Xy1)) ((x6 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_ref000:=(eq_ref00 (fun (x7:b)=> ((x4 Xx) Xy2))):(((x4 Xx) Xy2)->((x4 Xx) Xy2))
% Found (eq_ref00 (fun (x7:b)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x7:b)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x7:b)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x7:b)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->(P Xy1))
% Found (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x4 Xx) Xy2)))) as proof of (((x4 Xx) Xy2)->(P Xy1))
% Found (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x4 Xx) Xy2)))) as proof of (((x4 Xx) Xy1)->(((x4 Xx) Xy2)->(P Xy1)))
% Found (and_rect10 (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x4 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect1 (P Xy1)) (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x4 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x6:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x4 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x5:(P Xy2))=> (((fun (P0:Type) (x6:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x4 Xx) Xy2)))))) as proof of (P Xy1)
% Found eq_ref000:=(eq_ref00 P):((P Xy1)->(P Xy1))
% Found (eq_ref00 P) as proof of (P0 Xy1)
% Found ((eq_ref0 Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref b) Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref b) Xy1) P) as proof of (P0 Xy1)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq b) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq b) Xy1) Xy2)->(((eq b) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((eq_trans000 c) x7)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((eq_trans00 Xy1) c) x7)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((((eq_trans0 Xy1) Xy1) c) x7)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x7:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x7:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(((eq b) Xy1) Xy2)))
% Found (and_rect20 (fun (x7:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect2 (((eq b) Xy1) Xy2)) (fun (x7:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x7:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Type) (x7:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Type) (x7:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)))) as proof of (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq b) Xy1) Xy2)->(((eq b) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((eq_trans000 c) x7)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((eq_trans00 Xy1) c) x7)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((((eq_trans0 Xy1) Xy1) c) x7)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x7:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x7:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(((eq b) Xy1) Xy2)))
% Found (and_rect20 (fun (x7:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect2 (((eq b) Xy1) Xy2)) (fun (x7:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x7:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x7:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x7:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq b) Xy1) Xy2)->(((eq b) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x4 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((eq_trans000 c) x7)) Xy2) as proof of (((x4 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((eq_trans00 Xy1) c) x7)) Xy2) as proof of (((x4 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((((eq_trans0 Xy1) Xy1) c) x7)) Xy2) as proof of (((x4 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2) as proof of (((x4 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2) as proof of (((x4 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x7:((x4 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)) as proof of (((x4 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x7:((x4 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)) as proof of (((x4 Xx) Xy1)->(((x4 Xx) Xy2)->(((eq b) Xy1) Xy2)))
% Found (and_rect20 (fun (x7:((x4 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect2 (((eq b) Xy1) Xy2)) (fun (x7:((x4 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x7:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x4 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2)))=> (((fun (P:Type) (x7:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x4 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2)))=> (((fun (P:Type) (x7:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x4 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)))) as proof of (((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_ref000:=(eq_ref00 (fun (x7:b)=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x7:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy1)))
% Found (and_rect10 (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect1 (P Xy1)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x5:(P Xy2))=> (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2)))))) as proof of (P Xy1)
% Found eq_ref000:=(eq_ref00 (fun (x7:b)=> ((x2 Xx) Xy2))):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (fun (x7:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x7:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(P Xy1)))
% Found (and_rect10 (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect1 (P Xy1)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x6:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x5:(P Xy2))=> (((fun (P0:Type) (x6:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2)))))) as proof of (P Xy1)
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq b) Xy1) Xy2)->(((eq b) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((eq_trans000 c) x5)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((eq_trans00 Xy1) c) x5)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((((eq_trans0 Xy1) Xy1) c) x5)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x5:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2)) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x5:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2)) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(((eq b) Xy1) Xy2)))
% Found (and_rect10 (fun (x5:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect1 (((eq b) Xy1) Xy2)) (fun (x5:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x4:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x3:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (x4:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2)))) as proof of ((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))->(((eq b) Xy1) Xy2))
% Found (fun (x3:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (x4:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2)))) as proof of (((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))->((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))->(((eq b) Xy1) Xy2)))
% Found (and_rect00 (fun (x3:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (x4:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2))))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect0 (((eq b) Xy1) Xy2)) (fun (x3:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (x4:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2))))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x3:(((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy2))->(((eq a) Xy1) Xy2))))->((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))->P)))=> (((((and_rect ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))) P) x3) x2)) (((eq b) Xy1) Xy2)) (fun (x3:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (x4:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Type) (x3:(((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy2))->(((eq a) Xy1) Xy2))))->((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))->P)))=> (((((and_rect ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))) P) x3) x2)) (((eq b) Xy1) Xy2)) (fun (x3:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (x4:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Type) (x3:(((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy2))->(((eq a) Xy1) Xy2))))->((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))->P)))=> (((((and_rect ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))) P) x3) x2)) (((eq b) Xy1) Xy2)) (fun (x3:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (x4:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2)))))) as proof of (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_ref00:=(eq_ref0 Xy1):(((eq b) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq b) Xy1) Xy2)->(((eq b) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((eq_trans000 c) x5)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((eq_trans00 Xy1) c) x5)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((((eq_trans0 Xy1) Xy1) c) x5)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x5:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2)) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x5:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2)) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(((eq b) Xy1) Xy2)))
% Found (and_rect10 (fun (x5:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect1 (((eq b) Xy1) Xy2)) (fun (x5:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x4:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x3:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (x4:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2)))) as proof of ((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2)))->(((eq b) Xy1) Xy2))
% Found (fun (x3:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (x4:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2)))) as proof of (((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy20))->(((eq a) Xy10) Xy20))))->((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2)))->(((eq b) Xy1) Xy2)))
% Found (and_rect00 (fun (x3:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (x4:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2))))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect0 (((eq b) Xy1) Xy2)) (fun (x3:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (x4:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2))))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x3:(((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2))))->((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2)))->P)))=> (((((and_rect ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2)))) P) x3) x1)) (((eq b) Xy1) Xy2)) (fun (x3:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (x4:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x3:(((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2))))->((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2)))->P)))=> (((((and_rect ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2)))) P) x3) x1)) (((eq b) Xy1) Xy2)) (fun (x3:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (x4:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x3:(((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2))))->((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2)))->P)))=> (((((and_rect ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2)))) P) x3) x1)) (((eq b) Xy1) Xy2)) (fun (x3:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (x4:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x5)) Xy2)))))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_ref000:=(eq_ref00 (x0 Xx)):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy2)))
% Found (and_rect20 (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found ((and_rect2 (P Xy2)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x8:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found (fun (x7:(P Xy1))=> (((fun (P0:Type) (x8:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))))) as proof of (P Xy2)
% Found eq_ref000:=(eq_ref00 (x2 Xx)):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(P Xy2)))
% Found (and_rect20 (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))) as proof of (P Xy2)
% Found ((and_rect2 (P Xy2)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x8:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))) as proof of (P Xy2)
% Found (fun (x7:(P Xy1))=> (((fun (P0:Type) (x8:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))))) as proof of (P Xy2)
% Found eq_ref000:=(eq_ref00 (x4 Xx)):(((x4 Xx) Xy2)->((x4 Xx) Xy2))
% Found (eq_ref00 (x4 Xx)) as proof of (((x4 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x4 Xx)) as proof of (((x4 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x4 Xx)) as proof of (((x4 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x4 Xx)) as proof of (((x4 Xx) Xy2)->(P Xy2))
% Found (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx))) as proof of (((x4 Xx) Xy2)->(P Xy2))
% Found (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx))) as proof of (((x4 Xx) Xy1)->(((x4 Xx) Xy2)->(P Xy2)))
% Found (and_rect20 (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx)))) as proof of (P Xy2)
% Found ((and_rect2 (P Xy2)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x8:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx)))) as proof of (P Xy2)
% Found (fun (x7:(P Xy1))=> (((fun (P0:Type) (x8:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx))))) as proof of (P Xy2)
% Found eq_ref000:=(eq_ref00 (x0 Xx)):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy2)))
% Found (and_rect00 (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found ((and_rect0 (P Xy2)) (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x2:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x2) x00)) (P Xy2)) (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found (fun (x1:(P Xy1))=> (((fun (P0:Type) (x2:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x2) x00)) (P Xy2)) (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))))) as proof of (P Xy2)
% Found eq_ref000:=(eq_ref00 (x0 Xx)):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy2)))
% Found (and_rect20 (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found ((and_rect2 (P Xy2)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x8:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found (fun (x7:(forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy20))->(((eq a) Xy1) Xy20))))=> (((fun (P0:Type) (x8:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))))) as proof of (P Xy2)
% Found (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (x7:(forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy20))->(((eq a) Xy1) Xy20))))=> (((fun (P0:Type) (x8:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))))) as proof of ((forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy20))->(((eq a) Xy1) Xy20)))->(P Xy2))
% Found (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (x7:(forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy20))->(((eq a) Xy1) Xy20))))=> (((fun (P0:Type) (x8:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))))) as proof of ((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))->((forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy20))->(((eq a) Xy1) Xy20)))->(P Xy2)))
% Found (and_rect10 (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (x7:(forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy20))->(((eq a) Xy1) Xy20))))=> (((fun (P0:Type) (x8:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))))) as proof of (P Xy2)
% Found ((and_rect1 (P Xy2)) (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (x7:(forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy20))->(((eq a) Xy1) Xy20))))=> (((fun (P0:Type) (x8:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x6:((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))->((forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy2))->(((eq a) Xy1) Xy2)))->P0)))=> (((((and_rect (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy2))->(((eq a) Xy1) Xy2)))) P0) x6) x3)) (P Xy2)) (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (x7:(forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy20))->(((eq a) Xy1) Xy20))))=> (((fun (P0:Type) (x8:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))))) as proof of (P Xy2)
% Found (fun (x5:(P Xy1))=> (((fun (P0:Type) (x6:((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))->((forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy2))->(((eq a) Xy1) Xy2)))->P0)))=> (((((and_rect (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy2))->(((eq a) Xy1) Xy2)))) P0) x6) x3)) (P Xy2)) (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (x7:(forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy20))->(((eq a) Xy1) Xy20))))=> (((fun (P0:Type) (x8:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))))))) as proof of (P Xy2)
% Found eq_ref000:=(eq_ref00 (x2 Xx)):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(P Xy2)))
% Found (and_rect20 (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))) as proof of (P Xy2)
% Found ((and_rect2 (P Xy2)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x8:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))) as proof of (P Xy2)
% Found (fun (x7:(forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy20))->(((eq a) Xy1) Xy20))))=> (((fun (P0:Type) (x8:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))))) as proof of (P Xy2)
% Found (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x7:(forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy20))->(((eq a) Xy1) Xy20))))=> (((fun (P0:Type) (x8:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))))) as proof of ((forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy20))->(((eq a) Xy1) Xy20)))->(P Xy2))
