%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV254^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n108.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 13:33:57 EDT 2014

% Result   : Timeout 300.09s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV254^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n108.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 08:38:06 CDT 2014
% % CPUTime  : 300.09 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (forall (K:((fofType->Prop)->(fofType->Prop))), (((and (forall (Xx:(fofType->Prop)), ((forall (Xx0:fofType), ((Xx Xx0)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0))))))->(forall (Xx0:fofType), (((K Xx) Xx0)->((K (fun (Xx1:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx2:fofType), ((S Xx2)->((K S) Xx2)))) (S Xx1)))))) Xx0)))))) ((forall (Xx:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))->((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)))->(forall (Xx:fofType), (((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)->((K (K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0))))))) Xx)))))->(forall (Xx:fofType), ((iff ((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))))))) of role conjecture named cTHM2C_pme
% Conjecture to prove = (forall (K:((fofType->Prop)->(fofType->Prop))), (((and (forall (Xx:(fofType->Prop)), ((forall (Xx0:fofType), ((Xx Xx0)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0))))))->(forall (Xx0:fofType), (((K Xx) Xx0)->((K (fun (Xx1:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx2:fofType), ((S Xx2)->((K S) Xx2)))) (S Xx1)))))) Xx0)))))) ((forall (Xx:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))->((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)))->(forall (Xx:fofType), (((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)->((K (K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0))))))) Xx)))))->(forall (Xx:fofType), ((iff ((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))))))):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['(forall (K:((fofType->Prop)->(fofType->Prop))), (((and (forall (Xx:(fofType->Prop)), ((forall (Xx0:fofType), ((Xx Xx0)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0))))))->(forall (Xx0:fofType), (((K Xx) Xx0)->((K (fun (Xx1:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx2:fofType), ((S Xx2)->((K S) Xx2)))) (S Xx1)))))) Xx0)))))) ((forall (Xx:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))->((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)))->(forall (Xx:fofType), (((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)->((K (K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0))))))) Xx)))))->(forall (Xx:fofType), ((iff ((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))))))']
% Parameter fofType:Type.
% Trying to prove (forall (K:((fofType->Prop)->(fofType->Prop))), (((and (forall (Xx:(fofType->Prop)), ((forall (Xx0:fofType), ((Xx Xx0)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0))))))->(forall (Xx0:fofType), (((K Xx) Xx0)->((K (fun (Xx1:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx2:fofType), ((S Xx2)->((K S) Xx2)))) (S Xx1)))))) Xx0)))))) ((forall (Xx:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))->((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)))->(forall (Xx:fofType), (((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)->((K (K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0))))))) Xx)))))->(forall (Xx:fofType), ((iff ((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))))))
% Found x3:(Xx0 Xx00)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x3:(Xx0 Xx00))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)
% Found x2 as proof of (x3 Xx)
% Found x2 as proof of (x3 Xx)
% Found x3:(Xx0 Xx00)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x3:(Xx0 Xx00))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x3:(Xx0 Xx00)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x3:(Xx0 Xx00))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x3:(Xx0 Xx00)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x3:(Xx0 Xx00))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x3:(Xx00 Xx000)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x3:(Xx00 Xx000))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)
% Found x2 as proof of (x3 Xx)
% Found x2 as proof of (x3 Xx)
% Found x0:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)
% Found x0 as proof of (x3 Xx)
% Found x0 as proof of (x3 Xx)
% Found x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)
% Found x2 as proof of (x3 Xx)
% Found x2 as proof of (x3 Xx)
% Found x3:(Xx0 Xx00)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x3:(Xx0 Xx00))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x0:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)
% Found x0 as proof of (x1 Xx)
% Found x0 as proof of (x1 Xx)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (fun (x4:((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)))=> (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)))) as proof of ((K Xx0) Xx)
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x3:(Xx00 Xx000)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x3:(Xx00 Xx000))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x3:(Xx00 Xx000)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x3:(Xx00 Xx000))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x0:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)
% Found x0 as proof of (x3 Xx)
% Found x0 as proof of (x3 Xx)
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x10:(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))
% Instantiate: x3:=(K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))):(fofType->Prop)
% Found x10 as proof of (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))
% Found x0:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)
% Found x0 as proof of (x1 Xx)
% Found (fun (x3:((forall (Xx0:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx00:fofType), ((S Xx00)->((K S) Xx00)))) (S Xx0))))->((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)))->(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))))=> x0) as proof of (x1 Xx)
% Found (fun (x2:(forall (Xx0:(fofType->Prop)), ((forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))->(forall (Xx00:fofType), (((K Xx0) Xx00)->((K (fun (Xx1:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx2:fofType), ((S Xx2)->((K S) Xx2)))) (S Xx1)))))) Xx00)))))) (x3:((forall (Xx0:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx00:fofType), ((S Xx00)->((K S) Xx00)))) (S Xx0))))->((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)))->(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))))=> x0) as proof of (((forall (Xx0:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx00:fofType), ((S Xx00)->((K S) Xx00)))) (S Xx0))))->((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)))->(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0))))->(x1 Xx))
% Found (fun (x2:(forall (Xx0:(fofType->Prop)), ((forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))->(forall (Xx00:fofType), (((K Xx0) Xx00)->((K (fun (Xx1:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx2:fofType), ((S Xx2)->((K S) Xx2)))) (S Xx1)))))) Xx00)))))) (x3:((forall (Xx0:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx00:fofType), ((S Xx00)->((K S) Xx00)))) (S Xx0))))->((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)))->(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))))=> x0) as proof of ((forall (Xx0:(fofType->Prop)), ((forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))->(forall (Xx00:fofType), (((K Xx0) Xx00)->((K (fun (Xx1:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx2:fofType), ((S Xx2)->((K S) Xx2)))) (S Xx1)))))) Xx00)))))->(((forall (Xx0:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx00:fofType), ((S Xx00)->((K S) Xx00)))) (S Xx0))))->((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)))->(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0))))->(x1 Xx)))
% Found (and_rect00 (fun (x2:(forall (Xx0:(fofType->Prop)), ((forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))->(forall (Xx00:fofType), (((K Xx0) Xx00)->((K (fun (Xx1:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx2:fofType), ((S Xx2)->((K S) Xx2)))) (S Xx1)))))) Xx00)))))) (x3:((forall (Xx0:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx00:fofType), ((S Xx00)->((K S) Xx00)))) (S Xx0))))->((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)))->(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))))=> x0)) as proof of (x1 Xx)
% Found ((and_rect0 (x1 Xx)) (fun (x2:(forall (Xx0:(fofType->Prop)), ((forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))->(forall (Xx00:fofType), (((K Xx0) Xx00)->((K (fun (Xx1:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx2:fofType), ((S Xx2)->((K S) Xx2)))) (S Xx1)))))) Xx00)))))) (x3:((forall (Xx0:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx00:fofType), ((S Xx00)->((K S) Xx00)))) (S Xx0))))->((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)))->(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))))=> x0)) as proof of (x1 Xx)
% Found (((fun (P:Type) (x2:((forall (Xx:(fofType->Prop)), ((forall (Xx0:fofType), ((Xx Xx0)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0))))))->(forall (Xx0:fofType), (((K Xx) Xx0)->((K (fun (Xx1:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx2:fofType), ((S Xx2)->((K S) Xx2)))) (S Xx1)))))) Xx0)))))->(((forall (Xx:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))->((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)))->(forall (Xx:fofType), (((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)->((K (K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0))))))) Xx))))->P)))=> (((((and_rect (forall (Xx:(fofType->Prop)), ((forall (Xx0:fofType), ((Xx Xx0)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0))))))->(forall (Xx0:fofType), (((K Xx) Xx0)->((K (fun (Xx1:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx2:fofType), ((S Xx2)->((K S) Xx2)))) (S Xx1)))))) Xx0)))))) ((forall (Xx:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))->((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)))->(forall (Xx:fofType), (((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)->((K (K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0))))))) Xx))))) P) x2) x)) (x1 Xx)) (fun (x2:(forall (Xx0:(fofType->Prop)), ((forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))->(forall (Xx00:fofType), (((K Xx0) Xx00)->((K (fun (Xx1:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx2:fofType), ((S Xx2)->((K S) Xx2)))) (S Xx1)))))) Xx00)))))) (x3:((forall (Xx0:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx00:fofType), ((S Xx00)->((K S) Xx00)))) (S Xx0))))->((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)))->(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))))=> x0)) as proof of (x1 Xx)
% Found (((fun (P:Type) (x2:((forall (Xx:(fofType->Prop)), ((forall (Xx0:fofType), ((Xx Xx0)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0))))))->(forall (Xx0:fofType), (((K Xx) Xx0)->((K (fun (Xx1:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx2:fofType), ((S Xx2)->((K S) Xx2)))) (S Xx1)))))) Xx0)))))->(((forall (Xx:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))->((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)))->(forall (Xx:fofType), (((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)->((K (K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0))))))) Xx))))->P)))=> (((((and_rect (forall (Xx:(fofType->Prop)), ((forall (Xx0:fofType), ((Xx Xx0)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0))))))->(forall (Xx0:fofType), (((K Xx) Xx0)->((K (fun (Xx1:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx2:fofType), ((S Xx2)->((K S) Xx2)))) (S Xx1)))))) Xx0)))))) ((forall (Xx:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))->((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)))->(forall (Xx:fofType), (((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)->((K (K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0))))))) Xx))))) P) x2) x)) (x1 Xx)) (fun (x2:(forall (Xx0:(fofType->Prop)), ((forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))->(forall (Xx00:fofType), (((K Xx0) Xx00)->((K (fun (Xx1:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx2:fofType), ((S Xx2)->((K S) Xx2)))) (S Xx1)))))) Xx00)))))) (x3:((forall (Xx0:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx00:fofType), ((S Xx00)->((K S) Xx00)))) (S Xx0))))->((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)))->(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))))=> x0)) as proof of (x1 Xx)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (fun (x4:((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)))=> (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)))) as proof of ((K Xx0) Xx)
