TSTP Solution File: SYO517^1 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SYO517^1 : TPTP v7.5.0. Released v4.1.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n002.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 0s
% DateTime : Tue Mar 29 00:52:00 EDT 2022
% Result : Timeout 288.76s 289.05s
% Output : None
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----No solution output by system
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SYO517^1 : TPTP v7.5.0. Released v4.1.0.
% 0.00/0.11 % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.12/0.32 % Computer : n002.cluster.edu
% 0.12/0.32 % Model : x86_64 x86_64
% 0.12/0.32 % CPUModel : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.32 % RAMPerCPU : 8042.1875MB
% 0.12/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.32 % CPULimit : 300
% 0.12/0.32 % DateTime : Sun Mar 13 15:11:41 EDT 2022
% 0.12/0.32 % CPUTime :
% 0.12/0.33 ModuleCmd_Load.c(213):ERROR:105: Unable to locate a modulefile for 'python/python27'
% 0.12/0.33 Python 2.7.5
% 1.25/1.42 Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% 1.25/1.42 FOF formula ((ex ((fofType->Prop)->fofType)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) of role conjecture named descri
% 1.25/1.42 Conjecture to prove = ((ex ((fofType->Prop)->fofType)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))):Prop
% 1.25/1.42 Parameter fofType_DUMMY:fofType.
% 1.25/1.42 We need to prove ['((ex ((fofType->Prop)->fofType)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P))))))']
% 1.25/1.42 Parameter fofType:Type.
% 1.25/1.42 Trying to prove ((ex ((fofType->Prop)->fofType)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P))))))
% 1.25/1.42 Found eta_expansion000:=(eta_expansion00 (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))):(((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) (fun (x:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (x P))))))
% 1.25/1.42 Found (eta_expansion00 (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) b)
% 1.25/1.42 Found ((eta_expansion0 Prop) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) b)
% 1.25/1.42 Found (((eta_expansion ((fofType->Prop)->fofType)) Prop) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) b)
% 1.25/1.42 Found (((eta_expansion ((fofType->Prop)->fofType)) Prop) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) b)
% 1.25/1.42 Found (((eta_expansion ((fofType->Prop)->fofType)) Prop) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) b)
% 1.48/1.69 Found eq_ref00:=(eq_ref0 (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))):(((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P))))))
% 1.48/1.69 Found (eq_ref0 (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))) b)
% 1.48/1.69 Found ((eq_ref (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))) b)
% 1.48/1.69 Found ((eq_ref (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))) b)
% 1.48/1.69 Found ((eq_ref (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))) b)
% 1.48/1.69 Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% 1.48/1.69 Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (x P)))))
% 1.48/1.69 Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (x P)))))
% 1.48/1.69 Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (x P)))))
% 1.48/1.69 Found (fun (x:((fofType->Prop)->fofType))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (x P)))))
% 1.48/1.69 Found (fun (x:((fofType->Prop)->fofType))=> ((eq_ref Prop) (f x))) as proof of (forall (x:((fofType->Prop)->fofType)), (((eq Prop) (f x)) (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (x P))))))
% 1.48/1.69 Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% 1.48/1.69 Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (x P)))))
% 1.48/1.69 Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (x P)))))
% 1.48/1.69 Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (x P)))))
% 1.48/1.69 Found (fun (x:((fofType->Prop)->fofType))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (x P)))))
% 1.48/1.69 Found (fun (x:((fofType->Prop)->fofType))=> ((eq_ref Prop) (f x))) as proof of (forall (x:((fofType->Prop)->fofType)), (((eq Prop) (f x)) (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (x P))))))
% 2.16/2.39 Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% 2.16/2.39 Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (x P)))))
% 2.16/2.39 Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (x P)))))
% 2.16/2.39 Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (x P)))))
% 2.16/2.39 Found (fun (x:((fofType->Prop)->fofType))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (x P)))))
% 2.16/2.39 Found (fun (x:((fofType->Prop)->fofType))=> ((eq_ref Prop) (f x))) as proof of (forall (x:((fofType->Prop)->fofType)), (((eq Prop) (f x)) (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (x P))))))
% 2.16/2.39 Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% 2.16/2.39 Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (x P)))))
% 2.16/2.39 Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (x P)))))
% 2.16/2.39 Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (x P)))))
% 2.16/2.39 Found (fun (x:((fofType->Prop)->fofType))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (x P)))))
% 2.16/2.39 Found (fun (x:((fofType->Prop)->fofType))=> ((eq_ref Prop) (f x))) as proof of (forall (x:((fofType->Prop)->fofType)), (((eq Prop) (f x)) (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (x P))))))
% 2.16/2.39 Found fofType_DUMMY:fofType
% 2.16/2.39 Found fofType_DUMMY as proof of fofType
% 2.16/2.39 Found (choice_operator0 fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) b)
% 2.16/2.39 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) b)
% 2.16/2.39 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) b)
% 2.16/2.39 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) b)
% 2.16/2.39 Found ((choice_operator fofType) fofType_DUMMY) as proof of (P b)
% 2.16/2.39 Found fofType_DUMMY:fofType
% 2.16/2.39 Found fofType_DUMMY as proof of fofType
% 2.16/2.39 Found (choice_operator0 fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f)
% 2.16/2.39 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f)
% 2.16/2.39 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f)
% 2.16/2.39 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f)
% 2.16/2.39 Found ((choice_operator fofType) fofType_DUMMY) as proof of (P f)
% 2.16/2.39 Found fofType_DUMMY:fofType
% 2.16/2.39 Found fofType_DUMMY as proof of fofType
% 2.16/2.39 Found (choice_operator0 fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f)
% 2.16/2.39 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f)
% 2.16/2.39 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f)
% 2.16/2.39 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f)
% 2.16/2.39 Found ((choice_operator fofType) fofType_DUMMY) as proof of (P f)
% 2.16/2.39 Found fofType_DUMMY:fofType
% 2.16/2.39 Found fofType_DUMMY as proof of fofType
% 2.16/2.39 Found (choice_operator0 fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) b)
% 2.16/2.39 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) b)
% 2.16/2.39 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) b)
% 2.16/2.39 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) b)
% 2.16/2.39 Found ((choice_operator fofType) fofType_DUMMY) as proof of (P b)
% 2.16/2.39 Found fofType_DUMMY:fofType
% 2.16/2.39 Found fofType_DUMMY as proof of fofType
% 2.16/2.39 Found (choice_operator0 fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f)
% 2.16/2.39 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f)
% 2.45/2.60 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f)
% 2.45/2.60 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f)
% 2.45/2.60 Found ((choice_operator fofType) fofType_DUMMY) as proof of (P f)
% 2.45/2.60 Found fofType_DUMMY:fofType
% 2.45/2.60 Found fofType_DUMMY as proof of fofType
% 2.45/2.60 Found (choice_operator0 fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f)
% 2.45/2.60 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f)
% 2.45/2.60 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f)
% 2.45/2.60 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f)
% 2.45/2.60 Found ((choice_operator fofType) fofType_DUMMY) as proof of (P f)
% 2.45/2.60 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (((fofType->Prop)->fofType)->Prop)) a) (fun (x:((fofType->Prop)->fofType))=> (a x)))
% 2.45/2.60 Found (eta_expansion_dep00 a) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) a) b)
% 2.45/2.60 Found ((eta_expansion_dep0 (fun (x1:((fofType->Prop)->fofType))=> Prop)) a) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) a) b)
% 2.45/2.60 Found (((eta_expansion_dep ((fofType->Prop)->fofType)) (fun (x1:((fofType->Prop)->fofType))=> Prop)) a) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) a) b)
% 2.45/2.60 Found (((eta_expansion_dep ((fofType->Prop)->fofType)) (fun (x1:((fofType->Prop)->fofType))=> Prop)) a) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) a) b)
% 2.45/2.60 Found (((eta_expansion_dep ((fofType->Prop)->fofType)) (fun (x1:((fofType->Prop)->fofType))=> Prop)) a) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) a) b)
% 2.45/2.60 Found eta_expansion000:=(eta_expansion00 b):(((eq (((fofType->Prop)->fofType)->Prop)) b) (fun (x:((fofType->Prop)->fofType))=> (b x)))
% 2.45/2.60 Found (eta_expansion00 b) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) b) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P))))))
% 2.46/2.60 Found ((eta_expansion0 Prop) b) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) b) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P))))))
% 2.46/2.60 Found (((eta_expansion ((fofType->Prop)->fofType)) Prop) b) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) b) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P))))))
% 2.46/2.60 Found (((eta_expansion ((fofType->Prop)->fofType)) Prop) b) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) b) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P))))))
% 2.46/2.60 Found (((eta_expansion ((fofType->Prop)->fofType)) Prop) b) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) b) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P))))))
% 2.46/2.60 Found eta_expansion000:=(eta_expansion00 b):(((eq (((fofType->Prop)->fofType)->Prop)) b) (fun (x:((fofType->Prop)->fofType))=> (b x)))
% 2.46/2.60 Found (eta_expansion00 b) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) b) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P))))))
% 2.46/2.60 Found ((eta_expansion0 Prop) b) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) b) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P))))))
% 2.46/2.60 Found (((eta_expansion ((fofType->Prop)->fofType)) Prop) b) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) b) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P))))))
% 3.54/3.73 Found (((eta_expansion ((fofType->Prop)->fofType)) Prop) b) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) b) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P))))))
% 3.54/3.73 Found (((eta_expansion ((fofType->Prop)->fofType)) Prop) b) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) b) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P))))))
% 3.54/3.73 Found eq_ref00:=(eq_ref0 a):(((eq (((fofType->Prop)->fofType)->Prop)) a) a)
% 3.54/3.73 Found (eq_ref0 a) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) a) b)
% 3.54/3.73 Found ((eq_ref (((fofType->Prop)->fofType)->Prop)) a) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) a) b)
% 3.54/3.73 Found ((eq_ref (((fofType->Prop)->fofType)->Prop)) a) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) a) b)
% 3.54/3.73 Found ((eq_ref (((fofType->Prop)->fofType)->Prop)) a) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) a) b)
% 3.54/3.73 Found eta_expansion000:=(eta_expansion00 b):(((eq (((fofType->Prop)->fofType)->Prop)) b) (fun (x:((fofType->Prop)->fofType))=> (b x)))
% 3.54/3.73 Found (eta_expansion00 b) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) b) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P))))))
% 3.54/3.73 Found ((eta_expansion0 Prop) b) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) b) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P))))))
% 3.54/3.73 Found (((eta_expansion ((fofType->Prop)->fofType)) Prop) b) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) b) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P))))))
% 3.54/3.73 Found (((eta_expansion ((fofType->Prop)->fofType)) Prop) b) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) b) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P))))))
% 3.54/3.73 Found (((eta_expansion ((fofType->Prop)->fofType)) Prop) b) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) b) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P))))))
% 3.54/3.73 Found eq_ref00:=(eq_ref0 (x P)):(((eq fofType) (x P)) (x P))
% 3.54/3.73 Found (eq_ref0 (x P)) as proof of (((eq fofType) (x P)) b)
% 3.54/3.73 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b)
% 3.54/3.73 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b)
% 3.54/3.73 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b)
% 3.54/3.73 Found eq_ref00:=(eq_ref0 (x P)):(((eq fofType) (x P)) (x P))
% 3.54/3.73 Found (eq_ref0 (x P)) as proof of (((eq fofType) (x P)) b)
% 3.54/3.73 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b)
% 3.54/3.73 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b)
% 3.54/3.73 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b)
% 3.54/3.73 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 3.54/3.73 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) P)
% 3.54/3.73 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P)
% 3.54/3.73 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P)
% 3.54/3.73 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P)
% 3.54/3.73 Found x01:(P x00)
% 3.54/3.73 Instantiate: b:=x00:fofType
% 3.54/3.73 Found x01 as proof of (P0 b)
% 3.54/3.73 Found eq_ref00:=(eq_ref0 (x P)):(((eq fofType) (x P)) (x P))
% 3.54/3.73 Found (eq_ref0 (x P)) as proof of (((eq fofType) (x P)) b)
% 3.54/3.73 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b)
% 3.54/3.73 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b)
% 3.54/3.73 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b)
% 3.54/3.73 Found ((eq_sym0000 ((eq_ref fofType) (x P))) x01) as proof of (P (x P))
% 3.54/3.73 Found ((eq_sym0000 ((eq_ref fofType) (x P))) x01) as proof of (P (x P))
% 3.54/3.73 Found (((fun (x03:(((eq fofType) (x P)) b))=> ((eq_sym000 x03) P)) ((eq_ref fofType) (x P))) x01) as proof of (P (x P))
% 3.66/3.82 Found (((fun (x03:(((eq fofType) (x P)) x00))=> (((eq_sym00 x00) x03) P)) ((eq_ref fofType) (x P))) x01) as proof of (P (x P))
% 3.66/3.82 Found (((fun (x03:(((eq fofType) (x P)) x00))=> ((((eq_sym0 (x P)) x00) x03) P)) ((eq_ref fofType) (x P))) x01) as proof of (P (x P))
% 3.66/3.82 Found (((fun (x03:(((eq fofType) (x P)) x00))=> (((((eq_sym fofType) (x P)) x00) x03) P)) ((eq_ref fofType) (x P))) x01) as proof of (P (x P))
% 3.66/3.82 Found (fun (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x P)) x00))=> (((((eq_sym fofType) (x P)) x00) x03) P)) ((eq_ref fofType) (x P))) x01)) as proof of (P (x P))
% 3.66/3.82 Found (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x P)) x00))=> (((((eq_sym fofType) (x P)) x00) x03) P)) ((eq_ref fofType) (x P))) x01)) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P (x P)))
% 3.66/3.82 Found (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x P)) x00))=> (((((eq_sym fofType) (x P)) x00) x03) P)) ((eq_ref fofType) (x P))) x01)) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P (x P))))
% 3.66/3.82 Found (and_rect00 (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x P)) x00))=> (((((eq_sym fofType) (x P)) x00) x03) P)) ((eq_ref fofType) (x P))) x01))) as proof of (P (x P))
% 3.66/3.82 Found ((and_rect0 (P (x P))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x P)) x00))=> (((((eq_sym fofType) (x P)) x00) x03) P)) ((eq_ref fofType) (x P))) x01))) as proof of (P (x P))
% 3.66/3.82 Found (((fun (P0:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P0)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P0) x2) x1)) (P (x P))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x P)) x00))=> (((((eq_sym fofType) (x P)) x00) x03) P)) ((eq_ref fofType) (x P))) x01))) as proof of (P (x P))
% 3.66/3.82 Found (fun (x1:((and (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))))=> (((fun (P0:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P0)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P0) x2) x1)) (P (x P))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x P)) x00))=> (((((eq_sym fofType) (x P)) x00) x03) P)) ((eq_ref fofType) (x P))) x01)))) as proof of (P (x P))
% 3.66/3.82 Found x2:(P x00)
% 3.66/3.82 Instantiate: b:=x00:fofType
% 3.66/3.82 Found (fun (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of (P0 b)
% 3.66/3.82 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P0 b))
% 3.66/3.82 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P0 b)))
% 3.66/3.82 Found (and_rect00 (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P0 b)
% 3.66/3.82 Found ((and_rect0 (P0 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P0 b)
% 3.66/3.82 Found (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P0 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P0 b)
% 3.66/3.82 Found (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P0 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P0 b)
% 3.66/3.82 Found ((eq_sym0000 ((eq_ref fofType) (x P))) (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P0 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x P))
% 3.68/3.89 Found ((eq_sym0000 ((eq_ref fofType) (x P))) (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P0 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x P))
% 3.68/3.89 Found (((fun (x01:(((eq fofType) (x P)) b))=> ((eq_sym000 x01) P)) ((eq_ref fofType) (x P))) (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x P))
% 3.68/3.89 Found (((fun (x01:(((eq fofType) (x P)) x00))=> (((eq_sym00 x00) x01) P)) ((eq_ref fofType) (x P))) (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x P))
% 3.68/3.89 Found (((fun (x01:(((eq fofType) (x P)) x00))=> ((((eq_sym0 (x P)) x00) x01) P)) ((eq_ref fofType) (x P))) (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x P))
% 3.68/3.89 Found (((fun (x01:(((eq fofType) (x P)) x00))=> (((((eq_sym fofType) (x P)) x00) x01) P)) ((eq_ref fofType) (x P))) (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x P))
% 3.68/3.89 Found (fun (x1:((and (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))))=> (((fun (x01:(((eq fofType) (x P)) x00))=> (((((eq_sym fofType) (x P)) x00) x01) P)) ((eq_ref fofType) (x P))) (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)))) as proof of (P (x P))
% 3.68/3.89 Found x3:(P x1)
% 3.68/3.89 Instantiate: b:=x1:fofType
% 3.68/3.89 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P0 b)
% 3.68/3.89 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P0 b))
% 3.68/3.89 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P0 b)))
% 3.68/3.89 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 b)
% 3.68/3.89 Found ((and_rect0 (P0 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 b)
% 3.68/3.89 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P0 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 b)
% 3.68/3.89 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P0 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P0 b)
% 3.68/3.89 Found x3:(P x1)
% 3.68/3.89 Instantiate: b:=x1:fofType
% 3.94/4.09 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P0 b)
% 3.94/4.09 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P0 b))
% 3.94/4.09 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P0 b)))
% 3.94/4.09 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 b)
% 3.94/4.09 Found ((and_rect0 (P0 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 b)
% 3.94/4.09 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P0 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 b)
% 3.94/4.09 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P0 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P0 b)
% 3.94/4.09 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 3.94/4.09 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) P)
% 3.94/4.09 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P)
% 3.94/4.09 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P)
% 3.94/4.09 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P)
% 3.94/4.09 Found x01:(P x00)
% 3.94/4.09 Instantiate: b:=x00:fofType
% 3.94/4.09 Found x01 as proof of (P0 b)
% 3.94/4.09 Found eq_ref00:=(eq_ref0 (x P)):(((eq fofType) (x P)) (x P))
% 3.94/4.09 Found (eq_ref0 (x P)) as proof of (((eq fofType) (x P)) b)
% 3.94/4.09 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b)
% 3.94/4.09 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b)
% 3.94/4.09 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b)
% 3.94/4.09 Found ((eq_sym0000 ((eq_ref fofType) (x P))) x01) as proof of (P (x P))
% 3.94/4.09 Found ((eq_sym0000 ((eq_ref fofType) (x P))) x01) as proof of (P (x P))
% 3.94/4.09 Found (((fun (x03:(((eq fofType) (x P)) b))=> ((eq_sym000 x03) P)) ((eq_ref fofType) (x P))) x01) as proof of (P (x P))
% 3.94/4.09 Found (((fun (x03:(((eq fofType) (x P)) x00))=> (((eq_sym00 x00) x03) P)) ((eq_ref fofType) (x P))) x01) as proof of (P (x P))
% 3.94/4.09 Found (((fun (x03:(((eq fofType) (x P)) x00))=> ((((eq_sym0 (x P)) x00) x03) P)) ((eq_ref fofType) (x P))) x01) as proof of (P (x P))
% 3.94/4.09 Found (((fun (x03:(((eq fofType) (x P)) x00))=> (((((eq_sym fofType) (x P)) x00) x03) P)) ((eq_ref fofType) (x P))) x01) as proof of (P (x P))
% 3.94/4.09 Found (fun (x02:(forall (x':fofType), ((P x')->(((eq fofType) x00) x'))))=> (((fun (x03:(((eq fofType) (x P)) x00))=> (((((eq_sym fofType) (x P)) x00) x03) P)) ((eq_ref fofType) (x P))) x01)) as proof of (P (x P))
% 3.94/4.09 Found (fun (x01:(P x00)) (x02:(forall (x':fofType), ((P x')->(((eq fofType) x00) x'))))=> (((fun (x03:(((eq fofType) (x P)) x00))=> (((((eq_sym fofType) (x P)) x00) x03) P)) ((eq_ref fofType) (x P))) x01)) as proof of ((forall (x':fofType), ((P x')->(((eq fofType) x00) x')))->(P (x P)))
% 3.94/4.09 Found (fun (x01:(P x00)) (x02:(forall (x':fofType), ((P x')->(((eq fofType) x00) x'))))=> (((fun (x03:(((eq fofType) (x P)) x00))=> (((((eq_sym fofType) (x P)) x00) x03) P)) ((eq_ref fofType) (x P))) x01)) as proof of ((P x00)->((forall (x':fofType), ((P x')->(((eq fofType) x00) x')))->(P (x P))))
% 3.94/4.09 Found (and_rect00 (fun (x01:(P x00)) (x02:(forall (x':fofType), ((P x')->(((eq fofType) x00) x'))))=> (((fun (x03:(((eq fofType) (x P)) x00))=> (((((eq_sym fofType) (x P)) x00) x03) P)) ((eq_ref fofType) (x P))) x01))) as proof of (P (x P))
% 3.94/4.09 Found ((and_rect0 (P (x P))) (fun (x01:(P x00)) (x02:(forall (x':fofType), ((P x')->(((eq fofType) x00) x'))))=> (((fun (x03:(((eq fofType) (x P)) x00))=> (((((eq_sym fofType) (x P)) x00) x03) P)) ((eq_ref fofType) (x P))) x01))) as proof of (P (x P))
% 3.94/4.09 Found (((fun (P0:Type) (x2:((P x00)->((forall (x':fofType), ((P x')->(((eq fofType) x00) x')))->P0)))=> (((((and_rect (P x00)) (forall (x':fofType), ((P x')->(((eq fofType) x00) x')))) P0) x2) x1)) (P (x P))) (fun (x01:(P x00)) (x02:(forall (x':fofType), ((P x')->(((eq fofType) x00) x'))))=> (((fun (x03:(((eq fofType) (x P)) x00))=> (((((eq_sym fofType) (x P)) x00) x03) P)) ((eq_ref fofType) (x P))) x01))) as proof of (P (x P))
% 6.15/6.35 Found (fun (x1:(((unique fofType) P) x00))=> (((fun (P0:Type) (x2:((P x00)->((forall (x':fofType), ((P x')->(((eq fofType) x00) x')))->P0)))=> (((((and_rect (P x00)) (forall (x':fofType), ((P x')->(((eq fofType) x00) x')))) P0) x2) x1)) (P (x P))) (fun (x01:(P x00)) (x02:(forall (x':fofType), ((P x')->(((eq fofType) x00) x'))))=> (((fun (x03:(((eq fofType) (x P)) x00))=> (((((eq_sym fofType) (x P)) x00) x03) P)) ((eq_ref fofType) (x P))) x01)))) as proof of (P (x P))
% 6.15/6.35 Found fofType_DUMMY:fofType
% 6.15/6.35 Found fofType_DUMMY as proof of fofType
% 6.15/6.35 Found (choice_operator0 fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a)
% 6.15/6.35 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a)
% 6.15/6.35 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a)
% 6.15/6.35 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a)
% 6.15/6.35 Found ((choice_operator fofType) fofType_DUMMY) as proof of (P a)
% 6.15/6.35 Found fofType_DUMMY:fofType
% 6.15/6.35 Found fofType_DUMMY as proof of fofType
% 6.15/6.35 Found (choice_operator0 fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a)
% 6.15/6.35 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a)
% 6.15/6.35 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a)
% 6.15/6.35 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a)
% 6.15/6.35 Found ((choice_operator fofType) fofType_DUMMY) as proof of (P a)
% 6.15/6.35 Found fofType_DUMMY:fofType
% 6.15/6.35 Found fofType_DUMMY as proof of fofType
% 6.15/6.35 Found (choice_operator0 fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a)
% 6.15/6.35 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a)
% 6.15/6.35 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a)
% 6.15/6.35 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a)
% 6.15/6.35 Found ((choice_operator fofType) fofType_DUMMY) as proof of (P a)
% 6.15/6.35 Found fofType_DUMMY:fofType
% 6.15/6.35 Found fofType_DUMMY as proof of fofType
% 6.15/6.35 Found (choice_operator0 fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a)
% 6.15/6.35 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a)
% 6.15/6.35 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a)
% 6.15/6.35 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a)
% 6.15/6.35 Found ((choice_operator fofType) fofType_DUMMY) as proof of (P a)
% 6.15/6.35 Found x3:(P x1)
% 6.15/6.35 Instantiate: b:=x1:fofType
% 6.15/6.35 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b)
% 6.15/6.35 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b))
% 6.15/6.35 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b)))
% 6.15/6.35 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 6.15/6.35 Found ((and_rect0 (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 6.15/6.35 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 6.15/6.35 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 6.28/6.47 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P0 b)
% 6.28/6.47 Found x3:(P x1)
% 6.28/6.47 Instantiate: b:=x1:fofType
% 6.28/6.47 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b)
% 6.28/6.47 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b))
% 6.28/6.47 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b)))
% 6.28/6.47 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 6.28/6.47 Found ((and_rect0 (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 6.28/6.47 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 6.28/6.47 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b)
% 6.28/6.47 Found x3:(P x1)
% 6.28/6.47 Instantiate: b:=x1:fofType
% 6.28/6.47 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b)
% 6.28/6.47 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b))
% 6.28/6.47 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b)))
% 6.28/6.47 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 6.28/6.47 Found ((and_rect0 (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 6.28/6.47 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 6.28/6.47 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 6.28/6.47 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P0 b)
% 6.28/6.47 Found x3:(P x1)
% 6.28/6.47 Instantiate: b:=x1:fofType
% 6.28/6.47 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b)
% 6.28/6.47 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b))
% 6.28/6.47 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b)))
% 6.28/6.47 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 9.16/9.37 Found ((and_rect0 (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 9.16/9.37 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 9.16/9.37 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b)
% 9.16/9.37 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 9.16/9.37 Found (eq_ref0 b) as proof of (((eq fofType) b) (x P))
% 9.16/9.37 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x P))
% 9.16/9.37 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x P))
% 9.16/9.37 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x P))
% 9.16/9.37 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 9.16/9.37 Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% 9.16/9.37 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 9.16/9.37 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 9.16/9.37 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 9.16/9.37 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 9.16/9.37 Found (eq_ref0 b) as proof of (((eq fofType) b) (x P))
% 9.16/9.37 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x P))
% 9.16/9.37 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x P))
% 9.16/9.37 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x P))
% 9.16/9.37 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 9.16/9.37 Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% 9.16/9.37 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 9.16/9.37 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 9.16/9.37 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 9.16/9.37 Found x2:(P x00)
% 9.16/9.37 Instantiate: a:=x00:fofType
% 9.16/9.37 Found (fun (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of (P0 a)
% 9.16/9.37 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P0 a))
% 9.16/9.37 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P0 a)))
% 9.16/9.37 Found (and_rect00 (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P0 a)
% 9.16/9.37 Found ((and_rect0 (P0 a)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P0 a)
% 9.16/9.37 Found (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P0 a)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P0 a)
% 9.16/9.37 Found (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P0 a)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P0 a)
% 9.16/9.37 Found x3:(P x1)
% 9.16/9.37 Instantiate: a:=x1:fofType
% 9.16/9.37 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P0 a)
% 9.16/9.37 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P0 a))
% 9.16/9.37 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P0 a)))
% 9.16/9.37 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 a)
% 9.16/9.37 Found ((and_rect0 (P0 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 a)
% 9.16/9.37 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P0 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 a)
% 12.16/12.31 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P0 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P0 a)
% 12.16/12.31 Found x2:(P x00)
% 12.16/12.31 Instantiate: a:=x00:fofType
% 12.16/12.31 Found (fun (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of (P0 a)
% 12.16/12.31 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P0 a))
% 12.16/12.31 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P0 a)))
% 12.16/12.31 Found (and_rect00 (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P0 a)
% 12.16/12.31 Found ((and_rect0 (P0 a)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P0 a)
% 12.16/12.31 Found (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P0 a)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P0 a)
% 12.16/12.31 Found (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P0 a)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P0 a)
% 12.16/12.31 Found x3:(P x1)
% 12.16/12.31 Instantiate: a:=x1:fofType
% 12.16/12.31 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P0 a)
% 12.16/12.31 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P0 a))
% 12.16/12.31 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P0 a)))
% 12.16/12.31 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 a)
% 12.16/12.31 Found ((and_rect0 (P0 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 a)
% 12.16/12.31 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P0 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 a)
% 12.16/12.31 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P0 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P0 a)
% 12.16/12.31 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))):(((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) (fun (x:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (x P))))))
% 12.16/12.31 Found (eta_expansion_dep00 (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) b0)
% 12.46/12.67 Found ((eta_expansion_dep0 (fun (x1:((fofType->Prop)->fofType))=> Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) b0)
% 12.46/12.67 Found (((eta_expansion_dep ((fofType->Prop)->fofType)) (fun (x1:((fofType->Prop)->fofType))=> Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) b0)
% 12.46/12.67 Found (((eta_expansion_dep ((fofType->Prop)->fofType)) (fun (x1:((fofType->Prop)->fofType))=> Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) b0)
% 12.46/12.67 Found (((eta_expansion_dep ((fofType->Prop)->fofType)) (fun (x1:((fofType->Prop)->fofType))=> Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) b0)
% 12.46/12.67 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))):(((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))) (fun (x:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (x P))))))
% 12.46/12.67 Found (eta_expansion_dep00 (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))) b0)
% 12.46/12.67 Found ((eta_expansion_dep0 (fun (x1:((fofType->Prop)->fofType))=> Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))) b0)
% 12.46/12.67 Found (((eta_expansion_dep ((fofType->Prop)->fofType)) (fun (x1:((fofType->Prop)->fofType))=> Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))) b0)
% 12.46/12.67 Found (((eta_expansion_dep ((fofType->Prop)->fofType)) (fun (x1:((fofType->Prop)->fofType))=> Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))) b0)
% 13.28/13.45 Found (((eta_expansion_dep ((fofType->Prop)->fofType)) (fun (x1:((fofType->Prop)->fofType))=> Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))) b0)
% 13.28/13.45 Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% 13.28/13.45 Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% 13.28/13.45 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% 13.28/13.45 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% 13.28/13.45 Found (fun (x:((fofType->Prop)->fofType))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% 13.28/13.45 Found (fun (x:((fofType->Prop)->fofType))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:((fofType->Prop)->fofType)), (((eq Prop) (f0 x)) (f x)))
% 13.28/13.45 Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% 13.28/13.45 Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% 13.28/13.45 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% 13.28/13.45 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% 13.28/13.45 Found (fun (x:((fofType->Prop)->fofType))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% 13.28/13.45 Found (fun (x:((fofType->Prop)->fofType))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:((fofType->Prop)->fofType)), (((eq Prop) (f0 x)) (f x)))
% 13.28/13.45 Found x3:(P x1)
% 13.28/13.45 Instantiate: a:=x1:fofType
% 13.28/13.45 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a)
% 13.28/13.45 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a))
% 13.28/13.45 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a)))
% 13.28/13.45 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 13.28/13.45 Found ((and_rect0 (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 13.28/13.45 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 13.28/13.45 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 13.28/13.45 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P0 a)
% 13.28/13.45 Found x3:(P x1)
% 13.28/13.45 Instantiate: a:=x1:fofType
% 13.28/13.45 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a)
% 13.28/13.45 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a))
% 13.28/13.45 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a)))
% 13.28/13.45 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 13.28/13.45 Found ((and_rect0 (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 13.43/13.67 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 13.43/13.67 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 13.43/13.67 Found x3:(P x1)
% 13.43/13.67 Instantiate: a:=x1:fofType
% 13.43/13.67 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a)
% 13.43/13.67 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a))