% Found (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x7:(forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy20))->(((eq a) Xy1) Xy20))))=> (((fun (P0:Type) (x8:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))))) as proof of ((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))->((forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy20))->(((eq a) Xy1) Xy20)))->(P Xy2)))
% Found (and_rect10 (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x7:(forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy20))->(((eq a) Xy1) Xy20))))=> (((fun (P0:Type) (x8:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))))) as proof of (P Xy2)
% Found ((and_rect1 (P Xy2)) (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x7:(forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy20))->(((eq a) Xy1) Xy20))))=> (((fun (P0:Type) (x8:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x6:((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))->((forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2)))->P0)))=> (((((and_rect (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2)))) P0) x6) x3)) (P Xy2)) (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x7:(forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy20))->(((eq a) Xy1) Xy20))))=> (((fun (P0:Type) (x8:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))))) as proof of (P Xy2)
% Found (fun (x5:(P Xy1))=> (((fun (P0:Type) (x6:((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))->((forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2)))->P0)))=> (((((and_rect (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2)))) P0) x6) x3)) (P Xy2)) (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x7:(forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy20))->(((eq a) Xy1) Xy20))))=> (((fun (P0:Type) (x8:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))))))) as proof of (P Xy2)
% Found eq_ref000:=(eq_ref00 (x4 Xx)):(((x4 Xx) Xy2)->((x4 Xx) Xy2))
% Found (eq_ref00 (x4 Xx)) as proof of (((x4 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x4 Xx)) as proof of (((x4 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x4 Xx)) as proof of (((x4 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x4 Xx)) as proof of (((x4 Xx) Xy2)->(P Xy2))
% Found (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx))) as proof of (((x4 Xx) Xy2)->(P Xy2))
% Found (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx))) as proof of (((x4 Xx) Xy1)->(((x4 Xx) Xy2)->(P Xy2)))
% Found (and_rect20 (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx)))) as proof of (P Xy2)
% Found ((and_rect2 (P Xy2)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x8:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx)))) as proof of (P Xy2)
% Found (fun (x7:(forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy20))->(((eq a) Xy1) Xy20))))=> (((fun (P0:Type) (x8:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx))))) as proof of (P Xy2)
% Found (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x7:(forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy20))->(((eq a) Xy1) Xy20))))=> (((fun (P0:Type) (x8:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx))))) as proof of ((forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy20))->(((eq a) Xy1) Xy20)))->(P Xy2))
% Found (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x7:(forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy20))->(((eq a) Xy1) Xy20))))=> (((fun (P0:Type) (x8:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx))))) as proof of ((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))->((forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy20))->(((eq a) Xy1) Xy20)))->(P Xy2)))
% Found (and_rect10 (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x7:(forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy20))->(((eq a) Xy1) Xy20))))=> (((fun (P0:Type) (x8:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx)))))) as proof of (P Xy2)
% Found ((and_rect1 (P Xy2)) (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x7:(forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy20))->(((eq a) Xy1) Xy20))))=> (((fun (P0:Type) (x8:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx)))))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x6:((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))->((forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2)))->P0)))=> (((((and_rect (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2)))) P0) x6) x2)) (P Xy2)) (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x7:(forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy20))->(((eq a) Xy1) Xy20))))=> (((fun (P0:Type) (x8:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx)))))) as proof of (P Xy2)
% Found (fun (x5:(P Xy1))=> (((fun (P0:Type) (x6:((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))->((forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2)))->P0)))=> (((((and_rect (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2)))) P0) x6) x2)) (P Xy2)) (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x7:(forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy20))->(((eq a) Xy1) Xy20))))=> (((fun (P0:Type) (x8:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx))))))) as proof of (P Xy2)
% Found eq_ref000:=(eq_ref00 (fun (x5:b)=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x5:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy1)))
% Found (and_rect00 (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect0 (P Xy1)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy1)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x3:(P Xy2))=> (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy1)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2)))))) as proof of (P Xy1)
% Found eq_ref000:=(eq_ref00 (fun (x5:b)=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x5:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy1)))
% Found (and_rect00 (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect0 (P Xy1)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy1)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x3:((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x2 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x2 Xx) Xy10)) ((x2 Xx) Xy2))->(((eq a) Xy10) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x2 Xx1) Xy)) ((x2 Xx2) Xy))->(((eq b) Xx1) Xx2)))))=> (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy1)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2)))))) as proof of (P Xy1)
% Found (fun (x2:(b->(a->Prop))) (x3:((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x2 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x2 Xx) Xy10)) ((x2 Xx) Xy2))->(((eq a) Xy10) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x2 Xx1) Xy)) ((x2 Xx2) Xy))->(((eq b) Xx1) Xx2)))))=> (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy1)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2)))))) as proof of (((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x2 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x2 Xx) Xy10)) ((x2 Xx) Xy2))->(((eq a) Xy10) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x2 Xx1) Xy)) ((x2 Xx2) Xy))->(((eq b) Xx1) Xx2))))->(P Xy1))
% Found (fun (x2:(b->(a->Prop))) (x3:((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x2 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x2 Xx) Xy10)) ((x2 Xx) Xy2))->(((eq a) Xy10) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x2 Xx1) Xy)) ((x2 Xx2) Xy))->(((eq b) Xx1) Xx2)))))=> (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy1)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2)))))) as proof of (forall (x:(b->(a->Prop))), (((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x Xx) Xy10)) ((x Xx) Xy2))->(((eq a) Xy10) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x Xx1) Xy)) ((x Xx2) Xy))->(((eq b) Xx1) Xx2))))->(P Xy1)))
% Found (ex_ind00 (fun (x2:(b->(a->Prop))) (x3:((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x2 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x2 Xx) Xy10)) ((x2 Xx) Xy2))->(((eq a) Xy10) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x2 Xx1) Xy)) ((x2 Xx2) Xy))->(((eq b) Xx1) Xx2)))))=> (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy1)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))))))) as proof of (P Xy1)
% Found ((ex_ind0 (P Xy1)) (fun (x2:(b->(a->Prop))) (x3:((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x2 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x2 Xx) Xy10)) ((x2 Xx) Xy2))->(((eq a) Xy10) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x2 Xx1) Xy)) ((x2 Xx2) Xy))->(((eq b) Xx1) Xx2)))))=> (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy1)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))))))) as proof of (P Xy1)
% Found (((fun (P0:Prop) (x2:(forall (x:(b->(a->Prop))), (((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x Xx) Xy1)) ((x Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x Xx1) Xy)) ((x Xx2) Xy))->(((eq b) Xx1) Xx2))))->P0)))=> (((((ex_ind (b->(a->Prop))) (fun (S:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((S Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((S Xx) Xy1)) ((S Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((S Xx1) Xy)) ((S Xx2) Xy))->(((eq b) Xx1) Xx2)))))) P0) x2) x)) (P Xy1)) (fun (x2:(b->(a->Prop))) (x3:((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x2 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x2 Xx) Xy10)) ((x2 Xx) Xy2))->(((eq a) Xy10) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x2 Xx1) Xy)) ((x2 Xx2) Xy))->(((eq b) Xx1) Xx2)))))=> (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy1)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))))))) as proof of (P Xy1)
% Found (fun (x1:(P Xy2))=> (((fun (P0:Prop) (x2:(forall (x:(b->(a->Prop))), (((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x Xx) Xy1)) ((x Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x Xx1) Xy)) ((x Xx2) Xy))->(((eq b) Xx1) Xx2))))->P0)))=> (((((ex_ind (b->(a->Prop))) (fun (S:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((S Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((S Xx) Xy1)) ((S Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((S Xx1) Xy)) ((S Xx2) Xy))->(((eq b) Xx1) Xx2)))))) P0) x2) x)) (P Xy1)) (fun (x2:(b->(a->Prop))) (x3:((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x2 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x2 Xx) Xy10)) ((x2 Xx) Xy2))->(((eq a) Xy10) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x2 Xx1) Xy)) ((x2 Xx2) Xy))->(((eq b) Xx1) Xx2)))))=> (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy1)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2)))))))) as proof of (P Xy1)
% Found eq_ref00:=(eq_ref0 b0):(((eq ((a->(b->Prop))->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((a->(b->Prop))->Prop)) b0) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) b0) as proof of (((eq ((a->(b->Prop))->Prop)) b0) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) b0) as proof of (((eq ((a->(b->Prop))->Prop)) b0) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) b0) as proof of (((eq ((a->(b->Prop))->Prop)) b0) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq ((a->(b->Prop))->Prop)) a0) (fun (x:(a->(b->Prop)))=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x1:(a->(b->Prop)))=> Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found (((eta_expansion_dep (a->(b->Prop))) (fun (x1:(a->(b->Prop)))=> Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found (((eta_expansion_dep (a->(b->Prop))) (fun (x1:(a->(b->Prop)))=> Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found (((eta_expansion_dep (a->(b->Prop))) (fun (x1:(a->(b->Prop)))=> Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->(b->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found eq_ref000:=(eq_ref00 P):((P Xy1)->(P Xy1))
% Found (eq_ref00 P) as proof of (P0 Xy1)
% Found ((eq_ref0 Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref b) Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref b) Xy1) P) as proof of (P0 Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq b) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq b) Xy2) b0)
% Found ((eq_ref b) Xy2) as proof of (((eq b) Xy2) b0)
% Found ((eq_ref b) Xy2) as proof of (((eq b) Xy2) b0)
% Found ((eq_ref b) Xy2) as proof of (((eq b) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy1)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy1)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy1)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy1)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))):(((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))))
% Found (eq_ref0 (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq b) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found eq_ref000:=(eq_ref00 P):((P Xy1)->(P Xy1))
% Found (eq_ref00 P) as proof of (P0 Xy1)
% Found ((eq_ref0 Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref b) Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref b) Xy1) P) as proof of (P0 Xy1)
% Found eq_ref000:=(eq_ref00 P):((P Xy1)->(P Xy1))
% Found (eq_ref00 P) as proof of (P0 Xy1)
% Found ((eq_ref0 Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref b) Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref b) Xy1) P) as proof of (P0 Xy1)
% Found and_rect0000:=(and_rect000 (fun (x2:b)=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (and_rect000 (fun (x2:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(((eq b) b0) Xy2))
% Found ((and_rect00 eq_trans0000) (fun (x2:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(((eq b) b0) Xy2))
% Found ((and_rect00 eq_trans0000) (fun (x2:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(((eq b) b0) Xy2))
% Found (fun (x1:((x0 Xx) Xy1))=> ((and_rect00 eq_trans0000) (fun (x2:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->(((eq b) b0) Xy2))
% Found (fun (x1:((x0 Xx) Xy1))=> ((and_rect00 eq_trans0000) (fun (x2:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(((eq b) b0) Xy2)))
% Found (and_rect00 (fun (x1:((x0 Xx) Xy1))=> ((and_rect00 eq_trans0000) (fun (x2:b)=> ((x0 Xx) Xy2))))) as proof of (((eq b) b0) Xy2)
% Found ((and_rect0 (((eq b) b0) Xy2)) (fun (x1:((x0 Xx) Xy1))=> (((and_rect0 (((eq b) b0) Xy2)) eq_trans0000) (fun (x2:b)=> ((x0 Xx) Xy2))))) as proof of (((eq b) b0) Xy2)
% Found (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) b0) Xy2)) (fun (x1:((x0 Xx) Xy1))=> ((((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) b0) Xy2)) eq_trans0000) (fun (x2:b)=> ((x0 Xx) Xy2))))) as proof of (((eq b) b0) Xy2)
% Found (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) b0) Xy2)) (fun (x1:((x0 Xx) Xy1))=> ((((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) b0) Xy2)) eq_trans0000) (fun (x2:b)=> ((x0 Xx) Xy2))))) as proof of (((eq b) b0) Xy2)
% Found ((eq_trans0000 ((eq_ref b) Xy1)) (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) b0) Xy2)) (fun (x1:((x0 Xx) Xy1))=> ((((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) b0) Xy2)) eq_trans0000) (fun (x2:b)=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found (((eq_trans000 Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy1) Xy2)) (fun (x1:((x0 Xx) Xy1))=> ((((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy1) Xy2)) (eq_trans000 Xy1)) (fun (x2:b)=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found ((((fun (b0:b)=> ((eq_trans00 b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy1) Xy2)) (fun (x1:((x0 Xx) Xy1))=> ((((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> ((eq_trans00 b0) Xy2)) Xy1)) (fun (x2:b)=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found ((((fun (b0:b)=> (((eq_trans0 Xy1) b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy1) Xy2)) (fun (x1:((x0 Xx) Xy1))=> ((((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> (((eq_trans0 Xy1) b0) Xy2)) Xy1)) (fun (x2:b)=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found ((((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy1) Xy2)) (fun (x1:((x0 Xx) Xy1))=> ((((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1)) (fun (x2:b)=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> ((((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy1) Xy2)) (fun (x1:((x0 Xx) Xy1))=> ((((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1)) (fun (x2:b)=> ((x0 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> ((((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy1) Xy2)) (fun (x1:((x0 Xx) Xy1))=> ((((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1)) (fun (x2:b)=> ((x0 Xx) Xy2))))))) as proof of (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found conj10:=(conj1 ((x4 Xx) Xy2)):(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2))))
% Found (conj1 ((x4 Xx) Xy2)) as proof of (((x4 Xx) Xy1)->(((x4 Xx) Xy2)->((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))))
% Found ((conj ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) as proof of (((x4 Xx) Xy1)->(((x4 Xx) Xy2)->((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))))
% Found ((conj ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) as proof of (((x4 Xx) Xy1)->(((x4 Xx) Xy2)->((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))))
% Found ((conj ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) as proof of (((x4 Xx) Xy1)->(((x4 Xx) Xy2)->((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))))
% Found (and_rect10 ((conj ((x4 Xx) Xy1)) ((x4 Xx) Xy2))) as proof of ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))
% Found ((and_rect1 ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))) ((conj ((x4 Xx) Xy1)) ((x4 Xx) Xy2))) as proof of ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))
