cocATP---0.2.0.THM-Prf.s

%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEU546^2 : TPTP v6.1.0. Released v3.7.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n092.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 13:32:25 EDT 2014

% Result   : Theorem 136.94s
% Output   : Proof 136.94s
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----ERROR: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEU546^2 : TPTP v6.1.0. Released v3.7.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n092.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 10:37:36 CDT 2014
% % CPUTime  : 136.94 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x25b2b90>, <kernel.DependentProduct object at 0x25b2a70>) of role type named exu_type
% Using role type
% Declaring exu:((fofType->Prop)->Prop)
% FOF formula (((eq ((fofType->Prop)->Prop)) exu) (fun (Xphi:(fofType->Prop))=> ((ex fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))))) of role definition named exu
% A new definition: (((eq ((fofType->Prop)->Prop)) exu) (fun (Xphi:(fofType->Prop))=> ((ex fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))))
% Defined: exu:=(fun (Xphi:(fofType->Prop))=> ((ex fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))))
% FOF formula (<kernel.Constant object at 0x24971b8>, <kernel.Sort object at 0x2492ab8>) of role type named exuI1_type
% Using role type
% Declaring exuI1:Prop
% FOF formula (((eq Prop) exuI1) (forall (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))->(exu (fun (Xx:fofType)=> (Xphi Xx)))))) of role definition named exuI1
% A new definition: (((eq Prop) exuI1) (forall (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))->(exu (fun (Xx:fofType)=> (Xphi Xx))))))
% Defined: exuI1:=(forall (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))->(exu (fun (Xx:fofType)=> (Xphi Xx)))))
% FOF formula (exuI1->(forall (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))->(exu (fun (Xx:fofType)=> (Xphi Xx)))))) of role conjecture named exuI2
% Conjecture to prove = (exuI1->(forall (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))->(exu (fun (Xx:fofType)=> (Xphi Xx)))))):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['(exuI1->(forall (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))->(exu (fun (Xx:fofType)=> (Xphi Xx))))))']
% Parameter fofType:Type.
% Definition exu:=(fun (Xphi:(fofType->Prop))=> ((ex fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))):((fofType->Prop)->Prop).
% Definition exuI1:=(forall (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))->(exu (fun (Xx:fofType)=> (Xphi Xx))))):Prop.
% Trying to prove (exuI1->(forall (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))->(exu (fun (Xx:fofType)=> (Xphi Xx))))))
% Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq fofType) x1) Xy)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Xy)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Xy)
% Found (fun (x00:(Xphi Xy))=> ((eq_ref fofType) x1)) as proof of (((eq fofType) x1) Xy)
% Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% Found (eq_ref00 P) as proof of ((P x1)->(P Xy))
% Found ((eq_ref0 x1) P) as proof of ((P x1)->(P Xy))
% Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P Xy))
% Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of (((eq fofType) x1) Xy)
% Found x2:(P x1)
% Instantiate: x1:=Xy:fofType
% Found (fun (x2:(P x1))=> x2) as proof of (P Xy)
% Found (fun (P:(fofType->Prop)) (x2:(P x1))=> x2) as proof of ((P x1)->(P Xy))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop)) (x2:(P x1))=> x2) as proof of (((eq fofType) x1) Xy)
% Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq fofType) x1) Xy)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Xy)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Xy)
% Found (fun (x00:(Xphi Xy))=> ((eq_ref fofType) x1)) as proof of (((eq fofType) x1) Xy)
% Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% Found (eq_ref00 P) as proof of ((P x1)->(P Xy))
% Found ((eq_ref0 x1) P) as proof of ((P x1)->(P Xy))
% Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P Xy))
% Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of (((eq fofType) x1) Xy)
% Found x2:(P x1)
% Instantiate: x1:=Xy:fofType
% Found (fun (x2:(P x1))=> x2) as proof of (P Xy)
% Found (fun (P:(fofType->Prop)) (x2:(P x1))=> x2) as proof of ((P x1)->(P Xy))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop)) (x2:(P x1))=> x2) as proof of (((eq fofType) x1) Xy)
% Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq fofType) x3) Xy)
% Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Xy)
% Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Xy)
% Found (fun (x00:(Xphi Xy))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) Xy)
% Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq fofType) x1) Xy)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Xy)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Xy)
% Found (fun (x00:(Xphi Xy))=> ((eq_ref fofType) x1)) as proof of (((eq fofType) x1) Xy)
% Found x4:(P x3)
% Instantiate: x3:=Xy:fofType
% Found (fun (x4:(P x3))=> x4) as proof of (P Xy)
% Found (fun (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of ((P x3)->(P Xy))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of (((eq fofType) x3) Xy)
% Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% Found (eq_ref00 P) as proof of ((P x3)->(P Xy))
% Found ((eq_ref0 x3) P) as proof of ((P x3)->(P Xy))
% Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Xy))
% Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P Xy))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) Xy)
% Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq fofType) x3) Xy)
% Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Xy)
% Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Xy)
% Found (fun (x00:(Xphi Xy))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) Xy)
% Found x4:(P x1)
% Instantiate: x1:=Xy:fofType
% Found (fun (x4:(P x1))=> x4) as proof of (P Xy)
% Found (fun (P:(fofType->Prop)) (x4:(P x1))=> x4) as proof of ((P x1)->(P Xy))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop)) (x4:(P x1))=> x4) as proof of (((eq fofType) x1) Xy)
% Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% Found (eq_ref00 P) as proof of ((P x1)->(P Xy))
% Found ((eq_ref0 x1) P) as proof of ((P x1)->(P Xy))
% Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P Xy))
% Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of (((eq fofType) x1) Xy)
% Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq fofType) x1) Xy)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Xy)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Xy)
% Found (fun (x00:(Xphi Xy))=> ((eq_ref fofType) x1)) as proof of (((eq fofType) x1) Xy)
% Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% Found (eq_ref00 P) as proof of ((P x3)->(P Xy))
% Found ((eq_ref0 x3) P) as proof of ((P x3)->(P Xy))
% Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Xy))
% Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P Xy))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) Xy)
% Found x4:(P x3)
% Instantiate: x3:=Xy:fofType
% Found (fun (x4:(P x3))=> x4) as proof of (P Xy)
% Found (fun (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of ((P x3)->(P Xy))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of (((eq fofType) x3) Xy)
% Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% Found (eq_ref00 P) as proof of ((P x1)->(P Xy))
% Found ((eq_ref0 x1) P) as proof of ((P x1)->(P Xy))
% Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P Xy))
% Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of (((eq fofType) x1) Xy)
% Found x4:(P x1)
% Instantiate: x1:=Xy:fofType
% Found (fun (x4:(P x1))=> x4) as proof of (P Xy)
% Found (fun (P:(fofType->Prop)) (x4:(P x1))=> x4) as proof of ((P x1)->(P Xy))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop)) (x4:(P x1))=> x4) as proof of (((eq fofType) x1) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found (eq_sym000 ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found ((eq_sym00 x1) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found (((eq_sym0 Xy) x1) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found ((((eq_sym fofType) Xy) x1) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found (fun (x00:(Xphi Xy))=> ((((eq_sym fofType) Xy) x1) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x1) Xy)
% Found eq_ref00:=(eq_ref0 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found (eq_ref0 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found (eq_sym000 ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found ((eq_sym00 x1) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found (((eq_sym0 Xy) x1) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found ((((eq_sym fofType) Xy) x1) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found (fun (x00:(Xphi Xy))=> ((((eq_sym fofType) Xy) x1) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x1) Xy)
% Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq fofType) x1) Xy)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Xy)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Xy)
% Found (fun (x3:(forall (Xy0:fofType), ((iff (Xphi Xy0)) (((eq fofType) Xy0) x2))))=> ((eq_ref fofType) x1)) as proof of (((eq fofType) x1) Xy)
% Found (fun (x2:fofType) (x3:(forall (Xy0:fofType), ((iff (Xphi Xy0)) (((eq fofType) Xy0) x2))))=> ((eq_ref fofType) x1)) as proof of ((forall (Xy0:fofType), ((iff (Xphi Xy0)) (((eq fofType) Xy0) x2)))->(((eq fofType) x1) Xy))
% Found (fun (x2:fofType) (x3:(forall (Xy0:fofType), ((iff (Xphi Xy0)) (((eq fofType) Xy0) x2))))=> ((eq_ref fofType) x1)) as proof of (forall (x:fofType), ((forall (Xy0:fofType), ((iff (Xphi Xy0)) (((eq fofType) Xy0) x)))->(((eq fofType) x1) Xy)))
% Found (ex_ind00 (fun (x2:fofType) (x3:(forall (Xy0:fofType), ((iff (Xphi Xy0)) (((eq fofType) Xy0) x2))))=> ((eq_ref fofType) x1))) as proof of (((eq fofType) x1) Xy)
% Found ((ex_ind0 (((eq fofType) x1) Xy)) (fun (x2:fofType) (x3:(forall (Xy0:fofType), ((iff (Xphi Xy0)) (((eq fofType) Xy0) x2))))=> ((eq_ref fofType) x1))) as proof of (((eq fofType) x1) Xy)
% Found (((fun (P:Prop) (x2:(forall (x:fofType), ((forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x)))->P)))=> (((((ex_ind fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) P) x2) x0)) (((eq fofType) x1) Xy)) (fun (x2:fofType) (x3:(forall (Xy0:fofType), ((iff (Xphi Xy0)) (((eq fofType) Xy0) x2))))=> ((eq_ref fofType) x1))) as proof of (((eq fofType) x1) Xy)
% Found (fun (x00:(Xphi Xy))=> (((fun (P:Prop) (x2:(forall (x:fofType), ((forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x)))->P)))=> (((((ex_ind fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) P) x2) x0)) (((eq fofType) x1) Xy)) (fun (x2:fofType) (x3:(forall (Xy0:fofType), ((iff (Xphi Xy0)) (((eq fofType) Xy0) x2))))=> ((eq_ref fofType) x1)))) as proof of (((eq fofType) x1) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) (fun (x:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x)))))
% Found (eta_expansion00 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x1))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref fofType) Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref fofType) Xy) P) as proof of ((P Xy)->(P x1))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) Xy) P)) as proof of ((P Xy)->(P x1))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) Xy) P)) as proof of (((eq fofType) Xy) x1)
% Found x5:(P Xy)
% Instantiate: Xy:=x1:fofType
% Found (fun (x5:(P Xy))=> x5) as proof of (P x1)
% Found (fun (P:(fofType->Prop)) (x5:(P Xy))=> x5) as proof of ((P Xy)->(P x1))
% Found (fun (P:(fofType->Prop)) (x5:(P Xy))=> x5) as proof of (((eq fofType) Xy) x1)
% Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq fofType) x1) Xy)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Xy)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Xy)
% Found (fun (x3:(forall (Xy0:fofType), ((iff (Xphi Xy0)) (((eq fofType) Xy0) x2))))=> ((eq_ref fofType) x1)) as proof of (((eq fofType) x1) Xy)
% Found (fun (x2:fofType) (x3:(forall (Xy0:fofType), ((iff (Xphi Xy0)) (((eq fofType) Xy0) x2))))=> ((eq_ref fofType) x1)) as proof of ((forall (Xy0:fofType), ((iff (Xphi Xy0)) (((eq fofType) Xy0) x2)))->(((eq fofType) x1) Xy))
% Found (fun (x2:fofType) (x3:(forall (Xy0:fofType), ((iff (Xphi Xy0)) (((eq fofType) Xy0) x2))))=> ((eq_ref fofType) x1)) as proof of (forall (x:fofType), ((forall (Xy0:fofType), ((iff (Xphi Xy0)) (((eq fofType) Xy0) x)))->(((eq fofType) x1) Xy)))
% Found (ex_ind00 (fun (x2:fofType) (x3:(forall (Xy0:fofType), ((iff (Xphi Xy0)) (((eq fofType) Xy0) x2))))=> ((eq_ref fofType) x1))) as proof of (((eq fofType) x1) Xy)
% Found ((ex_ind0 (((eq fofType) x1) Xy)) (fun (x2:fofType) (x3:(forall (Xy0:fofType), ((iff (Xphi Xy0)) (((eq fofType) Xy0) x2))))=> ((eq_ref fofType) x1))) as proof of (((eq fofType) x1) Xy)
% Found (((fun (P:Prop) (x2:(forall (x:fofType), ((forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x)))->P)))=> (((((ex_ind fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) P) x2) x0)) (((eq fofType) x1) Xy)) (fun (x2:fofType) (x3:(forall (Xy0:fofType), ((iff (Xphi Xy0)) (((eq fofType) Xy0) x2))))=> ((eq_ref fofType) x1))) as proof of (((eq fofType) x1) Xy)
% Found (fun (x00:(Xphi Xy))=> (((fun (P:Prop) (x2:(forall (x:fofType), ((forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x)))->P)))=> (((((ex_ind fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) P) x2) x0)) (((eq fofType) x1) Xy)) (fun (x2:fofType) (x3:(forall (Xy0:fofType), ((iff (Xphi Xy0)) (((eq fofType) Xy0) x2))))=> ((eq_ref fofType) x1)))) as proof of (((eq fofType) x1) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x3)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x3)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x3)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x3)
% Found (eq_sym000 ((eq_ref fofType) Xy)) as proof of (((eq fofType) x3) Xy)
% Found ((eq_sym00 x3) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x3) Xy)
% Found (((eq_sym0 Xy) x3) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x3) Xy)
% Found ((((eq_sym fofType) Xy) x3) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x3) Xy)
% Found (fun (x00:(Xphi Xy))=> ((((eq_sym fofType) Xy) x3) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x3) Xy)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))->(P0 (fun (x:fofType)=> ((and (Xphi x)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x) Xy)))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found ((eta_expansion00 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found (((eta_expansion0 Prop) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found ((((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found ((((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found (x4 ((eq_ref fofType) Xy)) as proof of (Xphi Xy)
% Found (x4 ((eq_ref fofType) Xy)) as proof of (Xphi Xy)
% Found x5:(P Xy)
% Instantiate: Xy:=x1:fofType
% Found (fun (x5:(P Xy))=> x5) as proof of (P x1)
% Found (fun (P:(fofType->Prop)) (x5:(P Xy))=> x5) as proof of ((P Xy)->(P x1))