% 13.43/13.67 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a)))
% 13.43/13.67 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 13.43/13.67 Found ((and_rect0 (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 13.43/13.67 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 13.43/13.67 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 13.43/13.67 Found x3:(P x1)
% 13.43/13.67 Instantiate: a:=x1:fofType
% 13.43/13.67 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a)
% 13.43/13.67 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a))
% 13.43/13.67 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a)))
% 13.43/13.67 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 13.43/13.67 Found ((and_rect0 (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 13.43/13.67 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 13.43/13.67 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 13.43/13.67 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P0 a)
% 13.43/13.67 Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% 13.43/13.67 Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% 13.43/13.67 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% 13.43/13.67 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% 13.43/13.67 Found (fun (x:((fofType->Prop)->fofType))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% 15.96/16.15 Found (fun (x:((fofType->Prop)->fofType))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:((fofType->Prop)->fofType)), (((eq Prop) (f0 x)) (f x)))
% 15.96/16.15 Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% 15.96/16.15 Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% 15.96/16.15 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% 15.96/16.15 Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% 15.96/16.15 Found (fun (x:((fofType->Prop)->fofType))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% 15.96/16.15 Found (fun (x:((fofType->Prop)->fofType))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:((fofType->Prop)->fofType)), (((eq Prop) (f0 x)) (f x)))
% 15.96/16.15 Found fofType_DUMMY:fofType
% 15.96/16.15 Found fofType_DUMMY as proof of fofType
% 15.96/16.15 Found (choice_operator0 fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) b0)
% 15.96/16.15 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) b0)
% 15.96/16.15 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) b0)
% 15.96/16.15 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) b0)
% 15.96/16.15 Found ((choice_operator fofType) fofType_DUMMY) as proof of (P0 b0)
% 15.96/16.15 Found fofType_DUMMY:fofType
% 15.96/16.15 Found fofType_DUMMY as proof of fofType
% 15.96/16.15 Found (choice_operator0 fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) b0)
% 15.96/16.15 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) b0)
% 15.96/16.15 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) b0)
% 15.96/16.15 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) b0)
% 15.96/16.15 Found ((choice_operator fofType) fofType_DUMMY) as proof of (P0 b0)
% 15.96/16.15 Found fofType_DUMMY:fofType
% 15.96/16.15 Found fofType_DUMMY as proof of fofType
% 15.96/16.15 Found (choice_operator0 fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f0)
% 15.96/16.15 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f0)
% 15.96/16.15 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f0)
% 15.96/16.15 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f0)
% 15.96/16.15 Found ((choice_operator fofType) fofType_DUMMY) as proof of (P0 f0)
% 15.96/16.15 Found fofType_DUMMY:fofType
% 15.96/16.15 Found fofType_DUMMY as proof of fofType
% 15.96/16.15 Found (choice_operator0 fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f0)
% 15.96/16.15 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f0)
% 15.96/16.15 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f0)
% 15.96/16.15 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f0)
% 15.96/16.15 Found ((choice_operator fofType) fofType_DUMMY) as proof of (P0 f0)
% 15.96/16.15 Found eq_ref00:=(eq_ref0 (x a)):(((eq fofType) (x a)) (x a))
% 15.96/16.15 Found (eq_ref0 (x a)) as proof of (((eq fofType) (x a)) b)
% 15.96/16.15 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 15.96/16.15 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 15.96/16.15 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 15.96/16.15 Found eq_ref00:=(eq_ref0 (x a)):(((eq fofType) (x a)) (x a))
% 15.96/16.15 Found (eq_ref0 (x a)) as proof of (((eq fofType) (x a)) b)
% 15.96/16.15 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 15.96/16.15 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 15.96/16.15 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 15.96/16.15 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 15.96/16.15 Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% 15.96/16.15 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% 15.96/16.15 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% 15.96/16.15 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% 15.96/16.15 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 15.96/16.15 Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% 15.96/16.15 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% 15.96/16.15 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% 15.96/16.15 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% 15.96/16.15 Found fofType_DUMMY:fofType
% 15.96/16.15 Found fofType_DUMMY as proof of fofType
% 15.96/16.15 Found (choice_operator0 fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f0)
% 17.18/17.32 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f0)
% 17.18/17.32 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f0)
% 17.18/17.32 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f0)
% 17.18/17.32 Found ((choice_operator fofType) fofType_DUMMY) as proof of (P0 f0)
% 17.18/17.32 Found fofType_DUMMY:fofType
% 17.18/17.32 Found fofType_DUMMY as proof of fofType
% 17.18/17.32 Found (choice_operator0 fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f0)
% 17.18/17.32 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f0)
% 17.18/17.32 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f0)
% 17.18/17.32 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f0)
% 17.18/17.32 Found ((choice_operator fofType) fofType_DUMMY) as proof of (P0 f0)
% 17.18/17.32 Found eq_ref00:=(eq_ref0 (x P)):(((eq fofType) (x P)) (x P))
% 17.18/17.32 Found (eq_ref0 (x P)) as proof of (((eq fofType) (x P)) b0)
% 17.18/17.32 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b0)
% 17.18/17.32 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b0)
% 17.18/17.32 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b0)
% 17.18/17.32 Found eq_ref00:=(eq_ref0 (x a)):(((eq fofType) (x a)) (x a))
% 17.18/17.32 Found (eq_ref0 (x a)) as proof of (((eq fofType) (x a)) b)
% 17.18/17.32 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 17.18/17.32 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 17.18/17.32 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 17.18/17.32 Found x3:(P x1)
% 17.18/17.32 Instantiate: b0:=x1:fofType
% 17.18/17.32 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b0)
% 17.18/17.32 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0))
% 17.18/17.32 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0)))
% 17.18/17.32 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 17.18/17.32 Found ((and_rect0 (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 17.18/17.32 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 17.18/17.32 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 17.18/17.32 Found ((eq_sym0100 ((eq_ref fofType) b)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P0 b)
% 17.18/17.32 Found ((eq_sym0100 ((eq_ref fofType) b)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P0 b)
% 17.18/17.32 Found (((fun (x3:(((eq fofType) b) b0))=> ((eq_sym010 x3) P0)) ((eq_ref fofType) b)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P0 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P0 b)
% 17.18/17.32 Found (((fun (x3:(((eq fofType) b) x1))=> (((eq_sym01 x1) x3) P0)) ((eq_ref fofType) b)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P0 x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P0 b)
% 20.48/20.62 Found (((fun (x3:(((eq fofType) b) x1))=> ((((eq_sym0 b) x1) x3) P0)) ((eq_ref fofType) b)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P0 x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P0 b)
% 20.48/20.62 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (x3:(((eq fofType) b) x1))=> ((((eq_sym0 b) x1) x3) P0)) ((eq_ref fofType) b)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P0 x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)))) as proof of (P0 b)
% 20.48/20.62 Found eq_ref00:=(eq_ref0 (x a)):(((eq fofType) (x a)) (x a))
% 20.48/20.62 Found (eq_ref0 (x a)) as proof of (((eq fofType) (x a)) b)
% 20.48/20.62 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 20.48/20.62 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 20.48/20.62 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 20.48/20.62 Found x3:(P x1)
% 20.48/20.62 Instantiate: b0:=x1:fofType
% 20.48/20.62 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P00 b0)
% 20.48/20.62 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P00 b0))
% 20.48/20.62 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P00 b0)))
% 20.48/20.62 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 b0)
% 20.48/20.62 Found ((and_rect0 (P00 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 b0)
% 20.48/20.62 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P00 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 b0)
% 20.48/20.62 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P00 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 b0)
% 20.48/20.62 Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% 20.48/20.62 Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 20.48/20.62 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 20.48/20.62 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 20.48/20.62 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 20.48/20.62 Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% 20.48/20.62 Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 20.48/20.62 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 20.48/20.62 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 20.48/20.62 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 20.48/20.62 Found eq_ref00:=(eq_ref0 (x a)):(((eq fofType) (x a)) (x a))
% 20.48/20.62 Found (eq_ref0 (x a)) as proof of (((eq fofType) (x a)) b)
% 20.48/20.62 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 20.48/20.62 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 20.48/20.62 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 20.48/20.62 Found eq_ref00:=(eq_ref0 (x a)):(((eq fofType) (x a)) (x a))
% 20.48/20.62 Found (eq_ref0 (x a)) as proof of (((eq fofType) (x a)) b)
% 20.48/20.62 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 20.48/20.62 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 20.48/20.62 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 20.48/20.62 Found x01:(P x00)
% 20.48/20.62 Instantiate: b:=x00:fofType
% 20.48/20.62 Found x01 as proof of (P1 b)
% 20.56/20.71 Found eq_ref00:=(eq_ref0 (x a)):(((eq fofType) (x a)) (x a))
% 20.56/20.71 Found (eq_ref0 (x a)) as proof of (((eq fofType) (x a)) b)
% 20.56/20.71 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 20.56/20.71 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 20.56/20.71 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 20.56/20.71 Found ((eq_sym0000 ((eq_ref fofType) (x a))) x01) as proof of (P0 (x a))
% 20.56/20.71 Found ((eq_sym0000 ((eq_ref fofType) (x a))) x01) as proof of (P0 (x a))
% 20.56/20.71 Found (((fun (x03:(((eq fofType) (x a)) b))=> ((eq_sym000 x03) P0)) ((eq_ref fofType) (x a))) x01) as proof of (P0 (x a))
% 20.56/20.71 Found (((fun (x03:(((eq fofType) (x a)) x00))=> (((eq_sym00 x00) x03) P0)) ((eq_ref fofType) (x a))) x01) as proof of (P0 (x a))
% 20.56/20.71 Found (((fun (x03:(((eq fofType) (x a)) x00))=> ((((eq_sym0 (x a)) x00) x03) P0)) ((eq_ref fofType) (x a))) x01) as proof of (P0 (x a))
% 20.56/20.71 Found (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P0)) ((eq_ref fofType) (x a))) x01) as proof of (P0 (x a))
% 20.56/20.71 Found (fun (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P0)) ((eq_ref fofType) (x a))) x01)) as proof of (P0 (x a))
% 20.56/20.71 Found (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P0)) ((eq_ref fofType) (x a))) x01)) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P0 (x a)))
% 20.56/20.71 Found (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P0)) ((eq_ref fofType) (x a))) x01)) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P0 (x a))))
% 20.56/20.71 Found (and_rect00 (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P0)) ((eq_ref fofType) (x a))) x01))) as proof of (P0 (x a))
% 20.56/20.71 Found ((and_rect0 (P0 (x a))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P0)) ((eq_ref fofType) (x a))) x01))) as proof of (P0 (x a))
% 20.56/20.71 Found (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P0 (x a))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P0)) ((eq_ref fofType) (x a))) x01))) as proof of (P0 (x a))
% 20.56/20.71 Found (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P0 (x a))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P0)) ((eq_ref fofType) (x a))) x01))) as proof of (P0 (x a))
% 20.56/20.71 Found ((eq_substitution000000 ((eq_ref (fofType->Prop)) a)) (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P0 (x a))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P0)) ((eq_ref fofType) (x a))) x01)))) as proof of (P (x P))
% 20.56/20.71 Found ((eq_substitution000000 ((eq_ref (fofType->Prop)) a)) (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P0 (x a))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P0)) ((eq_ref fofType) (x a))) x01)))) as proof of (P (x P))
% 20.56/20.71 Found (((fun (x01:(((eq (fofType->Prop)) a) P))=> ((eq_substitution00000 x01) P)) ((eq_ref (fofType->Prop)) a)) (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P (x a))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01)))) as proof of (P (x P))
% 20.56/20.71 Found (((fun (x01:(((eq (fofType->Prop)) a) P))=> (((eq_substitution0000 P) x01) P)) ((eq_ref (fofType->Prop)) a)) (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P (x a))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01)))) as proof of (P (x P))
% 20.56/20.71 Found (((fun (x01:(((eq (fofType->Prop)) P) P))=> ((((eq_substitution000 P) P) x01) P)) ((eq_ref (fofType->Prop)) P)) (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P (x P))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x P)) x00))=> (((((eq_sym fofType) (x P)) x00) x03) P)) ((eq_ref fofType) (x P))) x01)))) as proof of (P (x P))
% 20.56/20.71 Found (((fun (x01:(((eq (fofType->Prop)) P) P))=> (((((fun (a:(fofType->Prop)) (b:(fofType->Prop))=> (((eq_substitution00 a) b) x)) P) P) x01) P)) ((eq_ref (fofType->Prop)) P)) (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P (x P))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x P)) x00))=> (((((eq_sym fofType) (x P)) x00) x03) P)) ((eq_ref fofType) (x P))) x01)))) as proof of (P (x P))
% 20.56/20.71 Found (((fun (x01:(((eq (fofType->Prop)) P) P))=> (((((fun (a:(fofType->Prop)) (b:(fofType->Prop))=> ((((eq_substitution0 fofType) a) b) x)) P) P) x01) P)) ((eq_ref (fofType->Prop)) P)) (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P (x P))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x P)) x00))=> (((((eq_sym fofType) (x P)) x00) x03) P)) ((eq_ref fofType) (x P))) x01)))) as proof of (P (x P))
% 20.56/20.71 Found (((fun (x01:(((eq (fofType->Prop)) P) P))=> (((((fun (a:(fofType->Prop)) (b:(fofType->Prop))=> (((((eq_substitution (fofType->Prop)) fofType) a) b) x)) P) P) x01) P)) ((eq_ref (fofType->Prop)) P)) (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P (x P))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x P)) x00))=> (((((eq_sym fofType) (x P)) x00) x03) P)) ((eq_ref fofType) (x P))) x01)))) as proof of (P (x P))
% 20.56/20.71 Found (fun (x1:((and (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))))=> (((fun (x01:(((eq (fofType->Prop)) P) P))=> (((((fun (a:(fofType->Prop)) (b:(fofType->Prop))=> (((((eq_substitution (fofType->Prop)) fofType) a) b) x)) P) P) x01) P)) ((eq_ref (fofType->Prop)) P)) (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P (x P))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x P)) x00))=> (((((eq_sym fofType) (x P)) x00) x03) P)) ((eq_ref fofType) (x P))) x01))))) as proof of (P (x P))
% 20.77/20.96 Found x01:(P x00)
% 20.77/20.96 Instantiate: b:=x00:fofType
% 20.77/20.96 Found x01 as proof of (P1 b)
% 20.77/20.96 Found eq_ref00:=(eq_ref0 (x a)):(((eq fofType) (x a)) (x a))
% 20.77/20.96 Found (eq_ref0 (x a)) as proof of (((eq fofType) (x a)) b)
% 20.77/20.96 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 20.77/20.96 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 20.77/20.96 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 20.77/20.96 Found ((eq_sym0000 ((eq_ref fofType) (x a))) x01) as proof of (P0 (x a))
% 20.77/20.96 Found ((eq_sym0000 ((eq_ref fofType) (x a))) x01) as proof of (P0 (x a))
% 20.77/20.96 Found (((fun (x03:(((eq fofType) (x a)) b))=> ((eq_sym000 x03) P0)) ((eq_ref fofType) (x a))) x01) as proof of (P0 (x a))
% 20.77/20.96 Found (((fun (x03:(((eq fofType) (x a)) x00))=> (((eq_sym00 x00) x03) P0)) ((eq_ref fofType) (x a))) x01) as proof of (P0 (x a))
% 20.77/20.96 Found (((fun (x03:(((eq fofType) (x a)) x00))=> ((((eq_sym0 (x a)) x00) x03) P0)) ((eq_ref fofType) (x a))) x01) as proof of (P0 (x a))
% 20.77/20.96 Found (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P0)) ((eq_ref fofType) (x a))) x01) as proof of (P0 (x a))
% 20.77/20.96 Found (fun (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P0)) ((eq_ref fofType) (x a))) x01)) as proof of (P0 (x a))
% 20.77/20.96 Found (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P0)) ((eq_ref fofType) (x a))) x01)) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P0 (x a)))
% 20.77/20.96 Found (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P0)) ((eq_ref fofType) (x a))) x01)) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P0 (x a))))
% 20.77/20.96 Found (and_rect00 (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P0)) ((eq_ref fofType) (x a))) x01))) as proof of (P0 (x a))
% 20.77/20.96 Found ((and_rect0 (P0 (x a))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P0)) ((eq_ref fofType) (x a))) x01))) as proof of (P0 (x a))
% 20.77/20.96 Found (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P0 (x a))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P0)) ((eq_ref fofType) (x a))) x01))) as proof of (P0 (x a))
% 20.77/20.96 Found (fun (x1:((and (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))))=> (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P0 (x a))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P0)) ((eq_ref fofType) (x a))) x01)))) as proof of (P0 (x a))
% 20.77/20.96 Found x3:(P x1)
% 20.77/20.96 Instantiate: b0:=x1:fofType
% 20.77/20.96 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b0)
% 20.77/20.96 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0))
% 20.77/20.96 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0)))
% 20.77/20.96 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 20.77/20.96 Found ((and_rect0 (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 20.77/20.96 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 20.86/21.06 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 20.86/21.06 Found x2:(P x00)
% 20.86/21.06 Instantiate: b:=x00:fofType
% 20.86/21.06 Found (fun (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of (P1 b)
% 20.86/21.06 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 b))
% 20.86/21.06 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 b)))
% 20.86/21.06 Found (and_rect00 (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b)
% 20.86/21.06 Found ((and_rect0 (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b)
% 20.86/21.06 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b)
% 20.86/21.06 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b)
% 20.86/21.06 Found ((eq_sym0000 ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P0 (x a))
% 20.86/21.06 Found ((eq_sym0000 ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P0 (x a))
% 20.86/21.06 Found (((fun (x01:(((eq fofType) (x a)) b))=> ((eq_sym000 x01) P0)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P0 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P0 (x a))
% 20.86/21.06 Found (((fun (x01:(((eq fofType) (x a)) x00))=> (((eq_sym00 x00) x01) P0)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P0 x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P0 (x a))
% 20.86/21.06 Found (((fun (x01:(((eq fofType) (x a)) x00))=> ((((eq_sym0 (x a)) x00) x01) P0)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P0 x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P0 (x a))
% 20.86/21.06 Found (((fun (x01:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x01) P0)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P0 x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P0 (x a))
% 21.08/21.25 Found (fun (x1:((and (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))))=> (((fun (x01:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x01) P0)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P0 x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)))) as proof of (P0 (x a))
% 21.08/21.25 Found x3:(P x1)
% 21.08/21.25 Instantiate: b:=x1:fofType
% 21.08/21.25 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b)
% 21.08/21.25 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b))
% 21.08/21.25 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b)))
% 21.08/21.25 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b)
% 21.08/21.25 Found ((and_rect0 (P1 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b)
% 21.08/21.25 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b)
% 21.08/21.25 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b)
% 21.08/21.25 Found x3:(P x1)
% 21.08/21.25 Instantiate: b0:=x1:fofType
% 21.08/21.25 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b0)
% 21.08/21.25 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0))
% 21.08/21.25 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0)))
% 21.08/21.25 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 21.08/21.25 Found ((and_rect0 (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 21.08/21.25 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 21.08/21.25 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 21.08/21.25 Found x01:(P x00)
% 21.08/21.25 Instantiate: b:=x00:fofType
% 21.08/21.25 Found x01 as proof of (P1 b)
% 21.08/21.25 Found eq_ref00:=(eq_ref0 (x a)):(((eq fofType) (x a)) (x a))
% 21.08/21.25 Found (eq_ref0 (x a)) as proof of (((eq fofType) (x a)) b)
% 21.08/21.25 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 21.08/21.25 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 21.08/21.25 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 21.08/21.25 Found ((eq_sym0000 ((eq_ref fofType) (x a))) x01) as proof of (P0 (x a))
% 21.08/21.25 Found ((eq_sym0000 ((eq_ref fofType) (x a))) x01) as proof of (P0 (x a))
% 21.08/21.25 Found (((fun (x03:(((eq fofType) (x a)) b))=> ((eq_sym000 x03) P0)) ((eq_ref fofType) (x a))) x01) as proof of (P0 (x a))
% 21.08/21.31 Found (((fun (x03:(((eq fofType) (x a)) x00))=> (((eq_sym00 x00) x03) P0)) ((eq_ref fofType) (x a))) x01) as proof of (P0 (x a))
% 21.08/21.31 Found (((fun (x03:(((eq fofType) (x a)) x00))=> ((((eq_sym0 (x a)) x00) x03) P0)) ((eq_ref fofType) (x a))) x01) as proof of (P0 (x a))
% 21.08/21.31 Found (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P0)) ((eq_ref fofType) (x a))) x01) as proof of (P0 (x a))
% 21.08/21.31 Found (fun (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P0)) ((eq_ref fofType) (x a))) x01)) as proof of (P0 (x a))
% 21.08/21.31 Found (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P0)) ((eq_ref fofType) (x a))) x01)) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P0 (x a)))
% 21.08/21.31 Found (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P0)) ((eq_ref fofType) (x a))) x01)) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P0 (x a))))
% 21.08/21.31 Found (and_rect00 (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P0)) ((eq_ref fofType) (x a))) x01))) as proof of (P0 (x a))
% 21.08/21.31 Found ((and_rect0 (P0 (x a))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P0)) ((eq_ref fofType) (x a))) x01))) as proof of (P0 (x a))
% 21.08/21.31 Found (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P0 (x a))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P0)) ((eq_ref fofType) (x a))) x01))) as proof of (P0 (x a))
% 21.08/21.31 Found (fun (x1:((and (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))))=> (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P0 (x a))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P0)) ((eq_ref fofType) (x a))) x01)))) as proof of (P0 (x a))
% 21.08/21.31 Found x2:(P x00)
% 21.08/21.31 Instantiate: b:=x00:fofType
% 21.08/21.31 Found (fun (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of (P1 b)
% 21.08/21.31 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 b))
% 21.08/21.31 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 b)))
% 21.08/21.31 Found (and_rect00 (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b)
% 21.08/21.31 Found ((and_rect0 (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b)
% 21.08/21.31 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b)
% 21.08/21.31 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b)
% 21.08/21.31 Found ((eq_sym0000 ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P0 (x a))
% 21.17/21.33 Found ((eq_sym0000 ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P0 (x a))
% 21.17/21.33 Found (((fun (x01:(((eq fofType) (x a)) b))=> ((eq_sym000 x01) P0)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P0 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P0 (x a))
% 21.17/21.33 Found (((fun (x01:(((eq fofType) (x a)) x00))=> (((eq_sym00 x00) x01) P0)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P0 x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P0 (x a))
% 21.17/21.33 Found (((fun (x01:(((eq fofType) (x a)) x00))=> ((((eq_sym0 (x a)) x00) x01) P0)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P0 x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P0 (x a))
% 21.17/21.33 Found (((fun (x01:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x01) P0)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P0 x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P0 (x a))
% 21.17/21.33 Found (fun (x1:((and (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))))=> (((fun (x01:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x01) P0)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P0 x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)))) as proof of (P0 (x a))
% 21.17/21.33 Found x3:(P x1)
% 21.17/21.33 Instantiate: b:=x1:fofType
% 21.17/21.33 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b)
% 21.17/21.33 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b))
% 21.17/21.33 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b)))
% 21.17/21.33 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b)
% 21.17/21.33 Found ((and_rect0 (P1 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b)
% 21.17/21.33 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b)
% 21.17/21.33 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b)
% 23.36/23.54 Found x3:(P x1)
% 23.36/23.54 Instantiate: b:=x1:fofType
% 23.36/23.54 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b)
% 23.36/23.54 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b))
% 23.36/23.54 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b)))
% 23.36/23.54 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b)
% 23.36/23.54 Found ((and_rect0 (P1 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b)
% 23.36/23.54 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b)
% 23.36/23.54 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b)
% 23.36/23.54 Found x3:(P x1)
% 23.36/23.54 Instantiate: b:=x1:fofType
% 23.36/23.54 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b)
% 23.36/23.54 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b))
% 23.36/23.54 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b)))
% 23.36/23.54 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b)
% 23.36/23.54 Found ((and_rect0 (P1 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b)
% 23.36/23.54 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b)
% 23.36/23.54 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b)
% 23.36/23.54 Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 23.36/23.54 Found (eta_expansion00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 23.36/23.54 Found ((eta_expansion0 Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 23.36/23.54 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 23.36/23.54 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 23.36/23.54 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 23.36/23.54 Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% 23.36/23.54 Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 23.36/23.54 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 23.36/23.54 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 23.36/23.54 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 23.36/23.54 Found eq_ref00:=(eq_ref0 (x P)):(((eq fofType) (x P)) (x P))
% 23.36/23.54 Found (eq_ref0 (x P)) as proof of (((eq fofType) (x P)) b0)
% 23.36/23.54 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b0)
% 23.36/23.54 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b0)
% 23.36/23.54 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b0)
% 23.36/23.54 Found eq_ref00:=(eq_ref0 (x a)):(((eq fofType) (x a)) (x a))
% 26.25/26.47 Found (eq_ref0 (x a)) as proof of (((eq fofType) (x a)) b)
% 26.25/26.47 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 26.25/26.47 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 26.25/26.47 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 26.25/26.47 Found eq_ref00:=(eq_ref0 (x a)):(((eq fofType) (x a)) (x a))
% 26.25/26.47 Found (eq_ref0 (x a)) as proof of (((eq fofType) (x a)) b)
% 26.25/26.47 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 26.25/26.47 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 26.25/26.47 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 26.25/26.47 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 26.25/26.47 Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% 26.25/26.47 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% 26.25/26.47 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% 26.25/26.47 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% 26.25/26.47 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 26.25/26.47 Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% 26.25/26.47 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% 26.25/26.47 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% 26.25/26.47 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% 26.25/26.47 Found eq_ref00:=(eq_ref0 (x P)):(((eq fofType) (x P)) (x P))
% 26.25/26.47 Found (eq_ref0 (x P)) as proof of (((eq fofType) (x P)) b0)
% 26.25/26.47 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b0)
% 26.25/26.47 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b0)
% 26.25/26.47 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b0)
% 26.25/26.47 Found eq_ref00:=(eq_ref0 (x P0)):(((eq fofType) (x P0)) (x P0))
% 26.25/26.47 Found (eq_ref0 (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 26.25/26.47 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 26.25/26.47 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 26.25/26.47 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 26.25/26.47 Found eq_ref00:=(eq_ref0 (x P0)):(((eq fofType) (x P0)) (x P0))
% 26.25/26.47 Found (eq_ref0 (x P0)) as proof of (((eq fofType) (x P0)) b)
% 26.25/26.47 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 26.25/26.47 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 26.25/26.47 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 26.25/26.47 Found eq_ref00:=(eq_ref0 (x P0)):(((eq fofType) (x P0)) (x P0))
% 26.25/26.47 Found (eq_ref0 (x P0)) as proof of (((eq fofType) (x P0)) b)
% 26.25/26.47 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 26.25/26.47 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 26.25/26.47 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 26.25/26.47 Found x3:(P x1)
% 26.25/26.47 Instantiate: b0:=x1:fofType
% 26.25/26.47 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b0)
% 26.25/26.47 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0))
% 26.25/26.47 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0)))
% 26.25/26.47 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 26.25/26.47 Found ((and_rect0 (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 26.25/26.47 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 26.25/26.47 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 26.25/26.47 Found ((eq_sym0100 ((eq_ref fofType) b)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b)
% 28.53/28.68 Found ((eq_sym0100 ((eq_ref fofType) b)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b)
% 28.53/28.68 Found (((fun (x3:(((eq fofType) b) b0))=> ((eq_sym010 x3) P)) ((eq_ref fofType) b)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b)
% 28.53/28.68 Found (((fun (x3:(((eq fofType) b) x1))=> (((eq_sym01 x1) x3) P)) ((eq_ref fofType) b)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b)
% 28.53/28.68 Found (((fun (x3:(((eq fofType) b) x1))=> ((((eq_sym0 b) x1) x3) P)) ((eq_ref fofType) b)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b)
% 28.53/28.68 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (x3:(((eq fofType) b) x1))=> ((((eq_sym0 b) x1) x3) P)) ((eq_ref fofType) b)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)))) as proof of (P b)
% 28.53/28.68 Found x3:(P x1)
% 28.53/28.68 Instantiate: b0:=x1:fofType
% 28.53/28.68 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P00 b0)
% 28.53/28.68 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P00 b0))
% 28.53/28.68 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P00 b0)))
% 28.53/28.68 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 b0)
% 28.53/28.68 Found ((and_rect0 (P00 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 b0)
% 28.53/28.68 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P00 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 b0)
% 28.53/28.68 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P00 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 b0)
% 28.53/28.68 Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% 28.53/28.68 Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 28.53/28.68 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 28.53/28.68 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 28.53/28.68 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 28.53/28.68 Found x01:(P0 x00)
% 28.53/28.68 Instantiate: b0:=x00:fofType
% 28.53/28.68 Found x01 as proof of (P1 b0)
% 28.53/28.68 Found eq_ref00:=(eq_ref0 (x P0)):(((eq fofType) (x P0)) (x P0))
% 28.53/28.68 Found (eq_ref0 (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 28.53/28.68 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 28.53/28.68 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 28.53/28.68 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 28.74/28.91 Found ((eq_sym1000 ((eq_ref fofType) (x P0))) x01) as proof of (P0 (x P0))
% 28.74/28.91 Found ((eq_sym1000 ((eq_ref fofType) (x P0))) x01) as proof of (P0 (x P0))
% 28.74/28.91 Found (((fun (x03:(((eq fofType) (x P0)) b0))=> ((eq_sym100 x03) P0)) ((eq_ref fofType) (x P0))) x01) as proof of (P0 (x P0))
% 28.74/28.91 Found (((fun (x03:(((eq fofType) (x P0)) x00))=> (((eq_sym10 x00) x03) P0)) ((eq_ref fofType) (x P0))) x01) as proof of (P0 (x P0))
% 28.74/28.91 Found (((fun (x03:(((eq fofType) (x P0)) x00))=> ((((eq_sym1 (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01) as proof of (P0 (x P0))
% 28.74/28.91 Found (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01) as proof of (P0 (x P0))
% 28.74/28.91 Found (fun (x02:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01)) as proof of (P0 (x P0))
% 28.74/28.91 Found (fun (x01:(P0 x00)) (x02:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01)) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->(P0 (x P0)))