% Found (((fun (P:Type) (x5:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x5) x00)) ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))) ((conj ((x4 Xx) Xy1)) ((x4 Xx) Xy2))) as proof of ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))
% Found (((fun (P:Type) (x5:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x5) x00)) ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))) ((conj ((x4 Xx) Xy1)) ((x4 Xx) Xy2))) as proof of ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))
% Found (x3000 (((fun (P:Type) (x5:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x5) x00)) ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))) ((conj ((x4 Xx) Xy1)) ((x4 Xx) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found eq_ref000:=(eq_ref00 (fun (x9:b)=> ((x6 Xx) Xy2))):(((x6 Xx) Xy2)->((x6 Xx) Xy2))
% Found (eq_ref00 (fun (x9:b)=> ((x6 Xx) Xy2))) as proof of (((x6 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x9:b)=> ((x6 Xx) Xy2))) as proof of (((x6 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x9:b)=> ((x6 Xx) Xy2))) as proof of (((x6 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x9:b)=> ((x6 Xx) Xy2))) as proof of (((x6 Xx) Xy2)->(P Xy1))
% Found (fun (x8:((x6 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x6 Xx) Xy2)))) as proof of (((x6 Xx) Xy2)->(P Xy1))
% Found (fun (x8:((x6 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x6 Xx) Xy2)))) as proof of (((x6 Xx) Xy1)->(((x6 Xx) Xy2)->(P Xy1)))
% Found (and_rect20 (fun (x8:((x6 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x6 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect2 (P Xy1)) (fun (x8:((x6 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x6 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x8:(((x6 Xx) Xy1)->(((x6 Xx) Xy2)->P0)))=> (((((and_rect ((x6 Xx) Xy1)) ((x6 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x6 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x6 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x7:(P Xy2))=> (((fun (P0:Type) (x8:(((x6 Xx) Xy1)->(((x6 Xx) Xy2)->P0)))=> (((((and_rect ((x6 Xx) Xy1)) ((x6 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x6 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x6 Xx) Xy2)))))) as proof of (P Xy1)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq b) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq b) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found eq_ref00:=(eq_ref0 b0):(((eq ((a->(b->Prop))->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((a->(b->Prop))->Prop)) b0) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) b0) as proof of (((eq ((a->(b->Prop))->Prop)) b0) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) b0) as proof of (((eq ((a->(b->Prop))->Prop)) b0) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) b0) as proof of (((eq ((a->(b->Prop))->Prop)) b0) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq ((a->(b->Prop))->Prop)) a0) (fun (x:(a->(b->Prop)))=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x3:(a->(b->Prop)))=> Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found (((eta_expansion_dep (a->(b->Prop))) (fun (x3:(a->(b->Prop)))=> Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found (((eta_expansion_dep (a->(b->Prop))) (fun (x3:(a->(b->Prop)))=> Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found (((eta_expansion_dep (a->(b->Prop))) (fun (x3:(a->(b->Prop)))=> Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->(b->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found conj10:=(conj1 ((x0 Xx) Xy2)):(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))))
% Found (conj1 ((x0 Xx) Xy2)) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->((and ((x1 Xy1) Xy)) ((x1 Xy2) Xy))))
% Found ((conj ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->((and ((x1 Xy1) Xy)) ((x1 Xy2) Xy))))
% Found ((conj ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->((and ((x1 Xy1) Xy)) ((x1 Xy2) Xy))))
% Found ((conj ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->((and ((x1 Xy1) Xy)) ((x1 Xy2) Xy))))
% Found (and_rect10 ((conj ((x0 Xx) Xy1)) ((x0 Xx) Xy2))) as proof of ((and ((x1 Xy1) Xy)) ((x1 Xy2) Xy))
% Found ((and_rect1 ((and ((x1 Xy1) Xy)) ((x1 Xy2) Xy))) ((conj ((x0 Xx) Xy1)) ((x0 Xx) Xy2))) as proof of ((and ((x1 Xy1) Xy)) ((x1 Xy2) Xy))
% Found (((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) ((and ((x1 Xy1) Xy)) ((x1 Xy2) Xy))) ((conj ((x0 Xx) Xy1)) ((x0 Xx) Xy2))) as proof of ((and ((x1 Xy1) Xy)) ((x1 Xy2) Xy))
% Found (((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) ((and ((x1 Xy1) Xy)) ((x1 Xy2) Xy))) ((conj ((x0 Xx) Xy1)) ((x0 Xx) Xy2))) as proof of ((and ((x1 Xy1) Xy)) ((x1 Xy2) Xy))
% Found (x4000 (((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) ((and ((x1 Xy1) Xy)) ((x1 Xy2) Xy))) ((conj ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found eq_ref000:=(eq_ref00 (fun (x3:b)=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x3:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P1 Xy1))
% Found ((eq_ref0 Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P1 Xy1))
% Found (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P1 Xy1))
% Found (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P1 Xy1))
% Found (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->(P1 Xy1))
% Found (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P1 Xy1)))
% Found (and_rect00 (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))))) as proof of (P1 Xy1)
% Found ((and_rect0 (P1 Xy1)) (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))))) as proof of (P1 Xy1)
% Found (((fun (P2:Type) (x2:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P2)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P2) x2) x00)) (P1 Xy1)) (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))))) as proof of (P1 Xy1)
% Found (fun (x1:(P1 Xy2))=> (((fun (P2:Type) (x2:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P2)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P2) x2) x00)) (P1 Xy1)) (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2)))))) as proof of (P1 Xy1)
% Found eq_ref000:=(eq_ref00 (fun (x3:b)=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x3:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P1 Xy1))
% Found ((eq_ref0 Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P1 Xy1))
% Found (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P1 Xy1))
% Found (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P1 Xy1))
% Found (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->(P1 Xy1))
% Found (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P1 Xy1)))
% Found (and_rect00 (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))))) as proof of (P1 Xy1)
% Found ((and_rect0 (P1 Xy1)) (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))))) as proof of (P1 Xy1)
% Found (((fun (P2:Type) (x2:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P2)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P2) x2) x00)) (P1 Xy1)) (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))))) as proof of (P1 Xy1)
% Found (fun (x1:(P1 Xy2))=> (((fun (P2:Type) (x2:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P2)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P2) x2) x00)) (P1 Xy1)) (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2)))))) as proof of (P1 Xy1)
% Found conj10:=(conj1 ((x2 Xx) Xy2)):(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))))
% Found (conj1 ((x2 Xx) Xy2)) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))))
% Found ((conj ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))))
% Found ((conj ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))))
% Found ((conj ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))))
% Found (and_rect10 ((conj ((x2 Xx) Xy1)) ((x2 Xx) Xy2))) as proof of ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))
% Found ((and_rect1 ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))) ((conj ((x2 Xx) Xy1)) ((x2 Xx) Xy2))) as proof of ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))
% Found (((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))) ((conj ((x2 Xx) Xy1)) ((x2 Xx) Xy2))) as proof of ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))
% Found (((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))) ((conj ((x2 Xx) Xy1)) ((x2 Xx) Xy2))) as proof of ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))
% Found (x4000 (((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))) ((conj ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found eq_ref000:=(eq_ref00 (x2 Xx)):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(P Xy2)))
% Found (and_rect00 (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))) as proof of (P Xy2)
% Found ((and_rect0 (P Xy2)) (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x4:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x4) x00)) (P Xy2)) (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))) as proof of (P Xy2)
% Found (fun (x3:(P Xy1))=> (((fun (P0:Type) (x4:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x4) x00)) (P Xy2)) (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))))) as proof of (P Xy2)
% Found eq_ref000:=(eq_ref00 (fun (x9:b)=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x9:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x9:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy1)))
% Found (and_rect20 (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect2 (P Xy1)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x8:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x7:(P Xy2))=> (((fun (P0:Type) (x8:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2)))))) as proof of (P Xy1)
% Found eq_ref000:=(eq_ref00 (fun (x9:b)=> ((x2 Xx) Xy2))):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (fun (x9:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x9:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(P Xy1)))
% Found (and_rect20 (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect2 (P Xy1)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x8:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x7:(P Xy2))=> (((fun (P0:Type) (x8:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2)))))) as proof of (P Xy1)
% Found eq_ref000:=(eq_ref00 (fun (x9:b)=> ((x4 Xx) Xy2))):(((x4 Xx) Xy2)->((x4 Xx) Xy2))
% Found (eq_ref00 (fun (x9:b)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x9:b)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->(P Xy1))
% Found (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2)))) as proof of (((x4 Xx) Xy2)->(P Xy1))
% Found (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2)))) as proof of (((x4 Xx) Xy1)->(((x4 Xx) Xy2)->(P Xy1)))
% Found (and_rect20 (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect2 (P Xy1)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x8:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x7:(P Xy2))=> (((fun (P0:Type) (x8:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2)))))) as proof of (P Xy1)
% 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: b0:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b0
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq b) Xy1) Xy2)->(((eq b) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((eq_trans000 c) x7)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((eq_trans00 Xy1) c) x7)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((((eq_trans0 Xy1) Xy1) c) x7)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x7:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x7:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(((eq b) Xy1) Xy2)))
% Found (and_rect20 (fun (x7:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect2 (((eq b) Xy1) Xy2)) (fun (x7:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x7:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x6:(forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))=> (((fun (P:Type) (x7:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (x6:(forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))=> (((fun (P:Type) (x7:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)))) as proof of ((forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20)))->(((eq b) Xy1) Xy2))
% Found (fun (x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (x6:(forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))=> (((fun (P:Type) (x7:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)))) as proof of ((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))->((forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20)))->(((eq b) Xy1) Xy2)))
% Found (and_rect10 (fun (x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (x6:(forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))=> (((fun (P:Type) (x7:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect1 (((eq b) Xy1) Xy2)) (fun (x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (x6:(forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))=> (((fun (P:Type) (x7:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x5:((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))->((forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy2))->(((eq a) Xy1) Xy2)))->P)))=> (((((and_rect (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy2))->(((eq a) Xy1) Xy2)))) P) x5) x3)) (((eq b) Xy1) Xy2)) (fun (x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (x6:(forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))=> (((fun (P:Type) (x7:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Type) (x5:((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))->((forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy2))->(((eq a) Xy1) Xy2)))->P)))=> (((((and_rect (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy2))->(((eq a) Xy1) Xy2)))) P) x5) x3)) (((eq b) Xy1) Xy2)) (fun (x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (x6:(forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))=> (((fun (P:Type) (x7:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Type) (x5:((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))->((forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy2))->(((eq a) Xy1) Xy2)))->P)))=> (((((and_rect (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy2))->(((eq a) Xy1) Xy2)))) P) x5) x3)) (((eq b) Xy1) Xy2)) (fun (x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (x6:(forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))=> (((fun (P:Type) (x7:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)))))) as proof of (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq b) Xy1) Xy2)->(((eq b) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((eq_trans000 c) x7)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((eq_trans00 Xy1) c) x7)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((((eq_trans0 Xy1) Xy1) c) x7)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x7:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x7:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(((eq b) Xy1) Xy2)))
% Found (and_rect20 (fun (x7:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect2 (((eq b) Xy1) Xy2)) (fun (x7:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x7:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x6:(forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy20))->(((eq a) Xy10) Xy20))))=> (((fun (P:Type) (x7:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x6:(forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy20))->(((eq a) Xy10) Xy20))))=> (((fun (P:Type) (x7:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)))) as proof of ((forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy20))->(((eq a) Xy10) Xy20)))->(((eq b) Xy1) Xy2))
% Found (fun (x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x6:(forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy20))->(((eq a) Xy10) Xy20))))=> (((fun (P:Type) (x7:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)))) as proof of ((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))->((forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy20))->(((eq a) Xy10) Xy20)))->(((eq b) Xy1) Xy2)))
% Found (and_rect10 (fun (x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x6:(forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy20))->(((eq a) Xy10) Xy20))))=> (((fun (P:Type) (x7:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect1 (((eq b) Xy1) Xy2)) (fun (x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x6:(forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy20))->(((eq a) Xy10) Xy20))))=> (((fun (P:Type) (x7:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x5:((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))->((forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2)))->P)))=> (((((and_rect (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2)))) P) x5) x3)) (((eq b) Xy1) Xy2)) (fun (x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x6:(forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy20))->(((eq a) Xy10) Xy20))))=> (((fun (P:Type) (x7:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x5:((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))->((forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2)))->P)))=> (((((and_rect (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2)))) P) x5) x3)) (((eq b) Xy1) Xy2)) (fun (x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x6:(forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy20))->(((eq a) Xy10) Xy20))))=> (((fun (P:Type) (x7:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x5:((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))->((forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2)))->P)))=> (((((and_rect (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2)))) P) x5) x3)) (((eq b) Xy1) Xy2)) (fun (x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x6:(forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy20))->(((eq a) Xy10) Xy20))))=> (((fun (P:Type) (x7:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)))))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq b) Xy1) Xy2)->(((eq b) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x4 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((eq_trans000 c) x7)) Xy2) as proof of (((x4 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((eq_trans00 Xy1) c) x7)) Xy2) as proof of (((x4 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((((eq_trans0 Xy1) Xy1) c) x7)) Xy2) as proof of (((x4 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2) as proof of (((x4 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2) as proof of (((x4 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x7:((x4 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)) as proof of (((x4 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x7:((x4 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)) as proof of (((x4 Xx) Xy1)->(((x4 Xx) Xy2)->(((eq b) Xy1) Xy2)))