% Found (fun (P:(fofType->Prop)) (x5:(P Xy))=> x5) as proof of (((eq fofType) Xy) x1)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x1))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref fofType) Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref fofType) Xy) P) as proof of ((P Xy)->(P x1))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) Xy) P)) as proof of ((P Xy)->(P x1))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) Xy) P)) as proof of (((eq fofType) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found (eq_sym000 ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found ((eq_sym00 x1) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found (((eq_sym0 Xy) x1) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found ((((eq_sym fofType) Xy) x1) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found (fun (x00:(Xphi Xy))=> ((((eq_sym fofType) Xy) x1) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x1) Xy)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) (fun (x:fofType)=> ((and (Xphi x)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x) Xy))))))
% Found (eta_expansion00 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x3)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x3)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x3)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x3)
% Found (eq_sym000 ((eq_ref fofType) Xy)) as proof of (((eq fofType) x3) Xy)
% Found ((eq_sym00 x3) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x3) Xy)
% Found (((eq_sym0 Xy) x3) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x3) Xy)
% Found ((((eq_sym fofType) Xy) x3) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x3) Xy)
% Found (fun (x00:(Xphi Xy))=> ((((eq_sym fofType) Xy) x3) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x3) Xy)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))->(P0 (fun (x:fofType)=> ((and (Xphi x)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x) Xy)))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found ((eta_expansion00 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found (((eta_expansion0 Prop) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found ((((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found ((((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))->(P0 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found ((eq_ref0 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found (((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found (((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))->(P0 (fun (x:fofType)=> ((and (Xphi x)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x) Xy)))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found ((eta_expansion00 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found (((eta_expansion0 Prop) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found ((((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found ((((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))->(P0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))))
% Found (eq_ref00 P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found (((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found (((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))->(P0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))))
% Found (eq_ref00 P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found (((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found (((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found (x4 ((eq_ref fofType) Xy)) as proof of (Xphi Xy)
% Found (x4 ((eq_ref fofType) Xy)) as proof of (Xphi Xy)
% Found eq_ref00:=(eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))):(((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found (eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found eq_ref00:=(eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))):(((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found (eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found (eq_sym000 ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found ((eq_sym00 x1) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found (((eq_sym0 Xy) x1) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found ((((eq_sym fofType) Xy) x1) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found (fun (x00:(Xphi Xy))=> ((((eq_sym fofType) Xy) x1) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x1) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found eq_ref00:=(eq_ref0 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found (eq_ref0 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))->(P0 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found ((eq_ref0 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found (((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found (((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))->(P0 (fun (x:fofType)=> ((and (Xphi x)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x) Xy)))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found ((eta_expansion00 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found (((eta_expansion0 Prop) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found ((((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found ((((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))->(P0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))))
% Found (eq_ref00 P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found (((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found (((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))->(P0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))))
% Found (eq_ref00 P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found (((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found (((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found eq_ref00:=(eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))):(((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found (eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found eq_ref00:=(eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))):(((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found (eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) (fun (x:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x)))))
% Found (eta_expansion00 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))->(P0 (fun (x:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found ((eta_expansion_dep00 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) P0) as proof of (P1 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found (((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) P0) as proof of (P1 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found ((((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) P0) as proof of (P1 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found ((((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) P0) as proof of (P1 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))->(P0 (fun (x:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found ((eta_expansion_dep00 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) P0) as proof of (P1 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found (((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) P0) as proof of (P1 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found ((((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) P0) as proof of (P1 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found ((((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) P0) as proof of (P1 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq fofType) x1) Xy)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Xy)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Xy)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Xy)
% Found (eq_sym000 ((eq_ref fofType) x1)) as proof of (((eq fofType) Xy) x1)
% Found ((eq_sym00 Xy) ((eq_ref fofType) x1)) as proof of (((eq fofType) Xy) x1)
% Found (((eq_sym0 x1) Xy) ((eq_ref fofType) x1)) as proof of (((eq fofType) Xy) x1)
% Found ((((eq_sym fofType) x1) Xy) ((eq_ref fofType) x1)) as proof of (((eq fofType) Xy) x1)
% Found ((((eq_sym fofType) x1) Xy) ((eq_ref fofType) x1)) as proof of (((eq fofType) Xy) x1)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) (fun (x:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x)))))
% Found (eta_expansion00 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))->(P0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))))
% Found (eq_ref00 P0) as proof of (P1 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) P0) as proof of (P1 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found (((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) P0) as proof of (P1 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found (((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) P0) as proof of (P1 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))->(P0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))))
% Found (eq_ref00 P0) as proof of (P1 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) P0) as proof of (P1 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found (((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) P0) as proof of (P1 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found (((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) P0) as proof of (P1 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))->(P0 (fun (x:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found ((eta_expansion_dep00 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) P0) as proof of (P1 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found (((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) P0) as proof of (P1 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found ((((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) P0) as proof of (P1 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found ((((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) P0) as proof of (P1 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))->(P0 (fun (x:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found ((eta_expansion_dep00 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) P0) as proof of (P1 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found (((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) P0) as proof of (P1 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found ((((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) P0) as proof of (P1 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found ((((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) P0) as proof of (P1 (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))
% Found eq_ref00:=(eq_ref0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))):(((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found (eq_ref0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found eq_ref00:=(eq_ref0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))):(((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found (eq_ref0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found eq_ref00:=(eq_ref0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))):(((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found (eq_ref0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found eq_ref00:=(eq_ref0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))):(((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found (eq_ref0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq fofType) x1) Xy)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Xy)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Xy)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Xy)
% Found (eq_sym000 ((eq_ref fofType) x1)) as proof of (((eq fofType) Xy) x1)
% Found ((eq_sym00 Xy) ((eq_ref fofType) x1)) as proof of (((eq fofType) Xy) x1)
% Found (((eq_sym0 x1) Xy) ((eq_ref fofType) x1)) as proof of (((eq fofType) Xy) x1)
% Found ((((eq_sym fofType) x1) Xy) ((eq_ref fofType) x1)) as proof of (((eq fofType) Xy) x1)
% Found ((((eq_sym fofType) x1) Xy) ((eq_ref fofType) x1)) as proof of (((eq fofType) Xy) x1)
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))->(P0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))))
% Found (eq_ref00 P0) as proof of (P1 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) P0) as proof of (P1 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found (((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) P0) as proof of (P1 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found (((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) P0) as proof of (P1 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))->(P0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))))
% Found (eq_ref00 P0) as proof of (P1 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) P0) as proof of (P1 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found (((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) P0) as proof of (P1 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found (((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) P0) as proof of (P1 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found eq_ref00:=(eq_ref0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))):(((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found (eq_ref0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found eq_ref00:=(eq_ref0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))):(((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found (eq_ref0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found eq_ref00:=(eq_ref0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))):(((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found (eq_ref0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found eq_ref00:=(eq_ref0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))):(((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found (eq_ref0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))->(P0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))))
% Found (eq_ref00 P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found (((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found (((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))->(P0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))))
% Found (eq_ref00 P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found (((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found (((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% Found (eq_ref0 x1) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found ((eq_ref fofType) x1) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found ((eq_ref fofType) x1) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found ((eq_ref fofType) x1) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found (fun (x00:(Xphi Xy))=> ((eq_ref fofType) x1)) as proof of (((eq fofType) x1) Xy)