% Found x0:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)
% Found x0 as proof of (x1 Xx)
% Found x0 as proof of (x1 Xx)
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (fun (x4:((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)))=> (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)))) as proof of ((K Xx0) Xx)
% Found x5:(Xx00 Xx000)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x5:(Xx00 Xx000))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (fun (x4:((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)))=> (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)))) as proof of ((K Xx0) Xx)
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x7:(Xx0 Xx00)
% Found x7 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x7:(Xx0 Xx00))=> x7) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x500:=(x50 x6):((K x3) Xx)
% Found (x50 x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x00:(x1 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x50:=(x5 Xx):((x1 Xx)->((K x1) Xx))
% Found (x5 Xx) as proof of ((x1 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x1 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x1 Xx00)->((K x1) Xx00))))=> (x5 Xx)) as proof of ((x1 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x1 Xx00)->((K x1) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x1 Xx00)->((K x1) Xx00)))->((x1 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x1 Xx00)->((K x1) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x1 Xx00)->((K x1) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x1 Xx0)->((K x1) Xx0)))->((x1 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x1 Xx0)->((K x1) Xx0)))) (x1 Xx)) P) x5) x2)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x1 Xx00)->((K x1) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x1 Xx0)->((K x1) Xx0)))->((x1 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x1 Xx0)->((K x1) Xx0)))) (x1 Xx)) P) x5) x2)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x1 Xx00)->((K x1) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x50:=(x5 Xx0):((x3 Xx0)->((K x3) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)) as proof of ((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->((K Xx00) Xx0)))
% Found (and_rect10 (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found ((and_rect1 ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx0)) P) x5) x4)) ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx0)) P) x5) x4)) ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found x50:=(x5 Xx0):((x3 Xx0)->((K x3) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)) as proof of ((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->((K Xx00) Xx0)))
% Found (and_rect10 (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found ((and_rect1 ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x5:((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))) (x3 Xx0)) P) x5) x4)) ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found (fun (x4:((and (forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))) (x3 Xx0)))=> (((fun (P:Type) (x5:((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))) (x3 Xx0)) P) x5) x4)) ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)))) as proof of ((K Xx00) Xx0)
% Found x00:(x1 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x00:(x1 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x3:(Xx0 Xx00)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x3:(Xx0 Xx00))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x10:(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))
% Instantiate: x3:=(K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))):(fofType->Prop)
% Found x10 as proof of (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))
% Found x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)
% Found x2 as proof of (x3 Xx)
% Found ((conj10 x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (((conj1 (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found x10:(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))
% Instantiate: x3:=(K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))):(fofType->Prop)
% Found x10 as proof of (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))
% Found x3:(Xx0 Xx00)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x3:(Xx0 Xx00))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x00:(x1 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x20:(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))
% Instantiate: x3:=(K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))):(fofType->Prop)
% Found x20 as proof of (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x10:(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))
% Instantiate: x3:=(K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))):(fofType->Prop)
% Found x10 as proof of (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x0:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)
% Found x0 as proof of (x1 Xx)
% Found (fun (x3:((forall (Xx0:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx00:fofType), ((S Xx00)->((K S) Xx00)))) (S Xx0))))->((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)))->(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))))=> x0) as proof of (x1 Xx)
% Found (fun (x2:(forall (Xx0:(fofType->Prop)), ((forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))->(forall (Xx00:fofType), (((K Xx0) Xx00)->((K (fun (Xx1:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx2:fofType), ((S Xx2)->((K S) Xx2)))) (S Xx1)))))) Xx00)))))) (x3:((forall (Xx0:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx00:fofType), ((S Xx00)->((K S) Xx00)))) (S Xx0))))->((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)))->(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))))=> x0) as proof of (((forall (Xx0:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx00:fofType), ((S Xx00)->((K S) Xx00)))) (S Xx0))))->((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)))->(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0))))->(x1 Xx))
% Found (fun (x2:(forall (Xx0:(fofType->Prop)), ((forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))->(forall (Xx00:fofType), (((K Xx0) Xx00)->((K (fun (Xx1:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx2:fofType), ((S Xx2)->((K S) Xx2)))) (S Xx1)))))) Xx00)))))) (x3:((forall (Xx0:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx00:fofType), ((S Xx00)->((K S) Xx00)))) (S Xx0))))->((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)))->(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))))=> x0) as proof of ((forall (Xx0:(fofType->Prop)), ((forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))->(forall (Xx00:fofType), (((K Xx0) Xx00)->((K (fun (Xx1:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx2:fofType), ((S Xx2)->((K S) Xx2)))) (S Xx1)))))) Xx00)))))->(((forall (Xx0:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx00:fofType), ((S Xx00)->((K S) Xx00)))) (S Xx0))))->((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)))->(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0))))->(x1 Xx)))
% Found (and_rect00 (fun (x2:(forall (Xx0:(fofType->Prop)), ((forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))->(forall (Xx00:fofType), (((K Xx0) Xx00)->((K (fun (Xx1:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx2:fofType), ((S Xx2)->((K S) Xx2)))) (S Xx1)))))) Xx00)))))) (x3:((forall (Xx0:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx00:fofType), ((S Xx00)->((K S) Xx00)))) (S Xx0))))->((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)))->(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))))=> x0)) as proof of (x1 Xx)
% Found ((and_rect0 (x1 Xx)) (fun (x2:(forall (Xx0:(fofType->Prop)), ((forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))->(forall (Xx00:fofType), (((K Xx0) Xx00)->((K (fun (Xx1:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx2:fofType), ((S Xx2)->((K S) Xx2)))) (S Xx1)))))) Xx00)))))) (x3:((forall (Xx0:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx00:fofType), ((S Xx00)->((K S) Xx00)))) (S Xx0))))->((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)))->(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))))=> x0)) as proof of (x1 Xx)
% Found (((fun (P:Type) (x2:((forall (Xx:(fofType->Prop)), ((forall (Xx0:fofType), ((Xx Xx0)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0))))))->(forall (Xx0:fofType), (((K Xx) Xx0)->((K (fun (Xx1:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx2:fofType), ((S Xx2)->((K S) Xx2)))) (S Xx1)))))) Xx0)))))->(((forall (Xx:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))->((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)))->(forall (Xx:fofType), (((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)->((K (K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0))))))) Xx))))->P)))=> (((((and_rect (forall (Xx:(fofType->Prop)), ((forall (Xx0:fofType), ((Xx Xx0)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0))))))->(forall (Xx0:fofType), (((K Xx) Xx0)->((K (fun (Xx1:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx2:fofType), ((S Xx2)->((K S) Xx2)))) (S Xx1)))))) Xx0)))))) ((forall (Xx:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))->((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)))->(forall (Xx:fofType), (((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)->((K (K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0))))))) Xx))))) P) x2) x)) (x1 Xx)) (fun (x2:(forall (Xx0:(fofType->Prop)), ((forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))->(forall (Xx00:fofType), (((K Xx0) Xx00)->((K (fun (Xx1:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx2:fofType), ((S Xx2)->((K S) Xx2)))) (S Xx1)))))) Xx00)))))) (x3:((forall (Xx0:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx00:fofType), ((S Xx00)->((K S) Xx00)))) (S Xx0))))->((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)))->(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))))=> x0)) as proof of (x1 Xx)
% Found (((fun (P:Type) (x2:((forall (Xx:(fofType->Prop)), ((forall (Xx0:fofType), ((Xx Xx0)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0))))))->(forall (Xx0:fofType), (((K Xx) Xx0)->((K (fun (Xx1:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx2:fofType), ((S Xx2)->((K S) Xx2)))) (S Xx1)))))) Xx0)))))->(((forall (Xx:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))->((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)))->(forall (Xx:fofType), (((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)->((K (K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0))))))) Xx))))->P)))=> (((((and_rect (forall (Xx:(fofType->Prop)), ((forall (Xx0:fofType), ((Xx Xx0)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0))))))->(forall (Xx0:fofType), (((K Xx) Xx0)->((K (fun (Xx1:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx2:fofType), ((S Xx2)->((K S) Xx2)))) (S Xx1)))))) Xx0)))))) ((forall (Xx:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))->((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)))->(forall (Xx:fofType), (((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)->((K (K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0))))))) Xx))))) P) x2) x)) (x1 Xx)) (fun (x2:(forall (Xx0:(fofType->Prop)), ((forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))->(forall (Xx00:fofType), (((K Xx0) Xx00)->((K (fun (Xx1:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx2:fofType), ((S Xx2)->((K S) Xx2)))) (S Xx1)))))) Xx00)))))) (x3:((forall (Xx0:fofType), (((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx00:fofType), ((S Xx00)->((K S) Xx00)))) (S Xx0))))->((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)))->(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))))=> x0)) as proof of (x1 Xx)
% Found x3:(Xx0 Xx00)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x3:(Xx0 Xx00))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x5:(Xx00 Xx000)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x5:(Xx00 Xx000))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x30:(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))
% Instantiate: x1:=(K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))):(fofType->Prop)
% Found x30 as proof of (forall (Xx0:fofType), ((x1 Xx0)->((K x1) Xx0)))
% Found x3:(Xx00 Xx000)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x3:(Xx00 Xx000))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x3:(Xx00 Xx000)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x3:(Xx00 Xx000))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x5:(Xx00 Xx000)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x5:(Xx00 Xx000))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x10:(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))