% 28.74/28.91 Found (fun (x01:(P0 x00)) (x02:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01)) as proof of ((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->(P0 (x P0))))
% 28.74/28.91 Found (and_rect00 (fun (x01:(P0 x00)) (x02:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01))) as proof of (P0 (x P0))
% 28.74/28.91 Found ((and_rect0 (P0 (x P0))) (fun (x01:(P0 x00)) (x02:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01))) as proof of (P0 (x P0))
% 28.74/28.91 Found (((fun (P1:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P0 (x P0))) (fun (x01:(P0 x00)) (x02:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01))) as proof of (P0 (x P0))
% 28.74/28.91 Found (fun (x1:((and (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))))=> (((fun (P1:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P0 (x P0))) (fun (x01:(P0 x00)) (x02:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01)))) as proof of (P0 (x P0))
% 28.74/28.91 Found x2:(P0 x00)
% 28.74/28.91 Instantiate: b0:=x00:fofType
% 28.74/28.91 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2) as proof of (P1 b0)
% 28.74/28.91 Found (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->(P1 b0))
% 28.74/28.91 Found (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->(P1 b0)))
% 28.74/28.91 Found (and_rect00 (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b0)
% 28.74/28.91 Found ((and_rect0 (P1 b0)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b0)
% 28.74/28.91 Found (((fun (P2:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b0)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b0)
% 28.74/28.91 Found (((fun (P2:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b0)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b0)
% 29.04/29.19 Found ((eq_sym1000 ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b0)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P0 (x P0))
% 29.04/29.19 Found ((eq_sym1000 ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b0)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P0 (x P0))
% 29.04/29.19 Found (((fun (x01:(((eq fofType) (x P0)) b0))=> ((eq_sym100 x01) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P0 b0)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P0 (x P0))
% 29.04/29.19 Found (((fun (x01:(((eq fofType) (x P0)) x00))=> (((eq_sym10 x00) x01) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P0 x00)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P0 (x P0))
% 29.04/29.19 Found (((fun (x01:(((eq fofType) (x P0)) x00))=> ((((eq_sym1 (x P0)) x00) x01) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P0 x00)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P0 (x P0))
% 29.04/29.19 Found (((fun (x01:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x01) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P0 x00)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P0 (x P0))
% 29.04/29.19 Found (fun (x1:((and (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))))=> (((fun (x01:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x01) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P0 x00)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2)))) as proof of (P0 (x P0))
% 29.04/29.19 Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% 29.04/29.19 Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 29.04/29.19 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 29.04/29.19 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 29.04/29.19 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 29.04/29.19 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 29.04/29.19 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) P0)
% 29.04/29.19 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P0)
% 29.04/29.19 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P0)
% 29.04/29.19 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P0)
% 29.04/29.19 Found eq_ref00:=(eq_ref0 (x a)):(((eq fofType) (x a)) (x a))
% 29.04/29.19 Found (eq_ref0 (x a)) as proof of (((eq fofType) (x a)) b)
% 29.04/29.19 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 29.04/29.19 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 29.77/29.94 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 29.77/29.94 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 29.77/29.94 Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% 29.77/29.94 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% 29.77/29.94 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% 29.77/29.94 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% 29.77/29.94 Found eq_ref00:=(eq_ref0 (x a)):(((eq fofType) (x a)) (x a))
% 29.77/29.94 Found (eq_ref0 (x a)) as proof of (((eq fofType) (x a)) b)
% 29.77/29.94 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 29.77/29.94 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 29.77/29.94 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 29.77/29.94 Found eq_ref00:=(eq_ref0 (x a)):(((eq fofType) (x a)) (x a))
% 29.77/29.94 Found (eq_ref0 (x a)) as proof of (((eq fofType) (x a)) b)
% 29.77/29.94 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 29.77/29.94 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 29.77/29.94 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 29.77/29.94 Found x10:(P0 x0)
% 29.77/29.94 Instantiate: b:=x0:fofType
% 29.77/29.94 Found x10 as proof of (P1 b)
% 29.77/29.94 Found eq_ref00:=(eq_ref0 (x P0)):(((eq fofType) (x P0)) (x P0))
% 29.77/29.94 Found (eq_ref0 (x P0)) as proof of (((eq fofType) (x P0)) b)
% 29.77/29.94 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 29.77/29.94 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 29.77/29.94 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 29.77/29.94 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 29.77/29.94 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 29.77/29.94 Found (((fun (x00:(((eq fofType) (x P0)) b))=> ((eq_sym000 x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 29.77/29.94 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> (((eq_sym00 x0) x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 29.77/29.94 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> ((((eq_sym0 (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 29.77/29.94 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 29.77/29.94 Found (fun (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of (P0 (x P0))
% 29.77/29.94 Found (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 (x P0)))
% 29.77/29.94 Found (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 (x P0))))
% 29.77/29.94 Found (and_rect00 (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 29.77/29.94 Found ((and_rect0 (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 29.77/29.94 Found (((fun (P1:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P1)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P1) x2) x1)) (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 29.77/29.94 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (P1:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P1)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P1) x2) x1)) (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)))) as proof of (P0 (x P0))
% 30.06/30.23 Found x10:(P0 x0)
% 30.06/30.23 Instantiate: b:=x0:fofType
% 30.06/30.23 Found x10 as proof of (P1 b)
% 30.06/30.23 Found eq_ref00:=(eq_ref0 (x P0)):(((eq fofType) (x P0)) (x P0))
% 30.06/30.23 Found (eq_ref0 (x P0)) as proof of (((eq fofType) (x P0)) b)
% 30.06/30.23 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 30.06/30.23 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 30.06/30.23 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 30.06/30.23 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 30.06/30.23 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 30.06/30.23 Found (((fun (x00:(((eq fofType) (x P0)) b))=> ((eq_sym000 x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 30.06/30.23 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> (((eq_sym00 x0) x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 30.06/30.23 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> ((((eq_sym0 (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 30.06/30.23 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 30.06/30.23 Found (fun (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of (P0 (x P0))
% 30.06/30.23 Found (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 (x P0)))
% 30.06/30.23 Found (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 (x P0))))
% 30.06/30.23 Found (and_rect00 (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 30.06/30.23 Found ((and_rect0 (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 30.06/30.23 Found (((fun (P1:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P1)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P1) x2) x1)) (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 30.06/30.23 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (P1:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P1)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P1) x2) x1)) (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)))) as proof of (P0 (x P0))
% 30.06/30.23 Found eq_ref00:=(eq_ref0 (x a)):(((eq fofType) (x a)) (x a))
% 30.06/30.23 Found (eq_ref0 (x a)) as proof of (((eq fofType) (x a)) b)
% 30.06/30.23 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 30.06/30.23 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 30.06/30.23 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 30.06/30.23 Found x2:(P0 x0)
% 30.06/30.23 Instantiate: b:=x0:fofType
% 30.06/30.23 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P1 b)
% 30.06/30.23 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 b))
% 30.06/30.24 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 b)))
% 30.06/30.24 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b)
% 30.06/30.24 Found ((and_rect0 (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b)
% 30.06/30.24 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b)
% 30.06/30.24 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b)
% 30.06/30.24 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 30.06/30.24 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 30.06/30.24 Found (((fun (x00:(((eq fofType) (x P0)) b))=> ((eq_sym000 x00) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 30.06/30.24 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> (((eq_sym00 x0) x00) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 x0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 30.06/30.24 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> ((((eq_sym0 (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 x0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 30.06/30.24 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 x0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 30.06/30.24 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 x0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)))) as proof of (P0 (x P0))
% 30.06/30.24 Found x2:(P0 x0)
% 30.06/30.24 Instantiate: b:=x0:fofType
% 30.06/30.24 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P1 b)
% 30.16/30.32 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 b))
% 30.16/30.32 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 b)))
% 30.16/30.32 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b)
% 30.16/30.32 Found ((and_rect0 (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b)
% 30.16/30.32 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b)
% 30.16/30.32 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b)
% 30.16/30.32 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 30.16/30.32 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 30.16/30.32 Found (((fun (x00:(((eq fofType) (x P0)) b))=> ((eq_sym000 x00) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 30.16/30.32 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> (((eq_sym00 x0) x00) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 x0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 30.16/30.32 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> ((((eq_sym0 (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 x0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 30.16/30.32 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 x0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 30.16/30.32 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 x0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)))) as proof of (P0 (x P0))
% 30.16/30.32 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 30.55/30.74 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) P0)
% 30.55/30.74 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P0)
% 30.55/30.74 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P0)
% 30.55/30.74 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P0)
% 30.55/30.74 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 30.55/30.74 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) P0)
% 30.55/30.74 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P0)
% 30.55/30.74 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P0)
% 30.55/30.74 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P0)
% 30.55/30.74 Found x3:(P x1)
% 30.55/30.74 Instantiate: b0:=x1:fofType
% 30.55/30.74 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P00 b0)
% 30.55/30.74 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P00 b0))
% 30.55/30.74 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P00 b0)))
% 30.55/30.74 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 b0)
% 30.55/30.74 Found ((and_rect0 (P00 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 b0)
% 30.55/30.74 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P00 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 b0)
% 30.55/30.74 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P00 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P00 b0)
% 30.55/30.74 Found x01:(P0 x00)
% 30.55/30.74 Instantiate: b0:=x00:fofType
% 30.55/30.74 Found x01 as proof of (P1 b0)
% 30.55/30.74 Found eq_ref00:=(eq_ref0 (x P0)):(((eq fofType) (x P0)) (x P0))
% 30.55/30.74 Found (eq_ref0 (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 30.55/30.74 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 30.55/30.74 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 30.55/30.74 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 30.55/30.74 Found ((eq_sym1000 ((eq_ref fofType) (x P0))) x01) as proof of (P0 (x P0))
% 30.55/30.74 Found ((eq_sym1000 ((eq_ref fofType) (x P0))) x01) as proof of (P0 (x P0))
% 30.55/30.74 Found (((fun (x03:(((eq fofType) (x P0)) b0))=> ((eq_sym100 x03) P0)) ((eq_ref fofType) (x P0))) x01) as proof of (P0 (x P0))
% 30.55/30.74 Found (((fun (x03:(((eq fofType) (x P0)) x00))=> (((eq_sym10 x00) x03) P0)) ((eq_ref fofType) (x P0))) x01) as proof of (P0 (x P0))
% 30.55/30.74 Found (((fun (x03:(((eq fofType) (x P0)) x00))=> ((((eq_sym1 (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01) as proof of (P0 (x P0))
% 30.55/30.74 Found (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01) as proof of (P0 (x P0))
% 30.55/30.74 Found (fun (x02:(forall (x':fofType), ((P0 x')->(((eq fofType) x00) x'))))=> (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01)) as proof of (P0 (x P0))
% 30.55/30.74 Found (fun (x01:(P0 x00)) (x02:(forall (x':fofType), ((P0 x')->(((eq fofType) x00) x'))))=> (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01)) as proof of ((forall (x':fofType), ((P0 x')->(((eq fofType) x00) x')))->(P0 (x P0)))
% 30.55/30.74 Found (fun (x01:(P0 x00)) (x02:(forall (x':fofType), ((P0 x')->(((eq fofType) x00) x'))))=> (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01)) as proof of ((P0 x00)->((forall (x':fofType), ((P0 x')->(((eq fofType) x00) x')))->(P0 (x P0))))
% 30.55/30.74 Found (and_rect00 (fun (x01:(P0 x00)) (x02:(forall (x':fofType), ((P0 x')->(((eq fofType) x00) x'))))=> (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01))) as proof of (P0 (x P0))
% 31.56/31.71 Found ((and_rect0 (P0 (x P0))) (fun (x01:(P0 x00)) (x02:(forall (x':fofType), ((P0 x')->(((eq fofType) x00) x'))))=> (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01))) as proof of (P0 (x P0))
% 31.56/31.71 Found (((fun (P1:Type) (x2:((P0 x00)->((forall (x':fofType), ((P0 x')->(((eq fofType) x00) x')))->P1)))=> (((((and_rect (P0 x00)) (forall (x':fofType), ((P0 x')->(((eq fofType) x00) x')))) P1) x2) x1)) (P0 (x P0))) (fun (x01:(P0 x00)) (x02:(forall (x':fofType), ((P0 x')->(((eq fofType) x00) x'))))=> (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01))) as proof of (P0 (x P0))
% 31.56/31.71 Found (fun (x1:(((unique fofType) P0) x00))=> (((fun (P1:Type) (x2:((P0 x00)->((forall (x':fofType), ((P0 x')->(((eq fofType) x00) x')))->P1)))=> (((((and_rect (P0 x00)) (forall (x':fofType), ((P0 x')->(((eq fofType) x00) x')))) P1) x2) x1)) (P0 (x P0))) (fun (x01:(P0 x00)) (x02:(forall (x':fofType), ((P0 x')->(((eq fofType) x00) x'))))=> (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01)))) as proof of (P0 (x P0))
% 31.56/31.71 Found x10:(P0 x0)
% 31.56/31.71 Instantiate: b:=x0:fofType
% 31.56/31.71 Found x10 as proof of (P1 b)
% 31.56/31.71 Found eq_ref00:=(eq_ref0 (x P0)):(((eq fofType) (x P0)) (x P0))
% 31.56/31.71 Found (eq_ref0 (x P0)) as proof of (((eq fofType) (x P0)) b)
% 31.56/31.71 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 31.56/31.71 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 31.56/31.71 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 31.56/31.71 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 31.56/31.71 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 31.56/31.71 Found (((fun (x00:(((eq fofType) (x P0)) b))=> ((eq_sym000 x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 31.56/31.71 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> (((eq_sym00 x0) x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 31.56/31.71 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> ((((eq_sym0 (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 31.56/31.71 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 31.56/31.71 Found (fun (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of (P0 (x P0))
% 31.56/31.71 Found (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of ((forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))->(P0 (x P0)))
% 31.56/31.71 Found (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of ((P0 x0)->((forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))->(P0 (x P0))))
% 31.56/31.71 Found (and_rect00 (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 31.56/31.71 Found ((and_rect0 (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 31.56/31.71 Found (((fun (P1:Type) (x2:((P0 x0)->((forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))->P1)))=> (((((and_rect (P0 x0)) (forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))) P1) x2) x1)) (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 34.16/34.35 Found (fun (x1:(((unique fofType) P0) x0))=> (((fun (P1:Type) (x2:((P0 x0)->((forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))->P1)))=> (((((and_rect (P0 x0)) (forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))) P1) x2) x1)) (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)))) as proof of (P0 (x P0))
% 34.16/34.35 Found x10:(P0 x0)
% 34.16/34.35 Instantiate: b:=x0:fofType
% 34.16/34.35 Found x10 as proof of (P1 b)
% 34.16/34.35 Found eq_ref00:=(eq_ref0 (x P0)):(((eq fofType) (x P0)) (x P0))
% 34.16/34.35 Found (eq_ref0 (x P0)) as proof of (((eq fofType) (x P0)) b)
% 34.16/34.35 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 34.16/34.35 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 34.16/34.35 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 34.16/34.35 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 34.16/34.35 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 34.16/34.35 Found (((fun (x00:(((eq fofType) (x P0)) b))=> ((eq_sym000 x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 34.16/34.35 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> (((eq_sym00 x0) x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 34.16/34.35 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> ((((eq_sym0 (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 34.16/34.35 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 34.16/34.35 Found (fun (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of (P0 (x P0))
% 34.16/34.35 Found (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of ((forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))->(P0 (x P0)))
% 34.16/34.35 Found (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of ((P0 x0)->((forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))->(P0 (x P0))))
% 34.16/34.35 Found (and_rect00 (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 34.16/34.35 Found ((and_rect0 (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 34.16/34.35 Found (((fun (P1:Type) (x2:((P0 x0)->((forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))->P1)))=> (((((and_rect (P0 x0)) (forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))) P1) x2) x1)) (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 34.16/34.35 Found (fun (x1:(((unique fofType) P0) x0))=> (((fun (P1:Type) (x2:((P0 x0)->((forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))->P1)))=> (((((and_rect (P0 x0)) (forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))) P1) x2) x1)) (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)))) as proof of (P0 (x P0))
% 34.16/34.35 Found x3:(P x1)
% 34.16/34.35 Instantiate: b0:=x1:fofType
% 34.16/34.35 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b0)
% 34.57/34.76 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0))
% 34.57/34.76 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0)))
% 34.57/34.76 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 34.57/34.76 Found ((and_rect0 (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 34.57/34.76 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 34.57/34.76 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 34.57/34.76 Found ((eq_sym0100 ((eq_ref fofType) b)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b)
% 34.57/34.76 Found ((eq_sym0100 ((eq_ref fofType) b)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b)
% 34.57/34.76 Found (((fun (x3:(((eq fofType) b) b0))=> ((eq_sym010 x3) P)) ((eq_ref fofType) b)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b)
% 34.57/34.76 Found (((fun (x3:(((eq fofType) b) x1))=> (((eq_sym01 x1) x3) P)) ((eq_ref fofType) b)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b)
% 34.57/34.76 Found (((fun (x3:(((eq fofType) b) x1))=> ((((eq_sym0 b) x1) x3) P)) ((eq_ref fofType) b)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b)
% 34.57/34.76 Found (((fun (x3:(((eq fofType) b) x1))=> ((((eq_sym0 b) x1) x3) P)) ((eq_ref fofType) b)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b)
% 34.57/34.76 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (x3:(((eq fofType) b) x1))=> ((((eq_sym0 b) x1) x3) P)) ((eq_ref fofType) b)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)))) as proof of (P0 b)
% 34.57/34.76 Found x01:(P x00)
% 34.57/34.76 Instantiate: b:=x00:fofType
% 34.57/34.76 Found x01 as proof of (P1 b)
% 34.57/34.76 Found eq_ref00:=(eq_ref0 (x a)):(((eq fofType) (x a)) (x a))
% 34.57/34.76 Found (eq_ref0 (x a)) as proof of (((eq fofType) (x a)) b)
% 34.57/34.76 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 34.57/34.76 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 34.66/34.80 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 34.66/34.80 Found ((eq_sym0000 ((eq_ref fofType) (x a))) x01) as proof of (P (x a))
% 34.66/34.80 Found ((eq_sym0000 ((eq_ref fofType) (x a))) x01) as proof of (P (x a))
% 34.66/34.80 Found (((fun (x03:(((eq fofType) (x a)) b))=> ((eq_sym000 x03) P)) ((eq_ref fofType) (x a))) x01) as proof of (P (x a))
% 34.66/34.80 Found (((fun (x03:(((eq fofType) (x a)) x00))=> (((eq_sym00 x00) x03) P)) ((eq_ref fofType) (x a))) x01) as proof of (P (x a))
% 34.66/34.80 Found (((fun (x03:(((eq fofType) (x a)) x00))=> ((((eq_sym0 (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01) as proof of (P (x a))
% 34.66/34.80 Found (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01) as proof of (P (x a))
% 34.66/34.80 Found (fun (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01)) as proof of (P (x a))
% 34.66/34.80 Found (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01)) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P (x a)))
% 34.66/34.80 Found (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01)) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P (x a))))
% 34.66/34.80 Found (and_rect00 (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01))) as proof of (P (x a))
% 34.66/34.80 Found ((and_rect0 (P (x a))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01))) as proof of (P (x a))
% 34.66/34.80 Found (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P (x a))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01))) as proof of (P (x a))
% 34.66/34.80 Found (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P (x a))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01))) as proof of (P (x a))
% 34.66/34.80 Found (fun (x1:((and (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))))=> (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P (x a))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01)))) as proof of (P0 (x a))
% 34.66/34.80 Found x3:(P x1)
% 34.66/34.80 Instantiate: b0:=x1:fofType
% 34.66/34.80 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b0)
% 34.66/34.80 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0))
% 34.66/34.80 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0)))
% 34.66/34.80 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 34.66/34.80 Found ((and_rect0 (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 34.86/35.01 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 34.86/35.01 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 34.86/35.01 Found x01:(P x00)
% 34.86/35.01 Instantiate: b:=x00:fofType
% 34.86/35.01 Found x01 as proof of (P1 b)
% 34.86/35.01 Found eq_ref00:=(eq_ref0 (x a)):(((eq fofType) (x a)) (x a))
% 34.86/35.01 Found (eq_ref0 (x a)) as proof of (((eq fofType) (x a)) b)
% 34.86/35.01 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 34.86/35.01 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 34.86/35.01 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 34.86/35.01 Found ((eq_sym0000 ((eq_ref fofType) (x a))) x01) as proof of (P (x a))
% 34.86/35.01 Found ((eq_sym0000 ((eq_ref fofType) (x a))) x01) as proof of (P (x a))
% 34.86/35.01 Found (((fun (x03:(((eq fofType) (x a)) b))=> ((eq_sym000 x03) P)) ((eq_ref fofType) (x a))) x01) as proof of (P (x a))
% 34.86/35.01 Found (((fun (x03:(((eq fofType) (x a)) x00))=> (((eq_sym00 x00) x03) P)) ((eq_ref fofType) (x a))) x01) as proof of (P (x a))
% 34.86/35.01 Found (((fun (x03:(((eq fofType) (x a)) x00))=> ((((eq_sym0 (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01) as proof of (P (x a))
% 34.86/35.01 Found (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01) as proof of (P (x a))
% 34.86/35.01 Found (fun (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01)) as proof of (P (x a))
% 34.86/35.01 Found (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01)) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P (x a)))
% 34.86/35.01 Found (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01)) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P (x a))))
% 34.86/35.01 Found (and_rect00 (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01))) as proof of (P (x a))
% 34.86/35.01 Found ((and_rect0 (P (x a))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01))) as proof of (P (x a))
% 34.86/35.01 Found (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P (x a))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01))) as proof of (P (x a))
% 34.86/35.01 Found (fun (x1:((and (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))))=> (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P (x a))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01)))) as proof of (P (x a))
% 34.86/35.01 Found x3:(P x1)
% 34.86/35.01 Instantiate: b0:=x1:fofType
% 34.86/35.01 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 35.17/35.38 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 35.17/35.38 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 35.17/35.38 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 35.17/35.38 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 35.17/35.38 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 35.17/35.38 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b0)
% 35.17/35.38 Found x3:(P x1)
% 35.17/35.38 Instantiate: b0:=x1:fofType
% 35.17/35.38 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 35.17/35.38 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 35.17/35.38 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 35.17/35.38 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 35.17/35.38 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 35.17/35.38 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 35.17/35.38 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 35.17/35.38 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 35.17/35.38 Found x2:(P x00)
% 35.17/35.38 Instantiate: b:=x00:fofType
% 35.17/35.38 Found (fun (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of (P1 b)
% 35.17/35.38 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 b))
% 35.17/35.38 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 b)))
% 35.17/35.38 Found (and_rect00 (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b)
% 35.17/35.38 Found ((and_rect0 (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b)
% 35.17/35.38 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b)
% 35.17/35.38 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b)
% 35.17/35.39 Found ((eq_sym0000 ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x a))
% 35.17/35.39 Found ((eq_sym0000 ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x a))
% 35.17/35.39 Found (((fun (x01:(((eq fofType) (x a)) b))=> ((eq_sym000 x01) P)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x a))
% 35.17/35.39 Found (((fun (x01:(((eq fofType) (x a)) x00))=> (((eq_sym00 x00) x01) P)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x a))
% 35.17/35.39 Found (((fun (x01:(((eq fofType) (x a)) x00))=> ((((eq_sym0 (x a)) x00) x01) P)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x a))
% 35.17/35.39 Found (((fun (x01:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x01) P)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x a))
% 35.17/35.39 Found (((fun (x01:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x01) P)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x a))
% 35.17/35.39 Found (fun (x1:((and (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))))=> (((fun (x01:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x01) P)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)))) as proof of (P0 (x a))
% 35.17/35.39 Found x2:(P x00)
% 35.17/35.39 Instantiate: b:=x00:fofType
% 35.17/35.39 Found (fun (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of (P1 b)
% 35.17/35.39 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 b))
% 35.17/35.39 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 b)))
% 35.17/35.39 Found (and_rect00 (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b)
% 35.17/35.39 Found ((and_rect0 (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b)
% 35.34/35.51 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b)
% 35.34/35.51 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b)
% 35.34/35.51 Found ((eq_sym0000 ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x a))
% 35.34/35.51 Found ((eq_sym0000 ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x a))
% 35.34/35.51 Found (((fun (x01:(((eq fofType) (x a)) b))=> ((eq_sym000 x01) P)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x a))
% 35.34/35.51 Found (((fun (x01:(((eq fofType) (x a)) x00))=> (((eq_sym00 x00) x01) P)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x a))
% 35.34/35.51 Found (((fun (x01:(((eq fofType) (x a)) x00))=> ((((eq_sym0 (x a)) x00) x01) P)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x a))
% 35.34/35.51 Found (((fun (x01:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x01) P)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x a))
% 35.34/35.51 Found (fun (x1:((and (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))))=> (((fun (x01:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x01) P)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)))) as proof of (P (x a))
% 35.34/35.51 Found x3:(P x1)
% 35.34/35.51 Instantiate: b0:=x1:fofType
% 35.34/35.51 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b0)
% 35.34/35.51 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0))
% 35.34/35.51 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0)))
% 35.34/35.51 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 35.57/35.76 Found ((and_rect0 (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 35.57/35.76 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 35.57/35.76 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 35.57/35.76 Found x3:(P x1)
% 35.57/35.76 Instantiate: b0:=x1:fofType
% 35.57/35.76 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 35.57/35.76 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 35.57/35.76 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 35.57/35.76 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 35.57/35.76 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 35.57/35.76 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 35.57/35.76 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 35.57/35.76 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 35.57/35.76 Found x3:(P x1)
% 35.57/35.76 Instantiate: b0:=x1:fofType
% 35.57/35.76 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 35.57/35.76 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 35.57/35.76 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 35.57/35.76 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 35.57/35.76 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 35.57/35.76 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 35.57/35.76 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b0)
% 35.57/35.76 Found x01:(P x00)
% 35.57/35.76 Instantiate: b:=x00:fofType
% 35.57/35.76 Found x01 as proof of (P1 b)
% 35.57/35.76 Found eq_ref00:=(eq_ref0 (x a)):(((eq fofType) (x a)) (x a))
% 35.57/35.76 Found (eq_ref0 (x a)) as proof of (((eq fofType) (x a)) b)
% 35.75/35.91 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 35.75/35.91 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 35.75/35.91 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 35.75/35.91 Found ((eq_sym0000 ((eq_ref fofType) (x a))) x01) as proof of (P (x a))
% 35.75/35.91 Found ((eq_sym0000 ((eq_ref fofType) (x a))) x01) as proof of (P (x a))
% 35.75/35.91 Found (((fun (x03:(((eq fofType) (x a)) b))=> ((eq_sym000 x03) P)) ((eq_ref fofType) (x a))) x01) as proof of (P (x a))
% 35.75/35.91 Found (((fun (x03:(((eq fofType) (x a)) x00))=> (((eq_sym00 x00) x03) P)) ((eq_ref fofType) (x a))) x01) as proof of (P (x a))
% 35.75/35.91 Found (((fun (x03:(((eq fofType) (x a)) x00))=> ((((eq_sym0 (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01) as proof of (P (x a))
% 35.75/35.91 Found (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01) as proof of (P (x a))
% 35.75/35.91 Found (fun (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01)) as proof of (P (x a))
% 35.75/35.91 Found (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01)) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P (x a)))
% 35.75/35.91 Found (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01)) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P (x a))))
% 35.75/35.91 Found (and_rect00 (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01))) as proof of (P (x a))
% 35.75/35.91 Found ((and_rect0 (P (x a))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01))) as proof of (P (x a))
% 35.75/35.91 Found (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P (x a))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01))) as proof of (P (x a))
% 35.75/35.91 Found (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P (x a))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01))) as proof of (P (x a))
% 35.75/35.91 Found (fun (x1:((and (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))))=> (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P (x a))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01)))) as proof of (P0 (x a))