% Found eq_ref00:=(eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))):(((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found (eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found eq_ref00:=(eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))):(((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found (eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% Found (eq_ref0 x5) as proof of (((eq fofType) x5) Xy0)
% Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Xy0)
% Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Xy0)
% Found (fun (x00:(Xphi Xy0))=> ((eq_ref fofType) x5)) as proof of (((eq fofType) x5) Xy0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found eq_ref00:=(eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))):(((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found (eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found eq_ref00:=(eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))):(((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found (eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% Found (eq_ref00 P) as proof of ((P x1)->(P Xy))
% Found ((eq_ref0 x1) P) as proof of ((P x1)->(P Xy))
% Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P Xy))
% Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of (((eq fofType) x1) Xy)
% Found x2:(P x1)
% Instantiate: x1:=Xy:fofType
% Found (fun (x2:(P x1))=> x2) as proof of (P Xy)
% Found (fun (P:(fofType->Prop)) (x2:(P x1))=> x2) as proof of ((P x1)->(P Xy))
% Found (fun (P:(fofType->Prop)) (x2:(P x1))=> x2) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop)) (x2:(P x1))=> x2) as proof of (((eq fofType) x1) Xy)
% Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq fofType) x3) Xy0)
% Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Xy0)
% Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Xy0)
% Found (fun (x00:(Xphi Xy0))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) Xy0)
% Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% Found (eq_ref0 x1) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found ((eq_ref fofType) x1) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found ((eq_ref fofType) x1) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found ((eq_ref fofType) x1) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found (fun (x00:(Xphi Xy))=> ((eq_ref fofType) x1)) as proof of (((eq fofType) x1) Xy)
% Found x40:=(x4 x30):(Xphi Xy)
% Found (x4 x30) as proof of (Xphi Xy)
% Found (x4 x30) as proof of (Xphi Xy)
% Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% Found (eq_ref00 P) as proof of (P0 x1)
% Found ((eq_ref0 x1) P) as proof of (P0 x1)
% Found (((eq_ref fofType) x1) P) as proof of (P0 x1)
% Found (((eq_ref fofType) x1) P) as proof of (P0 x1)
% Found x30:=(x3 x40):(((eq fofType) Xy) x1)
% Found (x3 x40) as proof of (((eq fofType) Xy) x1)
% Found (x3 x40) as proof of (((eq fofType) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found (eq_sym0000 ((eq_ref fofType) Xy)) as proof of ((P x1)->(P Xy))
% Found (eq_sym0000 ((eq_ref fofType) Xy)) as proof of ((P x1)->(P Xy))
% Found ((fun (x2:(((eq fofType) Xy) x1))=> ((eq_sym000 x2) P)) ((eq_ref fofType) Xy)) as proof of ((P x1)->(P Xy))
% Found ((fun (x2:(((eq fofType) Xy) x1))=> (((eq_sym00 x1) x2) P)) ((eq_ref fofType) Xy)) as proof of ((P x1)->(P Xy))
% Found ((fun (x2:(((eq fofType) Xy) x1))=> ((((eq_sym0 Xy) x1) x2) P)) ((eq_ref fofType) Xy)) as proof of ((P x1)->(P Xy))
% Found ((fun (x2:(((eq fofType) Xy) x1))=> (((((eq_sym fofType) Xy) x1) x2) P)) ((eq_ref fofType) Xy)) as proof of ((P x1)->(P Xy))
% Found (fun (P:(fofType->Prop))=> ((fun (x2:(((eq fofType) Xy) x1))=> (((((eq_sym fofType) Xy) x1) x2) P)) ((eq_ref fofType) Xy))) as proof of ((P x1)->(P Xy))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop))=> ((fun (x2:(((eq fofType) Xy) x1))=> (((((eq_sym fofType) Xy) x1) x2) P)) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x1) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_sym0000 ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found ((eq_sym0000 ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found (((fun (x2:(((eq fofType) Xy) x1))=> ((eq_sym000 x2) (fun (x4:fofType)=> ((P x1)->(P x4))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found (((fun (x2:(((eq fofType) Xy) x1))=> (((eq_sym00 x1) x2) (fun (x4:fofType)=> ((P x1)->(P x4))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found (((fun (x2:(((eq fofType) Xy) x1))=> ((((eq_sym0 Xy) x1) x2) (fun (x4:fofType)=> ((P x1)->(P x4))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found (((fun (x2:(((eq fofType) Xy) x1))=> (((((eq_sym fofType) Xy) x1) x2) (fun (x4:fofType)=> ((P x1)->(P x4))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((fun (x2:(((eq fofType) Xy) x1))=> (((((eq_sym fofType) Xy) x1) x2) (fun (x4:fofType)=> ((P x1)->(P x4))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P))) as proof of ((P x1)->(P Xy))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop))=> (((fun (x2:(((eq fofType) Xy) x1))=> (((((eq_sym fofType) Xy) x1) x2) (fun (x4:fofType)=> ((P x1)->(P x4))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P))) as proof of (((eq fofType) x1) Xy)
% Found eq_ref000:=(eq_ref00 P0):((P0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))->(P0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))))
% Found (eq_ref00 P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found (((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found (((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))->(P0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))))
% Found (eq_ref00 P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found (((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found (((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) P0) as proof of (P1 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found eq_ref00:=(eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))):(((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found (eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found eq_ref00:=(eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))):(((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found (eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found x6:(P x5)
% Instantiate: x5:=Xy0:fofType
% Found (fun (x6:(P x5))=> x6) as proof of (P Xy0)
% Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of ((P x5)->(P Xy0))
% Found (fun (x00:(Xphi Xy0)) (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) Xy0)
% Found eq_ref000:=(eq_ref00 P):((P x5)->(P x5))
% Found (eq_ref00 P) as proof of ((P x5)->(P Xy0))
% Found ((eq_ref0 x5) P) as proof of ((P x5)->(P Xy0))
% Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P Xy0))
% Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P Xy0))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of ((P x5)->(P Xy0))
% Found (fun (x00:(Xphi Xy0)) (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) Xy0)
% Found x40:=(x4 x30):(Xphi Xy)
% Found (x4 x30) as proof of (Xphi Xy)
% Found (x4 x30) as proof of (Xphi Xy)
% Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% Found (eq_ref0 x5) as proof of (((eq fofType) x5) Xy0)
% Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Xy0)
% Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Xy0)
% Found (fun (x00:(Xphi Xy0))=> ((eq_ref fofType) x5)) as proof of (((eq fofType) x5) Xy0)
% Found x5:(P Xy)
% Instantiate: Xy:=x1:fofType
% Found (fun (x5:(P Xy))=> x5) as proof of (P x1)
% Found (fun (x5:(P Xy))=> x5) as proof of ((P Xy)->(P x1))
% Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq fofType) x1) Xy0)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Xy0)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Xy0)
% Found (fun (x00:(Xphi Xy0))=> ((eq_ref fofType) x1)) as proof of (((eq fofType) x1) Xy0)
% Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% Found (eq_ref00 P) as proof of ((P x1)->(P Xy))
% Found ((eq_ref0 x1) P) as proof of ((P x1)->(P Xy))
% Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P Xy))
% Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of (((eq fofType) x1) Xy)
% Found x2:(P x1)
% Instantiate: x1:=Xy:fofType
% Found (fun (x2:(P x1))=> x2) as proof of (P Xy)
% Found (fun (P:(fofType->Prop)) (x2:(P x1))=> x2) as proof of ((P x1)->(P Xy))
% Found (fun (P:(fofType->Prop)) (x2:(P x1))=> x2) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop)) (x2:(P x1))=> x2) as proof of (((eq fofType) x1) Xy)
% Found eq_ref00:=(eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))):(((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found (eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found eq_ref00:=(eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))):(((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found (eq_ref0 (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) as proof of (((eq Prop) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0)))) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% Found (eq_ref00 P) as proof of ((P x3)->(P Xy0))
% Found ((eq_ref0 x3) P) as proof of ((P x3)->(P Xy0))
% Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Xy0))
% Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Xy0))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P Xy0))
% Found (fun (x00:(Xphi Xy0)) (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) Xy0)
% Found x6:(P x3)
% Instantiate: x3:=Xy0:fofType
% Found (fun (x6:(P x3))=> x6) as proof of (P Xy0)
% Found (fun (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of ((P x3)->(P Xy0))
% Found (fun (x00:(Xphi Xy0)) (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of (((eq fofType) x3) Xy0)
% Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq fofType) x1) b)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) b)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) b)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_trans0000 ((eq_ref fofType) x1)) ((eq_ref fofType) b)) as proof of (((eq fofType) x1) Xy)
% Found (((eq_trans000 Xy) ((eq_ref fofType) x1)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found ((((fun (b:fofType)=> ((eq_trans00 b) Xy)) Xy) ((eq_ref fofType) x1)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found ((((fun (b:fofType)=> (((eq_trans0 x1) b) Xy)) Xy) ((eq_ref fofType) x1)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found ((((fun (b:fofType)=> ((((eq_trans fofType) x1) b) Xy)) Xy) ((eq_ref fofType) x1)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found (fun (x00:(Xphi Xy))=> ((((fun (b:fofType)=> ((((eq_trans fofType) x1) b) Xy)) Xy) ((eq_ref fofType) x1)) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x1) Xy)
% Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% Found (eq_ref00 P) as proof of (P0 x1)
% Found ((eq_ref0 x1) P) as proof of (P0 x1)
% Found (((eq_ref fofType) x1) P) as proof of (P0 x1)
% Found (((eq_ref fofType) x1) P) as proof of (P0 x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_sym0000 ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found ((eq_sym0000 ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found (((fun (x2:(((eq fofType) Xy) x1))=> ((eq_sym000 x2) (fun (x4:fofType)=> ((P x1)->(P x4))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found (((fun (x2:(((eq fofType) Xy) x1))=> (((eq_sym00 x1) x2) (fun (x4:fofType)=> ((P x1)->(P x4))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found (((fun (x2:(((eq fofType) Xy) x1))=> ((((eq_sym0 Xy) x1) x2) (fun (x4:fofType)=> ((P x1)->(P x4))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found (((fun (x2:(((eq fofType) Xy) x1))=> (((((eq_sym fofType) Xy) x1) x2) (fun (x4:fofType)=> ((P x1)->(P x4))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((fun (x2:(((eq fofType) Xy) x1))=> (((((eq_sym fofType) Xy) x1) x2) (fun (x4:fofType)=> ((P x1)->(P x4))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P))) as proof of ((P x1)->(P Xy))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop))=> (((fun (x2:(((eq fofType) Xy) x1))=> (((((eq_sym fofType) Xy) x1) x2) (fun (x4:fofType)=> ((P x1)->(P x4))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P))) as proof of (((eq fofType) x1) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found (eq_sym0000 ((eq_ref fofType) Xy)) as proof of ((P x1)->(P Xy))
% Found (eq_sym0000 ((eq_ref fofType) Xy)) as proof of ((P x1)->(P Xy))
% Found ((fun (x2:(((eq fofType) Xy) x1))=> ((eq_sym000 x2) P)) ((eq_ref fofType) Xy)) as proof of ((P x1)->(P Xy))
% Found ((fun (x2:(((eq fofType) Xy) x1))=> (((eq_sym00 x1) x2) P)) ((eq_ref fofType) Xy)) as proof of ((P x1)->(P Xy))
% Found ((fun (x2:(((eq fofType) Xy) x1))=> ((((eq_sym0 Xy) x1) x2) P)) ((eq_ref fofType) Xy)) as proof of ((P x1)->(P Xy))
% Found ((fun (x2:(((eq fofType) Xy) x1))=> (((((eq_sym fofType) Xy) x1) x2) P)) ((eq_ref fofType) Xy)) as proof of ((P x1)->(P Xy))
% Found (fun (P:(fofType->Prop))=> ((fun (x2:(((eq fofType) Xy) x1))=> (((((eq_sym fofType) Xy) x1) x2) P)) ((eq_ref fofType) Xy))) as proof of ((P x1)->(P Xy))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop))=> ((fun (x2:(((eq fofType) Xy) x1))=> (((((eq_sym fofType) Xy) x1) x2) P)) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x1) Xy)
% Found x40:=(x4 x30):(Xphi Xy)
% Found (x4 x30) as proof of (Xphi Xy)
% Found (x4 x30) as proof of (Xphi Xy)
% Found x30:=(x3 x40):(((eq fofType) Xy) x1)
% Found (x3 x40) as proof of (((eq fofType) Xy) x1)
% Found (x3 x40) as proof of (((eq fofType) Xy) x1)
% Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq fofType) x3) Xy0)
% Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Xy0)
% Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Xy0)
% Found (fun (x00:(Xphi Xy0))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) Xy0)
% Found x6:(P Xy)
% Instantiate: x5:=x1:fofType
% Found (fun (x6:(P Xy))=> x6) as proof of (P x1)
% Found (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6) as proof of ((P Xy)->(P x1))
% Found (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6) as proof of (((eq fofType) Xy) x1)
% Found (x4 (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6)) as proof of (Xphi x5)
% Found (x4 (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6)) as proof of (Xphi x5)
% Found (x4 (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6)) as proof of (Xphi x5)
% Found eq_ref00:=(eq_ref0 Xy0):(((eq fofType) Xy0) Xy0)
% Found (eq_ref0 Xy0) as proof of (((eq fofType) Xy0) x5)
% Found ((eq_ref fofType) Xy0) as proof of (((eq fofType) Xy0) x5)
% Found ((eq_ref fofType) Xy0) as proof of (((eq fofType) Xy0) x5)
% Found ((eq_ref fofType) Xy0) as proof of (((eq fofType) Xy0) x5)
% Found (eq_sym000 ((eq_ref fofType) Xy0)) as proof of (((eq fofType) x5) Xy0)
% Found ((eq_sym00 x5) ((eq_ref fofType) Xy0)) as proof of (((eq fofType) x5) Xy0)
% Found (((eq_sym0 Xy0) x5) ((eq_ref fofType) Xy0)) as proof of (((eq fofType) x5) Xy0)
% Found ((((eq_sym fofType) Xy0) x5) ((eq_ref fofType) Xy0)) as proof of (((eq fofType) x5) Xy0)
% Found (fun (x00:(Xphi Xy0))=> ((((eq_sym fofType) Xy0) x5) ((eq_ref fofType) Xy0))) as proof of (((eq fofType) x5) Xy0)
% Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% Found (eq_ref0 x3) as proof of (forall (P:(fofType->Prop)), ((P x3)->(P Xy)))