% Found (and_rect20 (fun (x7:((x4 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect2 (((eq b) Xy1) Xy2)) (fun (x7:((x4 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x7:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x4 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x6:(forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy20))->(((eq a) Xy10) Xy20))))=> (((fun (P:Type) (x7:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x4 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x6:(forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy20))->(((eq a) Xy10) Xy20))))=> (((fun (P:Type) (x7:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x4 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)))) as proof of ((forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy20))->(((eq a) Xy10) Xy20)))->(((eq b) Xy1) Xy2))
% Found (fun (x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x6:(forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy20))->(((eq a) Xy10) Xy20))))=> (((fun (P:Type) (x7:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x4 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)))) as proof of ((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))->((forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy20))->(((eq a) Xy10) Xy20)))->(((eq b) Xy1) Xy2)))
% Found (and_rect10 (fun (x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x6:(forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy20))->(((eq a) Xy10) Xy20))))=> (((fun (P:Type) (x7:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x4 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect1 (((eq b) Xy1) Xy2)) (fun (x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x6:(forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy20))->(((eq a) Xy10) Xy20))))=> (((fun (P:Type) (x7:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x4 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x5:((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))->((forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2)))->P)))=> (((((and_rect (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2)))) P) x5) x2)) (((eq b) Xy1) Xy2)) (fun (x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x6:(forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy20))->(((eq a) Xy10) Xy20))))=> (((fun (P:Type) (x7:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x4 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2)))=> (((fun (P:Type) (x5:((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))->((forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2)))->P)))=> (((((and_rect (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2)))) P) x5) x2)) (((eq b) Xy1) Xy2)) (fun (x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x6:(forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy20))->(((eq a) Xy10) Xy20))))=> (((fun (P:Type) (x7:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x4 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2)))=> (((fun (P:Type) (x5:((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))->((forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2)))->P)))=> (((((and_rect (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2)))) P) x5) x2)) (((eq b) Xy1) Xy2)) (fun (x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x6:(forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy20))->(((eq a) Xy10) Xy20))))=> (((fun (P:Type) (x7:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x7) x00)) (((eq b) Xy1) Xy2)) (fun (x7:((x4 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x7)) Xy2)))))) as proof of (((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))):(((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))))
% Found (eq_ref0 (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found eq_ref000:=(eq_ref00 (x0 Xx)):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy2)))
% Found (and_rect00 (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found ((and_rect0 (P Xy2)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy2)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found (fun (x3:(P Xy1))=> (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy2)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))))) as proof of (P Xy2)
% Found x4:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))
% Instantiate: x8:=(fun (x12:a) (x110:b)=> ((x0 x110) x12)):(a->(b->Prop))
% Found x4 as proof of (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x8 Xx) Xy))))
% Found eq_ref000:=(eq_ref00 (fun (x7:b)=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x7:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy1)))
% Found (and_rect10 (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect1 (P Xy1)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x5:(P Xy2))=> (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2)))))) as proof of (P Xy1)
% Found eq_ref000:=(eq_ref00 (fun (x7:b)=> ((x2 Xx) Xy2))):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (fun (x7:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x7:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(P Xy1)))
% Found (and_rect10 (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect1 (P Xy1)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x6:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x5:(P Xy2))=> (((fun (P0:Type) (x6:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2)))))) as proof of (P Xy1)
% Found eq_ref000:=(eq_ref00 (fun (x7:b)=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x7:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy1)))
% Found (and_rect10 (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect1 (P Xy1)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2)))))) as proof of (P Xy1)
% Found (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy2))->(((eq a) Xy10) Xy2))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2)))))) as proof of ((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))->(P Xy1))
% Found (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy2))->(((eq a) Xy10) Xy2))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2)))))) as proof of (((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy2))->(((eq a) Xy10) Xy2))))->((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))->(P Xy1)))
% Found (and_rect00 (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy2))->(((eq a) Xy10) Xy2))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))))))) as proof of (P Xy1)
% Found ((and_rect0 (P Xy1)) (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy2))->(((eq a) Xy10) Xy2))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x4:(((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy2))->(((eq a) Xy1) Xy2))))->((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))->P0)))=> (((((and_rect ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))) P0) x4) x2)) (P Xy1)) (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy2))->(((eq a) Xy10) Xy2))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))))))) as proof of (P Xy1)
% Found (fun (x3:(P Xy2))=> (((fun (P0:Type) (x4:(((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy2))->(((eq a) Xy1) Xy2))))->((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))->P0)))=> (((((and_rect ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))) P0) x4) x2)) (P Xy1)) (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy2))->(((eq a) Xy10) Xy2))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2)))))))) as proof of (P Xy1)
% Found eq_ref000:=(eq_ref00 (fun (x7:b)=> ((x2 Xx) Xy2))):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (fun (x7:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x7:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(P Xy1)))
% Found (and_rect10 (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect1 (P Xy1)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x6:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2)))))) as proof of (P Xy1)
% Found (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy2))->(((eq a) Xy10) Xy2))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2)))))) as proof of ((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2)))->(P Xy1))
% Found (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy2))->(((eq a) Xy10) Xy2))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2)))))) as proof of (((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy2))->(((eq a) Xy10) Xy2))))->((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2)))->(P Xy1)))
% Found (and_rect00 (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy2))->(((eq a) Xy10) Xy2))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2))))))) as proof of (P Xy1)
% Found ((and_rect0 (P Xy1)) (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy2))->(((eq a) Xy10) Xy2))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2))))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x4:(((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2))))->((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2)))->P0)))=> (((((and_rect ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2)))) P0) x4) x1)) (P Xy1)) (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy2))->(((eq a) Xy10) Xy2))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2))))))) as proof of (P Xy1)
% Found (fun (x3:(P Xy2))=> (((fun (P0:Type) (x4:(((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2))))->((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2)))->P0)))=> (((((and_rect ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2)))) P0) x4) x1)) (P Xy1)) (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy2))->(((eq a) Xy10) Xy2))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x0 Xx1) Xy)) ((x0 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x2 Xx) Xy2)))))))) as proof of (P Xy1)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))):(((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))))
% Found (eq_ref0 (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found x4:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))
% Instantiate: x6:=(fun (x12:a) (x110:b)=> ((x0 x110) x12)):(a->(b->Prop))
% Found x4 as proof of (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x6 Xx) Xy))))
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy1)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy1)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy1)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq b) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq b) Xy2) b0)
% Found ((eq_ref b) Xy2) as proof of (((eq b) Xy2) b0)
% Found ((eq_ref b) Xy2) as proof of (((eq b) Xy2) b0)
% Found ((eq_ref b) Xy2) as proof of (((eq b) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq ((a->(b->Prop))->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((a->(b->Prop))->Prop)) b0) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) b0) as proof of (((eq ((a->(b->Prop))->Prop)) b0) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) b0) as proof of (((eq ((a->(b->Prop))->Prop)) b0) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) b0) as proof of (((eq ((a->(b->Prop))->Prop)) b0) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->(b->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))
% Instantiate: x0:=(fun (x12:a) (x110:b)=> ((x1 x110) x12)):(a->(b->Prop))
% Found x5 as proof of (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy))))
% Found x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))
% Instantiate: x2:=(fun (x12:a) (x110:b)=> ((x0 x110) x12)):(a->(b->Prop))
% Found x5 as proof of (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy))))
% Found and_rect0000:=(and_rect000 (fun (x4:b)=> ((x2 Xx) Xy2))):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (and_rect000 (fun (x4:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(((eq b) b0) Xy2))
% Found ((and_rect00 eq_trans0000) (fun (x4:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(((eq b) b0) Xy2))
% Found ((and_rect00 eq_trans0000) (fun (x4:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(((eq b) b0) Xy2))
% Found (fun (x3:((x2 Xx) Xy1))=> ((and_rect00 eq_trans0000) (fun (x4:b)=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy2)->(((eq b) b0) Xy2))
% Found (fun (x3:((x2 Xx) Xy1))=> ((and_rect00 eq_trans0000) (fun (x4:b)=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(((eq b) b0) Xy2)))
% Found (and_rect00 (fun (x3:((x2 Xx) Xy1))=> ((and_rect00 eq_trans0000) (fun (x4:b)=> ((x2 Xx) Xy2))))) as proof of (((eq b) b0) Xy2)
% Found ((and_rect0 (((eq b) b0) Xy2)) (fun (x3:((x2 Xx) Xy1))=> (((and_rect0 (((eq b) b0) Xy2)) eq_trans0000) (fun (x4:b)=> ((x2 Xx) Xy2))))) as proof of (((eq b) b0) Xy2)
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) b0) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) b0) Xy2)) eq_trans0000) (fun (x4:b)=> ((x2 Xx) Xy2))))) as proof of (((eq b) b0) Xy2)
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) b0) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) b0) Xy2)) eq_trans0000) (fun (x4:b)=> ((x2 Xx) Xy2))))) as proof of (((eq b) b0) Xy2)
% Found ((eq_trans0000 ((eq_ref b) Xy1)) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) b0) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) b0) Xy2)) eq_trans0000) (fun (x4:b)=> ((x2 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found (((eq_trans000 Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (eq_trans000 Xy1)) (fun (x4:b)=> ((x2 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found ((((fun (b0:b)=> ((eq_trans00 b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> ((eq_trans00 b0) Xy2)) Xy1)) (fun (x4:b)=> ((x2 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found ((((fun (b0:b)=> (((eq_trans0 Xy1) b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> (((eq_trans0 Xy1) b0) Xy2)) Xy1)) (fun (x4:b)=> ((x2 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found ((((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1)) (fun (x4:b)=> ((x2 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1)) (fun (x4:b)=> ((x2 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1)) (fun (x4:b)=> ((x2 Xx) Xy2))))))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))
% Instantiate: x4:=(fun (x12:a) (x110:b)=> ((x0 x110) x12)):(a->(b->Prop))
% Found x5 as proof of (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x4 Xx) Xy))))
% Found eq_ref000:=(eq_ref00 P):((P Xy1)->(P Xy1))
% Found (eq_ref00 P) as proof of (P0 Xy1)
% Found ((eq_ref0 Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref b) Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref b) Xy1) P) as proof of (P0 Xy1)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy1)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy1)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy1)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq b) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq b) Xy2) b0)
% Found ((eq_ref b) Xy2) as proof of (((eq b) Xy2) b0)
% Found ((eq_ref b) Xy2) as proof of (((eq b) Xy2) b0)
% Found ((eq_ref b) Xy2) as proof of (((eq b) Xy2) b0)
% Found ex_intro0:=(ex_intro (a->(b->Prop))):(forall (P:((a->(b->Prop))->Prop)) (x:(a->(b->Prop))), ((P x)->((ex (a->(b->Prop))) P)))
% Instantiate: b0:=(forall (P:((a->(b->Prop))->Prop)) (x:(a->(b->Prop))), ((P x)->((ex (a->(b->Prop))) P))):Prop
% Found ex_intro0 as proof of b0
% Found eq_ref00:=(eq_ref0 Xy1):(((eq b) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found eq_ref000:=(eq_ref00 P):((P Xy1)->(P Xy1))
% Found (eq_ref00 P) as proof of (P0 Xy1)
% Found ((eq_ref0 Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref b) Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref b) Xy1) P) as proof of (P0 Xy1)
% Found and_rect0000:=(and_rect000 (fun (x4:b)=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (and_rect000 (fun (x4:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(((eq b) b0) Xy2))
% Found ((and_rect00 eq_trans0000) (fun (x4:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(((eq b) b0) Xy2))
% Found ((and_rect00 eq_trans0000) (fun (x4:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(((eq b) b0) Xy2))
% Found (fun (x3:((x0 Xx) Xy1))=> ((and_rect00 eq_trans0000) (fun (x4:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->(((eq b) b0) Xy2))
% Found (fun (x3:((x0 Xx) Xy1))=> ((and_rect00 eq_trans0000) (fun (x4:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(((eq b) b0) Xy2)))