% Instantiate: x3:=(K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))):(fofType->Prop)
% Found x10 as proof of (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))
% Found x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)
% Found x2 as proof of (x3 Xx)
% Found ((conj10 x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (((conj1 (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (ex_intro000 ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found ((ex_intro00 (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found (((ex_intro0 (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found x7:(Xx0 Xx00)
% Found x7 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x7:(Xx0 Xx00))=> x7) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x500:=(x50 x6):((K x3) Xx)
% Found (x50 x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found x4:(Xx00 Xx000)
% Found x4 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x4:(Xx00 Xx000))=> x4) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (fun (x4:((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)))=> (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)))) as proof of ((K Xx0) Xx)
% Found x10:(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))
% Instantiate: x3:=(K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))):(fofType->Prop)
% Found x10 as proof of (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))
% Found x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)
% Found x2 as proof of (x3 Xx)
% Found ((conj10 x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (((conj1 (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (ex_intro000 ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found ((ex_intro00 (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found (((ex_intro0 (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found (fun (x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx))=> ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2))) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found (fun (x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx))=> ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2))) as proof of (((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))))
% Found x5:(Xx00 Xx000)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x5:(Xx00 Xx000))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x3:(Xx0 Xx00)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x3:(Xx0 Xx00))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x00:(x1 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x50:=(x5 Xx0):((x3 Xx0)->((K x3) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)) as proof of ((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->((K Xx00) Xx0)))
% Found (and_rect10 (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found ((and_rect1 ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx0)) P) x5) x4)) ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx0)) P) x5) x4)) ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found x50:=(x5 Xx0):((x3 Xx0)->((K x3) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)) as proof of ((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->((K Xx00) Xx0)))
% Found (and_rect10 (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found ((and_rect1 ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x5:((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))) (x3 Xx0)) P) x5) x4)) ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found (fun (x4:((and (forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))) (x3 Xx0)))=> (((fun (P:Type) (x5:((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))) (x3 Xx0)) P) x5) x4)) ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)))) as proof of ((K Xx00) Xx0)
% Found x7:(Xx0 Xx00)
% Found x7 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x7:(Xx0 Xx00))=> x7) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x500:=(x50 x6):((K x3) Xx)
% Found (x50 x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found x10:(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))
% Instantiate: x3:=(K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))):(fofType->Prop)
% Found x10 as proof of (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))
% Found x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)
% Found x2 as proof of (x3 Xx)
% Found ((conj10 x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (((conj1 (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found x7:(Xx0 Xx00)
% Found x7 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x7:(Xx0 Xx00))=> x7) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x500:=(x50 x6):((K x3) Xx)
% Found (x50 x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found x6:(Xx00 Xx000)
% Found x6 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x6:(Xx00 Xx000))=> x6) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x6:(Xx00 Xx000))=> x6) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x6:(Xx00 Xx000))=> x6) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x3:(Xx0 Xx00)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x3:(Xx0 Xx00))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x3:(Xx0 Xx00)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x3:(Xx0 Xx00))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x50:=(x5 Xx):((x1 Xx)->((K x1) Xx))
% Found (x5 Xx) as proof of ((x1 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x1 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x1 Xx00)->((K x1) Xx00))))=> (x5 Xx)) as proof of ((x1 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x1 Xx00)->((K x1) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x1 Xx00)->((K x1) Xx00)))->((x1 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x1 Xx00)->((K x1) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x1 Xx00)->((K x1) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x1 Xx0)->((K x1) Xx0)))->((x1 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x1 Xx0)->((K x1) Xx0)))) (x1 Xx)) P) x5) x2)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x1 Xx00)->((K x1) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x1 Xx0)->((K x1) Xx0)))->((x1 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x1 Xx0)->((K x1) Xx0)))) (x1 Xx)) P) x5) x2)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x1 Xx00)->((K x1) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x20:(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))
% Instantiate: x3:=(K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))):(fofType->Prop)
% Found x20 as proof of (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))
% Found x0:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)
% Found x0 as proof of (x3 Xx)
% Found ((conj10 x20) x0) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (((conj1 (x3 Xx)) x20) x0) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x20) x0) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x20) x0) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found x3:(Xx0 Xx00)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x3:(Xx0 Xx00))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x50:=(x5 Xx0):((x3 Xx0)->((K x3) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)) as proof of ((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->((K Xx00) Xx0)))
% Found (and_rect10 (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found ((and_rect1 ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx0)) P) x5) x4)) ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx0)) P) x5) x4)) ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found x50:=(x5 Xx0):((x3 Xx0)->((K x3) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)) as proof of ((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->((K Xx00) Xx0)))
% Found (and_rect10 (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found ((and_rect1 ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x5:((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))) (x3 Xx0)) P) x5) x4)) ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found (fun (x4:((and (forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))) (x3 Xx0)))=> (((fun (P:Type) (x5:((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))) (x3 Xx0)) P) x5) x4)) ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)))) as proof of ((K Xx00) Xx0)
% Found x00:(x1 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x10:(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))
% Instantiate: x3:=(K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))):(fofType->Prop)
% Found x10 as proof of (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))
% Found x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)
% Found x2 as proof of (x3 Xx)
% Found ((conj10 x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (((conj1 (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found x10:(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))
% Instantiate: x3:=(K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))):(fofType->Prop)
% Found x10 as proof of (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))
% Found x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)
% Found x2 as proof of (x3 Xx)
% Found ((conj10 x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (((conj1 (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (ex_intro000 ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found ((ex_intro00 (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found (((ex_intro0 (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found (fun (x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx))=> ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2))) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found (fun (x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx))=> ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2))) as proof of (((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))))
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x00:(x1 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x50:=(x5 Xx0):((x3 Xx0)->((K x3) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)) as proof of ((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->((K Xx00) Xx0)))
% Found (and_rect10 (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found ((and_rect1 ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx0)) P) x5) x4)) ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx0)) P) x5) x4)) ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found x50:=(x5 Xx0):((x3 Xx0)->((K x3) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)) as proof of ((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->((K Xx00) Xx0)))
% Found (and_rect10 (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found ((and_rect1 ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x5:((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))) (x3 Xx0)) P) x5) x4)) ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found (fun (x4:((and (forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))) (x3 Xx0)))=> (((fun (P:Type) (x5:((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))) (x3 Xx0)) P) x5) x4)) ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)))) as proof of ((K Xx00) Xx0)
% Found x3:(Xx0 Xx00)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x3:(Xx0 Xx00))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x7:(Xx0 Xx00)
% Found x7 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x7:(Xx0 Xx00))=> x7) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x300:=(x30 x4):((K x1) Xx)
% Found (x30 x4) as proof of ((K Xx0) Xx)
% Found ((x3 Xx) x4) as proof of ((K Xx0) Xx)
% Found ((x3 Xx) x4) as proof of ((K Xx0) Xx)
% Found ((x3 Xx) x4) as proof of ((K Xx0) Xx)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (fun (x4:((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)))=> (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)))) as proof of ((K Xx0) Xx)
% Found x7:(Xx0 Xx00)
% Found x7 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x7:(Xx0 Xx00))=> x7) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x500:=(x50 x6):((K x1) Xx)
% Found (x50 x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found x7:(Xx00 Xx000)
% Found x7 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x7:(Xx00 Xx000))=> x7) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x7:(Xx00 Xx000))=> x7) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x7:(Xx00 Xx000))=> x7) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x500:=(x50 x6):((K x3) Xx0)
% Found (x50 x6) as proof of ((K Xx00) Xx0)