% 35.75/35.91 Found x01:(P x00)
% 35.75/35.91 Instantiate: b:=x00:fofType
% 35.75/35.91 Found x01 as proof of (P1 b)
% 35.75/35.91 Found eq_ref00:=(eq_ref0 (x a)):(((eq fofType) (x a)) (x a))
% 35.75/35.91 Found (eq_ref0 (x a)) as proof of (((eq fofType) (x a)) b)
% 35.75/35.91 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 35.75/35.91 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 35.75/35.91 Found ((eq_ref fofType) (x a)) as proof of (((eq fofType) (x a)) b)
% 35.75/35.91 Found ((eq_sym0000 ((eq_ref fofType) (x a))) x01) as proof of (P (x a))
% 35.75/35.91 Found ((eq_sym0000 ((eq_ref fofType) (x a))) x01) as proof of (P (x a))
% 35.75/35.91 Found (((fun (x03:(((eq fofType) (x a)) b))=> ((eq_sym000 x03) P)) ((eq_ref fofType) (x a))) x01) as proof of (P (x a))
% 35.77/35.99 Found (((fun (x03:(((eq fofType) (x a)) x00))=> (((eq_sym00 x00) x03) P)) ((eq_ref fofType) (x a))) x01) as proof of (P (x a))
% 35.77/35.99 Found (((fun (x03:(((eq fofType) (x a)) x00))=> ((((eq_sym0 (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01) as proof of (P (x a))
% 35.77/35.99 Found (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01) as proof of (P (x a))
% 35.77/35.99 Found (fun (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01)) as proof of (P (x a))
% 35.77/35.99 Found (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01)) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P (x a)))
% 35.77/35.99 Found (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01)) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P (x a))))
% 35.77/35.99 Found (and_rect00 (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01))) as proof of (P (x a))
% 35.77/35.99 Found ((and_rect0 (P (x a))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01))) as proof of (P (x a))
% 35.77/35.99 Found (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P (x a))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01))) as proof of (P (x a))
% 35.77/35.99 Found (fun (x1:((and (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))))=> (((fun (P1:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P (x a))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x03) P)) ((eq_ref fofType) (x a))) x01)))) as proof of (P (x a))
% 35.77/35.99 Found x3:(P x1)
% 35.77/35.99 Instantiate: b:=x1:fofType
% 35.77/35.99 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b)
% 35.77/35.99 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b))
% 35.77/35.99 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b)))
% 35.77/35.99 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b)
% 35.77/35.99 Found ((and_rect0 (P1 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b)
% 35.77/35.99 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b)
% 35.77/35.99 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b)
% 35.77/35.99 Found x3:(P x1)
% 35.77/35.99 Instantiate: b:=x1:fofType
% 35.95/36.15 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b)
% 35.95/36.15 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b))
% 35.95/36.15 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b)))
% 35.95/36.15 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 35.95/36.15 Found ((and_rect0 (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 35.95/36.15 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 35.95/36.15 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b)
% 35.95/36.15 Found x3:(P x1)
% 35.95/36.15 Instantiate: b:=x1:fofType
% 35.95/36.15 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b)
% 35.95/36.15 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b))
% 35.95/36.15 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b)))
% 35.95/36.15 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 35.95/36.15 Found ((and_rect0 (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 35.95/36.15 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 35.95/36.15 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 35.95/36.15 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b)
% 35.95/36.15 Found x2:(P x00)
% 35.95/36.15 Instantiate: b:=x00:fofType
% 35.95/36.15 Found (fun (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of (P1 b)
% 35.95/36.15 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 b))
% 35.95/36.15 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 b)))
% 35.95/36.15 Found (and_rect00 (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b)
% 35.95/36.15 Found ((and_rect0 (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b)
% 35.95/36.15 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b)
% 35.95/36.15 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b)
% 35.95/36.16 Found ((eq_sym0000 ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x a))
% 35.95/36.16 Found ((eq_sym0000 ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x a))
% 35.95/36.16 Found (((fun (x01:(((eq fofType) (x a)) b))=> ((eq_sym000 x01) P)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x a))
% 35.95/36.16 Found (((fun (x01:(((eq fofType) (x a)) x00))=> (((eq_sym00 x00) x01) P)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x a))
% 35.95/36.16 Found (((fun (x01:(((eq fofType) (x a)) x00))=> ((((eq_sym0 (x a)) x00) x01) P)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x a))
% 35.95/36.16 Found (((fun (x01:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x01) P)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x a))
% 35.95/36.16 Found (((fun (x01:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x01) P)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x a))
% 35.95/36.16 Found (fun (x1:((and (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))))=> (((fun (x01:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x01) P)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)))) as proof of (P0 (x a))
% 35.95/36.16 Found x2:(P x00)
% 35.95/36.16 Instantiate: b:=x00:fofType
% 35.95/36.16 Found (fun (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of (P1 b)
% 35.95/36.16 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 b))
% 35.95/36.16 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 b)))
% 35.95/36.16 Found (and_rect00 (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b)
% 36.08/36.27 Found ((and_rect0 (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b)
% 36.08/36.27 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b)
% 36.08/36.27 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b)
% 36.08/36.27 Found ((eq_sym0000 ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x a))
% 36.08/36.27 Found ((eq_sym0000 ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x a))
% 36.08/36.27 Found (((fun (x01:(((eq fofType) (x a)) b))=> ((eq_sym000 x01) P)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P b)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x a))
% 36.08/36.27 Found (((fun (x01:(((eq fofType) (x a)) x00))=> (((eq_sym00 x00) x01) P)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x a))
% 36.08/36.27 Found (((fun (x01:(((eq fofType) (x a)) x00))=> ((((eq_sym0 (x a)) x00) x01) P)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x a))
% 36.08/36.27 Found (((fun (x01:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x01) P)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P (x a))
% 36.08/36.27 Found (fun (x1:((and (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))))=> (((fun (x01:(((eq fofType) (x a)) x00))=> (((((eq_sym fofType) (x a)) x00) x01) P)) ((eq_ref fofType) (x a))) (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P x00)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)))) as proof of (P (x a))
% 36.08/36.27 Found x3:(P x1)
% 36.08/36.27 Instantiate: b:=x1:fofType
% 36.08/36.27 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b)
% 36.08/36.27 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b))
% 36.08/36.27 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b)))
% 36.08/36.27 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b)
% 36.08/36.30 Found ((and_rect0 (P1 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b)
% 36.08/36.30 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b)
% 36.08/36.30 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b)
% 36.08/36.30 Found x3:(P x1)
% 36.08/36.30 Instantiate: b:=x1:fofType
% 36.08/36.30 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b)
% 36.08/36.30 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b))
% 36.08/36.30 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b)))
% 36.08/36.30 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 36.08/36.30 Found ((and_rect0 (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 36.08/36.30 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 36.08/36.30 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 36.08/36.30 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b)
% 36.08/36.30 Found x3:(P x1)
% 36.08/36.30 Instantiate: b:=x1:fofType
% 36.08/36.30 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b)
% 36.08/36.30 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b))
% 36.08/36.30 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b)))
% 36.08/36.30 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 36.08/36.30 Found ((and_rect0 (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 36.08/36.30 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 36.08/36.30 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b)
% 36.08/36.30 Found x3:(P x1)
% 36.08/36.30 Instantiate: b:=x1:fofType
% 36.08/36.30 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b)
% 36.08/36.30 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b))
% 36.24/36.42 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b)))
% 36.24/36.42 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b)
% 36.24/36.42 Found ((and_rect0 (P1 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b)
% 36.24/36.42 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b)
% 36.24/36.42 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b)
% 36.24/36.42 Found x3:(P x1)
% 36.24/36.42 Instantiate: b:=x1:fofType
% 36.24/36.42 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b)
% 36.24/36.42 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b))
% 36.24/36.42 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b)))
% 36.24/36.42 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 36.24/36.42 Found ((and_rect0 (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 36.24/36.42 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 36.24/36.42 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 36.24/36.42 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b)
% 36.24/36.42 Found x3:(P x1)
% 36.24/36.42 Instantiate: b:=x1:fofType
% 36.24/36.42 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b)
% 36.24/36.42 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b))
% 36.24/36.42 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b)))
% 36.24/36.42 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 36.24/36.42 Found ((and_rect0 (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 36.24/36.42 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 36.24/36.42 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b)
% 36.37/36.53 Found x3:(P x1)
% 36.37/36.53 Instantiate: b:=x1:fofType
% 36.37/36.53 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b)
% 36.37/36.53 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b))
% 36.37/36.53 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b)))
% 36.37/36.53 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b)
% 36.37/36.53 Found ((and_rect0 (P1 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b)
% 36.37/36.53 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b)
% 36.37/36.53 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b)
% 36.37/36.53 Found x3:(P x1)
% 36.37/36.53 Instantiate: b:=x1:fofType
% 36.37/36.53 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b)
% 36.37/36.53 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b))
% 36.37/36.53 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b)))
% 36.37/36.53 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 36.37/36.53 Found ((and_rect0 (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 36.37/36.53 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 36.37/36.53 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b)
% 36.37/36.53 Found x3:(P x1)
% 36.37/36.53 Instantiate: b:=x1:fofType
% 36.37/36.53 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b)
% 36.37/36.53 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b))
% 36.37/36.53 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b)))
% 36.37/36.53 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 36.37/36.53 Found ((and_rect0 (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 36.37/36.53 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 36.37/36.53 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 46.06/46.24 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b)
% 46.06/46.24 Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 46.06/46.24 Found (eta_expansion_dep00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 46.06/46.24 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 46.06/46.24 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 46.06/46.24 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 46.06/46.24 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 46.06/46.24 Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% 46.06/46.24 Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 46.06/46.24 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 46.06/46.24 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 46.06/46.24 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 46.06/46.24 Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 46.06/46.24 Found (eta_expansion_dep00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 46.06/46.24 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 46.06/46.24 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 46.06/46.24 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 46.06/46.24 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 46.06/46.24 Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% 46.06/46.24 Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 46.06/46.24 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 46.06/46.24 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 46.06/46.24 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 46.06/46.24 Found eq_ref00:=(eq_ref0 (x P)):(((eq fofType) (x P)) (x P))
% 46.06/46.24 Found (eq_ref0 (x P)) as proof of (((eq fofType) (x P)) b0)
% 46.06/46.24 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b0)
% 46.06/46.24 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b0)
% 46.06/46.24 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b0)
% 46.06/46.24 Found eq_ref00:=(eq_ref0 (x P)):(((eq fofType) (x P)) (x P))
% 46.06/46.24 Found (eq_ref0 (x P)) as proof of (((eq fofType) (x P)) b0)
% 46.06/46.24 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b0)
% 46.06/46.24 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b0)
% 46.06/46.24 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b0)
% 46.06/46.24 Found eq_ref00:=(eq_ref0 (x P0)):(((eq fofType) (x P0)) (x P0))
% 46.06/46.24 Found (eq_ref0 (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 46.06/46.24 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 46.06/46.24 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 46.06/46.24 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 46.06/46.24 Found eq_ref00:=(eq_ref0 (x P0)):(((eq fofType) (x P0)) (x P0))
% 46.06/46.24 Found (eq_ref0 (x P0)) as proof of (((eq fofType) (x P0)) b)
% 46.06/46.24 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 46.06/46.24 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 46.06/46.24 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 46.06/46.24 Found eq_ref00:=(eq_ref0 (x P0)):(((eq fofType) (x P0)) (x P0))
% 46.06/46.24 Found (eq_ref0 (x P0)) as proof of (((eq fofType) (x P0)) b)
% 46.06/46.24 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 46.06/46.24 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 46.06/46.24 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 46.28/46.51 Found x3:(P x1)
% 46.28/46.51 Instantiate: b0:=x1:fofType
% 46.28/46.51 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P00 b0)
% 46.28/46.51 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P00 b0))
% 46.28/46.51 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P00 b0)))
% 46.28/46.51 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 b0)
% 46.28/46.51 Found ((and_rect0 (P00 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 b0)
% 46.28/46.51 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P00 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 b0)
% 46.28/46.51 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P00 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 b0)
% 46.28/46.51 Found x3:(P x1)
% 46.28/46.51 Instantiate: b0:=x1:fofType
% 46.28/46.51 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P00 b0)
% 46.28/46.51 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P00 b0))
% 46.28/46.51 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P00 b0)))
% 46.28/46.51 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 b0)
% 46.28/46.51 Found ((and_rect0 (P00 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 b0)
% 46.28/46.51 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P00 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 b0)
% 46.28/46.51 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P00 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P00 b0)
% 46.28/46.51 Found x3:(P x1)
% 46.28/46.51 Instantiate: b0:=x1:fofType
% 46.28/46.51 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 46.28/46.51 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 46.28/46.51 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 46.28/46.51 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 46.28/46.51 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 46.28/46.51 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 46.28/46.51 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b0)
% 46.88/47.06 Found x3:(P x1)
% 46.88/47.06 Instantiate: b0:=x1:fofType
% 46.88/47.06 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 46.88/47.06 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 46.88/47.06 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 46.88/47.06 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 46.88/47.06 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 46.88/47.06 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 46.88/47.06 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 46.88/47.06 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P00 b0)
% 46.88/47.06 Found x01:(P0 x00)
% 46.88/47.06 Instantiate: b0:=x00:fofType
% 46.88/47.06 Found x01 as proof of (P1 b0)
% 46.88/47.06 Found eq_ref00:=(eq_ref0 (x P0)):(((eq fofType) (x P0)) (x P0))
% 46.88/47.06 Found (eq_ref0 (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 46.88/47.06 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 46.88/47.06 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 46.88/47.06 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 46.88/47.06 Found ((eq_sym1000 ((eq_ref fofType) (x P0))) x01) as proof of (P0 (x P0))
% 46.88/47.06 Found ((eq_sym1000 ((eq_ref fofType) (x P0))) x01) as proof of (P0 (x P0))
% 46.88/47.06 Found (((fun (x03:(((eq fofType) (x P0)) b0))=> ((eq_sym100 x03) P0)) ((eq_ref fofType) (x P0))) x01) as proof of (P0 (x P0))
% 46.88/47.06 Found (((fun (x03:(((eq fofType) (x P0)) x00))=> (((eq_sym10 x00) x03) P0)) ((eq_ref fofType) (x P0))) x01) as proof of (P0 (x P0))
% 46.88/47.06 Found (((fun (x03:(((eq fofType) (x P0)) x00))=> ((((eq_sym1 (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01) as proof of (P0 (x P0))
% 46.88/47.06 Found (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01) as proof of (P0 (x P0))
% 46.88/47.06 Found (fun (x02:(forall (x':fofType), ((P0 x')->(((eq fofType) x00) x'))))=> (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01)) as proof of (P0 (x P0))
% 46.88/47.06 Found (fun (x01:(P0 x00)) (x02:(forall (x':fofType), ((P0 x')->(((eq fofType) x00) x'))))=> (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01)) as proof of ((forall (x':fofType), ((P0 x')->(((eq fofType) x00) x')))->(P0 (x P0)))
% 46.88/47.06 Found (fun (x01:(P0 x00)) (x02:(forall (x':fofType), ((P0 x')->(((eq fofType) x00) x'))))=> (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01)) as proof of ((P0 x00)->((forall (x':fofType), ((P0 x')->(((eq fofType) x00) x')))->(P0 (x P0))))
% 46.88/47.06 Found (and_rect00 (fun (x01:(P0 x00)) (x02:(forall (x':fofType), ((P0 x')->(((eq fofType) x00) x'))))=> (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01))) as proof of (P0 (x P0))
% 46.88/47.06 Found ((and_rect0 (P0 (x P0))) (fun (x01:(P0 x00)) (x02:(forall (x':fofType), ((P0 x')->(((eq fofType) x00) x'))))=> (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01))) as proof of (P0 (x P0))
% 48.45/48.65 Found (((fun (P1:Type) (x2:((P0 x00)->((forall (x':fofType), ((P0 x')->(((eq fofType) x00) x')))->P1)))=> (((((and_rect (P0 x00)) (forall (x':fofType), ((P0 x')->(((eq fofType) x00) x')))) P1) x2) x1)) (P0 (x P0))) (fun (x01:(P0 x00)) (x02:(forall (x':fofType), ((P0 x')->(((eq fofType) x00) x'))))=> (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01))) as proof of (P0 (x P0))
% 48.45/48.65 Found (fun (x1:(((unique fofType) P0) x00))=> (((fun (P1:Type) (x2:((P0 x00)->((forall (x':fofType), ((P0 x')->(((eq fofType) x00) x')))->P1)))=> (((((and_rect (P0 x00)) (forall (x':fofType), ((P0 x')->(((eq fofType) x00) x')))) P1) x2) x1)) (P0 (x P0))) (fun (x01:(P0 x00)) (x02:(forall (x':fofType), ((P0 x')->(((eq fofType) x00) x'))))=> (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01)))) as proof of (P0 (x P0))
% 48.45/48.65 Found x10:(P0 x0)
% 48.45/48.65 Instantiate: b:=x0:fofType
% 48.45/48.65 Found x10 as proof of (P1 b)
% 48.45/48.65 Found eq_ref00:=(eq_ref0 (x P0)):(((eq fofType) (x P0)) (x P0))
% 48.45/48.65 Found (eq_ref0 (x P0)) as proof of (((eq fofType) (x P0)) b)
% 48.45/48.65 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 48.45/48.65 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 48.45/48.65 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 48.45/48.65 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 48.45/48.65 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 48.45/48.65 Found (((fun (x00:(((eq fofType) (x P0)) b))=> ((eq_sym000 x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 48.45/48.65 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> (((eq_sym00 x0) x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 48.45/48.65 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> ((((eq_sym0 (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 48.45/48.65 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 48.45/48.65 Found (fun (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of (P0 (x P0))
% 48.45/48.65 Found (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of ((forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))->(P0 (x P0)))
% 48.45/48.65 Found (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of ((P0 x0)->((forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))->(P0 (x P0))))
% 48.45/48.65 Found (and_rect00 (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 48.45/48.65 Found ((and_rect0 (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 48.45/48.65 Found (((fun (P1:Type) (x2:((P0 x0)->((forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))->P1)))=> (((((and_rect (P0 x0)) (forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))) P1) x2) x1)) (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 48.45/48.65 Found (fun (x1:(((unique fofType) P0) x0))=> (((fun (P1:Type) (x2:((P0 x0)->((forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))->P1)))=> (((((and_rect (P0 x0)) (forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))) P1) x2) x1)) (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)))) as proof of (P0 (x P0))
% 50.35/50.50 Found x10:(P0 x0)
% 50.35/50.50 Instantiate: b:=x0:fofType
% 50.35/50.50 Found x10 as proof of (P1 b)
% 50.35/50.50 Found eq_ref00:=(eq_ref0 (x P0)):(((eq fofType) (x P0)) (x P0))
% 50.35/50.50 Found (eq_ref0 (x P0)) as proof of (((eq fofType) (x P0)) b)
% 50.35/50.50 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 50.35/50.50 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 50.35/50.50 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 50.35/50.50 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 50.35/50.50 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 50.35/50.50 Found (((fun (x00:(((eq fofType) (x P0)) b))=> ((eq_sym000 x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 50.35/50.50 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> (((eq_sym00 x0) x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 50.35/50.50 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> ((((eq_sym0 (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 50.35/50.50 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 50.35/50.50 Found (fun (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of (P0 (x P0))
% 50.35/50.50 Found (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of ((forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))->(P0 (x P0)))
% 50.35/50.50 Found (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of ((P0 x0)->((forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))->(P0 (x P0))))
% 50.35/50.50 Found (and_rect00 (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 50.35/50.50 Found ((and_rect0 (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 50.35/50.50 Found (((fun (P1:Type) (x2:((P0 x0)->((forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))->P1)))=> (((((and_rect (P0 x0)) (forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))) P1) x2) x1)) (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 50.35/50.50 Found (fun (x1:(((unique fofType) P0) x0))=> (((fun (P1:Type) (x2:((P0 x0)->((forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))->P1)))=> (((((and_rect (P0 x0)) (forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))) P1) x2) x1)) (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)))) as proof of (P0 (x P0))
% 50.35/50.50 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 50.35/50.50 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) P0)
% 50.35/50.50 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P0)
% 50.35/50.50 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P0)
% 50.35/50.50 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P0)
% 50.35/50.50 Found x3:(P x1)
% 50.35/50.50 Instantiate: b0:=x1:fofType
% 50.35/50.50 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 51.24/51.45 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 51.24/51.45 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 51.24/51.45 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 51.24/51.45 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 51.24/51.45 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 51.24/51.45 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 51.24/51.45 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 51.24/51.45 Found x3:(P x1)
% 51.24/51.45 Instantiate: b0:=x1:fofType
% 51.24/51.45 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 51.24/51.45 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 51.24/51.45 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 51.24/51.45 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 51.24/51.45 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 51.24/51.45 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 51.24/51.45 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b0)
% 51.24/51.45 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 51.24/51.45 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 51.24/51.45 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 51.24/51.45 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 51.24/51.45 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 51.24/51.45 Found eq_ref00:=(eq_ref0 (x P0)):(((eq fofType) (x P0)) (x P0))
% 51.24/51.45 Found (eq_ref0 (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 51.24/51.45 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 51.24/51.45 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 51.24/51.45 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 51.24/51.45 Found x3:(P x1)
% 51.24/51.45 Instantiate: b0:=x1:fofType
% 51.24/51.45 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 51.24/51.45 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 51.24/51.45 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 51.24/51.45 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 51.85/52.07 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 51.85/52.07 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 51.85/52.07 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 51.85/52.07 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 51.85/52.07 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 51.85/52.07 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) P0)
% 51.85/52.07 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P0)
% 51.85/52.07 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P0)
% 51.85/52.07 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P0)
% 51.85/52.07 Found x3:(P x1)
% 51.85/52.07 Instantiate: b0:=x1:fofType
% 51.85/52.07 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 51.85/52.07 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 51.85/52.07 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 51.85/52.07 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 51.85/52.07 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 51.85/52.07 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 51.85/52.07 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b0)
% 51.85/52.07 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 51.85/52.07 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) P0)
% 51.85/52.07 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P0)
% 51.85/52.07 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P0)
% 51.85/52.07 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P0)
% 51.85/52.07 Found x3:(P x1)
% 51.85/52.07 Instantiate: b:=x1:fofType
% 51.85/52.07 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b)
% 51.85/52.07 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b))
% 51.85/52.07 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b)))
% 51.85/52.07 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 51.85/52.07 Found ((and_rect0 (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 51.85/52.07 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 52.45/52.60 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b)
% 52.45/52.60 Found x3:(P x1)
% 52.45/52.60 Instantiate: b:=x1:fofType
% 52.45/52.60 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b)
% 52.45/52.60 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b))
% 52.45/52.60 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b)))
% 52.45/52.60 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 52.45/52.60 Found ((and_rect0 (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 52.45/52.60 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 52.45/52.60 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 52.45/52.60 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b)
% 52.45/52.60 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 52.45/52.60 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 52.45/52.60 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 52.45/52.60 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 52.45/52.60 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 52.45/52.60 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 52.45/52.60 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 52.45/52.60 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 52.45/52.60 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 52.45/52.60 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 52.45/52.60 Found x3:(P x1)
% 52.45/52.60 Instantiate: b:=x1:fofType
% 52.45/52.60 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b)
% 52.45/52.60 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b))
% 52.45/52.60 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b)))
% 52.45/52.60 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 52.45/52.60 Found ((and_rect0 (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 52.45/52.60 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 52.45/52.60 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b)
% 52.45/52.62 Found x3:(P x1)
% 52.45/52.62 Instantiate: b:=x1:fofType
% 52.45/52.62 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b)
% 52.45/52.62 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b))
% 52.45/52.62 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b)))
% 52.45/52.62 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 52.45/52.62 Found ((and_rect0 (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 52.45/52.62 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 52.45/52.62 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 52.45/52.62 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b)
% 52.45/52.62 Found x3:(P x1)
% 52.45/52.62 Instantiate: b:=x1:fofType
% 52.45/52.62 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b)
% 52.45/52.62 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b))
% 52.45/52.62 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b)))
% 52.45/52.62 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 52.45/52.62 Found ((and_rect0 (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 52.45/52.62 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 52.45/52.62 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b)
% 52.45/52.62 Found x3:(P x1)
% 52.45/52.62 Instantiate: b:=x1:fofType
% 52.45/52.62 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b)
% 52.45/52.62 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b))
% 52.45/52.62 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b)))
% 52.45/52.62 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 52.45/52.62 Found ((and_rect0 (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 52.45/52.62 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 52.87/53.02 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 52.87/53.02 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b)
% 52.87/53.02 Found eq_ref00:=(eq_ref0 (x P0)):(((eq fofType) (x P0)) (x P0))
% 52.87/53.02 Found (eq_ref0 (x P0)) as proof of (((eq fofType) (x P0)) b)
% 52.87/53.02 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 52.87/53.02 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 52.87/53.02 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 52.87/53.02 Found eq_ref00:=(eq_ref0 (x P0)):(((eq fofType) (x P0)) (x P0))
% 52.87/53.02 Found (eq_ref0 (x P0)) as proof of (((eq fofType) (x P0)) b)
% 52.87/53.02 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 52.87/53.02 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 52.87/53.02 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 52.87/53.02 Found x3:(P x1)
% 52.87/53.02 Instantiate: b:=x1:fofType
% 52.87/53.02 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b)
% 52.87/53.02 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b))
% 52.87/53.02 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b)))
% 52.87/53.02 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 52.87/53.02 Found ((and_rect0 (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 52.87/53.02 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 52.87/53.02 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 52.87/53.02 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b)
% 52.87/53.02 Found x3:(P x1)
% 52.87/53.02 Instantiate: b:=x1:fofType
% 52.87/53.02 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b)
% 52.87/53.02 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b))
% 52.87/53.02 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b)))
% 52.87/53.02 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 52.87/53.02 Found ((and_rect0 (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 52.87/53.02 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b)
% 52.87/53.02 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b)