% Found (and_rect00 (fun (x3:((x0 Xx) Xy1))=> ((and_rect00 eq_trans0000) (fun (x4:b)=> ((x0 Xx) Xy2))))) as proof of (((eq b) b0) Xy2)
% Found ((and_rect0 (((eq b) b0) Xy2)) (fun (x3:((x0 Xx) Xy1))=> (((and_rect0 (((eq b) b0) Xy2)) eq_trans0000) (fun (x4:b)=> ((x0 Xx) Xy2))))) as proof of (((eq b) b0) Xy2)
% Found (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) b0) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) b0) Xy2)) eq_trans0000) (fun (x4:b)=> ((x0 Xx) Xy2))))) as proof of (((eq b) b0) Xy2)
% Found (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) b0) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) b0) Xy2)) eq_trans0000) (fun (x4:b)=> ((x0 Xx) Xy2))))) as proof of (((eq b) b0) Xy2)
% Found ((eq_trans0000 ((eq_ref b) Xy1)) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) b0) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) b0) Xy2)) eq_trans0000) (fun (x4:b)=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found (((eq_trans000 Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (eq_trans000 Xy1)) (fun (x4:b)=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found ((((fun (b0:b)=> ((eq_trans00 b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> ((eq_trans00 b0) Xy2)) Xy1)) (fun (x4:b)=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found ((((fun (b0:b)=> (((eq_trans0 Xy1) b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> (((eq_trans0 Xy1) b0) Xy2)) Xy1)) (fun (x4:b)=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found ((((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1)) (fun (x4:b)=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> ((((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1)) (fun (x4:b)=> ((x0 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> ((((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1)) (fun (x4:b)=> ((x0 Xx) Xy2))))))) as proof of (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_ref000:=(eq_ref00 P):((P Xy1)->(P Xy1))
% Found (eq_ref00 P) as proof of (P0 Xy1)
% Found ((eq_ref0 Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref b) Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref b) Xy1) P) as proof of (P0 Xy1)
% Found eq_ref000:=(eq_ref00 P):((P Xy1)->(P Xy1))
% Found (eq_ref00 P) as proof of (P0 Xy1)
% Found ((eq_ref0 Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref b) Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref b) Xy1) P) as proof of (P0 Xy1)
% Found ex_intro0:=(ex_intro (a->(b->Prop))):(forall (P:((a->(b->Prop))->Prop)) (x:(a->(b->Prop))), ((P x)->((ex (a->(b->Prop))) P)))
% Instantiate: b0:=(forall (P:((a->(b->Prop))->Prop)) (x:(a->(b->Prop))), ((P x)->((ex (a->(b->Prop))) P))):Prop
% Found ex_intro0 as proof of b0
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2))->(((eq b) Xy1) Xy2)))):(((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2))->(((eq b) Xy1) Xy2)))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2))->(((eq b) Xy1) Xy2))))
% Found (eq_ref0 (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found conj10:=(conj1 ((x6 Xx) Xy2)):(((x6 Xx) Xy1)->(((x6 Xx) Xy2)->((and ((x6 Xx) Xy1)) ((x6 Xx) Xy2))))
% Found (conj1 ((x6 Xx) Xy2)) as proof of (((x6 Xx) Xy1)->(((x6 Xx) Xy2)->((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))))
% Found ((conj ((x6 Xx) Xy1)) ((x6 Xx) Xy2)) as proof of (((x6 Xx) Xy1)->(((x6 Xx) Xy2)->((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))))
% Found ((conj ((x6 Xx) Xy1)) ((x6 Xx) Xy2)) as proof of (((x6 Xx) Xy1)->(((x6 Xx) Xy2)->((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))))
% Found ((conj ((x6 Xx) Xy1)) ((x6 Xx) Xy2)) as proof of (((x6 Xx) Xy1)->(((x6 Xx) Xy2)->((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))))
% Found (and_rect20 ((conj ((x6 Xx) Xy1)) ((x6 Xx) Xy2))) as proof of ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))
% Found ((and_rect2 ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))) ((conj ((x6 Xx) Xy1)) ((x6 Xx) Xy2))) as proof of ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))
% Found (((fun (P:Type) (x7:(((x6 Xx) Xy1)->(((x6 Xx) Xy2)->P)))=> (((((and_rect ((x6 Xx) Xy1)) ((x6 Xx) Xy2)) P) x7) x00)) ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))) ((conj ((x6 Xx) Xy1)) ((x6 Xx) Xy2))) as proof of ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))
% Found (((fun (P:Type) (x7:(((x6 Xx) Xy1)->(((x6 Xx) Xy2)->P)))=> (((((and_rect ((x6 Xx) Xy1)) ((x6 Xx) Xy2)) P) x7) x00)) ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))) ((conj ((x6 Xx) Xy1)) ((x6 Xx) Xy2))) as proof of ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))
% Found (x3000 (((fun (P:Type) (x7:(((x6 Xx) Xy1)->(((x6 Xx) Xy2)->P)))=> (((((and_rect ((x6 Xx) Xy1)) ((x6 Xx) Xy2)) P) x7) x00)) ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))) ((conj ((x6 Xx) Xy1)) ((x6 Xx) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq b) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found eq_ref000:=(eq_ref00 (fun (x5:b)=> ((x2 Xx) Xy2))):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (fun (x5:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P1 Xy1))
% Found ((eq_ref0 Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P1 Xy1))
% Found (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P1 Xy1))
% Found (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P1 Xy1))
% Found (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy2)->(P1 Xy1))
% Found (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(P1 Xy1)))
% Found (and_rect00 (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))))) as proof of (P1 Xy1)
% Found ((and_rect0 (P1 Xy1)) (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))))) as proof of (P1 Xy1)
% Found (((fun (P2:Type) (x4:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P2)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P2) x4) x00)) (P1 Xy1)) (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))))) as proof of (P1 Xy1)
% Found (fun (x3:(P1 Xy2))=> (((fun (P2:Type) (x4:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P2)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P2) x4) x00)) (P1 Xy1)) (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2)))))) as proof of (P1 Xy1)
% Found eq_ref000:=(eq_ref00 (fun (x5:b)=> ((x2 Xx) Xy2))):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (fun (x5:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P1 Xy1))
% Found ((eq_ref0 Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P1 Xy1))
% Found (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P1 Xy1))
% Found (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P1 Xy1))
% Found (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy2)->(P1 Xy1))
% Found (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(P1 Xy1)))
% Found (and_rect00 (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))))) as proof of (P1 Xy1)
% Found ((and_rect0 (P1 Xy1)) (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))))) as proof of (P1 Xy1)
% Found (((fun (P2:Type) (x4:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P2)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P2) x4) x00)) (P1 Xy1)) (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))))) as proof of (P1 Xy1)
% Found (fun (x3:(P1 Xy2))=> (((fun (P2:Type) (x4:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P2)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P2) x4) x00)) (P1 Xy1)) (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2)))))) as proof of (P1 Xy1)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq b) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq b) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found eq_ref000:=(eq_ref00 (x4 Xx)):(((x4 Xx) Xy2)->((x4 Xx) Xy2))
% Found (eq_ref00 (x4 Xx)) as proof of (((x4 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x4 Xx)) as proof of (((x4 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x4 Xx)) as proof of (((x4 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x4 Xx)) as proof of (((x4 Xx) Xy2)->(P Xy2))
% Found (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx))) as proof of (((x4 Xx) Xy2)->(P Xy2))
% Found (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx))) as proof of (((x4 Xx) Xy1)->(((x4 Xx) Xy2)->(P Xy2)))
% Found (and_rect10 (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx)))) as proof of (P Xy2)
% Found ((and_rect1 (P Xy2)) (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x6:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx)))) as proof of (P Xy2)
% Found (fun (x5:(P Xy1))=> (((fun (P0:Type) (x6:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy2) (x4 Xx))))) as proof of (P Xy2)
% Found eq_ref000:=(eq_ref00 (fun (x3:b)=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x3:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy1)))
% Found (and_rect00 (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect0 (P Xy1)) (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x2:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x2) x00)) (P Xy1)) (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x1:(P Xy2))=> (((fun (P0:Type) (x2:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x2) x00)) (P Xy1)) (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2)))))) as proof of (P Xy1)
% Found eq_ref000:=(eq_ref00 (fun (x3:b)=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x3:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy1)))
% Found (and_rect00 (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect0 (P Xy1)) (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x2:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x2) x00)) (P Xy1)) (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x1:(P Xy2))=> (((fun (P0:Type) (x2:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x2) x00)) (P Xy1)) (fun (x2:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x3:b)=> ((x0 Xx) Xy2)))))) as proof of (P Xy1)
% Found conj10:=(conj1 ((x0 Xx) Xy2)):(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))))
% Found (conj1 ((x0 Xx) Xy2)) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->((and ((x1 Xy1) Xy)) ((x1 Xy2) Xy))))
% Found ((conj ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->((and ((x1 Xy1) Xy)) ((x1 Xy2) Xy))))
% Found ((conj ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->((and ((x1 Xy1) Xy)) ((x1 Xy2) Xy))))
% Found ((conj ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->((and ((x1 Xy1) Xy)) ((x1 Xy2) Xy))))
% Found (and_rect20 ((conj ((x0 Xx) Xy1)) ((x0 Xx) Xy2))) as proof of ((and ((x1 Xy1) Xy)) ((x1 Xy2) Xy))
% Found ((and_rect2 ((and ((x1 Xy1) Xy)) ((x1 Xy2) Xy))) ((conj ((x0 Xx) Xy1)) ((x0 Xx) Xy2))) as proof of ((and ((x1 Xy1) Xy)) ((x1 Xy2) Xy))
% Found (((fun (P:Type) (x7:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x7) x00)) ((and ((x1 Xy1) Xy)) ((x1 Xy2) Xy))) ((conj ((x0 Xx) Xy1)) ((x0 Xx) Xy2))) as proof of ((and ((x1 Xy1) Xy)) ((x1 Xy2) Xy))
% Found (((fun (P:Type) (x7:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x7) x00)) ((and ((x1 Xy1) Xy)) ((x1 Xy2) Xy))) ((conj ((x0 Xx) Xy1)) ((x0 Xx) Xy2))) as proof of ((and ((x1 Xy1) Xy)) ((x1 Xy2) Xy))
% Found (x4000 (((fun (P:Type) (x7:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x7) x00)) ((and ((x1 Xy1) Xy)) ((x1 Xy2) Xy))) ((conj ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))):(((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))))
% Found (eq_ref0 (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found x50:=(x5 Xx):((ex a) (fun (Xy0:a)=> ((x1 Xx) Xy0)))
% Instantiate: x0:=(fun (x9:a) (x80:b)=> ((x1 Xx) x9)):(a->(b->Prop))
% Found conj10:=(conj1 ((x2 Xx) Xy2)):(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))))
% Found (conj1 ((x2 Xx) Xy2)) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))))
% Found ((conj ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))))
% Found ((conj ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))))
% Found ((conj ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))))
% Found (and_rect20 ((conj ((x2 Xx) Xy1)) ((x2 Xx) Xy2))) as proof of ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))
% Found ((and_rect2 ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))) ((conj ((x2 Xx) Xy1)) ((x2 Xx) Xy2))) as proof of ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))
% Found (((fun (P:Type) (x7:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x7) x00)) ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))) ((conj ((x2 Xx) Xy1)) ((x2 Xx) Xy2))) as proof of ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))
% Found (((fun (P:Type) (x7:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x7) x00)) ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))) ((conj ((x2 Xx) Xy1)) ((x2 Xx) Xy2))) as proof of ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))
% Found (x4000 (((fun (P:Type) (x7:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x7) x00)) ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))) ((conj ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))):(((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))))
% Found (eq_ref0 (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found conj10:=(conj1 ((x4 Xx) Xy2)):(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2))))
% Found (conj1 ((x4 Xx) Xy2)) as proof of (((x4 Xx) Xy1)->(((x4 Xx) Xy2)->((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))))
% Found ((conj ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) as proof of (((x4 Xx) Xy1)->(((x4 Xx) Xy2)->((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))))
% Found ((conj ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) as proof of (((x4 Xx) Xy1)->(((x4 Xx) Xy2)->((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))))
% Found ((conj ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) as proof of (((x4 Xx) Xy1)->(((x4 Xx) Xy2)->((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))))
% Found (and_rect20 ((conj ((x4 Xx) Xy1)) ((x4 Xx) Xy2))) as proof of ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))
% Found ((and_rect2 ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))) ((conj ((x4 Xx) Xy1)) ((x4 Xx) Xy2))) as proof of ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))
% Found (((fun (P:Type) (x7:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x7) x00)) ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))) ((conj ((x4 Xx) Xy1)) ((x4 Xx) Xy2))) as proof of ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))
% Found (((fun (P:Type) (x7:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x7) x00)) ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))) ((conj ((x4 Xx) Xy1)) ((x4 Xx) Xy2))) as proof of ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))
% Found (x3000 (((fun (P:Type) (x7:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x7) x00)) ((and ((x0 Xy1) Xy)) ((x0 Xy2) Xy))) ((conj ((x4 Xx) Xy1)) ((x4 Xx) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found x50:=(x5 Xx):((ex a) (fun (Xy0:a)=> ((x0 Xx) Xy0)))
% Instantiate: x2:=(fun (x9:a) (x80:b)=> ((x0 Xx) x9)):(a->(b->Prop))
% Found eq_ref000:=(eq_ref00 (fun (x5:b)=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x5:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P1 Xy1))
% Found ((eq_ref0 Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P1 Xy1))
% Found (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P1 Xy1))
% Found (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P1 Xy1))
% Found (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->(P1 Xy1))
% Found (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P1 Xy1)))
% Found (and_rect00 (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))))) as proof of (P1 Xy1)
% Found ((and_rect0 (P1 Xy1)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))))) as proof of (P1 Xy1)
% Found (((fun (P2:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P2)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P2) x4) x00)) (P1 Xy1)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))))) as proof of (P1 Xy1)
% Found (fun (x3:(P1 Xy2))=> (((fun (P2:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P2)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P2) x4) x00)) (P1 Xy1)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2)))))) as proof of (P1 Xy1)
% Found x50:=(x5 Xx):((ex a) (fun (Xy0:a)=> ((x0 Xx) Xy0)))
% Instantiate: x4:=(fun (x9:a) (x80:b)=> ((x0 Xx) x9)):(a->(b->Prop))
% Found eq_ref000:=(eq_ref00 (fun (x5:b)=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x5:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P1 Xy1))
% Found ((eq_ref0 Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P1 Xy1))
% Found (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P1 Xy1))
% Found (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P1 Xy1))
% Found (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->(P1 Xy1))
% Found (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P1 Xy1)))
% Found (and_rect00 (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))))) as proof of (P1 Xy1)
% Found ((and_rect0 (P1 Xy1)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))))) as proof of (P1 Xy1)
% Found (((fun (P2:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P2)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P2) x4) x00)) (P1 Xy1)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))))) as proof of (P1 Xy1)
% Found (fun (x3:(P1 Xy2))=> (((fun (P2:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P2)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P2) x4) x00)) (P1 Xy1)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2)))))) as proof of (P1 Xy1)
% Found eq_ref000:=(eq_ref00 (x0 Xx)):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy2)))