% Found ((x5 Xx0) x6) as proof of ((K Xx00) Xx0)
% Found ((x5 Xx0) x6) as proof of ((K Xx00) Xx0)
% Found ((x5 Xx0) x6) as proof of ((K Xx00) Xx0)
% Found x30:(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))
% Instantiate: x1:=(K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))):(fofType->Prop)
% Found x30 as proof of (forall (Xx0:fofType), ((x1 Xx0)->((K x1) Xx0)))
% Found x0:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)
% Found x0 as proof of (x1 Xx)
% Found ((conj10 x30) x0) as proof of ((and (forall (Xx0:fofType), ((x1 Xx0)->((K x1) Xx0)))) (x1 Xx))
% Found (((conj1 (x1 Xx)) x30) x0) as proof of ((and (forall (Xx0:fofType), ((x1 Xx0)->((K x1) Xx0)))) (x1 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x1 Xx0)->((K x1) Xx0)))) (x1 Xx)) x30) x0) as proof of ((and (forall (Xx0:fofType), ((x1 Xx0)->((K x1) Xx0)))) (x1 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x1 Xx0)->((K x1) Xx0)))) (x1 Xx)) x30) x0) as proof of ((and (forall (Xx0:fofType), ((x1 Xx0)->((K x1) Xx0)))) (x1 Xx))
% Found x3:(Xx0 Xx00)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x3:(Xx0 Xx00))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x20:(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))
% Instantiate: x3:=(K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))):(fofType->Prop)
% Found x20 as proof of (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))
% Found x00:(x1 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x3:(Xx00 Xx000)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x3:(Xx00 Xx000))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x3:(Xx00 Xx000)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x3:(Xx00 Xx000))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x60:=(x6 Xx0):((x4 Xx0)->((K x4) Xx0))
% Found (x6 Xx0) as proof of ((x4 Xx0)->((K Xx00) Xx0))
% Found (x6 Xx0) as proof of ((x4 Xx0)->((K Xx00) Xx0))
% Found (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0)) as proof of ((x4 Xx0)->((K Xx00) Xx0))
% Found (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0)) as proof of ((forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000)))->((x4 Xx0)->((K Xx00) Xx0)))
% Found (and_rect10 (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0))) as proof of ((K Xx00) Xx0)
% Found ((and_rect1 ((K Xx00) Xx0)) (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x6:((forall (Xx00:fofType), ((x4 Xx00)->((K x4) Xx00)))->((x4 Xx0)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x4 Xx00)->((K x4) Xx00)))) (x4 Xx0)) P) x6) x5)) ((K Xx00) Xx0)) (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x6:((forall (Xx00:fofType), ((x4 Xx00)->((K x4) Xx00)))->((x4 Xx0)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x4 Xx00)->((K x4) Xx00)))) (x4 Xx0)) P) x6) x5)) ((K Xx00) Xx0)) (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0))) as proof of ((K Xx00) Xx0)
% Found x3:(Xx0 Xx00)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x3:(Xx0 Xx00))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x30:(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))
% Instantiate: x1:=(K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))):(fofType->Prop)
% Found x30 as proof of (forall (Xx0:fofType), ((x1 Xx0)->((K x1) Xx0)))
% Found x3:(Xx00 Xx000)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x3:(Xx00 Xx000))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x3:(Xx00 Xx000)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x3:(Xx00 Xx000))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x10:(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))
% Instantiate: x3:=(K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))):(fofType->Prop)
% Found x10 as proof of (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))
% Found x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)
% Found x2 as proof of (x3 Xx)
% Found ((conj10 x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (((conj1 (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (ex_intro000 ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found ((ex_intro00 (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found (((ex_intro0 (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found x4:(Xx00 Xx000)
% Found x4 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x4:(Xx00 Xx000))=> x4) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x20:(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))
% Instantiate: x3:=(K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))):(fofType->Prop)
% Found x20 as proof of (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))
% Found x0:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)
% Found x0 as proof of (x3 Xx)
% Found ((conj10 x20) x0) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (((conj1 (x3 Xx)) x20) x0) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x20) x0) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x20) x0) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (ex_intro000 ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x20) x0)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found ((ex_intro00 (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x20) x0)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found (((ex_intro0 (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x20) x0)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x20) x0)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x20) x0)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found x10:(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))
% Instantiate: x3:=(K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))):(fofType->Prop)
% Found x10 as proof of (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))
% Found x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)
% Found x2 as proof of (x3 Xx)
% Found ((conj10 x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (((conj1 (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (ex_intro000 ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found ((ex_intro00 (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found (((ex_intro0 (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found (fun (x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx))=> ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2))) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found (fun (x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx))=> ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2))) as proof of (((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))))
% Found x5:(Xx00 Xx000)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x5:(Xx00 Xx000))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x10:(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))
% Instantiate: x3:=(K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))):(fofType->Prop)
% Found x10 as proof of (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))
% Found x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)
% Found x2 as proof of (x3 Xx)
% Found ((conj10 x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (((conj1 (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (ex_intro000 ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found ((ex_intro00 (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found (((ex_intro0 (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found x4:(Xx00 Xx000)
% Found x4 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x4:(Xx00 Xx000))=> x4) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x3:(Xx0 Xx00)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x3:(Xx0 Xx00))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x4:(Xx00 Xx000)
% Found x4 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x4:(Xx00 Xx000))=> x4) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x3:(Xx00 Xx000)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x3:(Xx00 Xx000))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x4:(Xx00 Xx000)
% Found x4 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x4:(Xx00 Xx000))=> x4) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x10:(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))
% Instantiate: x3:=(K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))):(fofType->Prop)
% Found x10 as proof of (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))
% Found x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)
% Found x2 as proof of (x3 Xx)
% Found ((conj10 x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (((conj1 (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (ex_intro000 ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found ((ex_intro00 (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found (((ex_intro0 (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found (fun (x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx))=> ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2))) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found (fun (x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx))=> ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2))) as proof of (((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))))
% Found x5:(Xx00 Xx000)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x5:(Xx00 Xx000))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x3:(Xx00 Xx000)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x3:(Xx00 Xx000))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x6:(Xx00 Xx000)
% Found x6 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x6:(Xx00 Xx000))=> x6) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x6:(Xx00 Xx000))=> x6) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x6:(Xx00 Xx000))=> x6) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x6:(Xx00 Xx000)
% Found x6 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x6:(Xx00 Xx000))=> x6) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x6:(Xx00 Xx000))=> x6) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x6:(Xx00 Xx000))=> x6) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x10:(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))
% Instantiate: x3:=(K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))):(fofType->Prop)
% Found x10 as proof of (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))
% Found x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)
% Found x2 as proof of (x3 Xx)
% Found ((conj10 x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (((conj1 (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (ex_intro000 ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found ((ex_intro00 (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found (((ex_intro0 (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found (fun (x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx))=> ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2))) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found (fun (x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx))=> ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2))) as proof of (((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))))
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x3:(Xx0 Xx00)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x3:(Xx0 Xx00))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x3:(Xx00 Xx000)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x3:(Xx00 Xx000))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (fun (x4:((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)))=> (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)))) as proof of ((K Xx0) Xx)
% Found (fun (x4:((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)))=> (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)))) as proof of (((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx))->((K Xx0) Xx))
% Found x20:(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))
% Instantiate: x3:=(K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))):(fofType->Prop)
% Found x20 as proof of (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))
% Found x0:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)
% Found x0 as proof of (x3 Xx)
% Found ((conj10 x20) x0) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (((conj1 (x3 Xx)) x20) x0) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x20) x0) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x20) x0) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found x6:(Xx00 Xx000)
% Found x6 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x6:(Xx00 Xx000))=> x6) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x6:(Xx00 Xx000))=> x6) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x6:(Xx00 Xx000))=> x6) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x5:(Xx00 Xx000)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x5:(Xx00 Xx000))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x10:(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))