% 56.16/56.32 Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% 56.16/56.32 Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% 56.16/56.32 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 56.16/56.32 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 56.16/56.32 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 56.16/56.32 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 56.16/56.32 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 56.16/56.32 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 56.16/56.32 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 56.16/56.32 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 56.16/56.32 Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% 56.16/56.32 Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% 56.16/56.32 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 56.16/56.32 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 56.16/56.32 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 56.16/56.32 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 56.16/56.32 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 56.16/56.32 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 56.16/56.32 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 56.16/56.32 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 56.16/56.32 Found x01:(P0 x00)
% 56.16/56.32 Instantiate: b0:=x00:fofType
% 56.16/56.32 Found x01 as proof of (P1 b0)
% 56.16/56.32 Found eq_ref00:=(eq_ref0 (x P0)):(((eq fofType) (x P0)) (x P0))
% 56.16/56.32 Found (eq_ref0 (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 56.16/56.32 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 56.16/56.32 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 56.16/56.32 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 56.16/56.32 Found ((eq_sym1000 ((eq_ref fofType) (x P0))) x01) as proof of (P0 (x P0))
% 56.16/56.32 Found ((eq_sym1000 ((eq_ref fofType) (x P0))) x01) as proof of (P0 (x P0))
% 56.16/56.32 Found (((fun (x03:(((eq fofType) (x P0)) b0))=> ((eq_sym100 x03) P0)) ((eq_ref fofType) (x P0))) x01) as proof of (P0 (x P0))
% 56.16/56.32 Found (((fun (x03:(((eq fofType) (x P0)) x00))=> (((eq_sym10 x00) x03) P0)) ((eq_ref fofType) (x P0))) x01) as proof of (P0 (x P0))
% 56.16/56.32 Found (((fun (x03:(((eq fofType) (x P0)) x00))=> ((((eq_sym1 (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01) as proof of (P0 (x P0))
% 56.16/56.32 Found (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01) as proof of (P0 (x P0))
% 56.16/56.32 Found (fun (x02:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01)) as proof of (P0 (x P0))
% 56.16/56.32 Found (fun (x01:(P0 x00)) (x02:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01)) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->(P0 (x P0)))
% 56.16/56.32 Found (fun (x01:(P0 x00)) (x02:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01)) as proof of ((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->(P0 (x P0))))
% 56.16/56.32 Found (and_rect00 (fun (x01:(P0 x00)) (x02:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01))) as proof of (P0 (x P0))
% 56.16/56.32 Found ((and_rect0 (P0 (x P0))) (fun (x01:(P0 x00)) (x02:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01))) as proof of (P0 (x P0))
% 56.16/56.32 Found (((fun (P1:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P0 (x P0))) (fun (x01:(P0 x00)) (x02:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01))) as proof of (P0 (x P0))
% 56.64/56.84 Found (fun (x1:((and (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))))=> (((fun (P1:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P1)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P1) x2) x1)) (P0 (x P0))) (fun (x01:(P0 x00)) (x02:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x03) P0)) ((eq_ref fofType) (x P0))) x01)))) as proof of (P0 (x P0))
% 56.64/56.84 Found x2:(P0 x00)
% 56.64/56.84 Instantiate: b0:=x00:fofType
% 56.64/56.84 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2) as proof of (P1 b0)
% 56.64/56.84 Found (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->(P1 b0))
% 56.64/56.84 Found (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->(P1 b0)))
% 56.64/56.84 Found (and_rect00 (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b0)
% 56.64/56.84 Found ((and_rect0 (P1 b0)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b0)
% 56.64/56.84 Found (((fun (P2:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b0)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b0)
% 56.64/56.84 Found (((fun (P2:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b0)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b0)
% 56.64/56.84 Found ((eq_sym1000 ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b0)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P0 (x P0))
% 56.64/56.84 Found ((eq_sym1000 ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b0)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P0 (x P0))
% 56.64/56.84 Found (((fun (x01:(((eq fofType) (x P0)) b0))=> ((eq_sym100 x01) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P0 b0)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P0 (x P0))
% 56.64/56.84 Found (((fun (x01:(((eq fofType) (x P0)) x00))=> (((eq_sym10 x00) x01) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P0 x00)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P0 (x P0))
% 56.64/56.84 Found (((fun (x01:(((eq fofType) (x P0)) x00))=> ((((eq_sym1 (x P0)) x00) x01) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P0 x00)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P0 (x P0))
% 57.47/57.62 Found (((fun (x01:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x01) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P0 x00)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2))) as proof of (P0 (x P0))
% 57.47/57.62 Found (fun (x1:((and (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))))=> (((fun (x01:(((eq fofType) (x P0)) x00))=> (((((eq_sym fofType) (x P0)) x00) x01) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P0 x00)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2)))) as proof of (P0 (x P0))
% 57.47/57.62 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 57.47/57.62 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) P0)
% 57.47/57.62 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P0)
% 57.47/57.62 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P0)
% 57.47/57.62 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P0)
% 57.47/57.62 Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% 57.47/57.62 Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) P)
% 57.47/57.62 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 57.47/57.62 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 57.47/57.62 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 57.47/57.62 Found x3:(P0 x1)
% 57.47/57.62 Instantiate: b0:=x1:fofType
% 57.47/57.62 Found (fun (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b0)
% 57.47/57.62 Found (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->(P1 b0))
% 57.47/57.62 Found (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->(P1 b0)))
% 57.47/57.62 Found (and_rect00 (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 57.47/57.62 Found ((and_rect0 (P1 b0)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 57.47/57.62 Found (((fun (P2:Type) (x3:((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 57.47/57.62 Found (fun (x2:((and (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 57.47/57.62 Found x3:(P0 x1)
% 57.47/57.62 Instantiate: b0:=x1:fofType
% 57.47/57.62 Found (fun (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b0)
% 57.47/57.62 Found (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->(P1 b0))
% 57.47/57.62 Found (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->(P1 b0)))
% 57.47/57.62 Found (and_rect00 (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 57.47/57.62 Found ((and_rect0 (P1 b0)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 57.47/57.62 Found (((fun (P2:Type) (x3:((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 58.25/58.45 Found (fun (x2:((and (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 58.25/58.45 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 58.25/58.45 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 58.25/58.45 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 58.25/58.45 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 58.25/58.45 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 58.25/58.45 Found x10:(P0 x0)
% 58.25/58.45 Instantiate: b:=x0:fofType
% 58.25/58.45 Found x10 as proof of (P1 b)
% 58.25/58.45 Found eq_ref00:=(eq_ref0 (x P0)):(((eq fofType) (x P0)) (x P0))
% 58.25/58.45 Found (eq_ref0 (x P0)) as proof of (((eq fofType) (x P0)) b)
% 58.25/58.45 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 58.25/58.45 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 58.25/58.45 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 58.25/58.45 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 58.25/58.45 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 58.25/58.45 Found (((fun (x00:(((eq fofType) (x P0)) b))=> ((eq_sym000 x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 58.25/58.45 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> (((eq_sym00 x0) x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 58.25/58.45 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> ((((eq_sym0 (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 58.25/58.45 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 58.25/58.45 Found (fun (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of (P0 (x P0))
% 58.25/58.45 Found (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 (x P0)))
% 58.25/58.45 Found (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 (x P0))))
% 58.25/58.45 Found (and_rect00 (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 58.25/58.45 Found ((and_rect0 (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 58.25/58.45 Found (((fun (P1:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P1)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P1) x2) x1)) (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 58.25/58.45 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (P1:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P1)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P1) x2) x1)) (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)))) as proof of (P0 (x P0))
% 58.25/58.45 Found x10:(P0 x0)
% 58.25/58.45 Instantiate: b:=x0:fofType
% 58.63/58.84 Found x10 as proof of (P1 b)
% 58.63/58.84 Found eq_ref00:=(eq_ref0 (x P0)):(((eq fofType) (x P0)) (x P0))
% 58.63/58.84 Found (eq_ref0 (x P0)) as proof of (((eq fofType) (x P0)) b)
% 58.63/58.84 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 58.63/58.84 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 58.63/58.84 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b)
% 58.63/58.84 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 58.63/58.84 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 58.63/58.84 Found (((fun (x00:(((eq fofType) (x P0)) b))=> ((eq_sym000 x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 58.63/58.84 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> (((eq_sym00 x0) x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 58.63/58.84 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> ((((eq_sym0 (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 58.63/58.84 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 58.63/58.84 Found (fun (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of (P0 (x P0))
% 58.63/58.84 Found (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 (x P0)))
% 58.63/58.84 Found (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 (x P0))))
% 58.63/58.84 Found (and_rect00 (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 58.63/58.84 Found ((and_rect0 (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 58.63/58.84 Found (((fun (P1:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P1)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P1) x2) x1)) (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 58.63/58.84 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (P1:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P1)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P1) x2) x1)) (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) x10)))) as proof of (P0 (x P0))
% 58.63/58.84 Found x2:(P0 x0)
% 58.63/58.84 Instantiate: b:=x0:fofType
% 58.63/58.84 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P1 b)
% 58.63/58.84 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 b))
% 58.63/58.84 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 b)))
% 58.63/58.84 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b)
% 58.63/58.84 Found ((and_rect0 (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b)
% 58.63/58.84 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b)
% 58.77/58.96 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b)
% 58.77/58.96 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 58.77/58.96 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 58.77/58.96 Found (((fun (x00:(((eq fofType) (x P0)) b))=> ((eq_sym000 x00) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 58.77/58.96 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> (((eq_sym00 x0) x00) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 x0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 58.77/58.96 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> ((((eq_sym0 (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 x0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 58.77/58.96 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 x0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 58.77/58.96 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 x0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)))) as proof of (P0 (x P0))
% 58.77/58.96 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 58.77/58.96 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) P0)
% 58.77/58.96 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P0)
% 58.77/58.96 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P0)
% 58.77/58.96 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P0)
% 58.77/58.96 Found x2:(P0 x0)
% 58.77/58.96 Instantiate: b:=x0:fofType
% 58.77/58.96 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P1 b)
% 58.77/58.96 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 b))
% 58.77/58.96 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 b)))
% 58.77/58.97 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b)
% 58.77/58.97 Found ((and_rect0 (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b)
% 58.77/58.97 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b)
% 58.77/58.97 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b)
% 58.77/58.97 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 58.77/58.97 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 58.77/58.97 Found (((fun (x00:(((eq fofType) (x P0)) b))=> ((eq_sym000 x00) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 58.77/58.97 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> (((eq_sym00 x0) x00) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 x0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 58.77/58.97 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> ((((eq_sym0 (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 x0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 58.77/58.97 Found (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 x0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 58.77/58.97 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (x00:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x00) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 x0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)))) as proof of (P0 (x P0))
% 58.77/58.97 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 58.77/58.97 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) P0)
% 58.77/58.97 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P0)
% 58.77/58.97 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P0)
% 59.34/59.49 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) P0)
% 59.34/59.49 Found x2:(P0 x0)
% 59.34/59.49 Instantiate: b:=x0:fofType
% 59.34/59.49 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P1 b)
% 59.34/59.49 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 b))
% 59.34/59.49 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 b)))
% 59.34/59.49 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b)
% 59.34/59.49 Found ((and_rect0 (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b)
% 59.34/59.49 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b)
% 59.34/59.49 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P1 b)
% 59.34/59.49 Found x2:(P0 x0)
% 59.34/59.49 Instantiate: b:=x0:fofType
% 59.34/59.49 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P1 b)
% 59.34/59.49 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 b))
% 59.34/59.49 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 b)))
% 59.34/59.49 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b)
% 59.34/59.49 Found ((and_rect0 (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b)
% 59.34/59.49 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b)
% 59.34/59.49 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P1 b)
% 59.34/59.49 Found x2:(P0 x0)
% 59.34/59.49 Instantiate: b:=x0:fofType
% 59.34/59.49 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P1 b)
% 59.34/59.49 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 b))
% 59.34/59.49 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 b)))
% 59.34/59.49 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b)
% 59.34/59.49 Found ((and_rect0 (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b)
% 59.34/59.49 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b)
% 59.34/59.49 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P1 b)
% 61.98/62.14 Found x2:(P0 x0)
% 61.98/62.14 Instantiate: b:=x0:fofType
% 61.98/62.14 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P1 b)
% 61.98/62.14 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 b))
% 61.98/62.14 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 b)))
% 61.98/62.14 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b)
% 61.98/62.14 Found ((and_rect0 (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b)
% 61.98/62.14 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b)
% 61.98/62.14 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P1 b)
% 61.98/62.14 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 61.98/62.14 Found (eq_ref0 b) as proof of (((eq fofType) b) (x a))
% 61.98/62.14 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 61.98/62.14 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 61.98/62.14 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 61.98/62.14 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 61.98/62.14 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b)
% 61.98/62.14 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 61.98/62.14 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 61.98/62.14 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 61.98/62.14 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 61.98/62.14 Found (eq_ref0 b) as proof of (((eq fofType) b) (x a))
% 61.98/62.14 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 61.98/62.14 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 61.98/62.14 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 61.98/62.14 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 61.98/62.14 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b)
% 61.98/62.14 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 61.98/62.14 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 61.98/62.14 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 61.98/62.14 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 61.98/62.14 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 61.98/62.14 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 61.98/62.14 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 61.98/62.14 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 61.98/62.14 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 61.98/62.14 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 61.98/62.14 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 61.98/62.14 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 61.98/62.14 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 61.98/62.14 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 61.98/62.14 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 61.98/62.14 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 61.98/62.14 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 61.98/62.14 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 61.98/62.14 Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% 61.98/62.14 Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% 61.98/62.14 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 61.98/62.14 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 61.98/62.14 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 61.98/62.14 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 61.98/62.14 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 61.98/62.14 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 66.76/66.92 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 66.76/66.92 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 66.76/66.92 Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% 66.76/66.92 Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% 66.76/66.92 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 66.76/66.92 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 66.76/66.92 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 66.76/66.92 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 66.76/66.92 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 66.76/66.92 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 66.76/66.92 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 66.76/66.92 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 66.76/66.92 Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% 66.76/66.92 Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) P)
% 66.76/66.92 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 66.76/66.92 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 66.76/66.92 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 66.76/66.92 Found x3:(P x1)
% 66.76/66.92 Instantiate: a:=x1:fofType
% 66.76/66.92 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a)
% 66.76/66.92 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a))
% 66.76/66.92 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a)))
% 66.76/66.92 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 66.76/66.92 Found ((and_rect0 (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 66.76/66.92 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 66.76/66.92 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 66.76/66.92 Found x3:(P x1)
% 66.76/66.92 Instantiate: b0:=x1:fofType
% 66.76/66.92 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b0)
% 66.76/66.92 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0))
% 66.76/66.92 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0)))
% 66.76/66.92 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 66.76/66.92 Found ((and_rect0 (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 66.76/66.92 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 66.76/66.92 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 66.76/66.92 Found ((eq_sym0000 ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P0 a)
% 66.76/66.92 Found ((eq_sym0000 ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P0 a)
% 67.25/67.48 Found (((fun (x3:(((eq fofType) a) b0))=> ((eq_sym000 x3) P0)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P0 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P0 a)
% 67.25/67.48 Found (((fun (x3:(((eq fofType) a) x1))=> (((eq_sym00 x1) x3) P0)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P0 x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P0 a)
% 67.25/67.48 Found (((fun (x3:(((eq fofType) a) x1))=> ((((eq_sym0 a) x1) x3) P0)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P0 x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P0 a)
% 67.25/67.48 Found (((fun (x3:(((eq fofType) a) x1))=> (((((eq_sym fofType) a) x1) x3) P0)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P0 x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P0 a)
% 67.25/67.48 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (x3:(((eq fofType) a) x1))=> (((((eq_sym fofType) a) x1) x3) P0)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P0 x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)))) as proof of (P0 a)
% 67.25/67.48 Found x2:(P x00)
% 67.25/67.48 Instantiate: b0:=x00:fofType
% 67.25/67.48 Found x2 as proof of (P1 b0)
% 67.25/67.48 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 67.25/67.48 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 67.25/67.48 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 67.25/67.48 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 67.25/67.48 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 67.25/67.48 Found x3:(P x1)
% 67.25/67.48 Instantiate: a:=x1:fofType
% 67.25/67.48 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a)
% 67.25/67.48 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a))
% 67.25/67.48 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a)))
% 67.25/67.48 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 67.25/67.48 Found ((and_rect0 (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 67.25/67.48 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 67.25/67.48 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a)
% 67.25/67.48 Found x3:(P x1)
% 67.25/67.48 Instantiate: b0:=x1:fofType
% 67.97/68.18 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b0)
% 67.97/68.18 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0))
% 67.97/68.18 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0)))
% 67.97/68.18 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 67.97/68.18 Found ((and_rect0 (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 67.97/68.18 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 67.97/68.18 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 67.97/68.18 Found x3:(P x1)
% 67.97/68.18 Instantiate: b0:=x1:fofType
% 67.97/68.18 Found x3 as proof of (P1 b0)
% 67.97/68.18 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 67.97/68.18 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 67.97/68.18 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 67.97/68.18 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 67.97/68.18 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 67.97/68.18 Found x2:(P x00)
% 67.97/68.18 Instantiate: b0:=x00:fofType
% 67.97/68.18 Found (fun (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of (P1 b0)
% 67.97/68.18 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 b0))
% 67.97/68.18 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 b0)))
% 67.97/68.18 Found (and_rect00 (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b0)
% 67.97/68.18 Found ((and_rect0 (P1 b0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b0)
% 67.97/68.18 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b0)
% 67.97/68.18 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b0)
% 67.97/68.18 Found x3:(P x1)
% 67.97/68.18 Instantiate: a:=x1:fofType
% 67.97/68.18 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a)
% 67.97/68.18 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a))
% 67.97/68.18 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a)))
% 67.97/68.18 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 67.97/68.18 Found ((and_rect0 (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 67.97/68.18 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 67.97/68.18 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 68.28/68.42 Found x3:(P x1)
% 68.28/68.42 Instantiate: a:=x1:fofType
% 68.28/68.42 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a)
% 68.28/68.42 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a))
% 68.28/68.42 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a)))
% 68.28/68.42 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 68.28/68.42 Found ((and_rect0 (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 68.28/68.42 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 68.28/68.42 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 68.28/68.42 Found x3:(P x1)
% 68.28/68.42 Instantiate: b0:=x1:fofType
% 68.28/68.42 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b0)
% 68.28/68.42 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0))
% 68.28/68.42 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0)))
% 68.28/68.42 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 68.28/68.42 Found ((and_rect0 (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 68.28/68.42 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 68.28/68.42 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 68.28/68.42 Found ((eq_sym0000 ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P0 a)
% 68.28/68.42 Found ((eq_sym0000 ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P0 a)
% 68.28/68.42 Found (((fun (x3:(((eq fofType) a) b0))=> ((eq_sym000 x3) P0)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P0 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P0 a)
% 68.28/68.42 Found (((fun (x3:(((eq fofType) a) x1))=> (((eq_sym00 x1) x3) P0)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P0 x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P0 a)
% 68.28/68.48 Found (((fun (x3:(((eq fofType) a) x1))=> ((((eq_sym0 a) x1) x3) P0)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P0 x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P0 a)
% 68.28/68.48 Found (((fun (x3:(((eq fofType) a) x1))=> (((((eq_sym fofType) a) x1) x3) P0)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P0 x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P0 a)
% 68.28/68.48 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (x3:(((eq fofType) a) x1))=> (((((eq_sym fofType) a) x1) x3) P0)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P0 x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)))) as proof of (P0 a)
% 68.28/68.48 Found x3:(P x1)
% 68.28/68.48 Instantiate: b0:=x1:fofType
% 68.28/68.48 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b0)
% 68.28/68.48 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0))
% 68.28/68.48 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0)))
% 68.28/68.48 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 68.28/68.48 Found ((and_rect0 (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 68.28/68.48 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 68.28/68.48 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 68.28/68.48 Found x3:(P x1)
% 68.28/68.48 Instantiate: a:=x1:fofType
% 68.28/68.48 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a)
% 68.28/68.48 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a))
% 68.28/68.48 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a)))
% 68.28/68.48 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 68.28/68.48 Found ((and_rect0 (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 68.28/68.48 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 68.28/68.48 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a)
% 68.28/68.49 Found x3:(P x1)
% 68.28/68.49 Instantiate: b0:=x1:fofType
% 68.28/68.49 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b0)
% 68.28/68.49 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0))
% 68.28/68.49 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0)))
% 68.28/68.49 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 68.28/68.49 Found ((and_rect0 (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 68.28/68.49 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 68.28/68.49 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 68.28/68.49 Found x3:(P x1)
% 68.28/68.49 Instantiate: a:=x1:fofType
% 68.28/68.49 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a)
% 68.28/68.49 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a))
% 68.28/68.49 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a)))
% 68.28/68.49 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 68.28/68.49 Found ((and_rect0 (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 68.28/68.49 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 68.28/68.49 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a)
% 68.28/68.49 Found x3:(P x1)
% 68.28/68.49 Instantiate: b0:=x1:fofType
% 68.28/68.49 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b0)
% 68.28/68.49 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0))
% 68.28/68.49 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0)))
% 68.28/68.49 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 68.28/68.49 Found ((and_rect0 (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 68.28/68.49 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 68.28/68.49 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 68.74/68.92 Found x3:(P x1)
% 68.74/68.92 Instantiate: b0:=x1:fofType
% 68.74/68.92 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b0)
% 68.74/68.92 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0))
% 68.74/68.92 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0)))
% 68.74/68.92 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 68.74/68.92 Found ((and_rect0 (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 68.74/68.92 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 68.74/68.92 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 68.74/68.92 Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% 68.74/68.92 Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) P)
% 68.74/68.92 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 68.74/68.92 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 68.74/68.92 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 68.74/68.92 Found x3:(P x1)
% 68.74/68.92 Instantiate: a:=x1:fofType
% 68.74/68.92 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a)
% 68.74/68.92 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a))
% 68.74/68.92 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a)))
% 68.74/68.92 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 68.74/68.92 Found ((and_rect0 (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 68.74/68.92 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 68.74/68.92 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 68.74/68.92 Found x3:(P x1)
% 68.74/68.92 Instantiate: b0:=x1:fofType
% 68.74/68.92 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b0)
% 68.74/68.92 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0))
% 68.74/68.92 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0)))
% 68.74/68.92 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 68.74/68.92 Found ((and_rect0 (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 68.74/68.92 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 73.37/73.55 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 73.37/73.55 Found x3:(P x1)
% 73.37/73.55 Instantiate: a:=x1:fofType
% 73.37/73.55 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a)
% 73.37/73.55 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a))
% 73.37/73.55 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a)))
% 73.37/73.55 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 73.37/73.55 Found ((and_rect0 (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 73.37/73.55 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 73.37/73.55 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a)
% 73.37/73.55 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 73.37/73.55 Found (eq_ref0 b) as proof of (((eq fofType) b) (x a))
% 73.37/73.55 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 73.37/73.55 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 73.37/73.55 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 73.37/73.55 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 73.37/73.55 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b)
% 73.37/73.55 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 73.37/73.55 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 73.37/73.55 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 73.37/73.55 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 73.37/73.55 Found (eq_ref0 b) as proof of (((eq fofType) b) (x a))
% 73.37/73.55 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 73.37/73.55 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 73.37/73.55 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 73.37/73.55 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 73.37/73.55 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b)