% Found ((eq_ref fofType) x3) as proof of (forall (P:(fofType->Prop)), ((P x3)->(P Xy)))
% Found ((eq_ref fofType) x3) as proof of (forall (P:(fofType->Prop)), ((P x3)->(P Xy)))
% Found ((eq_ref fofType) x3) as proof of (forall (P:(fofType->Prop)), ((P x3)->(P Xy)))
% Found (fun (x00:(Xphi Xy))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) Xy)
% Found x40:=(x4 x30):(Xphi Xy)
% Found (x4 x30) as proof of (Xphi Xy)
% Found (x4 x30) as proof of (Xphi Xy)
% Found eq_ref000:=(eq_ref00 P):((P x5)->(P x5))
% Found (eq_ref00 P) as proof of ((P x5)->(P Xy0))
% Found ((eq_ref0 x5) P) as proof of ((P x5)->(P Xy0))
% Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P Xy0))
% Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P Xy0))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of ((P x5)->(P Xy0))
% Found (fun (x00:(Xphi Xy0)) (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) Xy0)
% Found x6:(P x5)
% Instantiate: x5:=Xy0:fofType
% Found (fun (x6:(P x5))=> x6) as proof of (P Xy0)
% Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of ((P x5)->(P Xy0))
% Found (fun (x00:(Xphi Xy0)) (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) Xy0)
% Found x5:(P Xy)
% Instantiate: Xy:=x1:fofType
% Found (fun (x5:(P Xy))=> x5) as proof of (P x1)
% Found (fun (x5:(P Xy))=> x5) as proof of ((P Xy)->(P x1))
% Found x6:(P x1)
% Instantiate: x1:=Xy0:fofType
% Found (fun (x6:(P x1))=> x6) as proof of (P Xy0)
% Found (fun (P:(fofType->Prop)) (x6:(P x1))=> x6) as proof of ((P x1)->(P Xy0))
% Found (fun (x00:(Xphi Xy0)) (P:(fofType->Prop)) (x6:(P x1))=> x6) as proof of (((eq fofType) x1) Xy0)
% Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% Found (eq_ref00 P) as proof of ((P x1)->(P Xy0))
% Found ((eq_ref0 x1) P) as proof of ((P x1)->(P Xy0))
% Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P Xy0))
% Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P Xy0))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy0))
% Found (fun (x00:(Xphi Xy0)) (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of (((eq fofType) x1) Xy0)
% Found x6:(P Xy)
% Instantiate: x3:=x1:fofType
% Found (fun (x6:(P Xy))=> x6) as proof of (P x1)
% Found (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6) as proof of ((P Xy)->(P x1))
% Found (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6) as proof of (((eq fofType) Xy) x1)
% Found (x5 (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6)) as proof of (Xphi x3)
% Found (x5 (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6)) as proof of (Xphi x3)
% Found (x5 (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6)) as proof of (Xphi x3)
% Found eq_ref00:=(eq_ref0 Xy0):(((eq fofType) Xy0) Xy0)
% Found (eq_ref0 Xy0) as proof of (((eq fofType) Xy0) x3)
% Found ((eq_ref fofType) Xy0) as proof of (((eq fofType) Xy0) x3)
% Found ((eq_ref fofType) Xy0) as proof of (((eq fofType) Xy0) x3)
% Found ((eq_ref fofType) Xy0) as proof of (((eq fofType) Xy0) x3)
% Found (eq_sym000 ((eq_ref fofType) Xy0)) as proof of (((eq fofType) x3) Xy0)
% Found ((eq_sym00 x3) ((eq_ref fofType) Xy0)) as proof of (((eq fofType) x3) Xy0)
% Found (((eq_sym0 Xy0) x3) ((eq_ref fofType) Xy0)) as proof of (((eq fofType) x3) Xy0)
% Found ((((eq_sym fofType) Xy0) x3) ((eq_ref fofType) Xy0)) as proof of (((eq fofType) x3) Xy0)
% Found (fun (x00:(Xphi Xy0))=> ((((eq_sym fofType) Xy0) x3) ((eq_ref fofType) Xy0))) as proof of (((eq fofType) x3) Xy0)
% Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq fofType) x1) b)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) b)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) b)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_trans0000 ((eq_ref fofType) x1)) ((eq_ref fofType) b)) as proof of (((eq fofType) x1) Xy)
% Found (((eq_trans000 Xy) ((eq_ref fofType) x1)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found ((((fun (b:fofType)=> ((eq_trans00 b) Xy)) Xy) ((eq_ref fofType) x1)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found ((((fun (b:fofType)=> (((eq_trans0 x1) b) Xy)) Xy) ((eq_ref fofType) x1)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found ((((fun (b:fofType)=> ((((eq_trans fofType) x1) b) Xy)) Xy) ((eq_ref fofType) x1)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found (fun (x00:(Xphi Xy))=> ((((fun (b:fofType)=> ((((eq_trans fofType) x1) b) Xy)) Xy) ((eq_ref fofType) x1)) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x1) Xy)
% Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq fofType) x1) Xy0)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Xy0)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Xy0)
% Found (fun (x00:(Xphi Xy0))=> ((eq_ref fofType) x1)) as proof of (((eq fofType) x1) Xy0)
% Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% Found (eq_ref0 x1) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found ((eq_ref fofType) x1) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found ((eq_ref fofType) x1) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found ((eq_ref fofType) x1) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found (fun (x00:(Xphi Xy))=> ((eq_ref fofType) x1)) as proof of (((eq fofType) x1) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found x6:(P x3)
% Instantiate: x3:=Xy0:fofType
% Found (fun (x6:(P x3))=> x6) as proof of (P Xy0)
% Found (fun (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of ((P x3)->(P Xy0))
% Found (fun (x00:(Xphi Xy0)) (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of (((eq fofType) x3) Xy0)
% Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% Found (eq_ref00 P) as proof of ((P x3)->(P Xy0))
% Found ((eq_ref0 x3) P) as proof of ((P x3)->(P Xy0))
% Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Xy0))
% Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Xy0))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P Xy0))
% Found (fun (x00:(Xphi Xy0)) (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) Xy0)
% Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% Found (eq_ref00 P) as proof of ((P x3)->(P Xy))
% Found ((eq_ref0 x3) P) as proof of ((P x3)->(P Xy))
% Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Xy))
% Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (forall (P:(fofType->Prop)), ((P x3)->(P Xy)))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) Xy)
% Found x4:(P x3)
% Instantiate: x3:=Xy:fofType
% Found (fun (x4:(P x3))=> x4) as proof of (P Xy)
% Found (fun (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of ((P x3)->(P Xy))
% Found (fun (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of (forall (P:(fofType->Prop)), ((P x3)->(P Xy)))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of (((eq fofType) x3) Xy)
% Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% Found (eq_ref0 x3) as proof of (forall (P:(fofType->Prop)), ((P x3)->(P Xy)))
% Found ((eq_ref fofType) x3) as proof of (forall (P:(fofType->Prop)), ((P x3)->(P Xy)))
% Found ((eq_ref fofType) x3) as proof of (forall (P:(fofType->Prop)), ((P x3)->(P Xy)))
% Found ((eq_ref fofType) x3) as proof of (forall (P:(fofType->Prop)), ((P x3)->(P Xy)))
% Found (fun (x00:(Xphi Xy))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) Xy)
% Found x6:(P Xy)
% Instantiate: x5:=x1:fofType
% Found (fun (x6:(P Xy))=> x6) as proof of (P x1)
% Found (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6) as proof of ((P Xy)->(P x1))
% Found (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6) as proof of (((eq fofType) Xy) x1)
% Found (x4 (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6)) as proof of (Xphi x5)
% Found (x4 (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6)) as proof of (Xphi x5)
% Found (x4 (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6)) as proof of (Xphi x5)
% Found eq_ref00:=(eq_ref0 Xy0):(((eq fofType) Xy0) Xy0)
% Found (eq_ref0 Xy0) as proof of (((eq fofType) Xy0) x5)
% Found ((eq_ref fofType) Xy0) as proof of (((eq fofType) Xy0) x5)
% Found ((eq_ref fofType) Xy0) as proof of (((eq fofType) Xy0) x5)
% Found ((eq_ref fofType) Xy0) as proof of (((eq fofType) Xy0) x5)
% Found (eq_sym000 ((eq_ref fofType) Xy0)) as proof of (((eq fofType) x5) Xy0)
% Found ((eq_sym00 x5) ((eq_ref fofType) Xy0)) as proof of (((eq fofType) x5) Xy0)
% Found (((eq_sym0 Xy0) x5) ((eq_ref fofType) Xy0)) as proof of (((eq fofType) x5) Xy0)
% Found ((((eq_sym fofType) Xy0) x5) ((eq_ref fofType) Xy0)) as proof of (((eq fofType) x5) Xy0)
% Found (fun (x00:(Xphi Xy0))=> ((((eq_sym fofType) Xy0) x5) ((eq_ref fofType) Xy0))) as proof of (((eq fofType) x5) Xy0)
% Found x6:(P Xy)
% Instantiate: x1:=x2:fofType
% Found (fun (x6:(P Xy))=> x6) as proof of (P x2)
% Found (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6) as proof of ((P Xy)->(P x2))
% Found (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6) as proof of (((eq fofType) Xy) x2)
% Found (x5 (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6)) as proof of (Xphi x1)
% Found (x5 (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6)) as proof of (Xphi x1)
% Found (x5 (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6)) as proof of (Xphi x1)
% Found eq_ref00:=(eq_ref0 Xy0):(((eq fofType) Xy0) Xy0)
% Found (eq_ref0 Xy0) as proof of (((eq fofType) Xy0) x1)
% Found ((eq_ref fofType) Xy0) as proof of (((eq fofType) Xy0) x1)
% Found ((eq_ref fofType) Xy0) as proof of (((eq fofType) Xy0) x1)
% Found ((eq_ref fofType) Xy0) as proof of (((eq fofType) Xy0) x1)
% Found (eq_sym000 ((eq_ref fofType) Xy0)) as proof of (((eq fofType) x1) Xy0)
% Found ((eq_sym00 x1) ((eq_ref fofType) Xy0)) as proof of (((eq fofType) x1) Xy0)
% Found (((eq_sym0 Xy0) x1) ((eq_ref fofType) Xy0)) as proof of (((eq fofType) x1) Xy0)
% Found ((((eq_sym fofType) Xy0) x1) ((eq_ref fofType) Xy0)) as proof of (((eq fofType) x1) Xy0)
% Found (fun (x00:(Xphi Xy0))=> ((((eq_sym fofType) Xy0) x1) ((eq_ref fofType) Xy0))) as proof of (((eq fofType) x1) Xy0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% Found (eq_ref00 P) as proof of (P0 x3)
% Found ((eq_ref0 x3) P) as proof of (P0 x3)
% Found (((eq_ref fofType) x3) P) as proof of (P0 x3)
% Found (((eq_ref fofType) x3) P) as proof of (P0 x3)
% Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% Found (eq_ref00 P) as proof of ((P x1)->(P Xy0))
% Found ((eq_ref0 x1) P) as proof of ((P x1)->(P Xy0))
% Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P Xy0))
% Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P Xy0))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy0))
% Found (fun (x00:(Xphi Xy0)) (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of (((eq fofType) x1) Xy0)
% Found x6:(P x1)
% Instantiate: x1:=Xy0:fofType
% Found (fun (x6:(P x1))=> x6) as proof of (P Xy0)
% Found (fun (P:(fofType->Prop)) (x6:(P x1))=> x6) as proof of ((P x1)->(P Xy0))
% Found (fun (x00:(Xphi Xy0)) (P:(fofType->Prop)) (x6:(P x1))=> x6) as proof of (((eq fofType) x1) Xy0)
% Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% Found (eq_ref00 P) as proof of ((P x1)->(P Xy))
% Found ((eq_ref0 x1) P) as proof of ((P x1)->(P Xy))
% Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P Xy))
% Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of (((eq fofType) x1) Xy)
% Found x4:(P x1)
% Instantiate: x1:=Xy:fofType
% Found (fun (x4:(P x1))=> x4) as proof of (P Xy)
% Found (fun (P:(fofType->Prop)) (x4:(P x1))=> x4) as proof of ((P x1)->(P Xy))
% Found (fun (P:(fofType->Prop)) (x4:(P x1))=> x4) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop)) (x4:(P x1))=> x4) as proof of (((eq fofType) x1) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x3)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x3)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x3)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x3)
% Found (eq_sym0000 ((eq_ref fofType) Xy)) as proof of ((P x3)->(P Xy))
% Found (eq_sym0000 ((eq_ref fofType) Xy)) as proof of ((P x3)->(P Xy))
% Found ((fun (x4:(((eq fofType) Xy) x3))=> ((eq_sym000 x4) P)) ((eq_ref fofType) Xy)) as proof of ((P x3)->(P Xy))
% Found ((fun (x4:(((eq fofType) Xy) x3))=> (((eq_sym00 x3) x4) P)) ((eq_ref fofType) Xy)) as proof of ((P x3)->(P Xy))
% Found ((fun (x4:(((eq fofType) Xy) x3))=> ((((eq_sym0 Xy) x3) x4) P)) ((eq_ref fofType) Xy)) as proof of ((P x3)->(P Xy))
% Found ((fun (x4:(((eq fofType) Xy) x3))=> (((((eq_sym fofType) Xy) x3) x4) P)) ((eq_ref fofType) Xy)) as proof of ((P x3)->(P Xy))
% Found (fun (P:(fofType->Prop))=> ((fun (x4:(((eq fofType) Xy) x3))=> (((((eq_sym fofType) Xy) x3) x4) P)) ((eq_ref fofType) Xy))) as proof of ((P x3)->(P Xy))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop))=> ((fun (x4:(((eq fofType) Xy) x3))=> (((((eq_sym fofType) Xy) x3) x4) P)) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x3) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x3)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x3)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x3)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x3)
% Found ((eq_sym0000 ((eq_ref fofType) Xy)) (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P Xy))
% Found ((eq_sym0000 ((eq_ref fofType) Xy)) (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P Xy))
% Found (((fun (x4:(((eq fofType) Xy) x3))=> ((eq_sym000 x4) (fun (x6:fofType)=> ((P x3)->(P x6))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P Xy))
% Found (((fun (x4:(((eq fofType) Xy) x3))=> (((eq_sym00 x3) x4) (fun (x6:fofType)=> ((P x3)->(P x6))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P Xy))
% Found (((fun (x4:(((eq fofType) Xy) x3))=> ((((eq_sym0 Xy) x3) x4) (fun (x6:fofType)=> ((P x3)->(P x6))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P Xy))
% Found (((fun (x4:(((eq fofType) Xy) x3))=> (((((eq_sym fofType) Xy) x3) x4) (fun (x6:fofType)=> ((P x3)->(P x6))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((fun (x4:(((eq fofType) Xy) x3))=> (((((eq_sym fofType) Xy) x3) x4) (fun (x6:fofType)=> ((P x3)->(P x6))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x3) P))) as proof of ((P x3)->(P Xy))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop))=> (((fun (x4:(((eq fofType) Xy) x3))=> (((((eq_sym fofType) Xy) x3) x4) (fun (x6:fofType)=> ((P x3)->(P x6))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x3) P))) as proof of (((eq fofType) x3) Xy)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x1))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref fofType) Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref fofType) Xy) P) as proof of ((P Xy)->(P x1))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) Xy) P)) as proof of ((P Xy)->(P x1))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) Xy) P)) as proof of (((eq fofType) Xy) x1)
% Found x6:(P Xy)
% Instantiate: Xy:=x1:fofType
% Found (fun (x6:(P Xy))=> x6) as proof of (P x1)
% Found (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6) as proof of ((P Xy)->(P x1))
% Found (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6) as proof of (((eq fofType) Xy) x1)
% Found x6:(P Xy)
% Instantiate: x3:=x1:fofType
% Found (fun (x6:(P Xy))=> x6) as proof of (P x1)