% Instantiate: x3:=(K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))):(fofType->Prop)
% Found x10 as proof of (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))
% Found x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)
% Found x2 as proof of (x3 Xx)
% Found ((conj10 x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (((conj1 (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (ex_intro000 ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found ((ex_intro00 (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found (((ex_intro0 (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found (fun (x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx))=> ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2))) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found (fun (x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx))=> ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2))) as proof of (((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))))
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x7:(Xx0 Xx00)
% Found x7 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x7:(Xx0 Xx00))=> x7) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x500:=(x50 x6):((K x3) Xx)
% Found (x50 x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found x5:(Xx00 Xx000)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x5:(Xx00 Xx000))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x50:=(x5 Xx0):((x3 Xx0)->((K x3) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)) as proof of ((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->((K Xx00) Xx0)))
% Found (and_rect10 (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found ((and_rect1 ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x5:((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))) (x3 Xx0)) P) x5) x4)) ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found (fun (x4:((and (forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))) (x3 Xx0)))=> (((fun (P:Type) (x5:((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))) (x3 Xx0)) P) x5) x4)) ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)))) as proof of ((K Xx00) Xx0)
% Found x50:=(x5 Xx0):((x3 Xx0)->((K x3) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)) as proof of ((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->((K Xx00) Xx0)))
% Found (and_rect10 (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found ((and_rect1 ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx0)) P) x5) x4)) ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx0)) P) x5) x4)) ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (fun (x4:((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)))=> (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)))) as proof of ((K Xx0) Xx)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (fun (x4:((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)))=> (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)))) as proof of ((K Xx0) Xx)
% Found x7:(Xx00 Xx000)
% Found x7 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x7:(Xx00 Xx000))=> x7) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x7:(Xx00 Xx000))=> x7) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x7:(Xx00 Xx000))=> x7) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x500:=(x50 x6):((K x3) Xx0)
% Found (x50 x6) as proof of ((K Xx00) Xx0)
% Found ((x5 Xx0) x6) as proof of ((K Xx00) Xx0)
% Found ((x5 Xx0) x6) as proof of ((K Xx00) Xx0)
% Found ((x5 Xx0) x6) as proof of ((K Xx00) Xx0)
% Found x5:(Xx00 Xx000)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x5:(Xx00 Xx000))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x30:(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))
% Instantiate: x1:=(K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))):(fofType->Prop)
% Found x30 as proof of (forall (Xx0:fofType), ((x1 Xx0)->((K x1) Xx0)))
% Found x0:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)
% Found x0 as proof of (x1 Xx)
% Found ((conj10 x30) x0) as proof of ((and (forall (Xx0:fofType), ((x1 Xx0)->((K x1) Xx0)))) (x1 Xx))
% Found (((conj1 (x1 Xx)) x30) x0) as proof of ((and (forall (Xx0:fofType), ((x1 Xx0)->((K x1) Xx0)))) (x1 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x1 Xx0)->((K x1) Xx0)))) (x1 Xx)) x30) x0) as proof of ((and (forall (Xx0:fofType), ((x1 Xx0)->((K x1) Xx0)))) (x1 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x1 Xx0)->((K x1) Xx0)))) (x1 Xx)) x30) x0) as proof of ((and (forall (Xx0:fofType), ((x1 Xx0)->((K x1) Xx0)))) (x1 Xx))
% Found x50:=(x5 Xx0):((x3 Xx0)->((K x3) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)) as proof of ((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->((K Xx00) Xx0)))
% Found (and_rect10 (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found ((and_rect1 ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x5:((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))) (x3 Xx0)) P) x5) x4)) ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found (fun (x4:((and (forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))) (x3 Xx0)))=> (((fun (P:Type) (x5:((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))) (x3 Xx0)) P) x5) x4)) ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)))) as proof of ((K Xx00) Xx0)
% Found x50:=(x5 Xx0):((x3 Xx0)->((K x3) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)) as proof of ((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->((K Xx00) Xx0)))
% Found (and_rect10 (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found ((and_rect1 ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx0)) P) x5) x4)) ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx0)) P) x5) x4)) ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (fun (x4:((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)))=> (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)))) as proof of ((K Xx0) Xx)
% Found x7:(Xx0 Xx00)
% Found x7 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x7:(Xx0 Xx00))=> x7) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x300:=(x30 x4):((K x1) Xx)
% Found (x30 x4) as proof of ((K Xx0) Xx)
% Found ((x3 Xx) x4) as proof of ((K Xx0) Xx)
% Found ((x3 Xx) x4) as proof of ((K Xx0) Xx)
% Found ((x3 Xx) x4) as proof of ((K Xx0) Xx)
% Found x7:(Xx00 Xx000)
% Found x7 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x7:(Xx00 Xx000))=> x7) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x7:(Xx00 Xx000))=> x7) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x7:(Xx00 Xx000))=> x7) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x500:=(x50 x6):((K x3) Xx0)
% Found (x50 x6) as proof of ((K Xx00) Xx0)
% Found ((x5 Xx0) x6) as proof of ((K Xx00) Xx0)
% Found ((x5 Xx0) x6) as proof of ((K Xx00) Xx0)
% Found ((x5 Xx0) x6) as proof of ((K Xx00) Xx0)
% Found x7:(Xx0 Xx00)
% Found x7 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x7:(Xx0 Xx00))=> x7) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x500:=(x50 x6):((K x1) Xx)
% Found (x50 x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found x60:=(x6 Xx0):((x4 Xx0)->((K x4) Xx0))
% Found (x6 Xx0) as proof of ((x4 Xx0)->((K Xx00) Xx0))
% Found (x6 Xx0) as proof of ((x4 Xx0)->((K Xx00) Xx0))
% Found (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0)) as proof of ((x4 Xx0)->((K Xx00) Xx0))
% Found (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0)) as proof of ((forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000)))->((x4 Xx0)->((K Xx00) Xx0)))
% Found (and_rect10 (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0))) as proof of ((K Xx00) Xx0)
% Found ((and_rect1 ((K Xx00) Xx0)) (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x6:((forall (Xx00:fofType), ((x4 Xx00)->((K x4) Xx00)))->((x4 Xx0)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x4 Xx00)->((K x4) Xx00)))) (x4 Xx0)) P) x6) x5)) ((K Xx00) Xx0)) (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x6:((forall (Xx00:fofType), ((x4 Xx00)->((K x4) Xx00)))->((x4 Xx0)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x4 Xx00)->((K x4) Xx00)))) (x4 Xx0)) P) x6) x5)) ((K Xx00) Xx0)) (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0))) as proof of ((K Xx00) Xx0)
% Found x6:(Xx00 Xx000)
% Found x6 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x6:(Xx00 Xx000))=> x6) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x6:(Xx00 Xx000))=> x6) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x6:(Xx00 Xx000))=> x6) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x60:=(x6 Xx0):((x4 Xx0)->((K x4) Xx0))
% Found (x6 Xx0) as proof of ((x4 Xx0)->((K Xx00) Xx0))
% Found (x6 Xx0) as proof of ((x4 Xx0)->((K Xx00) Xx0))
% Found (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0)) as proof of ((x4 Xx0)->((K Xx00) Xx0))
% Found (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0)) as proof of ((forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000)))->((x4 Xx0)->((K Xx00) Xx0)))
% Found (and_rect10 (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0))) as proof of ((K Xx00) Xx0)
% Found ((and_rect1 ((K Xx00) Xx0)) (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x6:((forall (Xx00:fofType), ((x4 Xx00)->((K x4) Xx00)))->((x4 Xx0)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x4 Xx00)->((K x4) Xx00)))) (x4 Xx0)) P) x6) x5)) ((K Xx00) Xx0)) (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x6:((forall (Xx00:fofType), ((x4 Xx00)->((K x4) Xx00)))->((x4 Xx0)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x4 Xx00)->((K x4) Xx00)))) (x4 Xx0)) P) x6) x5)) ((K Xx00) Xx0)) (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0))) as proof of ((K Xx00) Xx0)
% Found x7:(Xx00 Xx000)
% Found x7 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x7:(Xx00 Xx000))=> x7) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x7:(Xx00 Xx000))=> x7) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x7:(Xx00 Xx000))=> x7) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x500:=(x50 x6):((K x3) Xx0)
% Found (x50 x6) as proof of ((K Xx00) Xx0)
% Found ((x5 Xx0) x6) as proof of ((K Xx00) Xx0)
% Found ((x5 Xx0) x6) as proof of ((K Xx00) Xx0)
% Found ((x5 Xx0) x6) as proof of ((K Xx00) Xx0)
% Found x10:(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))
% Instantiate: x3:=(K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))):(fofType->Prop)
% Found x10 as proof of (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))
% Found x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)
% Found x2 as proof of (x3 Xx)
% Found ((conj10 x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (((conj1 (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (ex_intro000 ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found ((ex_intro00 (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found (((ex_intro0 (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found (fun (x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx))=> ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2))) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found (fun (x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx))=> ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x10) x2))) as proof of (((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))))
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x7:(Xx0 Xx00)
% Found x7 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x7:(Xx0 Xx00))=> x7) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x500:=(x50 x6):((K x3) Xx)
% Found (x50 x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx)) as proof of ((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx)->((K Xx00) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found ((and_rect1 ((K Xx00) Xx)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx00) Xx)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx00) Xx)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x60:=(x6 Xx0):((x4 Xx0)->((K x4) Xx0))
% Found (x6 Xx0) as proof of ((x4 Xx0)->((K Xx00) Xx0))
% Found (x6 Xx0) as proof of ((x4 Xx0)->((K Xx00) Xx0))
% Found (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0)) as proof of ((x4 Xx0)->((K Xx00) Xx0))
% Found (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0)) as proof of ((forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000)))->((x4 Xx0)->((K Xx00) Xx0)))
% Found (and_rect10 (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0))) as proof of ((K Xx00) Xx0)