% 73.37/73.55 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 73.37/73.55 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 73.37/73.55 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 73.37/73.55 Found x2:(P x00)
% 73.37/73.55 Instantiate: a0:=x00:fofType
% 73.37/73.55 Found (fun (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of (P1 a0)
% 73.37/73.55 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 a0))
% 73.37/73.55 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 a0)))
% 73.37/73.55 Found (and_rect00 (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 73.37/73.55 Found ((and_rect0 (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 73.37/73.55 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 74.16/74.33 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 74.16/74.33 Found x3:(P x1)
% 74.16/74.33 Instantiate: a0:=x1:fofType
% 74.16/74.33 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a0)
% 74.16/74.33 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0))
% 74.16/74.33 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0)))
% 74.16/74.33 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 74.16/74.33 Found ((and_rect0 (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 74.16/74.33 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 74.16/74.33 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a0)
% 74.16/74.33 Found x2:(P x00)
% 74.16/74.33 Instantiate: a0:=x00:fofType
% 74.16/74.33 Found (fun (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of (P1 a0)
% 74.16/74.33 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 a0))
% 74.16/74.33 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 a0)))
% 74.16/74.33 Found (and_rect00 (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 74.16/74.33 Found ((and_rect0 (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 74.16/74.33 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 74.16/74.33 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 74.16/74.33 Found x2:(P x00)
% 74.16/74.33 Instantiate: a0:=x00:fofType
% 74.16/74.33 Found (fun (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of (P1 a0)
% 74.16/74.33 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 a0))
% 74.16/74.33 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 a0)))
% 74.16/74.33 Found (and_rect00 (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 74.16/74.33 Found ((and_rect0 (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 74.16/74.33 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 74.56/74.77 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 74.56/74.77 Found x3:(P x1)
% 74.56/74.77 Instantiate: a0:=x1:fofType
% 74.56/74.77 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a0)
% 74.56/74.77 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0))
% 74.56/74.77 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0)))
% 74.56/74.77 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 74.56/74.77 Found ((and_rect0 (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 74.56/74.77 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 74.56/74.77 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a0)
% 74.56/74.77 Found x3:(P x1)
% 74.56/74.77 Instantiate: a0:=x1:fofType
% 74.56/74.77 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a0)
% 74.56/74.77 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0))
% 74.56/74.77 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0)))
% 74.56/74.77 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 74.56/74.77 Found ((and_rect0 (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 74.56/74.77 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 74.56/74.77 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a0)
% 74.56/74.77 Found x2:(P x00)
% 74.56/74.77 Instantiate: a0:=x00:fofType
% 74.56/74.77 Found (fun (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of (P1 a0)
% 74.56/74.77 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 a0))
% 74.56/74.77 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 a0)))
% 74.56/74.77 Found (and_rect00 (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 74.56/74.77 Found ((and_rect0 (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 74.56/74.77 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 78.16/78.37 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 78.16/78.37 Found x3:(P x1)
% 78.16/78.37 Instantiate: a0:=x1:fofType
% 78.16/78.37 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a0)
% 78.16/78.37 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0))
% 78.16/78.37 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0)))
% 78.16/78.37 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 78.16/78.37 Found ((and_rect0 (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 78.16/78.37 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 78.16/78.37 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a0)
% 78.16/78.37 Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% 78.16/78.37 Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) P)
% 78.16/78.37 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 78.16/78.37 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 78.16/78.37 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 78.16/78.37 Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% 78.16/78.37 Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) P)
% 78.16/78.37 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 78.16/78.37 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 78.16/78.37 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 78.16/78.37 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 78.16/78.37 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 78.16/78.37 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 78.16/78.37 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 78.16/78.37 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 78.16/78.37 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 78.16/78.37 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 78.16/78.37 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 78.16/78.37 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 78.16/78.37 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 78.16/78.37 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 78.16/78.37 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 78.16/78.37 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 78.16/78.37 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 78.16/78.37 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 78.16/78.37 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 78.16/78.37 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 79.38/79.60 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 79.38/79.60 Found x3:(P x1)
% 79.38/79.60 Instantiate: b0:=x1:fofType
% 79.38/79.60 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 79.38/79.60 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 79.38/79.60 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 79.38/79.60 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 79.38/79.60 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 79.38/79.60 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 79.38/79.60 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 79.38/79.60 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P00 b0)
% 79.38/79.60 Found x3:(P x1)
% 79.38/79.60 Instantiate: b0:=x1:fofType
% 79.38/79.60 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 79.38/79.60 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 79.38/79.60 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 79.38/79.60 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 79.38/79.60 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 79.38/79.60 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 79.38/79.60 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b0)
% 79.38/79.60 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 79.38/79.60 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 79.38/79.60 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 79.38/79.60 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 79.38/79.60 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 79.38/79.60 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 79.38/79.60 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 79.38/79.60 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 79.38/79.60 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 79.38/79.60 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 79.38/79.60 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 83.87/84.05 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 83.87/84.05 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 83.87/84.05 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 83.87/84.05 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 83.87/84.05 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 83.87/84.05 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 83.87/84.05 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 83.87/84.05 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 83.87/84.05 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 83.87/84.05 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 83.87/84.05 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 83.87/84.05 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 83.87/84.05 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 83.87/84.05 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 83.87/84.05 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 83.87/84.05 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 83.87/84.05 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 83.87/84.05 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 83.87/84.05 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 83.87/84.05 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 83.87/84.05 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 83.87/84.05 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 83.87/84.05 Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (((fofType->Prop)->fofType)->Prop)) a0) (fun (x:((fofType->Prop)->fofType))=> (a0 x)))
% 83.87/84.05 Found (eta_expansion_dep00 a0) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) a0) b0)
% 83.87/84.05 Found ((eta_expansion_dep0 (fun (x1:((fofType->Prop)->fofType))=> Prop)) a0) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) a0) b0)
% 83.87/84.05 Found (((eta_expansion_dep ((fofType->Prop)->fofType)) (fun (x1:((fofType->Prop)->fofType))=> Prop)) a0) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) a0) b0)
% 83.87/84.05 Found (((eta_expansion_dep ((fofType->Prop)->fofType)) (fun (x1:((fofType->Prop)->fofType))=> Prop)) a0) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) a0) b0)
% 83.87/84.05 Found (((eta_expansion_dep ((fofType->Prop)->fofType)) (fun (x1:((fofType->Prop)->fofType))=> Prop)) a0) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) a0) b0)
% 83.87/84.05 Found eta_expansion000:=(eta_expansion00 b0):(((eq (((fofType->Prop)->fofType)->Prop)) b0) (fun (x:((fofType->Prop)->fofType))=> (b0 x)))
% 83.87/84.05 Found (eta_expansion00 b0) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) b0) a)
% 83.87/84.05 Found ((eta_expansion0 Prop) b0) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) b0) a)
% 83.87/84.05 Found (((eta_expansion ((fofType->Prop)->fofType)) Prop) b0) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) b0) a)
% 83.87/84.05 Found (((eta_expansion ((fofType->Prop)->fofType)) Prop) b0) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) b0) a)
% 83.87/84.05 Found (((eta_expansion ((fofType->Prop)->fofType)) Prop) b0) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) b0) a)
% 83.87/84.05 Found eta_expansion000:=(eta_expansion00 b0):(((eq (((fofType->Prop)->fofType)->Prop)) b0) (fun (x:((fofType->Prop)->fofType))=> (b0 x)))
% 83.87/84.05 Found (eta_expansion00 b0) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) b0) a)
% 83.87/84.05 Found ((eta_expansion0 Prop) b0) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) b0) a)
% 83.87/84.05 Found (((eta_expansion ((fofType->Prop)->fofType)) Prop) b0) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) b0) a)
% 83.87/84.05 Found (((eta_expansion ((fofType->Prop)->fofType)) Prop) b0) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) b0) a)
% 83.87/84.05 Found (((eta_expansion ((fofType->Prop)->fofType)) Prop) b0) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) b0) a)
% 83.87/84.05 Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% 83.87/84.05 Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% 83.87/84.05 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 83.87/84.05 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 83.87/84.05 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 88.97/89.17 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 88.97/89.17 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 88.97/89.17 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 88.97/89.17 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 88.97/89.17 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 88.97/89.17 Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% 88.97/89.17 Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% 88.97/89.17 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 88.97/89.17 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 88.97/89.17 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 88.97/89.17 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 88.97/89.17 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 88.97/89.17 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 88.97/89.17 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 88.97/89.17 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 88.97/89.17 Found eq_ref00:=(eq_ref0 b0):(((eq (((fofType->Prop)->fofType)->Prop)) b0) b0)
% 88.97/89.17 Found (eq_ref0 b0) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) b0) a)
% 88.97/89.17 Found ((eq_ref (((fofType->Prop)->fofType)->Prop)) b0) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) b0) a)
% 88.97/89.17 Found ((eq_ref (((fofType->Prop)->fofType)->Prop)) b0) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) b0) a)
% 88.97/89.17 Found ((eq_ref (((fofType->Prop)->fofType)->Prop)) b0) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) b0) a)
% 88.97/89.17 Found eq_ref00:=(eq_ref0 a0):(((eq (((fofType->Prop)->fofType)->Prop)) a0) a0)
% 88.97/89.17 Found (eq_ref0 a0) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) a0) b0)
% 88.97/89.17 Found ((eq_ref (((fofType->Prop)->fofType)->Prop)) a0) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) a0) b0)
% 88.97/89.17 Found ((eq_ref (((fofType->Prop)->fofType)->Prop)) a0) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) a0) b0)
% 88.97/89.17 Found ((eq_ref (((fofType->Prop)->fofType)->Prop)) a0) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) a0) b0)
% 88.97/89.17 Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% 88.97/89.17 Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) P)
% 88.97/89.17 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 88.97/89.17 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 88.97/89.17 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 88.97/89.17 Found eq_ref00:=(eq_ref0 (x P0)):(((eq fofType) (x P0)) (x P0))
% 88.97/89.17 Found (eq_ref0 (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 88.97/89.17 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 88.97/89.17 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 88.97/89.17 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 88.97/89.17 Found x3:(P0 x1)
% 88.97/89.17 Instantiate: b0:=x1:fofType
% 88.97/89.17 Found (fun (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P0 b0)
% 88.97/89.17 Found (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->(P0 b0))
% 88.97/89.17 Found (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->(P0 b0)))
% 88.97/89.17 Found (and_rect00 (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 b0)
% 88.97/89.17 Found ((and_rect0 (P0 b0)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 b0)
% 88.97/89.17 Found (((fun (P2:Type) (x3:((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P0 b0)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 b0)
% 88.97/89.17 Found (((fun (P2:Type) (x3:((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P0 b0)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 b0)
% 88.97/89.17 Found (fun (x2:((and (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P0 b0)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 90.45/90.67 Found x3:(P0 x1)
% 90.45/90.67 Instantiate: b0:=x1:fofType
% 90.45/90.67 Found (fun (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P0 b0)
% 90.45/90.67 Found (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->(P0 b0))
% 90.45/90.67 Found (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->(P0 b0)))
% 90.45/90.67 Found (and_rect00 (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 b0)
% 90.45/90.67 Found ((and_rect0 (P0 b0)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 b0)
% 90.45/90.67 Found (((fun (P2:Type) (x3:((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P0 b0)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 b0)
% 90.45/90.67 Found (fun (x2:((and (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P0 b0)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P0 b0)
% 90.45/90.67 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 90.45/90.67 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 90.45/90.67 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 90.45/90.67 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 90.45/90.67 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 90.45/90.67 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 90.45/90.67 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 90.45/90.67 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 90.45/90.67 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 90.45/90.67 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 90.45/90.67 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 90.45/90.67 Found x3:(P0 x1)
% 90.45/90.67 Instantiate: b0:=x1:fofType
% 90.45/90.67 Found (fun (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P0 b0)
% 90.45/90.67 Found (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->(P0 b0))
% 90.45/90.67 Found (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->(P0 b0)))
% 90.45/90.67 Found (and_rect00 (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 b0)
% 90.45/90.67 Found ((and_rect0 (P0 b0)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 b0)
% 90.45/90.67 Found (((fun (P2:Type) (x3:((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P0 b0)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 b0)
% 90.45/90.67 Found (fun (x2:((and (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P0 b0)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P0 b0)
% 90.45/90.67 Found x3:(P0 x1)
% 90.45/90.67 Instantiate: b0:=x1:fofType
% 90.45/90.67 Found (fun (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P0 b0)
% 90.45/90.67 Found (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->(P0 b0))
% 93.07/93.24 Found (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->(P0 b0)))
% 93.07/93.24 Found (and_rect00 (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 b0)
% 93.07/93.24 Found ((and_rect0 (P0 b0)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 b0)
% 93.07/93.24 Found (((fun (P2:Type) (x3:((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P0 b0)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 b0)
% 93.07/93.24 Found (((fun (P2:Type) (x3:((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P0 b0)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 b0)
% 93.07/93.24 Found (fun (x2:((and (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P0 b0)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 93.07/93.24 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 93.07/93.24 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 93.07/93.24 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 93.07/93.24 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 93.07/93.24 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 93.07/93.24 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 93.07/93.24 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 93.07/93.24 Found (eq_ref0 b) as proof of (((eq fofType) b) (x a))
% 93.07/93.24 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 93.07/93.24 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 93.07/93.24 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 93.07/93.24 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 93.07/93.24 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b)
% 93.07/93.24 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 93.07/93.24 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 93.07/93.24 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 93.07/93.24 Found x2:(P0 x0)
% 93.07/93.24 Instantiate: b:=x0:fofType
% 93.07/93.24 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P0 b)
% 93.07/93.24 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 b))
% 93.07/93.24 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 b)))
% 93.07/93.24 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b)
% 93.07/93.24 Found ((and_rect0 (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b)
% 93.07/93.24 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b)
% 93.07/93.24 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b)
% 93.07/93.24 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P1 b)
% 93.07/93.26 Found x2:(P0 x0)
% 93.07/93.26 Instantiate: b:=x0:fofType
% 93.07/93.26 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P0 b)
% 93.07/93.26 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 b))
% 93.07/93.26 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 b)))
% 93.07/93.26 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b)
% 93.07/93.26 Found ((and_rect0 (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b)
% 93.07/93.26 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b)
% 93.07/93.26 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 b)
% 93.07/93.26 Found x2:(P0 x0)
% 93.07/93.26 Instantiate: b:=x0:fofType
% 93.07/93.26 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P0 b)
% 93.07/93.26 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 b))
% 93.07/93.26 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 b)))
% 93.07/93.26 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b)
% 93.07/93.26 Found ((and_rect0 (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b)
% 93.07/93.26 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b)
% 93.07/93.26 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 b)
% 93.07/93.26 Found x2:(P0 x0)
% 93.07/93.26 Instantiate: b:=x0:fofType
% 93.07/93.26 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P0 b)
% 93.07/93.26 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 b))
% 93.07/93.26 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 b)))
% 93.07/93.26 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b)
% 93.07/93.26 Found ((and_rect0 (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b)
% 93.07/93.26 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b)
% 93.85/94.09 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b)
% 93.85/94.09 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P1 b)
% 93.85/94.09 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 93.85/94.09 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) a0)
% 93.85/94.09 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 93.85/94.09 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 93.85/94.09 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 93.85/94.09 Found x2:(P0 x0)
% 93.85/94.09 Instantiate: b:=x0:fofType
% 93.85/94.09 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P0 b)
% 93.85/94.09 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 b))
% 93.85/94.09 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 b)))
% 93.85/94.09 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b)
% 93.85/94.09 Found ((and_rect0 (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b)
% 93.85/94.09 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b)
% 93.85/94.09 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b)
% 93.85/94.09 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P1 b)
% 93.85/94.09 Found x2:(P0 x0)
% 93.85/94.09 Instantiate: b:=x0:fofType
% 93.85/94.09 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P0 b)
% 93.85/94.09 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 b))
% 93.85/94.09 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 b)))
% 93.85/94.09 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b)
% 93.85/94.09 Found ((and_rect0 (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b)
% 93.85/94.09 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b)
% 93.85/94.09 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b)
% 94.74/94.98 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P1 b)
% 94.74/94.98 Found x2:(P0 x0)
% 94.74/94.98 Instantiate: b:=x0:fofType
% 94.74/94.98 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P0 b)
% 94.74/94.98 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 b))
% 94.74/94.98 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 b)))
% 94.74/94.98 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b)
% 94.74/94.98 Found ((and_rect0 (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b)
% 94.74/94.98 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b)
% 94.74/94.98 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 b)
% 94.74/94.98 Found x2:(P0 x0)
% 94.74/94.98 Instantiate: b:=x0:fofType
% 94.74/94.98 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P0 b)
% 94.74/94.98 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 b))
% 94.74/94.98 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 b)))
% 94.74/94.98 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b)
% 94.74/94.98 Found ((and_rect0 (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b)
% 94.74/94.98 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b)
% 94.74/94.98 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 b)
% 94.74/94.98 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 94.74/94.98 Found (eq_ref0 b) as proof of (((eq fofType) b) (x a))
% 94.74/94.98 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 94.74/94.98 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 94.74/94.98 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 94.74/94.98 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 94.74/94.98 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b)
% 94.74/94.98 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 94.74/94.98 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 94.74/94.98 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 94.74/94.98 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 94.74/94.98 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 98.82/99.07 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 98.82/99.07 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 98.82/99.07 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 98.82/99.07 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 98.82/99.07 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 98.82/99.07 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 98.82/99.07 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 98.82/99.07 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 98.82/99.07 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 98.82/99.07 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 98.82/99.07 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 98.82/99.07 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 98.82/99.07 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 98.82/99.07 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 98.82/99.07 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 98.82/99.07 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 98.82/99.07 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 98.82/99.07 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 98.82/99.07 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 98.82/99.07 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 98.82/99.07 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 98.82/99.07 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 98.82/99.07 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 98.82/99.07 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 98.82/99.07 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 98.82/99.07 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 98.82/99.07 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 98.82/99.07 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 98.82/99.07 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 98.82/99.07 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 98.82/99.07 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 98.82/99.07 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 98.82/99.07 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 98.82/99.07 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 98.82/99.07 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 98.82/99.07 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 98.82/99.07 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 98.82/99.07 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 98.82/99.07 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 98.82/99.07 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 98.82/99.07 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 98.82/99.07 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 98.82/99.07 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 98.82/99.07 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 98.82/99.07 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 98.82/99.07 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 98.82/99.07 Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% 98.82/99.07 Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% 98.82/99.07 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 98.82/99.07 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 98.82/99.07 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 98.82/99.07 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 98.82/99.07 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 98.82/99.07 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 98.82/99.07 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 98.82/99.07 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 98.82/99.07 Found x10:(P0 x0)
% 98.82/99.07 Instantiate: b0:=x0:fofType
% 98.82/99.07 Found x10 as proof of (P1 b0)
% 98.82/99.07 Found eq_ref00:=(eq_ref0 (x P0)):(((eq fofType) (x P0)) (x P0))
% 98.82/99.07 Found (eq_ref0 (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 100.22/100.37 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 100.22/100.37 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 100.22/100.37 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 100.22/100.37 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 100.22/100.37 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 100.22/100.37 Found (((fun (x01:(((eq fofType) (x P0)) b0))=> ((eq_sym000 x01) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 100.22/100.37 Found (((fun (x01:(((eq fofType) (x P0)) x0))=> (((eq_sym00 x0) x01) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 100.22/100.37 Found (((fun (x01:(((eq fofType) (x P0)) x0))=> ((((eq_sym0 (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 100.22/100.37 Found (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 100.22/100.37 Found (fun (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of (P0 (x P0))
% 100.22/100.37 Found (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 (x P0)))
% 100.22/100.37 Found (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 (x P0))))
% 100.22/100.37 Found (and_rect00 (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 100.22/100.37 Found ((and_rect0 (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 100.22/100.37 Found (((fun (P1:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P1)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P1) x2) x1)) (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 100.22/100.37 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (P1:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P1)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P1) x2) x1)) (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10)))) as proof of (P0 (x P0))
% 100.22/100.37 Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% 100.22/100.37 Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% 100.22/100.37 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 100.22/100.37 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 100.22/100.37 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 100.22/100.37 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 100.22/100.37 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 100.22/100.37 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 100.22/100.37 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 100.22/100.37 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 100.22/100.37 Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% 100.22/100.37 Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% 100.22/100.37 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 100.22/100.37 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 100.22/100.37 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 100.22/100.37 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 100.22/100.37 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 100.57/100.75 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 100.57/100.75 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 100.57/100.75 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 100.57/100.75 Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% 100.57/100.75 Found (eq_ref0 b0) as proof of (((eq fofType) b0) (x P0))
% 100.57/100.75 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (x P0))
% 100.57/100.75 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (x P0))
% 100.57/100.75 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (x P0))
% 100.57/100.75 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 100.57/100.75 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 100.57/100.75 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 100.57/100.75 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 100.57/100.75 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 100.57/100.75 Found x2:(P0 x0)
% 100.57/100.75 Instantiate: b0:=x0:fofType
% 100.57/100.75 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P1 b0)
% 100.57/100.75 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 b0))
% 100.57/100.75 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 b0)))
% 100.57/100.75 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b0)
% 100.57/100.75 Found ((and_rect0 (P1 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b0)
% 100.57/100.75 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b0)
% 100.57/100.75 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b0)
% 100.57/100.75 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 100.57/100.75 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 100.57/100.75 Found (((fun (x01:(((eq fofType) (x P0)) b0))=> ((eq_sym000 x01) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 100.57/100.75 Found (((fun (x01:(((eq fofType) (x P0)) x0))=> (((eq_sym00 x0) x01) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 x0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 100.57/100.75 Found (((fun (x01:(((eq fofType) (x P0)) x0))=> ((((eq_sym0 (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 x0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 100.57/100.75 Found (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 x0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 109.23/109.41 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 x0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)))) as proof of (P0 (x P0))
% 109.23/109.41 Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% 109.23/109.41 Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) P0)
% 109.23/109.41 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P0)