% Found (and_rect10 (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found ((and_rect1 (P Xy2)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found (fun (x5:(P Xy1))=> (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))))) as proof of (P Xy2)
% Found eq_ref000:=(eq_ref00 (x2 Xx)):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x2 Xx)) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))) as proof of (((x2 Xx) Xy2)->(P Xy2))
% Found (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(P Xy2)))
% Found (and_rect10 (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))) as proof of (P Xy2)
% Found ((and_rect1 (P Xy2)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x6:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx)))) as proof of (P Xy2)
% Found (fun (x5:(P Xy1))=> (((fun (P0:Type) (x6:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x6) x00)) (P Xy2)) (fun (x6:((x2 Xx) Xy1))=> (((eq_ref b) Xy2) (x2 Xx))))) as proof of (P Xy2)
% 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)) b0)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b0)) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found (((eq_trans000 ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b0)) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found ((((eq_trans00 ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))))) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found (((((eq_trans0 (f x0)) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))))) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found ((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))))) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% 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)) b0)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b0)) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found (((eq_trans000 ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b0)) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found ((((eq_trans00 ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))))) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found (((((eq_trans0 (f x0)) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))))) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found ((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))))) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found ex_intro0:=(ex_intro (a->(b->Prop))):(forall (P:((a->(b->Prop))->Prop)) (x:(a->(b->Prop))), ((P x)->((ex (a->(b->Prop))) P)))
% Instantiate: b0:=(forall (P:((a->(b->Prop))->Prop)) (x:(a->(b->Prop))), ((P x)->((ex (a->(b->Prop))) P))):Prop
% Found ex_intro0 as proof of b0
% 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 b0):(((eq ((a->(b->Prop))->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((a->(b->Prop))->Prop)) b0) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) b0) as proof of (((eq ((a->(b->Prop))->Prop)) b0) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) b0) as proof of (((eq ((a->(b->Prop))->Prop)) b0) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) b0) as proof of (((eq ((a->(b->Prop))->Prop)) b0) (fun (R:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((R Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((R Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((R Xx) Xy1)) ((R Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq ((a->(b->Prop))->Prop)) a0) (fun (x:(a->(b->Prop)))=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x7:(a->(b->Prop)))=> Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found (((eta_expansion_dep (a->(b->Prop))) (fun (x7:(a->(b->Prop)))=> Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found (((eta_expansion_dep (a->(b->Prop))) (fun (x7:(a->(b->Prop)))=> Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found (((eta_expansion_dep (a->(b->Prop))) (fun (x7:(a->(b->Prop)))=> Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->(b->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) b0)
% Found eq_ref000:=(eq_ref00 (fun (x7:b)=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x7:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy1)))
% Found (and_rect10 (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect1 (P Xy1)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x5:(P Xy2))=> (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2)))))) as proof of (P Xy1)
% Found eq_ref000:=(eq_ref00 (fun (x7:b)=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x7:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy1)))
% Found (and_rect10 (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect1 (P Xy1)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2)))))) as proof of (P Xy1)
% Found (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2)))))) as proof of ((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))->(P Xy1))
% Found (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2)))))) as proof of (((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))->((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))->(P Xy1)))
% Found (and_rect00 (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))))))) as proof of (P Xy1)
% Found ((and_rect0 (P Xy1)) (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x4:(((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))->((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))->P0)))=> (((((and_rect ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))) P0) x4) x2)) (P Xy1)) (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2))))))) as proof of (P Xy1)
% Found (fun (x3:(P Xy2))=> (((fun (P0:Type) (x4:(((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))->((forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))->P0)))=> (((((and_rect ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2)))) P0) x4) x2)) (P Xy1)) (fun (x4:((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy10:a) (Xy20:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy20))->(((eq a) Xy10) Xy20))))) (x5:(forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x1 Xx1) Xy)) ((x1 Xx2) Xy))->(((eq b) Xx1) Xx2))))=> (((fun (P0:Type) (x6:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x6) x00)) (P Xy1)) (fun (x6:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x0 Xx) Xy2)))))))) as proof of (P Xy1)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy1)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy1)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy1)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq b) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq b) Xy2) b0)
% Found ((eq_ref b) Xy2) as proof of (((eq b) Xy2) b0)
% Found ((eq_ref b) Xy2) as proof of (((eq b) Xy2) b0)
% Found ((eq_ref b) Xy2) as proof of (((eq b) Xy2) b0)
% Found x4:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))
% Instantiate: x6:=(fun (x12:a) (x110:b)=> ((x0 x110) x12)):(a->(b->Prop))
% Found x4 as proof of (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x6 Xx) Xy))))
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy1)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy1)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy1)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq b) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq b) Xy2) b0)
% Found ((eq_ref b) Xy2) as proof of (((eq b) Xy2) b0)
% Found ((eq_ref b) Xy2) as proof of (((eq b) Xy2) b0)
% Found ((eq_ref b) Xy2) as proof of (((eq b) Xy2) b0)
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b0:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b0
% Found eq_ref00:=(eq_ref0 (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy))))):(((eq Prop) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy)))))
% Found (eq_ref0 (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy))))) as proof of (((eq Prop) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy))))) b0)
% Found ((eq_ref Prop) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy))))) as proof of (((eq Prop) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy))))) b0)
% Found ((eq_ref Prop) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy))))) as proof of (((eq Prop) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy))))) b0)
% Found ((eq_ref Prop) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy))))) as proof of (((eq Prop) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy))))) b0)
% Found eq_ref000:=(eq_ref00 (fun (x9:b)=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x9:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x9:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy1)))
% Found (and_rect20 (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect2 (P Xy1)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x8:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x7:(P Xy2))=> (((fun (P0:Type) (x8:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2)))))) as proof of (P Xy1)
% Found eq_ref000:=(eq_ref00 (fun (x9:b)=> ((x2 Xx) Xy2))):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (fun (x9:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x9:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(P Xy1)))
% Found (and_rect20 (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect2 (P Xy1)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x8:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x7:(P Xy2))=> (((fun (P0:Type) (x8:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2)))))) as proof of (P Xy1)
% Found eq_ref000:=(eq_ref00 (fun (x9:b)=> ((x4 Xx) Xy2))):(((x4 Xx) Xy2)->((x4 Xx) Xy2))
% Found (eq_ref00 (fun (x9:b)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x9:b)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->(P Xy1))
% Found (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2)))) as proof of (((x4 Xx) Xy2)->(P Xy1))
% Found (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2)))) as proof of (((x4 Xx) Xy1)->(((x4 Xx) Xy2)->(P Xy1)))
% Found (and_rect20 (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect2 (P Xy1)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x8:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x7:(P Xy2))=> (((fun (P0:Type) (x8:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2)))))) as proof of (P Xy1)
% Found eq_ref000:=(eq_ref00 (fun (x9:b)=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x9:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x9:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy1)))
% Found (and_rect20 (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect2 (P Xy1)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x8:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x7:(forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy2))->(((eq a) Xy10) Xy2))))=> (((fun (P0:Type) (x8:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2)))))) as proof of (P Xy1)
% Found (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (x7:(forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy2))->(((eq a) Xy10) Xy2))))=> (((fun (P0:Type) (x8:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2)))))) as proof of ((forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy2))->(((eq a) Xy10) Xy2)))->(P Xy1))
% Found (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (x7:(forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy2))->(((eq a) Xy10) Xy2))))=> (((fun (P0:Type) (x8:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2)))))) as proof of ((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))->((forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy2))->(((eq a) Xy10) Xy2)))->(P Xy1)))
% Found (and_rect10 (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (x7:(forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy2))->(((eq a) Xy10) Xy2))))=> (((fun (P0:Type) (x8:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2))))))) as proof of (P Xy1)
% Found ((and_rect1 (P Xy1)) (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (x7:(forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy2))->(((eq a) Xy10) Xy2))))=> (((fun (P0:Type) (x8:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2))))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x6:((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))->((forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy2))->(((eq a) Xy1) Xy2)))->P0)))=> (((((and_rect (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy2))->(((eq a) Xy1) Xy2)))) P0) x6) x3)) (P Xy1)) (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (x7:(forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy2))->(((eq a) Xy10) Xy2))))=> (((fun (P0:Type) (x8:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2))))))) as proof of (P Xy1)
% Found (fun (x5:(P Xy2))=> (((fun (P0:Type) (x6:((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))->((forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy2))->(((eq a) Xy1) Xy2)))->P0)))=> (((((and_rect (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x1 Xx) Xy1)) ((x1 Xx) Xy2))->(((eq a) Xy1) Xy2)))) P0) x6) x3)) (P Xy1)) (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))) (x7:(forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x1 Xx) Xy10)) ((x1 Xx) Xy2))->(((eq a) Xy10) Xy2))))=> (((fun (P0:Type) (x8:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x0 Xx) Xy2)))))))) as proof of (P Xy1)
% Found eq_ref000:=(eq_ref00 (fun (x9:b)=> ((x2 Xx) Xy2))):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (fun (x9:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x9:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(P Xy1)))
% Found (and_rect20 (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect2 (P Xy1)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x8:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x7:(forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy2))->(((eq a) Xy10) Xy2))))=> (((fun (P0:Type) (x8:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2)))))) as proof of (P Xy1)
% Found (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x7:(forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy2))->(((eq a) Xy10) Xy2))))=> (((fun (P0:Type) (x8:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2)))))) as proof of ((forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy2))->(((eq a) Xy10) Xy2)))->(P Xy1))
% Found (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x7:(forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy2))->(((eq a) Xy10) Xy2))))=> (((fun (P0:Type) (x8:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2)))))) as proof of ((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))->((forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy2))->(((eq a) Xy10) Xy2)))->(P Xy1)))
% Found (and_rect10 (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x7:(forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy2))->(((eq a) Xy10) Xy2))))=> (((fun (P0:Type) (x8:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2))))))) as proof of (P Xy1)
% Found ((and_rect1 (P Xy1)) (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x7:(forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy2))->(((eq a) Xy10) Xy2))))=> (((fun (P0:Type) (x8:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2))))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x6:((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))->((forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2)))->P0)))=> (((((and_rect (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2)))) P0) x6) x3)) (P Xy1)) (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x7:(forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy2))->(((eq a) Xy10) Xy2))))=> (((fun (P0:Type) (x8:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2))))))) as proof of (P Xy1)
% Found (fun (x5:(P Xy2))=> (((fun (P0:Type) (x6:((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))->((forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2)))->P0)))=> (((((and_rect (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2)))) P0) x6) x3)) (P Xy1)) (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x7:(forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy2))->(((eq a) Xy10) Xy2))))=> (((fun (P0:Type) (x8:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x2 Xx) Xy2)))))))) as proof of (P Xy1)
% Found eq_ref000:=(eq_ref00 (fun (x9:b)=> ((x4 Xx) Xy2))):(((x4 Xx) Xy2)->((x4 Xx) Xy2))
% Found (eq_ref00 (fun (x9:b)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x9:b)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->(P Xy1))
% Found (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2)))) as proof of (((x4 Xx) Xy2)->(P Xy1))
% Found (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2)))) as proof of (((x4 Xx) Xy1)->(((x4 Xx) Xy2)->(P Xy1)))
% Found (and_rect20 (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect2 (P Xy1)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x8:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x7:(forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy2))->(((eq a) Xy10) Xy2))))=> (((fun (P0:Type) (x8:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2)))))) as proof of (P Xy1)
% Found (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x7:(forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy2))->(((eq a) Xy10) Xy2))))=> (((fun (P0:Type) (x8:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2)))))) as proof of ((forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy2))->(((eq a) Xy10) Xy2)))->(P Xy1))
% Found (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x7:(forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy2))->(((eq a) Xy10) Xy2))))=> (((fun (P0:Type) (x8:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2)))))) as proof of ((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))->((forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy2))->(((eq a) Xy10) Xy2)))->(P Xy1)))