% Found ((and_rect1 ((K Xx00) Xx0)) (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x6:((forall (Xx00:fofType), ((x4 Xx00)->((K x4) Xx00)))->((x4 Xx0)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x4 Xx00)->((K x4) Xx00)))) (x4 Xx0)) P) x6) x5)) ((K Xx00) Xx0)) (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x6:((forall (Xx00:fofType), ((x4 Xx00)->((K x4) Xx00)))->((x4 Xx0)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x4 Xx00)->((K x4) Xx00)))) (x4 Xx0)) P) x6) x5)) ((K Xx00) Xx0)) (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0))) as proof of ((K Xx00) Xx0)
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x3:(Xx00 Xx000)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x3:(Xx00 Xx000))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (fun (x4:((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)))=> (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)))) as proof of ((K Xx0) Xx)
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x100:=(x10 Xx0):(((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0))
% Found (x10 Xx0) as proof of ((x3 Xx0)->((K x3) Xx0))
% Found (x10 Xx0) as proof of ((x3 Xx0)->((K x3) Xx0))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x3:(Xx0 Xx00)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x3:(Xx0 Xx00))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x20:(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))
% Instantiate: x3:=(K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))):(fofType->Prop)
% Found x20 as proof of (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))
% Found x0:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)
% Found x0 as proof of (x3 Xx)
% Found ((conj10 x20) x0) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (((conj1 (x3 Xx)) x20) x0) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x20) x0) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x20) x0) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (ex_intro000 ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x20) x0)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found ((ex_intro00 (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x20) x0)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found (((ex_intro0 (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x20) x0)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x20) x0)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found ((((ex_intro (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx)))) (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) ((((conj (forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))) ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx)) x20) x0)) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx0:fofType), ((S Xx0)->((K S) Xx0)))) (S Xx))))
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx)) as proof of ((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx)->((K Xx00) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found ((and_rect1 ((K Xx00) Xx)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx00) Xx)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx00) Xx)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found x8:(Xx00 Xx000)
% Found x8 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x8:(Xx00 Xx000))=> x8) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x8:(Xx00 Xx000))=> x8) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x8:(Xx00 Xx000))=> x8) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x600:=(x60 x7):((K x4) Xx0)
% Found (x60 x7) as proof of ((K Xx00) Xx0)
% Found ((x6 Xx0) x7) as proof of ((K Xx00) Xx0)
% Found ((x6 Xx0) x7) as proof of ((K Xx00) Xx0)
% Found ((x6 Xx0) x7) as proof of ((K Xx00) Xx0)
% Found x3:(Xx00 Xx000)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x3:(Xx00 Xx000))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x4:(Xx00 Xx000)
% Found x4 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x4:(Xx00 Xx000))=> x4) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x4:(Xx00 Xx000)
% Found x4 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x4:(Xx00 Xx000))=> x4) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (fun (x4:((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)))=> (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)))) as proof of ((K Xx0) Xx)
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x4:(Xx00 Xx000)
% Found x4 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x4:(Xx00 Xx000))=> x4) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x3:(Xx0 Xx00)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x3:(Xx0 Xx00))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x3:(Xx00 Xx000)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x3:(Xx00 Xx000))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x100:=(x10 Xx0):(((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0))
% Found (x10 Xx0) as proof of ((x3 Xx0)->((K x3) Xx0))
% Found (x10 Xx0) as proof of ((x3 Xx0)->((K x3) Xx0))
% Found x3:(Xx00 Xx000)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x3:(Xx00 Xx000))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x5:(Xx00 Xx000)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x5:(Xx00 Xx000))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x00:(x1 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x3:(Xx00 Xx000)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x3:(Xx00 Xx000))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x3:(Xx00 Xx000)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x3:(Xx00 Xx000))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x4:(Xx00 Xx000)
% Found x4 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x4:(Xx00 Xx000))=> x4) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x100:=(x10 Xx0):(((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0))
% Found (x10 Xx0) as proof of ((x3 Xx0)->((K x3) Xx0))
% Found (x10 Xx0) as proof of ((x3 Xx0)->((K x3) Xx0))
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x00:(x1 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x3:(Xx00 Xx000)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x3:(Xx00 Xx000))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x5:(Xx00 Xx000)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x5:(Xx00 Xx000))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (fun (x4:((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)))=> (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)))) as proof of ((K Xx0) Xx)
% Found (fun (x4:((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)))=> (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)))) as proof of (((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx))->((K Xx0) Xx))
% Found x6:(Xx00 Xx000)
% Found x6 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x6:(Xx00 Xx000))=> x6) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x6:(Xx00 Xx000))=> x6) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x6:(Xx00 Xx000))=> x6) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x5:(Xx00 Xx000)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x5:(Xx00 Xx000))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x00:(x1 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x4:(Xx00 Xx000)
% Found x4 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x4:(Xx00 Xx000))=> x4) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x3:(Xx00 Xx000)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x3:(Xx00 Xx000))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x00:(x1 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x6:(Xx00 Xx000)
% Found x6 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x6:(Xx00 Xx000))=> x6) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x6:(Xx00 Xx000))=> x6) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x6:(Xx00 Xx000))=> x6) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (fun (x4:((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)))=> (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)))) as proof of ((K Xx0) Xx)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x3:(Xx00 Xx000)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x3:(Xx00 Xx000))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x3:(Xx00 Xx000)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x3:(Xx00 Xx000))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x00:(x1 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x00:(x1 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x5:(Xx00 Xx000)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x5:(Xx00 Xx000))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (fun (x4:((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)))=> (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)))) as proof of ((K Xx0) Xx)
% Found (fun (x4:((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)))=> (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)))) as proof of (((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx))->((K Xx0) Xx))
% Found x5:(Xx00 Xx000)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x5:(Xx00 Xx000))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x5:(Xx00 Xx000)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x5:(Xx00 Xx000))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (fun (x4:((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)))=> (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)))) as proof of ((K Xx0) Xx)
% Found (fun (x4:((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)))=> (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)))) as proof of (((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx))->((K Xx0) Xx))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (fun (x4:((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)))=> (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)))) as proof of ((K Xx0) Xx)
% Found x00:(x1 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x5:(Xx00 Xx000)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x5:(Xx00 Xx000))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x00:(x1 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x6:(Xx00 Xx000)
% Found x6 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x6:(Xx00 Xx000))=> x6) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x6:(Xx00 Xx000))=> x6) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x6:(Xx00 Xx000))=> x6) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (fun (x4:((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)))=> (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)))) as proof of ((K Xx0) Xx)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (fun (x4:((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)))=> (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)))) as proof of ((K Xx0) Xx)
% Found x5:(Xx00 Xx000)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x5:(Xx00 Xx000))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x7:(Xx0 Xx00)
% Found x7 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x7:(Xx0 Xx00))=> x7) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x500:=(x50 x6):((K x3) Xx)
% Found (x50 x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found x6:(Xx00 Xx000)
% Found x6 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x6:(Xx00 Xx000))=> x6) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x6:(Xx00 Xx000))=> x6) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x6:(Xx00 Xx000))=> x6) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x5:(Xx00 Xx000)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x5:(Xx00 Xx000))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (fun (x4:((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)))=> (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)))) as proof of ((K Xx0) Xx)
% Found x00:(x1 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x4:(Xx00 Xx000)
% Found x4 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x4:(Xx00 Xx000))=> x4) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x3:(Xx000 Xx0000)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0000))))
% Found (fun (x3:(Xx000 Xx0000))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0000))))
% Found (fun (Xx0000:fofType) (x3:(Xx000 Xx0000))=> x3) as proof of ((Xx000 Xx0000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0000)))))
% Found (fun (Xx0000:fofType) (x3:(Xx000 Xx0000))=> x3) as proof of (forall (Xx0000:fofType), ((Xx000 Xx0000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0000))))))
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx)) as proof of ((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx)->((K Xx00) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found ((and_rect1 ((K Xx00) Xx)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx00) Xx)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx00) Xx)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found x7:(Xx0 Xx00)