% Found (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6) as proof of ((P Xy)->(P x1))
% Found (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6) as proof of (((eq fofType) Xy) x1)
% Found (x5 (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6)) as proof of (Xphi x3)
% Found (x5 (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6)) as proof of (Xphi x3)
% Found (x5 (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6)) as proof of (Xphi x3)
% Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% Found (eq_ref0 x1) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found ((eq_ref fofType) x1) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found ((eq_ref fofType) x1) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found ((eq_ref fofType) x1) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found (fun (x00:(Xphi Xy))=> ((eq_ref fofType) x1)) as proof of (((eq fofType) x1) Xy)
% Found eq_ref00:=(eq_ref0 Xy0):(((eq fofType) Xy0) Xy0)
% Found (eq_ref0 Xy0) as proof of (((eq fofType) Xy0) x3)
% Found ((eq_ref fofType) Xy0) as proof of (((eq fofType) Xy0) x3)
% Found ((eq_ref fofType) Xy0) as proof of (((eq fofType) Xy0) x3)
% Found ((eq_ref fofType) Xy0) as proof of (((eq fofType) Xy0) x3)
% Found (eq_sym000 ((eq_ref fofType) Xy0)) as proof of (((eq fofType) x3) Xy0)
% Found ((eq_sym00 x3) ((eq_ref fofType) Xy0)) as proof of (((eq fofType) x3) Xy0)
% Found (((eq_sym0 Xy0) x3) ((eq_ref fofType) Xy0)) as proof of (((eq fofType) x3) Xy0)
% Found ((((eq_sym fofType) Xy0) x3) ((eq_ref fofType) Xy0)) as proof of (((eq fofType) x3) Xy0)
% Found (fun (x00:(Xphi Xy0))=> ((((eq_sym fofType) Xy0) x3) ((eq_ref fofType) Xy0))) as proof of (((eq fofType) x3) Xy0)
% Found x4:(P x3)
% Instantiate: x3:=Xy:fofType
% Found (fun (x4:(P x3))=> x4) as proof of (P Xy)
% Found (fun (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of ((P x3)->(P Xy))
% Found (fun (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of (forall (P:(fofType->Prop)), ((P x3)->(P Xy)))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of (((eq fofType) x3) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found eq_ref00:=(eq_ref0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))):(((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found (eq_ref0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found eq_ref00:=(eq_ref0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))):(((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found (eq_ref0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% Found (eq_ref00 P) as proof of ((P x3)->(P Xy))
% Found ((eq_ref0 x3) P) as proof of ((P x3)->(P Xy))
% Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Xy))
% Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (forall (P:(fofType->Prop)), ((P x3)->(P Xy)))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% Found (eq_ref00 P) as proof of (P0 x1)
% Found ((eq_ref0 x1) P) as proof of (P0 x1)
% Found (((eq_ref fofType) x1) P) as proof of (P0 x1)
% Found (((eq_ref fofType) x1) P) as proof of (P0 x1)
% Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq fofType) x3) b)
% Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) b)
% Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) b)
% Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_trans0000 ((eq_ref fofType) x3)) ((eq_ref fofType) b)) as proof of (((eq fofType) x3) Xy)
% Found (((eq_trans000 Xy) ((eq_ref fofType) x3)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x3) Xy)
% Found ((((fun (b:fofType)=> ((eq_trans00 b) Xy)) Xy) ((eq_ref fofType) x3)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x3) Xy)
% Found ((((fun (b:fofType)=> (((eq_trans0 x3) b) Xy)) Xy) ((eq_ref fofType) x3)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x3) Xy)
% Found ((((fun (b:fofType)=> ((((eq_trans fofType) x3) b) Xy)) Xy) ((eq_ref fofType) x3)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x3) Xy)
% Found (fun (x00:(Xphi Xy))=> ((((fun (b:fofType)=> ((((eq_trans fofType) x3) b) Xy)) Xy) ((eq_ref fofType) x3)) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x3) Xy)
% Found x6:(P Xy)
% Instantiate: Xy:=x1:fofType
% Found (fun (x6:(P Xy))=> x6) as proof of (P x1)
% Found (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6) as proof of ((P Xy)->(P x1))
% Found (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6) as proof of (((eq fofType) Xy) x1)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x1))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref fofType) Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref fofType) Xy) P) as proof of ((P Xy)->(P x1))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) Xy) P)) as proof of ((P Xy)->(P x1))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) Xy) P)) as proof of (((eq fofType) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_sym0000 ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found ((eq_sym0000 ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found (((fun (x4:(((eq fofType) Xy) x1))=> ((eq_sym000 x4) (fun (x6:fofType)=> ((P x1)->(P x6))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found (((fun (x4:(((eq fofType) Xy) x1))=> (((eq_sym00 x1) x4) (fun (x6:fofType)=> ((P x1)->(P x6))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found (((fun (x4:(((eq fofType) Xy) x1))=> ((((eq_sym0 Xy) x1) x4) (fun (x6:fofType)=> ((P x1)->(P x6))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found (((fun (x4:(((eq fofType) Xy) x1))=> (((((eq_sym fofType) Xy) x1) x4) (fun (x6:fofType)=> ((P x1)->(P x6))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((fun (x4:(((eq fofType) Xy) x1))=> (((((eq_sym fofType) Xy) x1) x4) (fun (x6:fofType)=> ((P x1)->(P x6))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P))) as proof of ((P x1)->(P Xy))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop))=> (((fun (x4:(((eq fofType) Xy) x1))=> (((((eq_sym fofType) Xy) x1) x4) (fun (x6:fofType)=> ((P x1)->(P x6))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P))) as proof of (((eq fofType) x1) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found (eq_sym0000 ((eq_ref fofType) Xy)) as proof of ((P x1)->(P Xy))
% Found (eq_sym0000 ((eq_ref fofType) Xy)) as proof of ((P x1)->(P Xy))
% Found ((fun (x4:(((eq fofType) Xy) x1))=> ((eq_sym000 x4) P)) ((eq_ref fofType) Xy)) as proof of ((P x1)->(P Xy))
% Found ((fun (x4:(((eq fofType) Xy) x1))=> (((eq_sym00 x1) x4) P)) ((eq_ref fofType) Xy)) as proof of ((P x1)->(P Xy))
% Found ((fun (x4:(((eq fofType) Xy) x1))=> ((((eq_sym0 Xy) x1) x4) P)) ((eq_ref fofType) Xy)) as proof of ((P x1)->(P Xy))
% Found ((fun (x4:(((eq fofType) Xy) x1))=> (((((eq_sym fofType) Xy) x1) x4) P)) ((eq_ref fofType) Xy)) as proof of ((P x1)->(P Xy))
% Found (fun (P:(fofType->Prop))=> ((fun (x4:(((eq fofType) Xy) x1))=> (((((eq_sym fofType) Xy) x1) x4) P)) ((eq_ref fofType) Xy))) as proof of ((P x1)->(P Xy))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop))=> ((fun (x4:(((eq fofType) Xy) x1))=> (((((eq_sym fofType) Xy) x1) x4) P)) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x1) Xy)
% Found x6:(P Xy)
% Instantiate: x1:=x2:fofType
% Found (fun (x6:(P Xy))=> x6) as proof of (P x2)
% Found (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6) as proof of ((P Xy)->(P x2))
% Found (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6) as proof of (((eq fofType) Xy) x2)
% Found (x5 (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6)) as proof of (Xphi x1)
% Found (x5 (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6)) as proof of (Xphi x1)
% Found (x5 (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6)) as proof of (Xphi x1)
% Found eq_ref00:=(eq_ref0 Xy0):(((eq fofType) Xy0) Xy0)
% Found (eq_ref0 Xy0) as proof of (((eq fofType) Xy0) x1)
% Found ((eq_ref fofType) Xy0) as proof of (((eq fofType) Xy0) x1)
% Found ((eq_ref fofType) Xy0) as proof of (((eq fofType) Xy0) x1)
% Found ((eq_ref fofType) Xy0) as proof of (((eq fofType) Xy0) x1)
% Found (eq_sym000 ((eq_ref fofType) Xy0)) as proof of (((eq fofType) x1) Xy0)
% Found ((eq_sym00 x1) ((eq_ref fofType) Xy0)) as proof of (((eq fofType) x1) Xy0)
% Found (((eq_sym0 Xy0) x1) ((eq_ref fofType) Xy0)) as proof of (((eq fofType) x1) Xy0)
% Found ((((eq_sym fofType) Xy0) x1) ((eq_ref fofType) Xy0)) as proof of (((eq fofType) x1) Xy0)
% Found (fun (x00:(Xphi Xy0))=> ((((eq_sym fofType) Xy0) x1) ((eq_ref fofType) Xy0))) as proof of (((eq fofType) x1) Xy0)
% Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq fofType) x3) Xy)
% Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Xy)
% Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Xy)
% Found (fun (x5:((((eq fofType) Xy0) x1)->(Xphi Xy0)))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) Xy)
% Found (fun (x4:((Xphi Xy0)->(((eq fofType) Xy0) x1))) (x5:((((eq fofType) Xy0) x1)->(Xphi Xy0)))=> ((eq_ref fofType) x3)) as proof of (((((eq fofType) Xy0) x1)->(Xphi Xy0))->(((eq fofType) x3) Xy))
% Found (fun (x4:((Xphi Xy0)->(((eq fofType) Xy0) x1))) (x5:((((eq fofType) Xy0) x1)->(Xphi Xy0)))=> ((eq_ref fofType) x3)) as proof of (((Xphi Xy0)->(((eq fofType) Xy0) x1))->(((((eq fofType) Xy0) x1)->(Xphi Xy0))->(((eq fofType) x3) Xy)))
% Found (and_rect00 (fun (x4:((Xphi Xy0)->(((eq fofType) Xy0) x1))) (x5:((((eq fofType) Xy0) x1)->(Xphi Xy0)))=> ((eq_ref fofType) x3))) as proof of (((eq fofType) x3) Xy)
% Found ((and_rect0 (((eq fofType) x3) Xy)) (fun (x4:((Xphi Xy0)->(((eq fofType) Xy0) x1))) (x5:((((eq fofType) Xy0) x1)->(Xphi Xy0)))=> ((eq_ref fofType) x3))) as proof of (((eq fofType) x3) Xy)
% Found (((fun (P:Type) (x4:(((Xphi Xy0)->(((eq fofType) Xy0) x1))->(((((eq fofType) Xy0) x1)->(Xphi Xy0))->P)))=> (((((and_rect ((Xphi Xy0)->(((eq fofType) Xy0) x1))) ((((eq fofType) Xy0) x1)->(Xphi Xy0))) P) x4) x20)) (((eq fofType) x3) Xy)) (fun (x4:((Xphi Xy0)->(((eq fofType) Xy0) x1))) (x5:((((eq fofType) Xy0) x1)->(Xphi Xy0)))=> ((eq_ref fofType) x3))) as proof of (((eq fofType) x3) Xy)
% Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% Found (eq_ref00 P) as proof of (P0 x3)
% Found ((eq_ref0 x3) P) as proof of (P0 x3)
% Found (((eq_ref fofType) x3) P) as proof of (P0 x3)
% Found (((eq_ref fofType) x3) P) as proof of (P0 x3)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found x4:(P x1)
% Instantiate: x1:=Xy:fofType
% Found (fun (x4:(P x1))=> x4) as proof of (P Xy)
% Found (fun (P:(fofType->Prop)) (x4:(P x1))=> x4) as proof of ((P x1)->(P Xy))
% Found (fun (P:(fofType->Prop)) (x4:(P x1))=> x4) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop)) (x4:(P x1))=> x4) as proof of (((eq fofType) x1) Xy)
% Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% Found (eq_ref00 P) as proof of ((P x1)->(P Xy))
% Found ((eq_ref0 x1) P) as proof of ((P x1)->(P Xy))
% Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P Xy))
% Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of (((eq fofType) x1) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found (x4 ((eq_ref fofType) Xy)) as proof of (Xphi Xy)
% Found (x4 ((eq_ref fofType) Xy)) as proof of (Xphi Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x3)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x3)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x3)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x3)
% Found ((eq_sym0000 ((eq_ref fofType) Xy)) (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P Xy))
% Found ((eq_sym0000 ((eq_ref fofType) Xy)) (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P Xy))
% Found (((fun (x4:(((eq fofType) Xy) x3))=> ((eq_sym000 x4) (fun (x6:fofType)=> ((P x3)->(P x6))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P Xy))
% Found (((fun (x4:(((eq fofType) Xy) x3))=> (((eq_sym00 x3) x4) (fun (x6:fofType)=> ((P x3)->(P x6))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P Xy))
% Found (((fun (x4:(((eq fofType) Xy) x3))=> ((((eq_sym0 Xy) x3) x4) (fun (x6:fofType)=> ((P x3)->(P x6))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P Xy))
% Found (((fun (x4:(((eq fofType) Xy) x3))=> (((((eq_sym fofType) Xy) x3) x4) (fun (x6:fofType)=> ((P x3)->(P x6))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((fun (x4:(((eq fofType) Xy) x3))=> (((((eq_sym fofType) Xy) x3) x4) (fun (x6:fofType)=> ((P x3)->(P x6))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x3) P))) as proof of ((P x3)->(P Xy))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop))=> (((fun (x4:(((eq fofType) Xy) x3))=> (((((eq_sym fofType) Xy) x3) x4) (fun (x6:fofType)=> ((P x3)->(P x6))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x3) P))) as proof of (((eq fofType) x3) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x3)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x3)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x3)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x3)
% Found (eq_sym0000 ((eq_ref fofType) Xy)) as proof of ((P x3)->(P Xy))
% Found (eq_sym0000 ((eq_ref fofType) Xy)) as proof of ((P x3)->(P Xy))
% Found ((fun (x4:(((eq fofType) Xy) x3))=> ((eq_sym000 x4) P)) ((eq_ref fofType) Xy)) as proof of ((P x3)->(P Xy))
% Found ((fun (x4:(((eq fofType) Xy) x3))=> (((eq_sym00 x3) x4) P)) ((eq_ref fofType) Xy)) as proof of ((P x3)->(P Xy))
% Found ((fun (x4:(((eq fofType) Xy) x3))=> ((((eq_sym0 Xy) x3) x4) P)) ((eq_ref fofType) Xy)) as proof of ((P x3)->(P Xy))
% Found ((fun (x4:(((eq fofType) Xy) x3))=> (((((eq_sym fofType) Xy) x3) x4) P)) ((eq_ref fofType) Xy)) as proof of ((P x3)->(P Xy))
% Found (fun (P:(fofType->Prop))=> ((fun (x4:(((eq fofType) Xy) x3))=> (((((eq_sym fofType) Xy) x3) x4) P)) ((eq_ref fofType) Xy))) as proof of ((P x3)->(P Xy))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop))=> ((fun (x4:(((eq fofType) Xy) x3))=> (((((eq_sym fofType) Xy) x3) x4) P)) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x3) Xy)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x1))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref fofType) Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref fofType) Xy) P) as proof of ((P Xy)->(P x1))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) Xy) P)) as proof of ((P Xy)->(P x1))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) Xy) P)) as proof of (((eq fofType) Xy) x1)
% Found x6:(P Xy)
% Instantiate: Xy:=x1:fofType
% Found (fun (x6:(P Xy))=> x6) as proof of (P x1)
% Found (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6) as proof of ((P Xy)->(P x1))
% Found (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6) as proof of (((eq fofType) Xy) x1)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found eq_ref00:=(eq_ref0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))):(((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found (eq_ref0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x0))))
% Found eq_ref00:=(eq_ref0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))):(((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy)))))
% Found (eq_ref0 ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found ((eq_ref Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) as proof of (((eq Prop) ((and (Xphi x0)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x0) Xy))))) b)
% Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq fofType) x1) b)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) b)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) b)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_trans0000 ((eq_ref fofType) x1)) ((eq_ref fofType) b)) as proof of (((eq fofType) x1) Xy)
% Found (((eq_trans000 Xy) ((eq_ref fofType) x1)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found ((((fun (b:fofType)=> ((eq_trans00 b) Xy)) Xy) ((eq_ref fofType) x1)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found ((((fun (b:fofType)=> (((eq_trans0 x1) b) Xy)) Xy) ((eq_ref fofType) x1)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found ((((fun (b:fofType)=> ((((eq_trans fofType) x1) b) Xy)) Xy) ((eq_ref fofType) x1)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found (fun (x00:(Xphi Xy))=> ((((fun (b:fofType)=> ((((eq_trans fofType) x1) b) Xy)) Xy) ((eq_ref fofType) x1)) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x1) Xy)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x1))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref fofType) Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref fofType) Xy) P) as proof of ((P Xy)->(P x1))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) Xy) P)) as proof of ((P Xy)->(P x1))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) Xy) P)) as proof of (((eq fofType) Xy) x1)
% Found x6:(P Xy)
% Instantiate: Xy:=x2:fofType
% Found (fun (x6:(P Xy))=> x6) as proof of (P x2)
% Found (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6) as proof of ((P Xy)->(P x2))
% Found (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6) as proof of (((eq fofType) Xy) x2)
% Found x5:(P Xy)
% Instantiate: Xy:=x1:fofType
% Found (fun (x5:(P Xy))=> x5) as proof of (P x1)
% Found (fun (P:(fofType->Prop)) (x5:(P Xy))=> x5) as proof of ((P Xy)->(P x1))
% Found (fun (P:(fofType->Prop)) (x5:(P Xy))=> x5) as proof of (((eq fofType) Xy) x1)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x2))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x2))
% Found (((eq_ref fofType) Xy) P) as proof of ((P Xy)->(P x2))
% Found (((eq_ref fofType) Xy) P) as proof of ((P Xy)->(P x2))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) Xy) P)) as proof of ((P Xy)->(P x2))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) Xy) P)) as proof of (((eq fofType) Xy) x2)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found (eq_sym000 ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found ((eq_sym00 x1) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found (((eq_sym0 Xy) x1) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found ((((eq_sym fofType) Xy) x1) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found ((((eq_sym fofType) Xy) x1) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found (fun (x00:(Xphi Xy))=> ((((eq_sym fofType) Xy) x1) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x1) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found (x5 ((eq_ref fofType) Xy)) as proof of (Xphi Xy)
% Found (x5 ((eq_ref fofType) Xy)) as proof of (Xphi Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy0):(((eq fofType) Xy0) Xy0)
% Found (eq_ref0 Xy0) as proof of (((eq fofType) Xy0) x1)
% Found ((eq_ref fofType) Xy0) as proof of (((eq fofType) Xy0) x1)
% Found ((eq_ref fofType) Xy0) as proof of (((eq fofType) Xy0) x1)
% Found ((eq_ref fofType) Xy0) as proof of (((eq fofType) Xy0) x1)
% Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq fofType) x1) Xy)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Xy)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Xy)
% Found (fun (x5:((((eq fofType) Xy0) x2)->(Xphi Xy0)))=> ((eq_ref fofType) x1)) as proof of (((eq fofType) x1) Xy)
% Found (fun (x4:((Xphi Xy0)->(((eq fofType) Xy0) x2))) (x5:((((eq fofType) Xy0) x2)->(Xphi Xy0)))=> ((eq_ref fofType) x1)) as proof of (((((eq fofType) Xy0) x2)->(Xphi Xy0))->(((eq fofType) x1) Xy))
% Found (fun (x4:((Xphi Xy0)->(((eq fofType) Xy0) x2))) (x5:((((eq fofType) Xy0) x2)->(Xphi Xy0)))=> ((eq_ref fofType) x1)) as proof of (((Xphi Xy0)->(((eq fofType) Xy0) x2))->(((((eq fofType) Xy0) x2)->(Xphi Xy0))->(((eq fofType) x1) Xy)))
% Found (and_rect00 (fun (x4:((Xphi Xy0)->(((eq fofType) Xy0) x2))) (x5:((((eq fofType) Xy0) x2)->(Xphi Xy0)))=> ((eq_ref fofType) x1))) as proof of (((eq fofType) x1) Xy)
% Found ((and_rect0 (((eq fofType) x1) Xy)) (fun (x4:((Xphi Xy0)->(((eq fofType) Xy0) x2))) (x5:((((eq fofType) Xy0) x2)->(Xphi Xy0)))=> ((eq_ref fofType) x1))) as proof of (((eq fofType) x1) Xy)
% Found (((fun (P:Type) (x4:(((Xphi Xy0)->(((eq fofType) Xy0) x2))->(((((eq fofType) Xy0) x2)->(Xphi Xy0))->P)))=> (((((and_rect ((Xphi Xy0)->(((eq fofType) Xy0) x2))) ((((eq fofType) Xy0) x2)->(Xphi Xy0))) P) x4) x30)) (((eq fofType) x1) Xy)) (fun (x4:((Xphi Xy0)->(((eq fofType) Xy0) x2))) (x5:((((eq fofType) Xy0) x2)->(Xphi Xy0)))=> ((eq_ref fofType) x1))) as proof of (((eq fofType) x1) Xy)
% Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% Found (eq_ref00 P) as proof of (P0 x1)
% Found ((eq_ref0 x1) P) as proof of (P0 x1)
% Found (((eq_ref fofType) x1) P) as proof of (P0 x1)
% Found (((eq_ref fofType) x1) P) as proof of (P0 x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq fofType) x3) b)
% Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) b)
% Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) b)
% Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_trans0000 ((eq_ref fofType) x3)) ((eq_ref fofType) b)) as proof of (((eq fofType) x3) Xy)
% Found (((eq_trans000 Xy) ((eq_ref fofType) x3)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x3) Xy)
% Found ((((fun (b:fofType)=> ((eq_trans00 b) Xy)) Xy) ((eq_ref fofType) x3)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x3) Xy)
% Found ((((fun (b:fofType)=> (((eq_trans0 x3) b) Xy)) Xy) ((eq_ref fofType) x3)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x3) Xy)
% Found ((((fun (b:fofType)=> ((((eq_trans fofType) x3) b) Xy)) Xy) ((eq_ref fofType) x3)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x3) Xy)
% Found (fun (x00:(Xphi Xy))=> ((((fun (b:fofType)=> ((((eq_trans fofType) x3) b) Xy)) Xy) ((eq_ref fofType) x3)) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x3) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_sym0000 ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found ((eq_sym0000 ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found (((fun (x4:(((eq fofType) Xy) x1))=> ((eq_sym000 x4) (fun (x6:fofType)=> ((P x1)->(P x6))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found (((fun (x4:(((eq fofType) Xy) x1))=> (((eq_sym00 x1) x4) (fun (x6:fofType)=> ((P x1)->(P x6))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found (((fun (x4:(((eq fofType) Xy) x1))=> ((((eq_sym0 Xy) x1) x4) (fun (x6:fofType)=> ((P x1)->(P x6))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found (((fun (x4:(((eq fofType) Xy) x1))=> (((((eq_sym fofType) Xy) x1) x4) (fun (x6:fofType)=> ((P x1)->(P x6))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((fun (x4:(((eq fofType) Xy) x1))=> (((((eq_sym fofType) Xy) x1) x4) (fun (x6:fofType)=> ((P x1)->(P x6))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P))) as proof of ((P x1)->(P Xy))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop))=> (((fun (x4:(((eq fofType) Xy) x1))=> (((((eq_sym fofType) Xy) x1) x4) (fun (x6:fofType)=> ((P x1)->(P x6))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x1) P))) as proof of (((eq fofType) x1) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found (eq_sym0000 ((eq_ref fofType) Xy)) as proof of ((P x1)->(P Xy))
% Found (eq_sym0000 ((eq_ref fofType) Xy)) as proof of ((P x1)->(P Xy))
% Found ((fun (x4:(((eq fofType) Xy) x1))=> ((eq_sym000 x4) P)) ((eq_ref fofType) Xy)) as proof of ((P x1)->(P Xy))
% Found ((fun (x4:(((eq fofType) Xy) x1))=> (((eq_sym00 x1) x4) P)) ((eq_ref fofType) Xy)) as proof of ((P x1)->(P Xy))
% Found ((fun (x4:(((eq fofType) Xy) x1))=> ((((eq_sym0 Xy) x1) x4) P)) ((eq_ref fofType) Xy)) as proof of ((P x1)->(P Xy))
% Found ((fun (x4:(((eq fofType) Xy) x1))=> (((((eq_sym fofType) Xy) x1) x4) P)) ((eq_ref fofType) Xy)) as proof of ((P x1)->(P Xy))
% Found (fun (P:(fofType->Prop))=> ((fun (x4:(((eq fofType) Xy) x1))=> (((((eq_sym fofType) Xy) x1) x4) P)) ((eq_ref fofType) Xy))) as proof of ((P x1)->(P Xy))
% Found (fun (x00:(Xphi Xy)) (P:(fofType->Prop))=> ((fun (x4:(((eq fofType) Xy) x1))=> (((((eq_sym fofType) Xy) x1) x4) P)) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x1) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found x6:(P Xy)
% Instantiate: Xy:=x1:fofType
% Found (fun (x6:(P Xy))=> x6) as proof of (P x1)
% Found (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6) as proof of ((P Xy)->(P x1))
% Found (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6) as proof of (((eq fofType) Xy) x1)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x1))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref fofType) Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref fofType) Xy) P) as proof of ((P Xy)->(P x1))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) Xy) P)) as proof of ((P Xy)->(P x1))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) Xy) P)) as proof of (((eq fofType) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found x6:(P Xy)
% Instantiate: Xy:=x1:fofType
% Found (fun (x6:(P Xy))=> x6) as proof of (P x1)
% Found (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6) as proof of ((P Xy)->(P x1))
% Found (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6) as proof of (((eq fofType) Xy) x1)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x1))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref fofType) Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref fofType) Xy) P) as proof of ((P Xy)->(P x1))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) Xy) P)) as proof of ((P Xy)->(P x1))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) Xy) P)) as proof of (((eq fofType) Xy) x1)
% Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq fofType) x1) Xy)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Xy)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Xy)
% Found (fun (x00:(Xphi Xy))=> ((eq_ref fofType) x1)) as proof of (((eq fofType) x1) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq fofType) x3) Xy)
% Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Xy)
% Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Xy)
% Found (fun (x5:((((eq fofType) Xy0) x1)->(Xphi Xy0)))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) Xy)
% Found (fun (x4:((Xphi Xy0)->(((eq fofType) Xy0) x1))) (x5:((((eq fofType) Xy0) x1)->(Xphi Xy0)))=> ((eq_ref fofType) x3)) as proof of (((((eq fofType) Xy0) x1)->(Xphi Xy0))->(((eq fofType) x3) Xy))
% Found (fun (x4:((Xphi Xy0)->(((eq fofType) Xy0) x1))) (x5:((((eq fofType) Xy0) x1)->(Xphi Xy0)))=> ((eq_ref fofType) x3)) as proof of (((Xphi Xy0)->(((eq fofType) Xy0) x1))->(((((eq fofType) Xy0) x1)->(Xphi Xy0))->(((eq fofType) x3) Xy)))
% Found (and_rect00 (fun (x4:((Xphi Xy0)->(((eq fofType) Xy0) x1))) (x5:((((eq fofType) Xy0) x1)->(Xphi Xy0)))=> ((eq_ref fofType) x3))) as proof of (((eq fofType) x3) Xy)
% Found ((and_rect0 (((eq fofType) x3) Xy)) (fun (x4:((Xphi Xy0)->(((eq fofType) Xy0) x1))) (x5:((((eq fofType) Xy0) x1)->(Xphi Xy0)))=> ((eq_ref fofType) x3))) as proof of (((eq fofType) x3) Xy)
% Found (((fun (P:Type) (x4:(((Xphi Xy0)->(((eq fofType) Xy0) x1))->(((((eq fofType) Xy0) x1)->(Xphi Xy0))->P)))=> (((((and_rect ((Xphi Xy0)->(((eq fofType) Xy0) x1))) ((((eq fofType) Xy0) x1)->(Xphi Xy0))) P) x4) x20)) (((eq fofType) x3) Xy)) (fun (x4:((Xphi Xy0)->(((eq fofType) Xy0) x1))) (x5:((((eq fofType) Xy0) x1)->(Xphi Xy0)))=> ((eq_ref fofType) x3))) as proof of (((eq fofType) x3) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found (x4 ((eq_ref fofType) Xy)) as proof of (Xphi Xy)
% Found (x4 ((eq_ref fofType) Xy)) as proof of (Xphi Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found x00:(Xphi Xy0)
% Instantiate: Xy:=Xy0:fofType
% Found x00 as proof of (Xphi Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found (x4 ((eq_ref fofType) Xy)) as proof of (Xphi Xy)
% Found (x4 ((eq_ref fofType) Xy)) as proof of (Xphi Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found (x5 ((eq_ref fofType) Xy)) as proof of (Xphi Xy)
% Found (x5 ((eq_ref fofType) Xy)) as proof of (Xphi Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found (eq_sym000 ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found ((eq_sym00 x1) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found (((eq_sym0 Xy) x1) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found ((((eq_sym fofType) Xy) x1) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found ((((eq_sym fofType) Xy) x1) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x1)->(P Xy)))
% Found (fun (x00:(Xphi Xy))=> ((((eq_sym fofType) Xy) x1) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x1) Xy)
% Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq fofType) x1) b)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) b)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) b)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_trans0000 ((eq_ref fofType) x1)) ((eq_ref fofType) b)) as proof of (((eq fofType) x1) Xy)
% Found (((eq_trans000 Xy) ((eq_ref fofType) x1)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found ((((fun (b:fofType)=> ((eq_trans00 b) Xy)) Xy) ((eq_ref fofType) x1)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found ((((fun (b:fofType)=> (((eq_trans0 x1) b) Xy)) Xy) ((eq_ref fofType) x1)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found ((((fun (b:fofType)=> ((((eq_trans fofType) x1) b) Xy)) Xy) ((eq_ref fofType) x1)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x1) Xy)
% Found (fun (x00:(Xphi Xy))=> ((((fun (b:fofType)=> ((((eq_trans fofType) x1) b) Xy)) Xy) ((eq_ref fofType) x1)) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x1) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x1)
% Found x6:(P Xy)
% Instantiate: x3:=x1:fofType
% Found (fun (x6:(P Xy))=> x6) as proof of (P x1)
% Found (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6) as proof of ((P Xy)->(P x1))
% Found (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6) as proof of (((eq fofType) Xy) x1)
% Found (x5 (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6)) as proof of (Xphi x3)
% Found (x5 (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6)) as proof of (Xphi x3)
% Found (fun (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> (x5 (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6))) as proof of (Xphi x3)
% Found (fun (x4:((Xphi Xy)->(((eq fofType) Xy) x1))) (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> (x5 (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6))) as proof of (((((eq fofType) Xy) x1)->(Xphi Xy))->(Xphi x3))
% Found (fun (x4:((Xphi Xy)->(((eq fofType) Xy) x1))) (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> (x5 (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6))) as proof of (((Xphi Xy)->(((eq fofType) Xy) x1))->(((((eq fofType) Xy) x1)->(Xphi Xy))->(Xphi x3)))