% 109.23/109.41 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P0)
% 109.23/109.41 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P0)
% 109.23/109.41 Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% 109.23/109.41 Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% 109.23/109.41 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 109.23/109.41 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 109.23/109.41 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% 109.23/109.41 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 109.23/109.41 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 109.23/109.41 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 109.23/109.41 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 109.23/109.41 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 109.23/109.41 Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% 109.23/109.41 Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) P)
% 109.23/109.41 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 109.23/109.41 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 109.23/109.41 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 109.23/109.41 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 109.23/109.41 Found (eq_ref0 b) as proof of (((eq fofType) b) (x P0))
% 109.23/109.41 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x P0))
% 109.23/109.41 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x P0))
% 109.23/109.41 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x P0))
% 109.23/109.41 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 109.23/109.41 Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% 109.23/109.41 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 109.23/109.41 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 109.23/109.41 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 109.23/109.41 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 109.23/109.41 Found (eq_ref0 b) as proof of (((eq fofType) b) (x P0))
% 109.23/109.41 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x P0))
% 109.23/109.41 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x P0))
% 109.23/109.41 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x P0))
% 109.23/109.41 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 109.23/109.41 Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% 109.23/109.41 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 109.23/109.41 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 109.23/109.41 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 109.23/109.41 Found x10:(P0 x0)
% 109.23/109.41 Instantiate: b0:=x0:fofType
% 109.23/109.41 Found x10 as proof of (P1 b0)
% 109.23/109.41 Found eq_ref00:=(eq_ref0 (x P0)):(((eq fofType) (x P0)) (x P0))
% 109.23/109.41 Found (eq_ref0 (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 109.23/109.41 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 109.23/109.41 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 109.23/109.41 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 109.23/109.41 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 109.23/109.41 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 109.23/109.41 Found (((fun (x01:(((eq fofType) (x P0)) b0))=> ((eq_sym000 x01) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 109.23/109.41 Found (((fun (x01:(((eq fofType) (x P0)) x0))=> (((eq_sym00 x0) x01) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 111.45/111.68 Found (((fun (x01:(((eq fofType) (x P0)) x0))=> ((((eq_sym0 (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 111.45/111.68 Found (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 111.45/111.68 Found (fun (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of (P0 (x P0))
% 111.45/111.68 Found (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of ((forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))->(P0 (x P0)))
% 111.45/111.68 Found (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of ((P0 x0)->((forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))->(P0 (x P0))))
% 111.45/111.68 Found (and_rect00 (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 111.45/111.68 Found ((and_rect0 (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 111.45/111.68 Found (((fun (P1:Type) (x2:((P0 x0)->((forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))->P1)))=> (((((and_rect (P0 x0)) (forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))) P1) x2) x1)) (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 111.45/111.68 Found (fun (x1:(((unique fofType) P0) x0))=> (((fun (P1:Type) (x2:((P0 x0)->((forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))->P1)))=> (((((and_rect (P0 x0)) (forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))) P1) x2) x1)) (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10)))) as proof of (P0 (x P0))
% 111.45/111.68 Found x2:(P x00)
% 111.45/111.68 Instantiate: b0:=x00:fofType
% 111.45/111.68 Found x2 as proof of (P1 b0)
% 111.45/111.68 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 111.45/111.68 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 111.45/111.68 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 111.45/111.68 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 111.45/111.68 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 111.45/111.68 Found x3:(P x1)
% 111.45/111.68 Instantiate: a:=x1:fofType
% 111.45/111.68 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a)
% 111.45/111.68 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a))
% 111.45/111.68 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a)))
% 111.45/111.68 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 111.45/111.68 Found ((and_rect0 (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 111.45/111.68 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 111.45/111.68 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 112.04/112.24 Found x3:(P x1)
% 112.04/112.24 Instantiate: a:=x1:fofType
% 112.04/112.24 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a)
% 112.04/112.24 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a))
% 112.04/112.24 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a)))
% 112.04/112.24 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 112.04/112.24 Found ((and_rect0 (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 112.04/112.24 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 112.04/112.24 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 112.04/112.24 Found x3:(P x1)
% 112.04/112.24 Instantiate: b0:=x1:fofType
% 112.04/112.24 Found x3 as proof of (P1 b0)
% 112.04/112.24 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 112.04/112.24 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 112.04/112.24 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 112.04/112.24 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 112.04/112.24 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 112.04/112.24 Found x3:(P x1)
% 112.04/112.24 Instantiate: b0:=x1:fofType
% 112.04/112.24 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b0)
% 112.04/112.24 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0))
% 112.04/112.24 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0)))
% 112.04/112.24 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 112.04/112.24 Found ((and_rect0 (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 112.04/112.24 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 112.04/112.24 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 112.04/112.24 Found ((eq_sym0000 ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 112.04/112.24 Found ((eq_sym0000 ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 112.04/112.24 Found (((fun (x3:(((eq fofType) a) b0))=> ((eq_sym000 x3) P)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 112.23/112.41 Found (((fun (x3:(((eq fofType) a) x1))=> (((eq_sym00 x1) x3) P)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 112.23/112.41 Found (((fun (x3:(((eq fofType) a) x1))=> ((((eq_sym0 a) x1) x3) P)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 112.23/112.41 Found (((fun (x3:(((eq fofType) a) x1))=> (((((eq_sym fofType) a) x1) x3) P)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 112.23/112.41 Found (((fun (x3:(((eq fofType) a) x1))=> (((((eq_sym fofType) a) x1) x3) P)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 112.23/112.41 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (x3:(((eq fofType) a) x1))=> (((((eq_sym fofType) a) x1) x3) P)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)))) as proof of (P0 a)
% 112.23/112.41 Found x3:(P x1)
% 112.23/112.41 Instantiate: b0:=x1:fofType
% 112.23/112.41 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b0)
% 112.23/112.41 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0))
% 112.23/112.41 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0)))
% 112.23/112.41 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 112.23/112.41 Found ((and_rect0 (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 112.23/112.41 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 112.23/112.41 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 112.23/112.41 Found ((eq_sym0000 ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 112.23/112.41 Found ((eq_sym0000 ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 113.45/113.69 Found (((fun (x3:(((eq fofType) a) b0))=> ((eq_sym000 x3) P)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 113.45/113.69 Found (((fun (x3:(((eq fofType) a) x1))=> (((eq_sym00 x1) x3) P)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 113.45/113.69 Found (((fun (x3:(((eq fofType) a) x1))=> ((((eq_sym0 a) x1) x3) P)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 113.45/113.69 Found (((fun (x3:(((eq fofType) a) x1))=> (((((eq_sym fofType) a) x1) x3) P)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 113.45/113.69 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (x3:(((eq fofType) a) x1))=> (((((eq_sym fofType) a) x1) x3) P)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)))) as proof of (P a)
% 113.45/113.69 Found x2:(P x00)
% 113.45/113.69 Instantiate: b0:=x00:fofType
% 113.45/113.69 Found x2 as proof of (P1 b0)
% 113.45/113.69 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 113.45/113.69 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 113.45/113.69 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 113.45/113.69 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 113.45/113.69 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 113.45/113.69 Found x3:(P x1)
% 113.45/113.69 Instantiate: a:=x1:fofType
% 113.45/113.69 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a)
% 113.45/113.69 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a))
% 113.45/113.69 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a)))
% 113.45/113.69 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 113.45/113.69 Found ((and_rect0 (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 113.45/113.69 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 113.45/113.69 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a)
% 113.45/113.69 Found x3:(P x1)
% 113.45/113.69 Instantiate: b0:=x1:fofType
% 113.45/113.69 Found x3 as proof of (P1 b0)
% 113.45/113.69 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 113.45/113.69 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 113.45/113.69 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 113.45/113.69 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 113.45/113.69 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 113.92/114.11 Found x3:(P x1)
% 113.92/114.11 Instantiate: b0:=x1:fofType
% 113.92/114.11 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b0)
% 113.92/114.11 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0))
% 113.92/114.11 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0)))
% 113.92/114.11 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 113.92/114.11 Found ((and_rect0 (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 113.92/114.11 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 113.92/114.11 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 113.92/114.11 Found x3:(P x1)
% 113.92/114.11 Instantiate: b0:=x1:fofType
% 113.92/114.11 Found x3 as proof of (P1 b0)
% 113.92/114.11 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 113.92/114.11 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 113.92/114.11 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 113.92/114.11 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 113.92/114.11 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 113.92/114.11 Found x3:(P x1)
% 113.92/114.11 Instantiate: a:=x1:fofType
% 113.92/114.11 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a)
% 113.92/114.11 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a))
% 113.92/114.11 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a)))
% 113.92/114.11 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 113.92/114.11 Found ((and_rect0 (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 113.92/114.11 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 113.92/114.11 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 113.92/114.11 Found x3:(P x1)
% 113.92/114.11 Instantiate: a:=x1:fofType
% 113.92/114.11 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a)
% 113.92/114.11 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a))
% 113.92/114.11 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a)))
% 113.92/114.11 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 113.92/114.11 Found ((and_rect0 (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 113.92/114.11 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 114.33/114.50 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 114.33/114.50 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a)
% 114.33/114.50 Found x2:(P0 x00)
% 114.33/114.50 Instantiate: a:=x00:fofType
% 114.33/114.50 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2) as proof of (P1 a)
% 114.33/114.50 Found (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->(P1 a))
% 114.33/114.50 Found (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->(P1 a)))
% 114.33/114.50 Found (and_rect00 (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a)
% 114.33/114.50 Found ((and_rect0 (P1 a)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a)
% 114.33/114.50 Found (((fun (P2:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a)
% 114.33/114.50 Found (((fun (P2:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a)
% 114.33/114.50 Found x3:(P x1)
% 114.33/114.50 Instantiate: b0:=x1:fofType
% 114.33/114.50 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 114.33/114.50 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 114.33/114.50 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 114.33/114.50 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 114.33/114.50 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 114.33/114.50 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 114.33/114.50 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 114.33/114.50 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 114.33/114.50 Found x3:(P x1)
% 114.33/114.50 Instantiate: b0:=x1:fofType
% 114.33/114.50 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 114.33/114.50 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 115.12/115.28 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 115.12/115.28 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 115.12/115.28 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 115.12/115.28 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 115.12/115.28 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b0)
% 115.12/115.28 Found x3:(P x1)
% 115.12/115.28 Instantiate: a:=x1:fofType
% 115.12/115.28 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a)
% 115.12/115.28 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a))
% 115.12/115.28 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a)))
% 115.12/115.28 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 115.12/115.28 Found ((and_rect0 (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 115.12/115.28 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 115.12/115.28 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 115.12/115.28 Found x3:(P x1)
% 115.12/115.28 Instantiate: a:=x1:fofType
% 115.12/115.28 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a)
% 115.12/115.28 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a))
% 115.12/115.28 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a)))
% 115.12/115.28 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 115.12/115.28 Found ((and_rect0 (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 115.12/115.28 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 115.12/115.28 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 115.12/115.28 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 115.12/115.28 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 115.12/115.28 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 115.12/115.28 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 115.12/115.28 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 115.43/115.62 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 115.43/115.62 Found x2:(P x00)
% 115.43/115.62 Instantiate: b0:=x00:fofType
% 115.43/115.62 Found (fun (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of (P1 b0)
% 115.43/115.62 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 b0))
% 115.43/115.62 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 b0)))
% 115.43/115.62 Found (and_rect00 (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b0)
% 115.43/115.62 Found ((and_rect0 (P1 b0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b0)
% 115.43/115.62 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b0)
% 115.43/115.62 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 b0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 b0)
% 115.43/115.62 Found x3:(P x1)
% 115.43/115.62 Instantiate: a:=x1:fofType
% 115.43/115.62 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a)
% 115.43/115.62 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a))
% 115.43/115.62 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a)))
% 115.43/115.62 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 115.43/115.62 Found ((and_rect0 (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 115.43/115.62 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 115.43/115.62 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 115.43/115.62 Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% 115.43/115.62 Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) P)
% 115.43/115.62 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 115.43/115.62 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 115.43/115.62 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 115.43/115.62 Found x3:(P x1)
% 115.43/115.62 Instantiate: a:=x1:fofType
% 115.43/115.62 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a)
% 115.43/115.62 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a))
% 115.43/115.62 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a)))
% 115.43/115.62 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 115.43/115.62 Found ((and_rect0 (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 115.43/115.62 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 115.84/116.00 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 115.84/116.00 Found x3:(P x1)
% 115.84/116.00 Instantiate: b0:=x1:fofType
% 115.84/116.00 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b0)
% 115.84/116.00 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0))
% 115.84/116.00 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0)))
% 115.84/116.00 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 115.84/116.00 Found ((and_rect0 (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 115.84/116.00 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 115.84/116.00 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 115.84/116.00 Found ((eq_sym0000 ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 115.84/116.00 Found ((eq_sym0000 ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 115.84/116.00 Found (((fun (x3:(((eq fofType) a) b0))=> ((eq_sym000 x3) P)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 115.84/116.00 Found (((fun (x3:(((eq fofType) a) x1))=> (((eq_sym00 x1) x3) P)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 115.84/116.00 Found (((fun (x3:(((eq fofType) a) x1))=> ((((eq_sym0 a) x1) x3) P)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 115.84/116.00 Found (((fun (x3:(((eq fofType) a) x1))=> (((((eq_sym fofType) a) x1) x3) P)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 115.84/116.00 Found (((fun (x3:(((eq fofType) a) x1))=> (((((eq_sym fofType) a) x1) x3) P)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 115.85/116.02 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (x3:(((eq fofType) a) x1))=> (((((eq_sym fofType) a) x1) x3) P)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)))) as proof of (P0 a)
% 115.85/116.02 Found x3:(P x1)
% 115.85/116.02 Instantiate: b0:=x1:fofType
% 115.85/116.02 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b0)
% 115.85/116.02 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0))
% 115.85/116.02 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0)))
% 115.85/116.02 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 115.85/116.02 Found ((and_rect0 (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 115.85/116.02 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 115.85/116.02 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 115.85/116.02 Found ((eq_sym0000 ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 115.85/116.02 Found ((eq_sym0000 ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 115.85/116.02 Found (((fun (x3:(((eq fofType) a) b0))=> ((eq_sym000 x3) P)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 115.85/116.02 Found (((fun (x3:(((eq fofType) a) x1))=> (((eq_sym00 x1) x3) P)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 115.85/116.02 Found (((fun (x3:(((eq fofType) a) x1))=> ((((eq_sym0 a) x1) x3) P)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 115.85/116.02 Found (((fun (x3:(((eq fofType) a) x1))=> (((((eq_sym fofType) a) x1) x3) P)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 115.94/116.17 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (x3:(((eq fofType) a) x1))=> (((((eq_sym fofType) a) x1) x3) P)) ((eq_ref fofType) a)) (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)))) as proof of (P a)
% 115.94/116.17 Found x3:(P x1)
% 115.94/116.17 Instantiate: b0:=x1:fofType
% 115.94/116.17 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b0)
% 115.94/116.17 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0))
% 115.94/116.17 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0)))
% 115.94/116.17 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 115.94/116.17 Found ((and_rect0 (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 115.94/116.17 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 115.94/116.17 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 115.94/116.17 Found x3:(P x1)
% 115.94/116.17 Instantiate: b0:=x1:fofType
% 115.94/116.17 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b0)
% 115.94/116.17 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0))
% 115.94/116.17 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0)))
% 115.94/116.17 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 115.94/116.17 Found ((and_rect0 (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 115.94/116.17 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 115.94/116.17 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 115.94/116.17 Found x3:(P x1)
% 115.94/116.17 Instantiate: a:=x1:fofType
% 115.94/116.17 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a)
% 115.94/116.17 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a))
% 115.94/116.17 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a)))
% 115.94/116.17 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 115.94/116.17 Found ((and_rect0 (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 115.94/116.17 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 116.14/116.37 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a)
% 116.14/116.37 Found x3:(P x1)
% 116.14/116.37 Instantiate: a:=x1:fofType
% 116.14/116.37 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a)
% 116.14/116.37 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a))
% 116.14/116.37 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a)))
% 116.14/116.37 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 116.14/116.37 Found ((and_rect0 (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 116.14/116.37 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 116.14/116.37 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a)
% 116.14/116.37 Found x3:(P x1)
% 116.14/116.37 Instantiate: b0:=x1:fofType
% 116.14/116.37 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b0)
% 116.14/116.37 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0))
% 116.14/116.37 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0)))
% 116.14/116.37 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 116.14/116.37 Found ((and_rect0 (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 116.14/116.37 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 116.14/116.37 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 116.14/116.37 Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% 116.14/116.37 Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) P)
% 116.14/116.37 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 116.14/116.37 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 116.14/116.37 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 116.14/116.37 Found x3:(P x1)
% 116.14/116.37 Instantiate: b0:=x1:fofType
% 116.14/116.37 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 116.14/116.37 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 116.14/116.37 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 116.14/116.37 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 116.23/116.46 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 116.23/116.46 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 116.23/116.46 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 116.23/116.46 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 116.23/116.46 Found x2:(P0 x00)
% 116.23/116.46 Instantiate: a:=x00:fofType
% 116.23/116.46 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2) as proof of (P1 a)
% 116.23/116.46 Found (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->(P1 a))
% 116.23/116.46 Found (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->(P1 a)))
% 116.23/116.46 Found (and_rect00 (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a)
% 116.23/116.46 Found ((and_rect0 (P1 a)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a)
% 116.23/116.46 Found (((fun (P2:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a)
% 116.23/116.46 Found (((fun (P2:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a)
% 116.23/116.46 Found x3:(P x1)
% 116.23/116.46 Instantiate: b0:=x1:fofType
% 116.23/116.46 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 116.23/116.46 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 116.23/116.46 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 116.23/116.46 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 116.23/116.46 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 116.23/116.46 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 116.23/116.46 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b0)
% 116.23/116.46 Found x3:(P x1)
% 116.23/116.46 Instantiate: b0:=x1:fofType
% 116.23/116.46 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b0)
% 116.23/116.46 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0))
% 116.33/116.58 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0)))
% 116.33/116.58 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 116.33/116.58 Found ((and_rect0 (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 116.33/116.58 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 116.33/116.58 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 116.33/116.58 Found x3:(P x1)
% 116.33/116.58 Instantiate: b0:=x1:fofType
% 116.33/116.58 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b0)
% 116.33/116.58 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0))
% 116.33/116.58 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0)))
% 116.33/116.58 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 116.33/116.58 Found ((and_rect0 (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 116.33/116.58 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 116.33/116.58 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 116.33/116.58 Found x3:(P x1)
% 116.33/116.58 Instantiate: a:=x1:fofType
% 116.33/116.58 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a)
% 116.33/116.58 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a))
% 116.33/116.58 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a)))
% 116.33/116.58 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 116.33/116.58 Found ((and_rect0 (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 116.33/116.58 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 116.33/116.58 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 116.33/116.58 Found x3:(P x1)
% 116.33/116.58 Instantiate: a:=x1:fofType
% 116.33/116.58 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a)
% 116.43/116.63 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a))
% 116.43/116.63 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a)))
% 116.43/116.63 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 116.43/116.63 Found ((and_rect0 (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 116.43/116.63 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 116.43/116.63 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 116.43/116.63 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a)
% 116.43/116.63 Found x3:(P x1)
% 116.43/116.63 Instantiate: a:=x1:fofType
% 116.43/116.63 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a)
% 116.43/116.63 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a))
% 116.43/116.63 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a)))
% 116.43/116.63 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 116.43/116.63 Found ((and_rect0 (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 116.43/116.63 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 116.43/116.63 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 116.43/116.63 Found x3:(P x1)
% 116.43/116.63 Instantiate: a:=x1:fofType
% 116.43/116.63 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a)
% 116.43/116.63 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a))
% 116.43/116.63 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a)))
% 116.43/116.63 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 116.43/116.63 Found ((and_rect0 (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 116.43/116.63 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 116.43/116.63 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 116.53/116.76 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a)
% 116.53/116.76 Found x3:(P x1)
% 116.53/116.76 Instantiate: b0:=x1:fofType
% 116.53/116.76 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 116.53/116.76 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 116.53/116.76 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 116.53/116.76 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 116.53/116.76 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 116.53/116.76 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 116.53/116.76 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b0)
% 116.53/116.76 Found x3:(P x1)
% 116.53/116.76 Instantiate: b0:=x1:fofType
% 116.53/116.76 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 116.53/116.76 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 116.53/116.76 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 116.53/116.76 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 116.53/116.76 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 116.53/116.76 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 116.53/116.76 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 116.53/116.76 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 116.53/116.76 Found x3:(P x1)
% 116.53/116.76 Instantiate: b0:=x1:fofType
% 116.53/116.76 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 116.53/116.76 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 116.53/116.76 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 117.05/117.23 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 117.05/117.23 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 117.05/117.23 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 117.05/117.23 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b0)
% 117.05/117.23 Found x3:(P x1)
% 117.05/117.23 Instantiate: b0:=x1:fofType
% 117.05/117.23 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 117.05/117.23 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 117.05/117.23 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 117.05/117.23 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 117.05/117.23 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 117.05/117.23 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 117.05/117.23 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 117.05/117.23 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 117.05/117.23 Found x3:(P x1)
% 117.05/117.23 Instantiate: a:=x1:fofType
% 117.05/117.23 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a)
% 117.05/117.23 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a))
% 117.05/117.23 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a)))
% 117.05/117.23 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 117.05/117.23 Found ((and_rect0 (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 117.05/117.23 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 117.05/117.23 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 117.05/117.23 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 117.05/117.23 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 118.05/118.29 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 118.05/118.29 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 118.05/118.29 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 118.05/118.29 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 118.05/118.29 Found x3:(P x1)
% 118.05/118.29 Instantiate: a:=x1:fofType
% 118.05/118.29 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a)
% 118.05/118.29 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a))
% 118.05/118.29 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a)))
% 118.05/118.29 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 118.05/118.29 Found ((and_rect0 (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 118.05/118.29 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 118.05/118.29 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 118.05/118.29 Found x3:(P x1)
% 118.05/118.29 Instantiate: a:=x1:fofType
% 118.05/118.29 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a)
% 118.05/118.29 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a))
% 118.05/118.29 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a)))
% 118.05/118.29 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 118.05/118.29 Found ((and_rect0 (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 118.05/118.29 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 118.05/118.29 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a)
% 118.05/118.29 Found x3:(P x1)
% 118.05/118.29 Instantiate: b0:=x1:fofType
% 118.05/118.29 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 b0)
% 118.05/118.29 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0))
% 118.05/118.29 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 b0)))
% 118.05/118.29 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 118.05/118.29 Found ((and_rect0 (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 118.05/118.29 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 b0)
% 118.05/118.29 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 118.35/118.52 Found x3:(P x1)
% 118.35/118.52 Instantiate: b0:=x1:fofType