% Found x7 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x7:(Xx0 Xx00))=> x7) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x500:=(x50 x6):((K x3) Xx)
% Found (x50 x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found x7:(Xx00 Xx000)
% Found x7 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x7:(Xx00 Xx000))=> x7) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x7:(Xx00 Xx000))=> x7) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x7:(Xx00 Xx000))=> x7) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x500:=(x50 x6):((K x3) Xx0)
% Found (x50 x6) as proof of ((K Xx00) Xx0)
% Found ((x5 Xx0) x6) as proof of ((K Xx00) Xx0)
% Found ((x5 Xx0) x6) as proof of ((K Xx00) Xx0)
% Found ((x5 Xx0) x6) as proof of ((K Xx00) Xx0)
% Found x5:(Xx00 Xx000)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x5:(Xx00 Xx000))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x7:(Xx0 Xx00)
% Found x7 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x7:(Xx0 Xx00))=> x7) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x500:=(x50 x6):((K x3) Xx)
% Found (x50 x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found x10:(forall (Xx0:fofType), (((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0)))
% Instantiate: x3:=(K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))):(fofType->Prop)
% Found x10 as proof of (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))
% Found x2:((K (fun (Xx0:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0)))))) Xx)
% Found x2 as proof of (x3 Xx)
% Found ((conj10 x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found (((conj1 (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found ((((conj (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) x10) x2) as proof of ((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx))
% Found x5:(Xx00 Xx000)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x5:(Xx00 Xx000))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x6:(Xx00 Xx000)
% Found x6 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x6:(Xx00 Xx000))=> x6) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x6:(Xx00 Xx000))=> x6) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x6:(Xx00 Xx000))=> x6) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x3:(Xx0 Xx00)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x3:(Xx0 Xx00))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x60:=(x6 Xx0):((x4 Xx0)->((K x4) Xx0))
% Found (x6 Xx0) as proof of ((x4 Xx0)->((K Xx00) Xx0))
% Found (x6 Xx0) as proof of ((x4 Xx0)->((K Xx00) Xx0))
% Found (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0)) as proof of ((x4 Xx0)->((K Xx00) Xx0))
% Found (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0)) as proof of ((forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000)))->((x4 Xx0)->((K Xx00) Xx0)))
% Found (and_rect10 (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0))) as proof of ((K Xx00) Xx0)
% Found ((and_rect1 ((K Xx00) Xx0)) (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x6:((forall (Xx00:fofType), ((x4 Xx00)->((K x4) Xx00)))->((x4 Xx0)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x4 Xx00)->((K x4) Xx00)))) (x4 Xx0)) P) x6) x5)) ((K Xx00) Xx0)) (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x6:((forall (Xx00:fofType), ((x4 Xx00)->((K x4) Xx00)))->((x4 Xx0)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x4 Xx00)->((K x4) Xx00)))) (x4 Xx0)) P) x6) x5)) ((K Xx00) Xx0)) (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0))) as proof of ((K Xx00) Xx0)
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x5:(Xx000 Xx0000)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0000))))
% Found (fun (x5:(Xx000 Xx0000))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0000))))
% Found (fun (Xx0000:fofType) (x5:(Xx000 Xx0000))=> x5) as proof of ((Xx000 Xx0000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0000)))))
% Found (fun (Xx0000:fofType) (x5:(Xx000 Xx0000))=> x5) as proof of (forall (Xx0000:fofType), ((Xx000 Xx0000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0000))))))
% Found x6:(Xx00 Xx000)
% Found x6 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x6:(Xx00 Xx000))=> x6) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x6:(Xx00 Xx000))=> x6) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x6:(Xx00 Xx000))=> x6) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x3:(Xx000 Xx0000)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0000))))
% Found (fun (x3:(Xx000 Xx0000))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0000))))
% Found (fun (Xx0000:fofType) (x3:(Xx000 Xx0000))=> x3) as proof of ((Xx000 Xx0000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0000)))))
% Found (fun (Xx0000:fofType) (x3:(Xx000 Xx0000))=> x3) as proof of (forall (Xx0000:fofType), ((Xx000 Xx0000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0000))))))
% Found x100:=(x10 Xx0):(((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0))
% Found (x10 Xx0) as proof of ((x3 Xx0)->((K x3) Xx0))
% Found (x10 Xx0) as proof of ((x3 Xx0)->((K x3) Xx0))
% Found x6:(Xx00 Xx000)
% Found x6 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x6:(Xx00 Xx000))=> x6) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x6:(Xx00 Xx000))=> x6) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x6:(Xx00 Xx000))=> x6) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x7:(Xx0 Xx00)
% Found x7 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x7:(Xx0 Xx00))=> x7) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x500:=(x50 x6):((K x3) Xx)
% Found (x50 x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found x6:(Xx00 Xx000)
% Found x6 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x6:(Xx00 Xx000))=> x6) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x6:(Xx00 Xx000))=> x6) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x6:(Xx00 Xx000))=> x6) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx)) as proof of ((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx)->((K Xx00) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found ((and_rect1 ((K Xx00) Xx)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx00) Xx)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx00) Xx)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found x200:=(x20 Xx0):(((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0))
% Found (x20 Xx0) as proof of ((x3 Xx0)->((K x3) Xx0))
% Found (x20 Xx0) as proof of ((x3 Xx0)->((K x3) Xx0))
% Found x7:(Xx00 Xx000)
% Found x7 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x7:(Xx00 Xx000))=> x7) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x7:(Xx00 Xx000))=> x7) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x7:(Xx00 Xx000))=> x7) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x500:=(x50 x6):((K x3) Xx0)
% Found (x50 x6) as proof of ((K Xx00) Xx0)
% Found ((x5 Xx0) x6) as proof of ((K Xx00) Xx0)
% Found ((x5 Xx0) x6) as proof of ((K Xx00) Xx0)
% Found ((x5 Xx0) x6) as proof of ((K Xx00) Xx0)
% Found x7:(Xx0 Xx00)
% Found x7 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x7:(Xx0 Xx00))=> x7) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x500:=(x50 x6):((K x3) Xx)
% Found (x50 x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x3:(Xx0 Xx00)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x3:(Xx0 Xx00))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x50:=(x5 Xx0):((x3 Xx0)->((K x3) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)) as proof of ((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->((K Xx00) Xx0)))
% Found (and_rect10 (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found ((and_rect1 ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x5:((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))) (x3 Xx0)) P) x5) x4)) ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found (fun (x4:((and (forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))) (x3 Xx0)))=> (((fun (P:Type) (x5:((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))) (x3 Xx0)) P) x5) x4)) ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)))) as proof of ((K Xx00) Xx0)
% Found x50:=(x5 Xx0):((x3 Xx0)->((K x3) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)) as proof of ((x3 Xx0)->((K Xx00) Xx0))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0)) as proof of ((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->((K Xx00) Xx0)))
% Found (and_rect10 (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found ((and_rect1 ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx0)) P) x5) x4)) ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx0)) P) x5) x4)) ((K Xx00) Xx0)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx0))) as proof of ((K Xx00) Xx0)
% Found x8:(Xx00 Xx000)
% Found x8 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x8:(Xx00 Xx000))=> x8) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x8:(Xx00 Xx000))=> x8) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x8:(Xx00 Xx000))=> x8) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x600:=(x60 x7):((K x4) Xx0)
% Found (x60 x7) as proof of ((K Xx00) Xx0)
% Found ((x6 Xx0) x7) as proof of ((K Xx00) Xx0)
% Found ((x6 Xx0) x7) as proof of ((K Xx00) Xx0)
% Found ((x6 Xx0) x7) as proof of ((K Xx00) Xx0)
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx)) as proof of ((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx)->((K Xx00) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found ((and_rect1 ((K Xx00) Xx)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx00) Xx)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx00) Xx)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x3:(Xx0 Xx00)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x3:(Xx0 Xx00))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx)) as proof of ((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx)->((K Xx00) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found ((and_rect1 ((K Xx00) Xx)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx00) Xx)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx00) Xx)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x5:(Xx000 Xx0000)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0000))))
% Found (fun (x5:(Xx000 Xx0000))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0000))))
% Found (fun (Xx0000:fofType) (x5:(Xx000 Xx0000))=> x5) as proof of ((Xx000 Xx0000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0000)))))
% Found (fun (Xx0000:fofType) (x5:(Xx000 Xx0000))=> x5) as proof of (forall (Xx0000:fofType), ((Xx000 Xx0000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx0000))))))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (fun (x4:((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)))=> (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)))) as proof of ((K Xx0) Xx)
% Found x100:=(x10 Xx0):(((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0))
% Found (x10 Xx0) as proof of ((x3 Xx0)->((K x3) Xx0))
% Found (x10 Xx0) as proof of ((x3 Xx0)->((K x3) Xx0))
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x4:(Xx00 Xx000)
% Found x4 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x4:(Xx00 Xx000))=> x4) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x8:(Xx00 Xx000)
% Found x8 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x8:(Xx00 Xx000))=> x8) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x8:(Xx00 Xx000))=> x8) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x8:(Xx00 Xx000))=> x8) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x600:=(x60 x7):((K x4) Xx0)
% Found (x60 x7) as proof of ((K Xx00) Xx0)
% Found ((x6 Xx0) x7) as proof of ((K Xx00) Xx0)
% Found ((x6 Xx0) x7) as proof of ((K Xx00) Xx0)
% Found ((x6 Xx0) x7) as proof of ((K Xx00) Xx0)
% Found x4:(Xx00 Xx000)
% Found x4 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x4:(Xx00 Xx000))=> x4) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x100:=(x10 Xx0):(((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0))
% Found (x10 Xx0) as proof of ((x3 Xx0)->((K x3) Xx0))
% Found (x10 Xx0) as proof of ((x3 Xx0)->((K x3) Xx0))
% Found x5:(Xx00 Xx000)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x5:(Xx00 Xx000))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x200:=(x20 Xx0):(((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0))
% Found (x20 Xx0) as proof of ((x3 Xx0)->((K x3) Xx0))
% Found (x20 Xx0) as proof of ((x3 Xx0)->((K x3) Xx0))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x3:(Xx0 Xx00)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x3:(Xx0 Xx00))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x3:(Xx0 Xx00)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x3:(Xx0 Xx00))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x300:=(x30 Xx0):(((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0))