% Found (and_rect00 (fun (x4:((Xphi Xy)->(((eq fofType) Xy) x1))) (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> (x5 (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6)))) as proof of (Xphi x3)
% Found ((and_rect0 (Xphi x3)) (fun (x4:((Xphi Xy)->(((eq fofType) Xy) x1))) (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> (x5 (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6)))) as proof of (Xphi x3)
% Found (((fun (P:Type) (x4:(((Xphi Xy)->(((eq fofType) Xy) x1))->(((((eq fofType) Xy) x1)->(Xphi Xy))->P)))=> (((((and_rect ((Xphi Xy)->(((eq fofType) Xy) x1))) ((((eq fofType) Xy) x1)->(Xphi Xy))) P) x4) x20)) (Xphi x3)) (fun (x4:((Xphi Xy)->(((eq fofType) Xy) x1))) (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> (x5 (fun (P:(fofType->Prop)) (x6:(P Xy))=> x6)))) as proof of (Xphi x3)
% Found (((fun (P:Type) (x4:(((Xphi x3)->(((eq fofType) x3) x1))->(((((eq fofType) x3) x1)->(Xphi x3))->P)))=> (((((and_rect ((Xphi x3)->(((eq fofType) x3) x1))) ((((eq fofType) x3) x1)->(Xphi x3))) P) x4) (x2 x3))) (Xphi x3)) (fun (x4:((Xphi x3)->(((eq fofType) x3) x1))) (x5:((((eq fofType) x3) x1)->(Xphi x3)))=> (x5 (fun (P:(fofType->Prop)) (x6:(P x3))=> x6)))) as proof of (Xphi x3)
% Found (((fun (P:Type) (x4:(((Xphi x3)->(((eq fofType) x3) x1))->(((((eq fofType) x3) x1)->(Xphi x3))->P)))=> (((((and_rect ((Xphi x3)->(((eq fofType) x3) x1))) ((((eq fofType) x3) x1)->(Xphi x3))) P) x4) (x2 x3))) (Xphi x3)) (fun (x4:((Xphi x3)->(((eq fofType) x3) x1))) (x5:((((eq fofType) x3) x1)->(Xphi x3)))=> (x5 (fun (P:(fofType->Prop)) (x6:(P x3))=> x6)))) as proof of (Xphi x3)
% Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% Found (eq_ref00 P) as proof of (P0 x3)
% Found ((eq_ref0 x3) P) as proof of (P0 x3)
% Found (((eq_ref fofType) x3) P) as proof of (P0 x3)
% Found (((eq_ref fofType) x3) P) as proof of (P0 x3)
% Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq fofType) x3) b)
% Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) b)
% Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) b)
% Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x3)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x3)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x3)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x3)
% Found (eq_sym000 ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x3)->(P Xy)))
% Found ((eq_sym00 x3) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x3)->(P Xy)))
% Found (((eq_sym0 Xy) x3) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x3)->(P Xy)))
% Found ((((eq_sym fofType) Xy) x3) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x3)->(P Xy)))
% Found ((((eq_sym fofType) Xy) x3) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x3)->(P Xy)))
% Found (fun (x00:(Xphi Xy))=> ((((eq_sym fofType) Xy) x3) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x3) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) x3)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x3)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x3)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x3)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found x40:=(x4 x00):(((eq fofType) Xy0) x1)
% Found (x4 x00) as proof of (((eq fofType) Xy) x3)
% Found (x4 x00) as proof of (((eq fofType) Xy) x3)
% Found (x4 x00) as proof of (((eq fofType) Xy) x3)
% Found (eq_sym000 (x4 x00)) as proof of (((eq fofType) x3) Xy)
% Found ((eq_sym00 x3) (x4 x00)) as proof of (((eq fofType) x3) Xy)
% Found (((eq_sym0 Xy) x3) (x4 x00)) as proof of (((eq fofType) x3) Xy)
% Found ((((eq_sym fofType) Xy) x3) (x4 x00)) as proof of (((eq fofType) x3) Xy)
% Found (fun (x5:((((eq fofType) Xy0) x1)->(Xphi Xy0)))=> ((((eq_sym fofType) Xy) x3) (x4 x00))) as proof of (((eq fofType) x3) Xy)
% Found (fun (x4:((Xphi Xy0)->(((eq fofType) Xy0) x1))) (x5:((((eq fofType) Xy0) x1)->(Xphi Xy0)))=> ((((eq_sym fofType) Xy) x3) (x4 x00))) as proof of (((((eq fofType) Xy0) x1)->(Xphi Xy0))->(((eq fofType) x3) Xy))
% Found (fun (x4:((Xphi Xy0)->(((eq fofType) Xy0) x1))) (x5:((((eq fofType) Xy0) x1)->(Xphi Xy0)))=> ((((eq_sym fofType) Xy) x3) (x4 x00))) as proof of (((Xphi Xy0)->(((eq fofType) Xy0) x1))->(((((eq fofType) Xy0) x1)->(Xphi Xy0))->(((eq fofType) x3) Xy)))
% Found (and_rect00 (fun (x4:((Xphi Xy0)->(((eq fofType) Xy0) x1))) (x5:((((eq fofType) Xy0) x1)->(Xphi Xy0)))=> ((((eq_sym fofType) Xy) x3) (x4 x00)))) as proof of (((eq fofType) x3) Xy)
% Found ((and_rect0 (((eq fofType) x3) Xy)) (fun (x4:((Xphi Xy0)->(((eq fofType) Xy0) x1))) (x5:((((eq fofType) Xy0) x1)->(Xphi Xy0)))=> ((((eq_sym fofType) Xy) x3) (x4 x00)))) as proof of (((eq fofType) x3) Xy)
% Found (((fun (P:Type) (x4:(((Xphi Xy0)->(((eq fofType) Xy0) x1))->(((((eq fofType) Xy0) x1)->(Xphi Xy0))->P)))=> (((((and_rect ((Xphi Xy0)->(((eq fofType) Xy0) x1))) ((((eq fofType) Xy0) x1)->(Xphi Xy0))) P) x4) x20)) (((eq fofType) x3) Xy)) (fun (x4:((Xphi Xy0)->(((eq fofType) Xy0) x1))) (x5:((((eq fofType) Xy0) x1)->(Xphi Xy0)))=> ((((eq_sym fofType) Xy) x3) (x4 x00)))) as proof of (((eq fofType) x3) Xy)
% Found (((fun (P:Type) (x4:(((Xphi Xy)->(((eq fofType) Xy) x1))->(((((eq fofType) Xy) x1)->(Xphi Xy))->P)))=> (((((and_rect ((Xphi Xy)->(((eq fofType) Xy) x1))) ((((eq fofType) Xy) x1)->(Xphi Xy))) P) x4) (x2 Xy))) (((eq fofType) x3) Xy)) (fun (x4:((Xphi Xy)->(((eq fofType) Xy) x1))) (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> ((((eq_sym fofType) Xy) x3) (x4 x00)))) as proof of (((eq fofType) x3) Xy)
% Found (fun (x00:(Xphi Xy))=> (((fun (P:Type) (x4:(((Xphi Xy)->(((eq fofType) Xy) x1))->(((((eq fofType) Xy) x1)->(Xphi Xy))->P)))=> (((((and_rect ((Xphi Xy)->(((eq fofType) Xy) x1))) ((((eq fofType) Xy) x1)->(Xphi Xy))) P) x4) (x2 Xy))) (((eq fofType) x3) Xy)) (fun (x4:((Xphi Xy)->(((eq fofType) Xy) x1))) (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> ((((eq_sym fofType) Xy) x3) (x4 x00))))) as proof of (((eq fofType) x3) Xy)
% Found (fun (Xy:fofType) (x00:(Xphi Xy))=> (((fun (P:Type) (x4:(((Xphi Xy)->(((eq fofType) Xy) x1))->(((((eq fofType) Xy) x1)->(Xphi Xy))->P)))=> (((((and_rect ((Xphi Xy)->(((eq fofType) Xy) x1))) ((((eq fofType) Xy) x1)->(Xphi Xy))) P) x4) (x2 Xy))) (((eq fofType) x3) Xy)) (fun (x4:((Xphi Xy)->(((eq fofType) Xy) x1))) (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> ((((eq_sym fofType) Xy) x3) (x4 x00))))) as proof of ((Xphi Xy)->(((eq fofType) x3) Xy))
% Found (fun (Xy:fofType) (x00:(Xphi Xy))=> (((fun (P:Type) (x4:(((Xphi Xy)->(((eq fofType) Xy) x1))->(((((eq fofType) Xy) x1)->(Xphi Xy))->P)))=> (((((and_rect ((Xphi Xy)->(((eq fofType) Xy) x1))) ((((eq fofType) Xy) x1)->(Xphi Xy))) P) x4) (x2 Xy))) (((eq fofType) x3) Xy)) (fun (x4:((Xphi Xy)->(((eq fofType) Xy) x1))) (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> ((((eq_sym fofType) Xy) x3) (x4 x00))))) as proof of (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x3) Xy)))
% Found ((conj00 (((fun (P:Type) (x4:(((Xphi x3)->(((eq fofType) x3) x1))->(((((eq fofType) x3) x1)->(Xphi x3))->P)))=> (((((and_rect ((Xphi x3)->(((eq fofType) x3) x1))) ((((eq fofType) x3) x1)->(Xphi x3))) P) x4) (x2 x3))) (Xphi x3)) (fun (x4:((Xphi x3)->(((eq fofType) x3) x1))) (x5:((((eq fofType) x3) x1)->(Xphi x3)))=> (x5 (fun (P:(fofType->Prop)) (x6:(P x3))=> x6))))) (fun (Xy:fofType) (x00:(Xphi Xy))=> (((fun (P:Type) (x4:(((Xphi Xy)->(((eq fofType) Xy) x1))->(((((eq fofType) Xy) x1)->(Xphi Xy))->P)))=> (((((and_rect ((Xphi Xy)->(((eq fofType) Xy) x1))) ((((eq fofType) Xy) x1)->(Xphi Xy))) P) x4) (x2 Xy))) (((eq fofType) x3) Xy)) (fun (x4:((Xphi Xy)->(((eq fofType) Xy) x1))) (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> ((((eq_sym fofType) Xy) x3) (x4 x00)))))) as proof of ((and (Xphi x3)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x3) Xy))))
% Found (((conj0 (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x3) Xy)))) (((fun (P:Type) (x4:(((Xphi x3)->(((eq fofType) x3) x1))->(((((eq fofType) x3) x1)->(Xphi x3))->P)))=> (((((and_rect ((Xphi x3)->(((eq fofType) x3) x1))) ((((eq fofType) x3) x1)->(Xphi x3))) P) x4) (x2 x3))) (Xphi x3)) (fun (x4:((Xphi x3)->(((eq fofType) x3) x1))) (x5:((((eq fofType) x3) x1)->(Xphi x3)))=> (x5 (fun (P:(fofType->Prop)) (x6:(P x3))=> x6))))) (fun (Xy:fofType) (x00:(Xphi Xy))=> (((fun (P:Type) (x4:(((Xphi Xy)->(((eq fofType) Xy) x1))->(((((eq fofType) Xy) x1)->(Xphi Xy))->P)))=> (((((and_rect ((Xphi Xy)->(((eq fofType) Xy) x1))) ((((eq fofType) Xy) x1)->(Xphi Xy))) P) x4) (x2 Xy))) (((eq fofType) x3) Xy)) (fun (x4:((Xphi Xy)->(((eq fofType) Xy) x1))) (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> ((((eq_sym fofType) Xy) x3) (x4 x00)))))) as proof of ((and (Xphi x3)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x3) Xy))))
% Found ((((conj (Xphi x3)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x3) Xy)))) (((fun (P:Type) (x4:(((Xphi x3)->(((eq fofType) x3) x1))->(((((eq fofType) x3) x1)->(Xphi x3))->P)))=> (((((and_rect ((Xphi x3)->(((eq fofType) x3) x1))) ((((eq fofType) x3) x1)->(Xphi x3))) P) x4) (x2 x3))) (Xphi x3)) (fun (x4:((Xphi x3)->(((eq fofType) x3) x1))) (x5:((((eq fofType) x3) x1)->(Xphi x3)))=> (x5 (fun (P:(fofType->Prop)) (x6:(P x3))=> x6))))) (fun (Xy:fofType) (x00:(Xphi Xy))=> (((fun (P:Type) (x4:(((Xphi Xy)->(((eq fofType) Xy) x1))->(((((eq fofType) Xy) x1)->(Xphi Xy))->P)))=> (((((and_rect ((Xphi Xy)->(((eq fofType) Xy) x1))) ((((eq fofType) Xy) x1)->(Xphi Xy))) P) x4) (x2 Xy))) (((eq fofType) x3) Xy)) (fun (x4:((Xphi Xy)->(((eq fofType) Xy) x1))) (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> ((((eq_sym fofType) Xy) x3) (x4 x00)))))) as proof of ((and (Xphi x3)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x3) Xy))))
% Found ((((conj (Xphi x3)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x3) Xy)))) (((fun (P:Type) (x4:(((Xphi x3)->(((eq fofType) x3) x1))->(((((eq fofType) x3) x1)->(Xphi x3))->P)))=> (((((and_rect ((Xphi x3)->(((eq fofType) x3) x1))) ((((eq fofType) x3) x1)->(Xphi x3))) P) x4) (x2 x3))) (Xphi x3)) (fun (x4:((Xphi x3)->(((eq fofType) x3) x1))) (x5:((((eq fofType) x3) x1)->(Xphi x3)))=> (x5 (fun (P:(fofType->Prop)) (x6:(P x3))=> x6))))) (fun (Xy:fofType) (x00:(Xphi Xy))=> (((fun (P:Type) (x4:(((Xphi Xy)->(((eq fofType) Xy) x1))->(((((eq fofType) Xy) x1)->(Xphi Xy))->P)))=> (((((and_rect ((Xphi Xy)->(((eq fofType) Xy) x1))) ((((eq fofType) Xy) x1)->(Xphi Xy))) P) x4) (x2 Xy))) (((eq fofType) x3) Xy)) (fun (x4:((Xphi Xy)->(((eq fofType) Xy) x1))) (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> ((((eq_sym fofType) Xy) x3) (x4 x00)))))) as proof of ((and (Xphi x3)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x3) Xy))))
% Found (ex_intro000 ((((conj (Xphi x3)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x3) Xy)))) (((fun (P:Type) (x4:(((Xphi x3)->(((eq fofType) x3) x1))->(((((eq fofType) x3) x1)->(Xphi x3))->P)))=> (((((and_rect ((Xphi x3)->(((eq fofType) x3) x1))) ((((eq fofType) x3) x1)->(Xphi x3))) P) x4) (x2 x3))) (Xphi x3)) (fun (x4:((Xphi x3)->(((eq fofType) x3) x1))) (x5:((((eq fofType) x3) x1)->(Xphi x3)))=> (x5 (fun (P:(fofType->Prop)) (x6:(P x3))=> x6))))) (fun (Xy:fofType) (x00:(Xphi Xy))=> (((fun (P:Type) (x4:(((Xphi Xy)->(((eq fofType) Xy) x1))->(((((eq fofType) Xy) x1)->(Xphi Xy))->P)))=> (((((and_rect ((Xphi Xy)->(((eq fofType) Xy) x1))) ((((eq fofType) Xy) x1)->(Xphi Xy))) P) x4) (x2 Xy))) (((eq fofType) x3) Xy)) (fun (x4:((Xphi Xy)->(((eq fofType) Xy) x1))) (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> ((((eq_sym fofType) Xy) x3) (x4 x00))))))) as proof of (exu (fun (Xx:fofType)=> (Xphi Xx)))
% Found ((ex_intro00 x1) ((((conj (Xphi x1)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x1) Xy)))) (((fun (P:Type) (x4:(((Xphi x1)->(((eq fofType) x1) x1))->(((((eq fofType) x1) x1)->(Xphi x1))->P)))=> (((((and_rect ((Xphi x1)->(((eq fofType) x1) x1))) ((((eq fofType) x1) x1)->(Xphi x1))) P) x4) (x2 x1))) (Xphi x1)) (fun (x4:((Xphi x1)->(((eq fofType) x1) x1))) (x5:((((eq fofType) x1) x1)->(Xphi x1)))=> (x5 (fun (P:(fofType->Prop)) (x6:(P x1))=> x6))))) (fun (Xy:fofType) (x00:(Xphi Xy))=> (((fun (P:Type) (x4:(((Xphi Xy)->(((eq fofType) Xy) x1))->(((((eq fofType) Xy) x1)->(Xphi Xy))->P)))=> (((((and_rect ((Xphi Xy)->(((eq fofType) Xy) x1))) ((((eq fofType) Xy) x1)->(Xphi Xy))) P) x4) (x2 Xy))) (((eq fofType) x1) Xy)) (fun (x4:((Xphi Xy)->(((eq fofType) Xy) x1))) (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> ((((eq_sym fofType) Xy) x1) (x4 x00))))))) as proof of (exu (fun (Xx:fofType)=> (Xphi Xx)))
% Found (((ex_intro0 (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) x1) ((((conj (Xphi x1)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x1) Xy)))) (((fun (P:Type) (x4:(((Xphi x1)->(((eq fofType) x1) x1))->(((((eq fofType) x1) x1)->(Xphi x1))->P)))=> (((((and_rect ((Xphi x1)->(((eq fofType) x1) x1))) ((((eq fofType) x1) x1)->(Xphi x1))) P) x4) (x2 x1))) (Xphi x1)) (fun (x4:((Xphi x1)->(((eq fofType) x1) x1))) (x5:((((eq fofType) x1) x1)->(Xphi x1)))=> (x5 (fun (P:(fofType->Prop)) (x6:(P x1))=> x6))))) (fun (Xy:fofType) (x00:(Xphi Xy))=> (((fun (P:Type) (x4:(((Xphi Xy)->(((eq fofType) Xy) x1))->(((((eq fofType) Xy) x1)->(Xphi Xy))->P)))=> (((((and_rect ((Xphi Xy)->(((eq fofType) Xy) x1))) ((((eq fofType) Xy) x1)->(Xphi Xy))) P) x4) (x2 Xy))) (((eq fofType) x1) Xy)) (fun (x4:((Xphi Xy)->(((eq fofType) Xy) x1))) (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> ((((eq_sym fofType) Xy) x1) (x4 x00))))))) as proof of (exu (fun (Xx:fofType)=> (Xphi Xx)))
% Found ((((ex_intro fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) x1) ((((conj (Xphi x1)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x1) Xy)))) (((fun (P:Type) (x4:(((Xphi x1)->(((eq fofType) x1) x1))->(((((eq fofType) x1) x1)->(Xphi x1))->P)))=> (((((and_rect ((Xphi x1)->(((eq fofType) x1) x1))) ((((eq fofType) x1) x1)->(Xphi x1))) P) x4) (x2 x1))) (Xphi x1)) (fun (x4:((Xphi x1)->(((eq fofType) x1) x1))) (x5:((((eq fofType) x1) x1)->(Xphi x1)))=> (x5 (fun (P:(fofType->Prop)) (x6:(P x1))=> x6))))) (fun (Xy:fofType) (x00:(Xphi Xy))=> (((fun (P:Type) (x4:(((Xphi Xy)->(((eq fofType) Xy) x1))->(((((eq fofType) Xy) x1)->(Xphi Xy))->P)))=> (((((and_rect ((Xphi Xy)->(((eq fofType) Xy) x1))) ((((eq fofType) Xy) x1)->(Xphi Xy))) P) x4) (x2 Xy))) (((eq fofType) x1) Xy)) (fun (x4:((Xphi Xy)->(((eq fofType) Xy) x1))) (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> ((((eq_sym fofType) Xy) x1) (x4 x00))))))) as proof of (exu (fun (Xx:fofType)=> (Xphi Xx)))
% Found (fun (x2:(forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x1))))=> ((((ex_intro fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) x1) ((((conj (Xphi x1)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x1) Xy)))) (((fun (P:Type) (x4:(((Xphi x1)->(((eq fofType) x1) x1))->(((((eq fofType) x1) x1)->(Xphi x1))->P)))=> (((((and_rect ((Xphi x1)->(((eq fofType) x1) x1))) ((((eq fofType) x1) x1)->(Xphi x1))) P) x4) (x2 x1))) (Xphi x1)) (fun (x4:((Xphi x1)->(((eq fofType) x1) x1))) (x5:((((eq fofType) x1) x1)->(Xphi x1)))=> (x5 (fun (P:(fofType->Prop)) (x6:(P x1))=> x6))))) (fun (Xy:fofType) (x00:(Xphi Xy))=> (((fun (P:Type) (x4:(((Xphi Xy)->(((eq fofType) Xy) x1))->(((((eq fofType) Xy) x1)->(Xphi Xy))->P)))=> (((((and_rect ((Xphi Xy)->(((eq fofType) Xy) x1))) ((((eq fofType) Xy) x1)->(Xphi Xy))) P) x4) (x2 Xy))) (((eq fofType) x1) Xy)) (fun (x4:((Xphi Xy)->(((eq fofType) Xy) x1))) (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> ((((eq_sym fofType) Xy) x1) (x4 x00)))))))) as proof of (exu (fun (Xx:fofType)=> (Xphi Xx)))