% Found (and_rect10 (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x7:(forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy2))->(((eq a) Xy10) Xy2))))=> (((fun (P0:Type) (x8:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2))))))) as proof of (P Xy1)
% Found ((and_rect1 (P Xy1)) (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x7:(forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy2))->(((eq a) Xy10) Xy2))))=> (((fun (P0:Type) (x8:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2))))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x6:((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))->((forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2)))->P0)))=> (((((and_rect (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2)))) P0) x6) x2)) (P Xy1)) (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x7:(forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy2))->(((eq a) Xy10) Xy2))))=> (((fun (P0:Type) (x8:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2))))))) as proof of (P Xy1)
% Found (fun (x5:(P Xy2))=> (((fun (P0:Type) (x6:((forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))->((forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2)))->P0)))=> (((((and_rect (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2)))) P0) x6) x2)) (P Xy1)) (fun (x6:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))) (x7:(forall (Xx:b) (Xy10:a) (Xy2:a), (((and ((x0 Xx) Xy10)) ((x0 Xx) Xy2))->(((eq a) Xy10) Xy2))))=> (((fun (P0:Type) (x8:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P0)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P0) x8) x00)) (P Xy1)) (fun (x8:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x9:b)=> ((x4 Xx) Xy2)))))))) as proof of (P Xy1)
% Found eq_sym00000:=(eq_sym0000 (fun (x2:b)=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_sym0000 (fun (x2:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(((eq b) b0) Xy1))
% Found ((eq_sym000 x1) (fun (x2:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(((eq b) b0) Xy1))
% Found ((eq_sym000 x1) (fun (x2:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(((eq b) b0) Xy1))
% Found (fun (x1:((x0 Xx) Xy1))=> ((eq_sym000 x1) (fun (x2:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->(((eq b) b0) Xy1))
% Found (fun (x1:((x0 Xx) Xy1))=> ((eq_sym000 x1) (fun (x2:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(((eq b) b0) Xy1)))
% Found (and_rect00 (fun (x1:((x0 Xx) Xy1))=> ((eq_sym000 x1) (fun (x2:b)=> ((x0 Xx) Xy2))))) as proof of (((eq b) b0) Xy1)
% Found ((and_rect0 (((eq b) b0) Xy1)) (fun (x1:((x0 Xx) Xy1))=> ((eq_sym000 x1) (fun (x2:b)=> ((x0 Xx) Xy2))))) as proof of (((eq b) b0) Xy1)
% Found (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) b0) Xy1)) (fun (x1:((x0 Xx) Xy1))=> ((eq_sym000 x1) (fun (x2:b)=> ((x0 Xx) Xy2))))) as proof of (((eq b) b0) Xy1)
% Found (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) b0) Xy1)) (fun (x1:((x0 Xx) Xy1))=> ((eq_sym000 x1) (fun (x2:b)=> ((x0 Xx) Xy2))))) as proof of (((eq b) b0) Xy1)
% Found ((eq_trans0000 ((eq_ref b) Xy2)) (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) b0) Xy1)) (fun (x1:((x0 Xx) Xy1))=> ((eq_sym000 x1) (fun (x2:b)=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy2) Xy1)
% Found (((eq_trans000 Xy2) ((eq_ref b) Xy2)) (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy2) Xy1)) (fun (x1:((x0 Xx) Xy1))=> ((eq_sym000 x1) (fun (x2:b)=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy2) Xy1)
% Found ((((fun (b0:b)=> ((eq_trans00 b0) Xy1)) Xy2) ((eq_ref b) Xy2)) (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy2) Xy1)) (fun (x1:((x0 Xx) Xy1))=> ((eq_sym000 x1) (fun (x2:b)=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy2) Xy1)
% Found ((((fun (b0:b)=> (((eq_trans0 Xy2) b0) Xy1)) Xy2) ((eq_ref b) Xy2)) (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy2) Xy1)) (fun (x1:((x0 Xx) Xy1))=> ((eq_sym000 x1) (fun (x2:b)=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy2) Xy1)
% Found ((((fun (b0:b)=> ((((eq_trans b) Xy2) b0) Xy1)) Xy2) ((eq_ref b) Xy2)) (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy2) Xy1)) (fun (x1:((x0 Xx) Xy1))=> ((eq_sym000 x1) (fun (x2:b)=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy2) Xy1)
% Found ((((fun (b0:b)=> ((((eq_trans b) Xy2) b0) Xy1)) Xy2) ((eq_ref b) Xy2)) (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy2) Xy1)) (fun (x1:((x0 Xx) Xy1))=> ((eq_sym000 x1) (fun (x2:b)=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy2) Xy1)
% Found (eq_sym000 ((((fun (b0:b)=> ((((eq_trans b) Xy2) b0) Xy1)) Xy2) ((eq_ref b) Xy2)) (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy2) Xy1)) (fun (x1:((x0 Xx) Xy1))=> ((eq_sym000 x1) (fun (x2:b)=> ((x0 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found ((eq_sym00 Xy1) ((((fun (b0:b)=> ((((eq_trans b) Xy2) b0) Xy1)) Xy2) ((eq_ref b) Xy2)) (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy2) Xy1)) (fun (x1:((x0 Xx) Xy1))=> (((eq_sym00 Xy1) x1) (fun (x2:b)=> ((x0 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found (((eq_sym0 Xy2) Xy1) ((((fun (b0:b)=> ((((eq_trans b) Xy2) b0) Xy1)) Xy2) ((eq_ref b) Xy2)) (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy2) Xy1)) (fun (x1:((x0 Xx) Xy1))=> ((((eq_sym0 Xy2) Xy1) x1) (fun (x2:b)=> ((x0 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found ((((eq_sym b) Xy2) Xy1) ((((fun (b0:b)=> ((((eq_trans b) Xy2) b0) Xy1)) Xy2) ((eq_ref b) Xy2)) (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy2) Xy1)) (fun (x1:((x0 Xx) Xy1))=> (((((eq_sym b) Xy2) Xy1) x1) (fun (x2:b)=> ((x0 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> ((((eq_sym b) Xy2) Xy1) ((((fun (b0:b)=> ((((eq_trans b) Xy2) b0) Xy1)) Xy2) ((eq_ref b) Xy2)) (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy2) Xy1)) (fun (x1:((x0 Xx) Xy1))=> (((((eq_sym b) Xy2) Xy1) x1) (fun (x2:b)=> ((x0 Xx) Xy2)))))))) as proof of (((eq b) Xy1) Xy2)
% Found eq_ref000:=(eq_ref00 (x0 Xx)):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x0 Xx)) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))) as proof of (((x0 Xx) Xy2)->(P Xy2))
% Found (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy2)))
% Found (and_rect00 (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found ((and_rect0 (P Xy2)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy2)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))) as proof of (P Xy2)
% Found (fun (x3:((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x2 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy20))->(((eq a) Xy1) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x2 Xx1) Xy)) ((x2 Xx2) Xy))->(((eq b) Xx1) Xx2)))))=> (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy2)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))))) as proof of (P Xy2)
% Found (fun (x2:(b->(a->Prop))) (x3:((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x2 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy20))->(((eq a) Xy1) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x2 Xx1) Xy)) ((x2 Xx2) Xy))->(((eq b) Xx1) Xx2)))))=> (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy2)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))))) as proof of (((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x2 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy20))->(((eq a) Xy1) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x2 Xx1) Xy)) ((x2 Xx2) Xy))->(((eq b) Xx1) Xx2))))->(P Xy2))
% Found (fun (x2:(b->(a->Prop))) (x3:((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x2 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy20))->(((eq a) Xy1) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x2 Xx1) Xy)) ((x2 Xx2) Xy))->(((eq b) Xx1) Xx2)))))=> (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy2)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))))) as proof of (forall (x:(b->(a->Prop))), (((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x Xx) Xy1)) ((x Xx) Xy20))->(((eq a) Xy1) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x Xx1) Xy)) ((x Xx2) Xy))->(((eq b) Xx1) Xx2))))->(P Xy2)))
% Found (ex_ind00 (fun (x2:(b->(a->Prop))) (x3:((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x2 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy20))->(((eq a) Xy1) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x2 Xx1) Xy)) ((x2 Xx2) Xy))->(((eq b) Xx1) Xx2)))))=> (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy2)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))))) as proof of (P Xy2)
% Found ((ex_ind0 (P Xy2)) (fun (x2:(b->(a->Prop))) (x3:((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x2 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy20))->(((eq a) Xy1) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x2 Xx1) Xy)) ((x2 Xx2) Xy))->(((eq b) Xx1) Xx2)))))=> (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy2)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))))) as proof of (P Xy2)
% Found (((fun (P0:Prop) (x2:(forall (x:(b->(a->Prop))), (((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x Xx) Xy1)) ((x Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x Xx1) Xy)) ((x Xx2) Xy))->(((eq b) Xx1) Xx2))))->P0)))=> (((((ex_ind (b->(a->Prop))) (fun (S:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((S Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((S Xx) Xy1)) ((S Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((S Xx1) Xy)) ((S Xx2) Xy))->(((eq b) Xx1) Xx2)))))) P0) x2) x)) (P Xy2)) (fun (x2:(b->(a->Prop))) (x3:((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x2 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy20))->(((eq a) Xy1) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x2 Xx1) Xy)) ((x2 Xx2) Xy))->(((eq b) Xx1) Xx2)))))=> (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy2)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx)))))) as proof of (P Xy2)
% Found (fun (x1:(P Xy1))=> (((fun (P0:Prop) (x2:(forall (x:(b->(a->Prop))), (((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x Xx) Xy1)) ((x Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x Xx1) Xy)) ((x Xx2) Xy))->(((eq b) Xx1) Xx2))))->P0)))=> (((((ex_ind (b->(a->Prop))) (fun (S:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((S Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((S Xx) Xy1)) ((S Xx) Xy2))->(((eq a) Xy1) Xy2))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((S Xx1) Xy)) ((S Xx2) Xy))->(((eq b) Xx1) Xx2)))))) P0) x2) x)) (P Xy2)) (fun (x2:(b->(a->Prop))) (x3:((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x2 Xx) Xy))))) (forall (Xx:b) (Xy1:a) (Xy20:a), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy20))->(((eq a) Xy1) Xy20))))) (forall (Xx1:b) (Xx2:b) (Xy:a), (((and ((x2 Xx1) Xy)) ((x2 Xx2) Xy))->(((eq b) Xx1) Xx2)))))=> (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy2)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy2) (x0 Xx))))))) as proof of (P Xy2)
% Found x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x1 Xx) Xy))))
% Instantiate: x0:=(fun (x12:a) (x110:b)=> ((x1 x110) x12)):(a->(b->Prop))
% Found x5 as proof of (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x0 Xx) Xy))))
% Found x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))
% Instantiate: x2:=(fun (x12:a) (x110:b)=> ((x0 x110) x12)):(a->(b->Prop))
% Found x5 as proof of (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy))))
% Found x5:(forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x0 Xx) Xy))))
% Instantiate: x4:=(fun (x12:a) (x110:b)=> ((x0 x110) x12)):(a->(b->Prop))
% Found x5 as proof of (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x4 Xx) Xy))))
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy1)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy1)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy1)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq b) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq b) Xy2) b0)
% Found ((eq_ref b) Xy2) as proof of (((eq b) Xy2) b0)
% Found ((eq_ref b) Xy2) as proof of (((eq b) Xy2) b0)
% Found ((eq_ref b) Xy2) as proof of (((eq b) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy1)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy1)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy1)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq b) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq b) Xy2) b0)
% Found ((eq_ref b) Xy2) as proof of (((eq b) Xy2) b0)
% Found ((eq_ref b) Xy2) as proof of (((eq b) Xy2) b0)
% Found ((eq_ref b) Xy2) as proof of (((eq b) Xy2) b0)
% Found and_rect1000:=(and_rect100 (fun (x6:b)=> ((x4 Xx) Xy2))):(((x4 Xx) Xy2)->((x4 Xx) Xy2))
% Found (and_rect100 (fun (x6:b)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->(((eq b) b0) Xy2))
% Found ((and_rect10 eq_trans0000) (fun (x6:b)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->(((eq b) b0) Xy2))
% Found ((and_rect10 eq_trans0000) (fun (x6:b)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->(((eq b) b0) Xy2))
% Found (fun (x5:((x4 Xx) Xy1))=> ((and_rect10 eq_trans0000) (fun (x6:b)=> ((x4 Xx) Xy2)))) as proof of (((x4 Xx) Xy2)->(((eq b) b0) Xy2))
% Found (fun (x5:((x4 Xx) Xy1))=> ((and_rect10 eq_trans0000) (fun (x6:b)=> ((x4 Xx) Xy2)))) as proof of (((x4 Xx) Xy1)->(((x4 Xx) Xy2)->(((eq b) b0) Xy2)))
% Found (and_rect10 (fun (x5:((x4 Xx) Xy1))=> ((and_rect10 eq_trans0000) (fun (x6:b)=> ((x4 Xx) Xy2))))) as proof of (((eq b) b0) Xy2)
% Found ((and_rect1 (((eq b) b0) Xy2)) (fun (x5:((x4 Xx) Xy1))=> (((and_rect1 (((eq b) b0) Xy2)) eq_trans0000) (fun (x6:b)=> ((x4 Xx) Xy2))))) as proof of (((eq b) b0) Xy2)
% Found (((fun (P:Type) (x5:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x5) x00)) (((eq b) b0) Xy2)) (fun (x5:((x4 Xx) Xy1))=> ((((fun (P:Type) (x5:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x5) x00)) (((eq b) b0) Xy2)) eq_trans0000) (fun (x6:b)=> ((x4 Xx) Xy2))))) as proof of (((eq b) b0) Xy2)
% Found (((fun (P:Type) (x5:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x5) x00)) (((eq b) b0) Xy2)) (fun (x5:((x4 Xx) Xy1))=> ((((fun (P:Type) (x5:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x5) x00)) (((eq b) b0) Xy2)) eq_trans0000) (fun (x6:b)=> ((x4 Xx) Xy2))))) as proof of (((eq b) b0) Xy2)
% Found ((eq_trans0000 ((eq_ref b) Xy1)) (((fun (P:Type) (x5:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x5) x00)) (((eq b) b0) Xy2)) (fun (x5:((x4 Xx) Xy1))=> ((((fun (P:Type) (x5:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x5) x00)) (((eq b) b0) Xy2)) eq_trans0000) (fun (x6:b)=> ((x4 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found (((eq_trans000 Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x5:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x4 Xx) Xy1))=> ((((fun (P:Type) (x5:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (eq_trans000 Xy1)) (fun (x6:b)=> ((x4 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found ((((fun (b0:b)=> ((eq_trans00 b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x5:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x4 Xx) Xy1))=> ((((fun (P:Type) (x5:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> ((eq_trans00 b0) Xy2)) Xy1)) (fun (x6:b)=> ((x4 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found ((((fun (b0:b)=> (((eq_trans0 Xy1) b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x5:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x4 Xx) Xy1))=> ((((fun (P:Type) (x5:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> (((eq_trans0 Xy1) b0) Xy2)) Xy1)) (fun (x6:b)=> ((x4 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found ((((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x5:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x4 Xx) Xy1))=> ((((fun (P:Type) (x5:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1)) (fun (x6:b)=> ((x4 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2)))=> ((((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x5:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x4 Xx) Xy1))=> ((((fun (P:Type) (x5:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1)) (fun (x6:b)=> ((x4 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2)))=> ((((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x5:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x4 Xx) Xy1))=> ((((fun (P:Type) (x5:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1)) (fun (x6:b)=> ((x4 Xx) Xy2))))))) as proof of (((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b0:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b0
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b0:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b0
% 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)) b0)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b0)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b0)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b0)) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found (((eq_trans000 ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b0)) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found ((((eq_trans00 ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))))) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found (((((eq_trans0 (f x2)) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))))) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found ((((((eq_trans Prop) (f x2)) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))))) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% 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)) b0)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b0)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b0)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b0)) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found (((eq_trans000 ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b0)) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found ((((eq_trans00 ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))))) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found (((((eq_trans0 (f x2)) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))))) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found ((((((eq_trans Prop) (f x2)) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))))) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x2 Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x2 Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))))