% Found (x30 Xx0) as proof of ((x1 Xx0)->((K x1) Xx0))
% Found (x30 Xx0) as proof of ((x1 Xx0)->((K x1) Xx0))
% Found x3:(Xx00 Xx000)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x3:(Xx00 Xx000))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x7:(Xx00 Xx000)
% Found x7 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x7:(Xx00 Xx000))=> x7) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x7:(Xx00 Xx000))=> x7) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x7:(Xx00 Xx000))=> x7) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx)) as proof of ((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx)->((K Xx00) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found ((and_rect1 ((K Xx00) Xx)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx00) Xx)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx00) Xx)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x100:=(x10 Xx0):(((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0))
% Found (x10 Xx0) as proof of ((x3 Xx0)->((K x3) Xx0))
% Found (x10 Xx0) as proof of ((x3 Xx0)->((K x3) Xx0))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (fun (x4:((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)))=> (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)))) as proof of ((K Xx0) Xx)
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx)) as proof of ((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx)->((K Xx00) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found ((and_rect1 ((K Xx00) Xx)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx00) Xx)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx00) Xx)) (fun (x5:(forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x200:=(x20 Xx0):(((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0))
% Found (x20 Xx0) as proof of ((x3 Xx0)->((K x3) Xx0))
% Found (x20 Xx0) as proof of ((x3 Xx0)->((K x3) Xx0))
% Found x8:(Xx00 Xx000)
% Found x8 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x8:(Xx00 Xx000))=> x8) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x8:(Xx00 Xx000))=> x8) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x8:(Xx00 Xx000))=> x8) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x600:=(x60 x7):((K x4) Xx0)
% Found (x60 x7) as proof of ((K Xx00) Xx0)
% Found ((x6 Xx0) x7) as proof of ((K Xx00) Xx0)
% Found ((x6 Xx0) x7) as proof of ((K Xx00) Xx0)
% Found ((x6 Xx0) x7) as proof of ((K Xx00) Xx0)
% Found x3:(Xx0 Xx00)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x3:(Xx0 Xx00))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x4:(Xx00 Xx000)
% Found x4 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x4:(Xx00 Xx000))=> x4) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x100:=(x10 Xx0):(((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0))
% Found (x10 Xx0) as proof of ((x3 Xx0)->((K x3) Xx0))
% Found (x10 Xx0) as proof of ((x3 Xx0)->((K x3) Xx0))
% Found x50:=(x5 Xx0):((x3 Xx0)->((K x3) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx000) Xx0))
% Found (x5 Xx0) as proof of ((x3 Xx0)->((K Xx000) Xx0))
% Found (fun (x5:(forall (Xx0000:fofType), ((x3 Xx0000)->((K x3) Xx0000))))=> (x5 Xx0)) as proof of ((x3 Xx0)->((K Xx000) Xx0))
% Found (fun (x5:(forall (Xx0000:fofType), ((x3 Xx0000)->((K x3) Xx0000))))=> (x5 Xx0)) as proof of ((forall (Xx0000:fofType), ((x3 Xx0000)->((K x3) Xx0000)))->((x3 Xx0)->((K Xx000) Xx0)))
% Found (and_rect10 (fun (x5:(forall (Xx0000:fofType), ((x3 Xx0000)->((K x3) Xx0000))))=> (x5 Xx0))) as proof of ((K Xx000) Xx0)
% Found ((and_rect1 ((K Xx000) Xx0)) (fun (x5:(forall (Xx0000:fofType), ((x3 Xx0000)->((K x3) Xx0000))))=> (x5 Xx0))) as proof of ((K Xx000) Xx0)
% Found (((fun (P:Type) (x5:((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))) (x3 Xx0)) P) x5) x4)) ((K Xx000) Xx0)) (fun (x5:(forall (Xx0000:fofType), ((x3 Xx0000)->((K x3) Xx0000))))=> (x5 Xx0))) as proof of ((K Xx000) Xx0)
% Found (((fun (P:Type) (x5:((forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))->((x3 Xx0)->P)))=> (((((and_rect (forall (Xx000:fofType), ((x3 Xx000)->((K x3) Xx000)))) (x3 Xx0)) P) x5) x4)) ((K Xx000) Xx0)) (fun (x5:(forall (Xx0000:fofType), ((x3 Xx0000)->((K x3) Xx0000))))=> (x5 Xx0))) as proof of ((K Xx000) Xx0)
% Found x5:(Xx00 Xx000)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x5:(Xx00 Xx000))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x5:(Xx00 Xx000))=> x5) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x00:(x3 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x60:=(x6 Xx0):((x4 Xx0)->((K x4) Xx0))
% Found (x6 Xx0) as proof of ((x4 Xx0)->((K Xx00) Xx0))
% Found (x6 Xx0) as proof of ((x4 Xx0)->((K Xx00) Xx0))
% Found (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0)) as proof of ((x4 Xx0)->((K Xx00) Xx0))
% Found (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0)) as proof of ((forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000)))->((x4 Xx0)->((K Xx00) Xx0)))
% Found (and_rect10 (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0))) as proof of ((K Xx00) Xx0)
% Found ((and_rect1 ((K Xx00) Xx0)) (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x6:((forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000)))->((x4 Xx0)->P)))=> (((((and_rect (forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000)))) (x4 Xx0)) P) x6) x5)) ((K Xx00) Xx0)) (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0))) as proof of ((K Xx00) Xx0)
% Found (fun (x5:((and (forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000)))) (x4 Xx0)))=> (((fun (P:Type) (x6:((forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000)))->((x4 Xx0)->P)))=> (((((and_rect (forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000)))) (x4 Xx0)) P) x6) x5)) ((K Xx00) Xx0)) (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0)))) as proof of ((K Xx00) Xx0)
% Found x4:(Xx00 Xx000)
% Found x4 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x4:(Xx00 Xx000))=> x4) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x4:(Xx00 Xx000))=> x4) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x00:(x1 Xx0)
% Found x00 as proof of ((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)
% Found x60:=(x6 Xx0):((x4 Xx0)->((K x4) Xx0))
% Found (x6 Xx0) as proof of ((x4 Xx0)->((K Xx00) Xx0))
% Found (x6 Xx0) as proof of ((x4 Xx0)->((K Xx00) Xx0))
% Found (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0)) as proof of ((x4 Xx0)->((K Xx00) Xx0))
% Found (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0)) as proof of ((forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000)))->((x4 Xx0)->((K Xx00) Xx0)))
% Found (and_rect10 (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0))) as proof of ((K Xx00) Xx0)
% Found ((and_rect1 ((K Xx00) Xx0)) (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x6:((forall (Xx00:fofType), ((x4 Xx00)->((K x4) Xx00)))->((x4 Xx0)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x4 Xx00)->((K x4) Xx00)))) (x4 Xx0)) P) x6) x5)) ((K Xx00) Xx0)) (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0))) as proof of ((K Xx00) Xx0)
% Found (((fun (P:Type) (x6:((forall (Xx00:fofType), ((x4 Xx00)->((K x4) Xx00)))->((x4 Xx0)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x4 Xx00)->((K x4) Xx00)))) (x4 Xx0)) P) x6) x5)) ((K Xx00) Xx0)) (fun (x6:(forall (Xx000:fofType), ((x4 Xx000)->((K x4) Xx000))))=> (x6 Xx0))) as proof of ((K Xx00) Xx0)
% Found x50:=(x5 Xx):((x1 Xx)->((K x1) Xx))
% Found (x5 Xx) as proof of ((x1 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x1 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x1 Xx00)->((K x1) Xx00))))=> (x5 Xx)) as proof of ((x1 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x1 Xx00)->((K x1) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x1 Xx00)->((K x1) Xx00)))->((x1 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x1 Xx00)->((K x1) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x1 Xx00)->((K x1) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x1 Xx0)->((K x1) Xx0)))->((x1 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x1 Xx0)->((K x1) Xx0)))) (x1 Xx)) P) x5) x2)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x1 Xx00)->((K x1) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x1 Xx0)->((K x1) Xx0)))->((x1 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x1 Xx0)->((K x1) Xx0)))) (x1 Xx)) P) x5) x2)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x1 Xx00)->((K x1) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found x7:(Xx0 Xx00)
% Found x7 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x7:(Xx0 Xx00))=> x7) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x7:(Xx0 Xx00))=> x7) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x500:=(x50 x6):((K x3) Xx)
% Found (x50 x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found ((x5 Xx) x6) as proof of ((K Xx0) Xx)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx0) Xx))
% Found (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)) as proof of ((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->((K Xx0) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found ((and_rect1 ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx))) as proof of ((K Xx0) Xx)
% Found (fun (x4:((and (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)))=> (((fun (P:Type) (x5:((forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00)))) (x3 Xx)) P) x5) x4)) ((K Xx0) Xx)) (fun (x5:(forall (Xx00:fofType), ((x3 Xx00)->((K x3) Xx00))))=> (x5 Xx)))) as proof of ((K Xx0) Xx)
% Found x300:=(x30 Xx0):(((K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))) Xx0)->((K (K (fun (Xx00:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))) Xx0))
% Found (x30 Xx0) as proof of ((x1 Xx0)->((K x1) Xx0))
% Found (x30 Xx0) as proof of ((x1 Xx0)->((K x1) Xx0))
% Found x3:(Xx00 Xx000)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x3:(Xx00 Xx000))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x3:(Xx00 Xx000)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (x3:(Xx00 Xx000))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000)))))
% Found (fun (Xx000:fofType) (x3:(Xx00 Xx000))=> x3) as proof of (forall (Xx000:fofType), ((Xx00 Xx000)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx000))))))
% Found x3:(Xx0 Xx00)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x3:(Xx0 Xx00))=> x3) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x3:(Xx0 Xx00))=> x3) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x5:(Xx0 Xx00)
% Found x5 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (x5:(Xx0 Xx00))=> x5) as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00)))))
% Found (fun (Xx00:fofType) (x5:(Xx0 Xx00))=> x5) as proof of (forall (Xx00:fofType), ((Xx0 Xx00)->((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)->((K S) Xx1)))) (S Xx00))))))
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (fun (x5:(forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (fun (x5:(forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0))))=> (x5 Xx)) as proof of ((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->((K Xx00) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found ((and_rect1 ((K Xx00) Xx)) (fun (x5:(forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx00) Xx)) (fun (x5:(forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx00) Xx)) (fun (x5:(forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found x50:=(x5 Xx):((x3 Xx)->((K x3) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (x5 Xx) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (fun (x5:(forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0))))=> (x5 Xx)) as proof of ((x3 Xx)->((K Xx00) Xx))
% Found (fun (x5:(forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0))))=> (x5 Xx)) as proof of ((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->((K Xx00) Xx)))
% Found (and_rect10 (fun (x5:(forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found ((and_rect1 ((K Xx00) Xx)) (fun (x5:(forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx00) Xx)) (fun (x5:(forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0))))=> (x5 Xx))) as proof of ((K Xx00) Xx)
% Found (fun (x4:((and (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)))=> (((fun (P:Type) (x5:((forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))->((x3 Xx)->P)))=> (((((and_rect (forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0)))) (x3 Xx)) P) x5) x4)) ((K Xx00) Xx)) (fun (x5:(forall (Xx0:fofType), ((x3 Xx0)->((K x3) Xx0))))=> (x5 Xx)))) as proof of ((K Xx00) Xx)
% Found x3:(Xx0 Xx00)
% Found x3 as proof of ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (forall (Xx1:fofType), ((S Xx1)-
% EOF
%------------------------------------------------------------------------------