% Found (fun (x1:fofType) (x2:(forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x1))))=> ((((ex_intro fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) x1) ((((conj (Xphi x1)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x1) Xy)))) (((fun (P:Type) (x4:(((Xphi x1)->(((eq fofType) x1) x1))->(((((eq fofType) x1) x1)->(Xphi x1))->P)))=> (((((and_rect ((Xphi x1)->(((eq fofType) x1) x1))) ((((eq fofType) x1) x1)->(Xphi x1))) P) x4) (x2 x1))) (Xphi x1)) (fun (x4:((Xphi x1)->(((eq fofType) x1) x1))) (x5:((((eq fofType) x1) x1)->(Xphi x1)))=> (x5 (fun (P:(fofType->Prop)) (x6:(P x1))=> x6))))) (fun (Xy:fofType) (x00:(Xphi Xy))=> (((fun (P:Type) (x4:(((Xphi Xy)->(((eq fofType) Xy) x1))->(((((eq fofType) Xy) x1)->(Xphi Xy))->P)))=> (((((and_rect ((Xphi Xy)->(((eq fofType) Xy) x1))) ((((eq fofType) Xy) x1)->(Xphi Xy))) P) x4) (x2 Xy))) (((eq fofType) x1) Xy)) (fun (x4:((Xphi Xy)->(((eq fofType) Xy) x1))) (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> ((((eq_sym fofType) Xy) x1) (x4 x00)))))))) as proof of ((forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x1)))->(exu (fun (Xx:fofType)=> (Xphi Xx))))
% Found (fun (x1:fofType) (x2:(forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x1))))=> ((((ex_intro fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) x1) ((((conj (Xphi x1)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x1) Xy)))) (((fun (P:Type) (x4:(((Xphi x1)->(((eq fofType) x1) x1))->(((((eq fofType) x1) x1)->(Xphi x1))->P)))=> (((((and_rect ((Xphi x1)->(((eq fofType) x1) x1))) ((((eq fofType) x1) x1)->(Xphi x1))) P) x4) (x2 x1))) (Xphi x1)) (fun (x4:((Xphi x1)->(((eq fofType) x1) x1))) (x5:((((eq fofType) x1) x1)->(Xphi x1)))=> (x5 (fun (P:(fofType->Prop)) (x6:(P x1))=> x6))))) (fun (Xy:fofType) (x00:(Xphi Xy))=> (((fun (P:Type) (x4:(((Xphi Xy)->(((eq fofType) Xy) x1))->(((((eq fofType) Xy) x1)->(Xphi Xy))->P)))=> (((((and_rect ((Xphi Xy)->(((eq fofType) Xy) x1))) ((((eq fofType) Xy) x1)->(Xphi Xy))) P) x4) (x2 Xy))) (((eq fofType) x1) Xy)) (fun (x4:((Xphi Xy)->(((eq fofType) Xy) x1))) (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> ((((eq_sym fofType) Xy) x1) (x4 x00)))))))) as proof of (forall (x:fofType), ((forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x)))->(exu (fun (Xx:fofType)=> (Xphi Xx)))))
% Found (ex_ind00 (fun (x1:fofType) (x2:(forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x1))))=> ((((ex_intro fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) x1) ((((conj (Xphi x1)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x1) Xy)))) (((fun (P:Type) (x4:(((Xphi x1)->(((eq fofType) x1) x1))->(((((eq fofType) x1) x1)->(Xphi x1))->P)))=> (((((and_rect ((Xphi x1)->(((eq fofType) x1) x1))) ((((eq fofType) x1) x1)->(Xphi x1))) P) x4) (x2 x1))) (Xphi x1)) (fun (x4:((Xphi x1)->(((eq fofType) x1) x1))) (x5:((((eq fofType) x1) x1)->(Xphi x1)))=> (x5 (fun (P:(fofType->Prop)) (x6:(P x1))=> x6))))) (fun (Xy:fofType) (x00:(Xphi Xy))=> (((fun (P:Type) (x4:(((Xphi Xy)->(((eq fofType) Xy) x1))->(((((eq fofType) Xy) x1)->(Xphi Xy))->P)))=> (((((and_rect ((Xphi Xy)->(((eq fofType) Xy) x1))) ((((eq fofType) Xy) x1)->(Xphi Xy))) P) x4) (x2 Xy))) (((eq fofType) x1) Xy)) (fun (x4:((Xphi Xy)->(((eq fofType) Xy) x1))) (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> ((((eq_sym fofType) Xy) x1) (x4 x00))))))))) as proof of (exu (fun (Xx:fofType)=> (Xphi Xx)))
% Found ((ex_ind0 (exu (fun (Xx:fofType)=> (Xphi Xx)))) (fun (x1:fofType) (x2:(forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x1))))=> ((((ex_intro fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) x1) ((((conj (Xphi x1)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x1) Xy)))) (((fun (P:Type) (x4:(((Xphi x1)->(((eq fofType) x1) x1))->(((((eq fofType) x1) x1)->(Xphi x1))->P)))=> (((((and_rect ((Xphi x1)->(((eq fofType) x1) x1))) ((((eq fofType) x1) x1)->(Xphi x1))) P) x4) (x2 x1))) (Xphi x1)) (fun (x4:((Xphi x1)->(((eq fofType) x1) x1))) (x5:((((eq fofType) x1) x1)->(Xphi x1)))=> (x5 (fun (P:(fofType->Prop)) (x6:(P x1))=> x6))))) (fun (Xy:fofType) (x00:(Xphi Xy))=> (((fun (P:Type) (x4:(((Xphi Xy)->(((eq fofType) Xy) x1))->(((((eq fofType) Xy) x1)->(Xphi Xy))->P)))=> (((((and_rect ((Xphi Xy)->(((eq fofType) Xy) x1))) ((((eq fofType) Xy) x1)->(Xphi Xy))) P) x4) (x2 Xy))) (((eq fofType) x1) Xy)) (fun (x4:((Xphi Xy)->(((eq fofType) Xy) x1))) (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> ((((eq_sym fofType) Xy) x1) (x4 x00))))))))) as proof of (exu (fun (Xx:fofType)=> (Xphi Xx)))
% Found (((fun (P:Prop) (x1:(forall (x:fofType), ((forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x)))->P)))=> (((((ex_ind fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) P) x1) x0)) (exu (fun (Xx:fofType)=> (Xphi Xx)))) (fun (x1:fofType) (x2:(forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x1))))=> ((((ex_intro fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) x1) ((((conj (Xphi x1)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x1) Xy)))) (((fun (P:Type) (x4:(((Xphi x1)->(((eq fofType) x1) x1))->(((((eq fofType) x1) x1)->(Xphi x1))->P)))=> (((((and_rect ((Xphi x1)->(((eq fofType) x1) x1))) ((((eq fofType) x1) x1)->(Xphi x1))) P) x4) (x2 x1))) (Xphi x1)) (fun (x4:((Xphi x1)->(((eq fofType) x1) x1))) (x5:((((eq fofType) x1) x1)->(Xphi x1)))=> (x5 (fun (P:(fofType->Prop)) (x6:(P x1))=> x6))))) (fun (Xy:fofType) (x00:(Xphi Xy))=> (((fun (P:Type) (x4:(((Xphi Xy)->(((eq fofType) Xy) x1))->(((((eq fofType) Xy) x1)->(Xphi Xy))->P)))=> (((((and_rect ((Xphi Xy)->(((eq fofType) Xy) x1))) ((((eq fofType) Xy) x1)->(Xphi Xy))) P) x4) (x2 Xy))) (((eq fofType) x1) Xy)) (fun (x4:((Xphi Xy)->(((eq fofType) Xy) x1))) (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> ((((eq_sym fofType) Xy) x1) (x4 x00))))))))) as proof of (exu (fun (Xx:fofType)=> (Xphi Xx)))
% Found (fun (x0:((ex fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))))=> (((fun (P:Prop) (x1:(forall (x:fofType), ((forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x)))->P)))=> (((((ex_ind fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) P) x1) x0)) (exu (fun (Xx:fofType)=> (Xphi Xx)))) (fun (x1:fofType) (x2:(forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x1))))=> ((((ex_intro fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) x1) ((((conj (Xphi x1)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x1) Xy)))) (((fun (P:Type) (x4:(((Xphi x1)->(((eq fofType) x1) x1))->(((((eq fofType) x1) x1)->(Xphi x1))->P)))=> (((((and_rect ((Xphi x1)->(((eq fofType) x1) x1))) ((((eq fofType) x1) x1)->(Xphi x1))) P) x4) (x2 x1))) (Xphi x1)) (fun (x4:((Xphi x1)->(((eq fofType) x1) x1))) (x5:((((eq fofType) x1) x1)->(Xphi x1)))=> (x5 (fun (P:(fofType->Prop)) (x6:(P x1))=> x6))))) (fun (Xy:fofType) (x00:(Xphi Xy))=> (((fun (P:Type) (x4:(((Xphi Xy)->(((eq fofType) Xy) x1))->(((((eq fofType) Xy) x1)->(Xphi Xy))->P)))=> (((((and_rect ((Xphi Xy)->(((eq fofType) Xy) x1))) ((((eq fofType) Xy) x1)->(Xphi Xy))) P) x4) (x2 Xy))) (((eq fofType) x1) Xy)) (fun (x4:((Xphi Xy)->(((eq fofType) Xy) x1))) (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> ((((eq_sym fofType) Xy) x1) (x4 x00)))))))))) as proof of (exu (fun (Xx:fofType)=> (Xphi Xx)))
% Found (fun (Xphi:(fofType->Prop)) (x0:((ex fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))))=> (((fun (P:Prop) (x1:(forall (x:fofType), ((forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x)))->P)))=> (((((ex_ind fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) P) x1) x0)) (exu (fun (Xx:fofType)=> (Xphi Xx)))) (fun (x1:fofType) (x2:(forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x1))))=> ((((ex_intro fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) x1) ((((conj (Xphi x1)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x1) Xy)))) (((fun (P:Type) (x4:(((Xphi x1)->(((eq fofType) x1) x1))->(((((eq fofType) x1) x1)->(Xphi x1))->P)))=> (((((and_rect ((Xphi x1)->(((eq fofType) x1) x1))) ((((eq fofType) x1) x1)->(Xphi x1))) P) x4) (x2 x1))) (Xphi x1)) (fun (x4:((Xphi x1)->(((eq fofType) x1) x1))) (x5:((((eq fofType) x1) x1)->(Xphi x1)))=> (x5 (fun (P:(fofType->Prop)) (x6:(P x1))=> x6))))) (fun (Xy:fofType) (x00:(Xphi Xy))=> (((fun (P:Type) (x4:(((Xphi Xy)->(((eq fofType) Xy) x1))->(((((eq fofType) Xy) x1)->(Xphi Xy))->P)))=> (((((and_rect ((Xphi Xy)->(((eq fofType) Xy) x1))) ((((eq fofType) Xy) x1)->(Xphi Xy))) P) x4) (x2 Xy))) (((eq fofType) x1) Xy)) (fun (x4:((Xphi Xy)->(((eq fofType) Xy) x1))) (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> ((((eq_sym fofType) Xy) x1) (x4 x00)))))))))) as proof of (((ex fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))->(exu (fun (Xx:fofType)=> (Xphi Xx))))
% Found (fun (x:exuI1) (Xphi:(fofType->Prop)) (x0:((ex fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))))=> (((fun (P:Prop) (x1:(forall (x:fofType), ((forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x)))->P)))=> (((((ex_ind fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) P) x1) x0)) (exu (fun (Xx:fofType)=> (Xphi Xx)))) (fun (x1:fofType) (x2:(forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x1))))=> ((((ex_intro fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) x1) ((((conj (Xphi x1)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x1) Xy)))) (((fun (P:Type) (x4:(((Xphi x1)->(((eq fofType) x1) x1))->(((((eq fofType) x1) x1)->(Xphi x1))->P)))=> (((((and_rect ((Xphi x1)->(((eq fofType) x1) x1))) ((((eq fofType) x1) x1)->(Xphi x1))) P) x4) (x2 x1))) (Xphi x1)) (fun (x4:((Xphi x1)->(((eq fofType) x1) x1))) (x5:((((eq fofType) x1) x1)->(Xphi x1)))=> (x5 (fun (P:(fofType->Prop)) (x6:(P x1))=> x6))))) (fun (Xy:fofType) (x00:(Xphi Xy))=> (((fun (P:Type) (x4:(((Xphi Xy)->(((eq fofType) Xy) x1))->(((((eq fofType) Xy) x1)->(Xphi Xy))->P)))=> (((((and_rect ((Xphi Xy)->(((eq fofType) Xy) x1))) ((((eq fofType) Xy) x1)->(Xphi Xy))) P) x4) (x2 Xy))) (((eq fofType) x1) Xy)) (fun (x4:((Xphi Xy)->(((eq fofType) Xy) x1))) (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> ((((eq_sym fofType) Xy) x1) (x4 x00)))))))))) as proof of (forall (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))->(exu (fun (Xx:fofType)=> (Xphi Xx)))))
% Found (fun (x:exuI1) (Xphi:(fofType->Prop)) (x0:((ex fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))))=> (((fun (P:Prop) (x1:(forall (x:fofType), ((forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x)))->P)))=> (((((ex_ind fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) P) x1) x0)) (exu (fun (Xx:fofType)=> (Xphi Xx)))) (fun (x1:fofType) (x2:(forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x1))))=> ((((ex_intro fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) x1) ((((conj (Xphi x1)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x1) Xy)))) (((fun (P:Type) (x4:(((Xphi x1)->(((eq fofType) x1) x1))->(((((eq fofType) x1) x1)->(Xphi x1))->P)))=> (((((and_rect ((Xphi x1)->(((eq fofType) x1) x1))) ((((eq fofType) x1) x1)->(Xphi x1))) P) x4) (x2 x1))) (Xphi x1)) (fun (x4:((Xphi x1)->(((eq fofType) x1) x1))) (x5:((((eq fofType) x1) x1)->(Xphi x1)))=> (x5 (fun (P:(fofType->Prop)) (x6:(P x1))=> x6))))) (fun (Xy:fofType) (x00:(Xphi Xy))=> (((fun (P:Type) (x4:(((Xphi Xy)->(((eq fofType) Xy) x1))->(((((eq fofType) Xy) x1)->(Xphi Xy))->P)))=> (((((and_rect ((Xphi Xy)->(((eq fofType) Xy) x1))) ((((eq fofType) Xy) x1)->(Xphi Xy))) P) x4) (x2 Xy))) (((eq fofType) x1) Xy)) (fun (x4:((Xphi Xy)->(((eq fofType) Xy) x1))) (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> ((((eq_sym fofType) Xy) x1) (x4 x00)))))))))) as proof of (exuI1->(forall (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))->(exu (fun (Xx:fofType)=> (Xphi Xx))))))
% Got proof (fun (x:exuI1) (Xphi:(fofType->Prop)) (x0:((ex fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))))=> (((fun (P:Prop) (x1:(forall (x:fofType), ((forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x)))->P)))=> (((((ex_ind fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) P) x1) x0)) (exu (fun (Xx:fofType)=> (Xphi Xx)))) (fun (x1:fofType) (x2:(forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x1))))=> ((((ex_intro fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) x1) ((((conj (Xphi x1)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x1) Xy)))) (((fun (P:Type) (x4:(((Xphi x1)->(((eq fofType) x1) x1))->(((((eq fofType) x1) x1)->(Xphi x1))->P)))=> (((((and_rect ((Xphi x1)->(((eq fofType) x1) x1))) ((((eq fofType) x1) x1)->(Xphi x1))) P) x4) (x2 x1))) (Xphi x1)) (fun (x4:((Xphi x1)->(((eq fofType) x1) x1))) (x5:((((eq fofType) x1) x1)->(Xphi x1)))=> (x5 (fun (P:(fofType->Prop)) (x6:(P x1))=> x6))))) (fun (Xy:fofType) (x00:(Xphi Xy))=> (((fun (P:Type) (x4:(((Xphi Xy)->(((eq fofType) Xy) x1))->(((((eq fofType) Xy) x1)->(Xphi Xy))->P)))=> (((((and_rect ((Xphi Xy)->(((eq fofType) Xy) x1))) ((((eq fofType) Xy) x1)->(Xphi Xy))) P) x4) (x2 Xy))) (((eq fofType) x1) Xy)) (fun (x4:((Xphi Xy)->(((eq fofType) Xy) x1))) (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> ((((eq_sym fofType) Xy) x1) (x4 x00))))))))))
% Time elapsed = 135.684571s
% node=19624 cost=1922.000000 depth=34
% ::::::::::::::::::::::
% % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p
% % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% (fun (x:exuI1) (Xphi:(fofType->Prop)) (x0:((ex fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))))=> (((fun (P:Prop) (x1:(forall (x:fofType), ((forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x)))->P)))=> (((((ex_ind fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))) P) x1) x0)) (exu (fun (Xx:fofType)=> (Xphi Xx)))) (fun (x1:fofType) (x2:(forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) x1))))=> ((((ex_intro fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))) x1) ((((conj (Xphi x1)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) x1) Xy)))) (((fun (P:Type) (x4:(((Xphi x1)->(((eq fofType) x1) x1))->(((((eq fofType) x1) x1)->(Xphi x1))->P)))=> (((((and_rect ((Xphi x1)->(((eq fofType) x1) x1))) ((((eq fofType) x1) x1)->(Xphi x1))) P) x4) (x2 x1))) (Xphi x1)) (fun (x4:((Xphi x1)->(((eq fofType) x1) x1))) (x5:((((eq fofType) x1) x1)->(Xphi x1)))=> (x5 (fun (P:(fofType->Prop)) (x6:(P x1))=> x6))))) (fun (Xy:fofType) (x00:(Xphi Xy))=> (((fun (P:Type) (x4:(((Xphi Xy)->(((eq fofType) Xy) x1))->(((((eq fofType) Xy) x1)->(Xphi Xy))->P)))=> (((((and_rect ((Xphi Xy)->(((eq fofType) Xy) x1))) ((((eq fofType) Xy) x1)->(Xphi Xy))) P) x4) (x2 Xy))) (((eq fofType) x1) Xy)) (fun (x4:((Xphi Xy)->(((eq fofType) Xy) x1))) (x5:((((eq fofType) Xy) x1)->(Xphi Xy)))=> ((((eq_sym fofType) Xy) x1) (x4 x00))))))))))
% % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% EOF
%------------------------------------------------------------------------------