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x6 Xx) Xy1)) ((x6 Xx) Xy2))->(((eq b) Xy1) Xy2)))):(((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x6 Xx) Xy1)) ((x6 Xx) Xy2))->(((eq b) Xy1) Xy2)))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x6 Xx) Xy1)) ((x6 Xx) Xy2))->(((eq b) Xy1) Xy2))))
% Found (eq_ref0 (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x6 Xx) Xy1)) ((x6 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x6 Xx) Xy1)) ((x6 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x6 Xx) Xy1)) ((x6 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x6 Xx) Xy1)) ((x6 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x6 Xx) Xy1)) ((x6 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x6 Xx) Xy1)) ((x6 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x6 Xx) Xy1)) ((x6 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x6 Xx) Xy1)) ((x6 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found and_rect1000:=(and_rect100 (fun (x6:b)=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (and_rect100 (fun (x6:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(((eq b) b0) Xy2))
% Found ((and_rect10 eq_trans0000) (fun (x6:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(((eq b) b0) Xy2))
% Found ((and_rect10 eq_trans0000) (fun (x6:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(((eq b) b0) Xy2))
% Found (fun (x5:((x0 Xx) Xy1))=> ((and_rect10 eq_trans0000) (fun (x6:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->(((eq b) b0) Xy2))
% Found (fun (x5:((x0 Xx) Xy1))=> ((and_rect10 eq_trans0000) (fun (x6:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(((eq b) b0) Xy2)))
% Found (and_rect10 (fun (x5:((x0 Xx) Xy1))=> ((and_rect10 eq_trans0000) (fun (x6:b)=> ((x0 Xx) Xy2))))) as proof of (((eq b) b0) Xy2)
% Found ((and_rect1 (((eq b) b0) Xy2)) (fun (x5:((x0 Xx) Xy1))=> (((and_rect1 (((eq b) b0) Xy2)) eq_trans0000) (fun (x6:b)=> ((x0 Xx) Xy2))))) as proof of (((eq b) b0) Xy2)
% Found (((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) b0) Xy2)) (fun (x5:((x0 Xx) Xy1))=> ((((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) b0) Xy2)) eq_trans0000) (fun (x6:b)=> ((x0 Xx) Xy2))))) as proof of (((eq b) b0) Xy2)
% Found (((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) b0) Xy2)) (fun (x5:((x0 Xx) Xy1))=> ((((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) b0) Xy2)) eq_trans0000) (fun (x6:b)=> ((x0 Xx) Xy2))))) as proof of (((eq b) b0) Xy2)
% Found ((eq_trans0000 ((eq_ref b) Xy1)) (((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) b0) Xy2)) (fun (x5:((x0 Xx) Xy1))=> ((((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) b0) Xy2)) eq_trans0000) (fun (x6:b)=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found (((eq_trans000 Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x0 Xx) Xy1))=> ((((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (eq_trans000 Xy1)) (fun (x6:b)=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found ((((fun (b0:b)=> ((eq_trans00 b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x0 Xx) Xy1))=> ((((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> ((eq_trans00 b0) Xy2)) Xy1)) (fun (x6:b)=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found ((((fun (b0:b)=> (((eq_trans0 Xy1) b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x0 Xx) Xy1))=> ((((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> (((eq_trans0 Xy1) b0) Xy2)) Xy1)) (fun (x6:b)=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found ((((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x0 Xx) Xy1))=> ((((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1)) (fun (x6:b)=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> ((((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x0 Xx) Xy1))=> ((((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1)) (fun (x6:b)=> ((x0 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> ((((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x0 Xx) Xy1))=> ((((fun (P:Type) (x5:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1)) (fun (x6:b)=> ((x0 Xx) Xy2))))))) as proof of (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found x50:=(x5 Xx):((ex a) (fun (Xy0:a)=> ((x1 Xx) Xy0)))
% Instantiate: x0:=(fun (x9:a) (x80:b)=> ((x1 Xx) x9)):(a->(b->Prop))
% Found and_rect1000:=(and_rect100 (fun (x6:b)=> ((x2 Xx) Xy2))):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (and_rect100 (fun (x6:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(((eq b) b0) Xy2))
% Found ((and_rect10 eq_trans0000) (fun (x6:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(((eq b) b0) Xy2))
% Found ((and_rect10 eq_trans0000) (fun (x6:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(((eq b) b0) Xy2))
% Found (fun (x5:((x2 Xx) Xy1))=> ((and_rect10 eq_trans0000) (fun (x6:b)=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy2)->(((eq b) b0) Xy2))
% Found (fun (x5:((x2 Xx) Xy1))=> ((and_rect10 eq_trans0000) (fun (x6:b)=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(((eq b) b0) Xy2)))
% Found (and_rect10 (fun (x5:((x2 Xx) Xy1))=> ((and_rect10 eq_trans0000) (fun (x6:b)=> ((x2 Xx) Xy2))))) as proof of (((eq b) b0) Xy2)
% Found ((and_rect1 (((eq b) b0) Xy2)) (fun (x5:((x2 Xx) Xy1))=> (((and_rect1 (((eq b) b0) Xy2)) eq_trans0000) (fun (x6:b)=> ((x2 Xx) Xy2))))) as proof of (((eq b) b0) Xy2)
% Found (((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) b0) Xy2)) (fun (x5:((x2 Xx) Xy1))=> ((((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) b0) Xy2)) eq_trans0000) (fun (x6:b)=> ((x2 Xx) Xy2))))) as proof of (((eq b) b0) Xy2)
% Found (((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) b0) Xy2)) (fun (x5:((x2 Xx) Xy1))=> ((((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) b0) Xy2)) eq_trans0000) (fun (x6:b)=> ((x2 Xx) Xy2))))) as proof of (((eq b) b0) Xy2)
% Found ((eq_trans0000 ((eq_ref b) Xy1)) (((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) b0) Xy2)) (fun (x5:((x2 Xx) Xy1))=> ((((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) b0) Xy2)) eq_trans0000) (fun (x6:b)=> ((x2 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found (((eq_trans000 Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x2 Xx) Xy1))=> ((((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (eq_trans000 Xy1)) (fun (x6:b)=> ((x2 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found ((((fun (b0:b)=> ((eq_trans00 b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x2 Xx) Xy1))=> ((((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> ((eq_trans00 b0) Xy2)) Xy1)) (fun (x6:b)=> ((x2 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found ((((fun (b0:b)=> (((eq_trans0 Xy1) b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x2 Xx) Xy1))=> ((((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> (((eq_trans0 Xy1) b0) Xy2)) Xy1)) (fun (x6:b)=> ((x2 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found ((((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x2 Xx) Xy1))=> ((((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1)) (fun (x6:b)=> ((x2 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x2 Xx) Xy1))=> ((((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1)) (fun (x6:b)=> ((x2 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1) ((eq_ref b) Xy1)) (((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) (fun (x5:((x2 Xx) Xy1))=> ((((fun (P:Type) (x5:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x5) x00)) (((eq b) Xy1) Xy2)) ((fun (b0:b)=> ((((eq_trans b) Xy1) b0) Xy2)) Xy1)) (fun (x6:b)=> ((x2 Xx) Xy2))))))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found x50:=(x5 Xx):((ex a) (fun (Xy0:a)=> ((x0 Xx) Xy0)))
% Instantiate: x2:=(fun (x9:a) (x80:b)=> ((x0 Xx) x9)):(a->(b->Prop))
% 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 (x5:b)=> ((x2 Xx) Xy2))):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (fun (x5:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(P Xy1)))
% Found (and_rect00 (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect0 (P Xy1)) (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x4:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x4) x00)) (P Xy1)) (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x3:(P Xy2))=> (((fun (P0:Type) (x4:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x4) x00)) (P Xy1)) (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2)))))) as proof of (P Xy1)
% Found eq_ref000:=(eq_ref00 (fun (x5:b)=> ((x2 Xx) Xy2))):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (fun (x5:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy2)->(P Xy1))
% Found (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(P Xy1)))
% Found (and_rect00 (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect0 (P Xy1)) (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x4:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x4) x00)) (P Xy1)) (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x3:(P Xy2))=> (((fun (P0:Type) (x4:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P0)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P0) x4) x00)) (P Xy1)) (fun (x4:((x2 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x2 Xx) Xy2)))))) as proof of (P Xy1)
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b0:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b0
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b0:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b0
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b0:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b0
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))):(((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))))
% Found (eq_ref0 (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))):(((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))))
% Found (eq_ref0 (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2))->(((eq b) Xy1) Xy2)))):(((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2))->(((eq b) Xy1) Xy2)))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2))->(((eq b) Xy1) Xy2))))
% Found (eq_ref0 (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2))->(((eq b) Xy1) Xy2)))) as proof of (((eq Prop) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x4 Xx) Xy1)) ((x4 Xx) Xy2))->(((eq b) Xy1) Xy2)))) b0)
% Found eq_ref000:=(eq_ref00 (fun (x7:b)=> ((x4 Xx) Xy2))):(((x4 Xx) Xy2)->((x4 Xx) Xy2))
% Found (eq_ref00 (fun (x7:b)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->(P1 Xy1))
% Found ((eq_ref0 Xy1) (fun (x7:b)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->(P1 Xy1))
% Found (((eq_ref b) Xy1) (fun (x7:b)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->(P1 Xy1))
% Found (((eq_ref b) Xy1) (fun (x7:b)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->(P1 Xy1))
% Found (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x4 Xx) Xy2)))) as proof of (((x4 Xx) Xy2)->(P1 Xy1))
% Found (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x4 Xx) Xy2)))) as proof of (((x4 Xx) Xy1)->(((x4 Xx) Xy2)->(P1 Xy1)))
% Found (and_rect10 (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x4 Xx) Xy2))))) as proof of (P1 Xy1)
% Found ((and_rect1 (P1 Xy1)) (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x4 Xx) Xy2))))) as proof of (P1 Xy1)
% Found (((fun (P2:Type) (x6:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P2)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P2) x6) x00)) (P1 Xy1)) (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x4 Xx) Xy2))))) as proof of (P1 Xy1)
% Found (fun (x5:(P1 Xy2))=> (((fun (P2:Type) (x6:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P2)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P2) x6) x00)) (P1 Xy1)) (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x4 Xx) Xy2)))))) as proof of (P1 Xy1)
% Found eq_ref000:=(eq_ref00 P):((P Xy1)->(P Xy1))
% Found (eq_ref00 P) as proof of (P0 Xy1)
% Found ((eq_ref0 Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref b) Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref b) Xy1) P) as proof of (P0 Xy1)
% Found eq_ref000:=(eq_ref00 (fun (x7:b)=> ((x4 Xx) Xy2))):(((x4 Xx) Xy2)->((x4 Xx) Xy2))
% Found (eq_ref00 (fun (x7:b)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->(P1 Xy1))
% Found ((eq_ref0 Xy1) (fun (x7:b)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->(P1 Xy1))
% Found (((eq_ref b) Xy1) (fun (x7:b)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->(P1 Xy1))
% Found (((eq_ref b) Xy1) (fun (x7:b)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->(P1 Xy1))
% Found (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x4 Xx) Xy2)))) as proof of (((x4 Xx) Xy2)->(P1 Xy1))
% Found (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x4 Xx) Xy2)))) as proof of (((x4 Xx) Xy1)->(((x4 Xx) Xy2)->(P1 Xy1)))
% Found (and_rect10 (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x4 Xx) Xy2))))) as proof of (P1 Xy1)
% Found ((and_rect1 (P1 Xy1)) (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x4 Xx) Xy2))))) as proof of (P1 Xy1)
% Found (((fun (P2:Type) (x6:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P2)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P2) x6) x00)) (P1 Xy1)) (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x4 Xx) Xy2))))) as proof of (P1 Xy1)
% Found (fun (x5:(P1 Xy2))=> (((fun (P2:Type) (x6:(((x4 Xx) Xy1)->(((x4 Xx) Xy2)->P2)))=> (((((and_rect ((x4 Xx) Xy1)) ((x4 Xx) Xy2)) P2) x6) x00)) (P1 Xy1)) (fun (x6:((x4 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x7:b)=> ((x4 Xx) Xy2)))))) as proof of (P1 Xy1)
% Found eq_ref000:=(eq_ref00 (x6 Xx)):(((x6 Xx) Xy2)->((x6 Xx) Xy2))
% Found (eq_ref00 (x6 Xx)) as proof of (((x6 Xx) Xy2)->(P Xy2))
% Found ((eq_ref0 Xy2) (x6 Xx)) as proof of (((x6 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x6 Xx)) as proof of (((x6 Xx) Xy2)->(P Xy2))
% Found (((eq_ref b) Xy2) (x6 Xx)) as proof of (((x6 Xx) Xy2)->(P Xy2))
% Found (fun (x8:((x6 Xx) Xy1))=> (((eq_ref b) Xy2) (x6 Xx))) as proof of (((x6 Xx) Xy2)->(P Xy2))
% Found (fun (x8:((x6 Xx) Xy1))=> (((eq_ref b) Xy2) (x6 Xx))) as proof of (((x6 Xx) Xy1)->(((x6 Xx) Xy2)->(P Xy2)))
% Found (and_rect20 (fun (x8:((x6 Xx) Xy1))=> (((eq_ref b) Xy2) (x6 Xx)))) as proof of (P Xy2)
% Found ((and_rect2 (P Xy2)) (fun (x8:((x6 Xx) Xy1))=> (((eq_ref b) Xy2) (x6 Xx)))) as proof of (P Xy2)
% Found (((fun (P0:Type) (x8:(((x6 Xx) Xy1)->(((x6 Xx) Xy2)->P0)))=> (((((and_rect ((x6 Xx) Xy1)) ((x6 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x6 Xx) Xy1))=> (((eq_ref b) Xy2) (x6 Xx)))) as proof of (P Xy2)
% Found (fun (x7:(P Xy1))=> (((fun (P0:Type) (x8:(((x6 Xx) Xy1)->(((x6 Xx) Xy2)->P0)))=> (((((and_rect ((x6 Xx) Xy1)) ((x6 Xx) Xy2)) P0) x8) x00)) (P Xy2)) (fun (x8:((x6 Xx) Xy1))=> (((eq_ref b) Xy2) (x6 Xx))))) as proof of (P Xy2)
% Found eq_ref000:=(eq_ref00 (fun (x5:b)=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x5:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy1)))
% Found (and_rect00 (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect0 (P Xy1)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy1)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x3:(P Xy2))=> (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy1)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2)))))) as proof of (P Xy1)
% Found eq_ref000:=(eq_ref00 (fun (x5:b)=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x5:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found ((eq_ref0 Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->(P Xy1))
% Found (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(P Xy1)))
% Found (and_rect00 (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found ((and_rect0 (P Xy1)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy1)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2))))) as proof of (P Xy1)
% Found (fun (x3:(P Xy2))=> (((fun (P0:Type) (x4:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P0)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P0) x4) x00)) (P Xy1)) (fun (x4:((x0 Xx) Xy1))=> (((eq_ref b) Xy1) (fun (x5:b)=> ((x0 Xx) Xy2)))))) as proof of (P Xy1)
% Found eq_ref000:=(eq_ref00 (fun (x6:Prop)=> ((x4 Xx) Xy2))):(((x4 Xx) Xy2)->((x4 Xx) Xy2))
% Found (eq_ref00 (fun (x6:Prop)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->((and ((x0 Xy2) Xy)) ((x0 Xy1) Xy)))
% Found ((eq_ref0 ((x0 Xy1) Xy)) (fun (x6:Prop)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->((and ((x0 Xy2) Xy)) ((x0 Xy1) Xy)))
% Found (((eq_ref Prop) ((x0 Xy1) Xy)) (fun (x6:Prop)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->((and ((x0 Xy2) Xy)) ((x0 Xy1) Xy)))
% Found (((eq_ref Prop) ((x0 Xy1) Xy)) (fun (x6:Prop)=> ((x4 Xx) Xy2))) as proof of (((x4 Xx) Xy2)->((and ((x0 Xy2) Xy)) ((x0 Xy1) Xy)))
% Found (fun (x5:((x4 Xx) Xy1))=> (((eq_ref Prop) ((x0 Xy1) Xy)) (fun (x6:Prop)=> ((x4 Xx) Xy2)))
% EOF
%------------------------------------------------------------------------------