% 118.35/118.52 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 118.35/118.52 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 118.35/118.52 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 118.35/118.52 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 118.35/118.52 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 118.35/118.52 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 118.35/118.52 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 118.35/118.52 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 118.35/118.52 Found x3:(P x1)
% 118.35/118.52 Instantiate: b0:=x1:fofType
% 118.35/118.52 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 118.35/118.52 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 118.35/118.52 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 118.35/118.52 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 118.35/118.52 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 118.35/118.52 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 118.35/118.52 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b0)
% 118.35/118.52 Found x3:(P x1)
% 118.35/118.52 Instantiate: a:=x1:fofType
% 118.35/118.52 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a)
% 118.35/118.52 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a))
% 118.35/118.52 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a)))
% 118.35/118.52 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 118.35/118.52 Found ((and_rect0 (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 118.72/118.89 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 118.72/118.89 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 118.72/118.89 Found x3:(P x1)
% 118.72/118.89 Instantiate: a:=x1:fofType
% 118.72/118.89 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a)
% 118.72/118.89 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a))
% 118.72/118.89 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a)))
% 118.72/118.89 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 118.72/118.89 Found ((and_rect0 (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 118.72/118.89 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 118.72/118.89 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 118.72/118.89 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a)
% 118.72/118.89 Found x2:(P0 x0)
% 118.72/118.89 Instantiate: a:=x0:fofType
% 118.72/118.89 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P1 a)
% 118.72/118.89 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 a))
% 118.72/118.89 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 a)))
% 118.72/118.89 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 118.72/118.89 Found ((and_rect0 (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 118.72/118.89 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 118.72/118.89 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 118.72/118.89 Found x2:(P0 x0)
% 118.72/118.89 Instantiate: a:=x0:fofType
% 118.72/118.89 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P1 a)
% 118.72/118.89 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 a))
% 118.72/118.89 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 a)))
% 119.44/119.61 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 119.44/119.61 Found ((and_rect0 (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 119.44/119.61 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 119.44/119.61 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 119.44/119.61 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 119.44/119.61 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) a0)
% 119.44/119.61 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 119.44/119.61 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 119.44/119.61 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 119.44/119.61 Found x2:(P0 x0)
% 119.44/119.61 Instantiate: a:=x0:fofType
% 119.44/119.61 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P1 a)
% 119.44/119.61 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 a))
% 119.44/119.61 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 a)))
% 119.44/119.61 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 119.44/119.61 Found ((and_rect0 (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 119.44/119.61 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 119.44/119.61 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 119.44/119.61 Found x2:(P0 x0)
% 119.44/119.61 Instantiate: a:=x0:fofType
% 119.44/119.61 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P1 a)
% 119.44/119.61 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 a))
% 119.44/119.61 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 a)))
% 119.44/119.61 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 119.44/119.61 Found ((and_rect0 (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 119.44/119.61 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 119.44/119.61 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 119.44/119.61 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 123.04/123.23 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 123.04/123.23 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 123.04/123.23 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 123.04/123.23 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 123.04/123.23 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 123.04/123.23 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 123.04/123.23 Found (eq_ref0 b) as proof of (((eq fofType) b) (x a))
% 123.04/123.23 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 123.04/123.23 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 123.04/123.23 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 123.04/123.23 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 123.04/123.23 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b)
% 123.04/123.23 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 123.04/123.23 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 123.04/123.23 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 123.04/123.23 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 123.04/123.23 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 123.04/123.23 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 123.04/123.23 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 123.04/123.23 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 123.04/123.23 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 123.04/123.23 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 123.04/123.23 Found (eq_ref0 b) as proof of (((eq fofType) b) (x a))
% 123.04/123.23 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 123.04/123.23 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 123.04/123.23 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 123.04/123.23 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 123.04/123.23 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b)
% 123.04/123.23 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 123.04/123.23 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 123.04/123.23 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 123.04/123.23 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 123.04/123.23 Found (eq_ref0 b) as proof of (((eq fofType) b) (x a))
% 123.04/123.23 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 123.04/123.23 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 123.04/123.23 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 123.04/123.23 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 123.04/123.23 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b)
% 123.04/123.23 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 123.04/123.23 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 123.04/123.23 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 123.04/123.23 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 123.04/123.23 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) a0)
% 123.04/123.23 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 123.04/123.23 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 123.04/123.23 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 123.04/123.23 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 123.04/123.23 Found (eq_ref0 b) as proof of (((eq fofType) b) (x a))
% 123.04/123.23 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 123.04/123.23 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 123.04/123.23 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x a))
% 123.04/123.23 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 123.04/123.23 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b)
% 123.04/123.23 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 123.04/123.23 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 123.04/123.23 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% 123.04/123.23 Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 123.04/123.23 Found (eta_expansion00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 123.04/123.23 Found ((eta_expansion0 Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 123.04/123.23 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 128.04/128.29 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 128.04/128.29 Found (eq_sym000 (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 128.04/128.29 Found ((eq_sym00 a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 128.04/128.29 Found (((eq_sym0 a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 128.04/128.29 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 128.04/128.29 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 128.04/128.29 Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 128.04/128.29 Found (eta_expansion_dep00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 128.04/128.29 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 128.04/128.29 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 128.04/128.29 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 128.04/128.29 Found (eq_sym000 (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 128.04/128.29 Found ((eq_sym00 a) (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 128.04/128.29 Found (((eq_sym0 a0) a) (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 128.04/128.29 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 128.04/128.29 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 128.04/128.29 Found x2:(P x00)
% 128.04/128.29 Instantiate: a0:=x00:fofType
% 128.04/128.29 Found (fun (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of (P1 a0)
% 128.04/128.29 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 a0))
% 128.04/128.29 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 a0)))
% 128.04/128.29 Found (and_rect00 (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 128.04/128.29 Found ((and_rect0 (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 128.04/128.29 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 128.04/128.29 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 128.04/128.29 Found x2:(P x00)
% 128.04/128.29 Instantiate: a0:=x00:fofType
% 128.04/128.29 Found (fun (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of (P1 a0)
% 128.04/128.29 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 a0))
% 128.04/128.29 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 a0)))
% 128.04/128.29 Found (and_rect00 (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 128.04/128.29 Found ((and_rect0 (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 128.04/128.29 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 129.20/129.40 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 129.20/129.40 Found x3:(P x1)
% 129.20/129.40 Instantiate: a0:=x1:fofType
% 129.20/129.40 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a0)
% 129.20/129.40 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0))
% 129.20/129.40 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0)))
% 129.20/129.40 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 129.20/129.40 Found ((and_rect0 (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 129.20/129.40 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 129.20/129.40 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a0)
% 129.20/129.40 Found x3:(P x1)
% 129.20/129.40 Instantiate: a0:=x1:fofType
% 129.20/129.40 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a0)
% 129.20/129.40 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0))
% 129.20/129.40 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0)))
% 129.20/129.40 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 129.20/129.40 Found ((and_rect0 (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 129.20/129.40 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 129.20/129.40 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a0)
% 129.20/129.40 Found x3:(P x1)
% 129.20/129.40 Instantiate: a0:=x1:fofType
% 129.20/129.40 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a0)
% 129.20/129.40 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0))
% 129.20/129.40 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0)))
% 129.20/129.40 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 129.20/129.40 Found ((and_rect0 (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 129.20/129.40 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 130.25/130.48 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 130.25/130.48 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a0)
% 130.25/130.48 Found x2:(P x00)
% 130.25/130.48 Instantiate: a0:=x00:fofType
% 130.25/130.48 Found (fun (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of (P1 a0)
% 130.25/130.48 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 a0))
% 130.25/130.48 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 a0)))
% 130.25/130.48 Found (and_rect00 (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 130.25/130.48 Found ((and_rect0 (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 130.25/130.48 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 130.25/130.48 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 130.25/130.48 Found x2:(P x00)
% 130.25/130.48 Instantiate: a0:=x00:fofType
% 130.25/130.48 Found (fun (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of (P1 a0)
% 130.25/130.48 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 a0))
% 130.25/130.48 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 a0)))
% 130.25/130.48 Found (and_rect00 (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 130.25/130.48 Found ((and_rect0 (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 130.25/130.48 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 130.25/130.48 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 130.25/130.48 Found x2:(P x00)
% 130.25/130.48 Instantiate: a0:=x00:fofType
% 130.25/130.48 Found (fun (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of (P1 a0)
% 130.25/130.48 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 a0))
% 130.25/130.48 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 a0)))
% 130.25/130.48 Found (and_rect00 (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 130.83/131.06 Found ((and_rect0 (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 130.83/131.06 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 130.83/131.06 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 130.83/131.06 Found x2:(P x00)
% 130.83/131.06 Instantiate: a0:=x00:fofType
% 130.83/131.06 Found (fun (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of (P1 a0)
% 130.83/131.06 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 a0))
% 130.83/131.06 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 a0)))
% 130.83/131.06 Found (and_rect00 (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 130.83/131.06 Found ((and_rect0 (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 130.83/131.06 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 130.83/131.06 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 130.83/131.06 Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 130.83/131.06 Found (eta_expansion00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 130.83/131.06 Found ((eta_expansion0 Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 130.83/131.06 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 130.83/131.06 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 130.83/131.06 Found (eq_sym000 (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 130.83/131.06 Found ((eq_sym00 a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 130.83/131.06 Found (((eq_sym0 a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 130.83/131.06 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 130.83/131.06 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 130.83/131.06 Found x3:(P x1)
% 130.83/131.06 Instantiate: a0:=x1:fofType
% 130.83/131.06 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a0)
% 130.83/131.06 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0))
% 130.83/131.06 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0)))
% 130.83/131.06 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 130.83/131.06 Found ((and_rect0 (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 130.83/131.06 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 131.04/131.26 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a0)
% 131.04/131.26 Found x3:(P x1)
% 131.04/131.26 Instantiate: a0:=x1:fofType
% 131.04/131.26 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a0)
% 131.04/131.26 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0))
% 131.04/131.26 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0)))
% 131.04/131.26 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 131.04/131.26 Found ((and_rect0 (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 131.04/131.26 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 131.04/131.26 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a0)
% 131.04/131.26 Found x3:(P x1)
% 131.04/131.26 Instantiate: a0:=x1:fofType
% 131.04/131.26 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a0)
% 131.04/131.26 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0))
% 131.04/131.26 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0)))
% 131.04/131.26 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 131.04/131.26 Found ((and_rect0 (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 131.04/131.26 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 131.04/131.26 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a0)
% 131.04/131.26 Found x3:(P x1)
% 131.04/131.26 Instantiate: a0:=x1:fofType
% 131.04/131.26 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a0)
% 131.04/131.26 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0))
% 131.04/131.26 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0)))
% 131.04/131.26 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 131.04/131.26 Found ((and_rect0 (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 131.04/131.26 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 131.64/131.84 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 131.64/131.84 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a0)
% 131.64/131.84 Found x3:(P x1)
% 131.64/131.84 Instantiate: a0:=x1:fofType
% 131.64/131.84 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a0)
% 131.64/131.84 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0))
% 131.64/131.84 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0)))
% 131.64/131.84 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 131.64/131.84 Found ((and_rect0 (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 131.64/131.84 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 131.64/131.84 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 131.64/131.84 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a0)
% 131.64/131.84 Found x3:(P x1)
% 131.64/131.84 Instantiate: a0:=x1:fofType
% 131.64/131.84 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a0)
% 131.64/131.84 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0))
% 131.64/131.84 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0)))
% 131.64/131.84 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 131.64/131.84 Found ((and_rect0 (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 131.64/131.84 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 131.64/131.84 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a0)
% 131.64/131.84 Found x2:(P x00)
% 131.64/131.84 Instantiate: a0:=x00:fofType
% 131.64/131.84 Found (fun (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of (P1 a0)
% 131.64/131.84 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 a0))
% 132.55/132.78 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 a0)))
% 132.55/132.78 Found (and_rect00 (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 132.55/132.78 Found ((and_rect0 (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 132.55/132.78 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 132.55/132.78 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 132.55/132.78 Found x2:(P x00)
% 132.55/132.78 Instantiate: a0:=x00:fofType
% 132.55/132.78 Found (fun (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of (P1 a0)
% 132.55/132.78 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 a0))
% 132.55/132.78 Found (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 a0)))
% 132.55/132.78 Found (and_rect00 (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 132.55/132.78 Found ((and_rect0 (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 132.55/132.78 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 132.55/132.78 Found (((fun (P2:Type) (x2:((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P x00)) (forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a0)) (fun (x2:(P x00)) (x3:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a0)
% 132.55/132.78 Found x3:(P x1)
% 132.55/132.78 Instantiate: a0:=x1:fofType
% 132.55/132.78 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a0)
% 132.55/132.78 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0))
% 132.55/132.78 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0)))
% 132.55/132.78 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 132.55/132.78 Found ((and_rect0 (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 132.55/132.78 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 132.55/132.78 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a0)
% 132.55/132.78 Found x3:(P x1)
% 132.55/132.78 Instantiate: a0:=x1:fofType
% 132.55/132.78 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a0)
% 132.55/132.78 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0))
% 133.36/133.60 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0)))
% 133.36/133.60 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 133.36/133.60 Found ((and_rect0 (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 133.36/133.60 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 133.36/133.60 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a0)
% 133.36/133.60 Found x3:(P x1)
% 133.36/133.60 Instantiate: a0:=x1:fofType
% 133.36/133.60 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a0)
% 133.36/133.60 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0))
% 133.36/133.60 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0)))
% 133.36/133.60 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 133.36/133.60 Found ((and_rect0 (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 133.36/133.60 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 133.36/133.60 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 133.36/133.60 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a0)
% 133.36/133.60 Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% 133.36/133.60 Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) P)
% 133.36/133.60 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 133.36/133.60 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 133.36/133.60 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 133.36/133.60 Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% 133.36/133.60 Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) P)
% 133.36/133.60 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 133.36/133.60 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 133.36/133.60 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 133.36/133.60 Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% 133.36/133.60 Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) P)
% 133.36/133.60 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 133.36/133.60 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 133.36/133.60 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 133.36/133.60 Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% 133.36/133.60 Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) P)
% 133.36/133.60 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 139.42/139.61 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 139.42/139.61 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 139.42/139.61 Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% 139.42/139.61 Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) P)
% 139.42/139.61 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 139.42/139.61 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 139.42/139.61 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P)
% 139.42/139.61 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 139.42/139.61 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 139.42/139.61 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 139.42/139.61 Found (eta_expansion00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 139.42/139.61 Found ((eta_expansion0 Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 139.42/139.61 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 139.42/139.61 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 139.42/139.61 Found (eq_sym000 (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found ((eq_sym00 a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found (((eq_sym0 a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 139.42/139.61 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 139.42/139.61 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 139.42/139.61 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 139.42/139.61 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 139.42/139.61 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 142.03/142.20 Found (eta_expansion_dep00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 142.03/142.20 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 142.03/142.20 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 142.03/142.20 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 142.03/142.20 Found (eq_sym000 (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found ((eq_sym00 a) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found (((eq_sym0 a0) a) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 142.03/142.20 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 142.03/142.20 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 142.03/142.20 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 142.03/142.20 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 142.03/142.20 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 142.03/142.20 Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 142.03/142.20 Found (eta_expansion00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 145.42/145.68 Found ((eta_expansion0 Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 145.42/145.68 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 145.42/145.68 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 145.42/145.68 Found (eq_sym000 (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 145.42/145.68 Found ((eq_sym00 a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 145.42/145.68 Found (((eq_sym0 a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 145.42/145.68 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 145.42/145.68 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 145.42/145.68 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 145.42/145.68 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 145.42/145.68 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 145.42/145.68 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 145.42/145.68 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 145.42/145.68 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 145.42/145.68 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 145.42/145.68 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 145.42/145.68 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 145.42/145.68 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 145.42/145.68 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 145.42/145.68 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 145.42/145.68 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 145.42/145.68 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 145.42/145.68 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 145.42/145.68 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 145.42/145.68 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 145.42/145.68 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 145.42/145.68 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 145.42/145.68 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 145.42/145.68 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 145.42/145.68 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 145.42/145.68 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 145.42/145.68 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 145.42/145.68 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 145.42/145.68 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) a0)
% 145.42/145.68 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 145.42/145.68 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 145.42/145.68 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 145.42/145.68 Found fofType_DUMMY:fofType
% 145.42/145.68 Found fofType_DUMMY as proof of fofType
% 145.42/145.68 Found (choice_operator0 fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a0)
% 145.42/145.68 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a0)
% 145.42/145.68 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a0)
% 145.42/145.68 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a0)
% 145.42/145.68 Found ((choice_operator fofType) fofType_DUMMY) as proof of (P0 a0)
% 145.42/145.68 Found fofType_DUMMY:fofType
% 145.42/145.68 Found fofType_DUMMY as proof of fofType
% 145.42/145.68 Found (choice_operator0 fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a0)
% 145.42/145.68 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a0)
% 145.42/145.68 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a0)
% 152.32/152.55 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a0)
% 152.32/152.55 Found ((choice_operator fofType) fofType_DUMMY) as proof of (P0 a0)
% 152.32/152.55 Found fofType_DUMMY:fofType
% 152.32/152.55 Found fofType_DUMMY as proof of fofType
% 152.32/152.55 Found (choice_operator0 fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a0)
% 152.32/152.55 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a0)
% 152.32/152.55 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a0)
% 152.32/152.55 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a0)
% 152.32/152.55 Found ((choice_operator fofType) fofType_DUMMY) as proof of (P0 a0)
% 152.32/152.55 Found fofType_DUMMY:fofType
% 152.32/152.55 Found fofType_DUMMY as proof of fofType
% 152.32/152.55 Found (choice_operator0 fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a0)
% 152.32/152.55 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a0)
% 152.32/152.55 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a0)
% 152.32/152.55 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) a0)
% 152.32/152.55 Found ((choice_operator fofType) fofType_DUMMY) as proof of (P0 a0)
% 152.32/152.55 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 152.32/152.55 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 152.32/152.55 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 152.32/152.55 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 152.32/152.55 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 152.32/152.55 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 152.32/152.55 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 152.32/152.55 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 152.32/152.55 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 152.32/152.55 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 152.32/152.55 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 152.32/152.55 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 152.32/152.55 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 152.32/152.55 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) a0)
% 152.32/152.55 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 152.32/152.55 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 152.32/152.55 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 152.32/152.55 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 152.32/152.55 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) a0)
% 152.32/152.55 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 152.32/152.55 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 152.32/152.55 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 152.32/152.55 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 152.32/152.55 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 152.32/152.55 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 152.32/152.55 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 152.32/152.55 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 152.32/152.55 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 152.32/152.55 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 152.32/152.55 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 152.32/152.55 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 152.32/152.55 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 152.32/152.55 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 152.32/152.55 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 159.31/159.54 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 159.31/159.54 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 159.31/159.54 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 159.31/159.54 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 159.31/159.54 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 159.31/159.54 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found eq_ref00:=(eq_ref0 (x P0)):(((eq fofType) (x P0)) (x P0))
% 159.31/159.54 Found (eq_ref0 (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 159.31/159.54 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 159.31/159.54 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 159.31/159.54 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 159.31/159.54 Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 159.31/159.54 Found (eta_expansion_dep00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 159.31/159.54 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 159.31/159.54 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 159.31/159.54 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 159.31/159.54 Found (eq_sym000 (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found ((eq_sym00 a) (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found (((eq_sym0 a0) a) (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 159.31/159.54 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 171.77/171.98 Found eq_ref00:=(eq_ref0 (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))):(((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P))))))
% 171.77/171.98 Found (eq_ref0 (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) b1)
% 171.77/171.98 Found ((eq_ref (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) b1)
% 171.77/171.98 Found ((eq_ref (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) b1)
% 171.77/171.98 Found ((eq_ref (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> ((and (P X)) (forall (Y:fofType), ((P Y)->(((eq fofType) X) Y))))))->(P (D P)))))) b1)
% 171.77/171.98 Found x10:(P0 x0)
% 171.77/171.98 Instantiate: b0:=x0:fofType
% 171.77/171.98 Found x10 as proof of (P1 b0)
% 171.77/171.98 Found eq_ref00:=(eq_ref0 (x P0)):(((eq fofType) (x P0)) (x P0))
% 171.77/171.98 Found (eq_ref0 (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 171.77/171.98 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 171.77/171.98 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 171.77/171.98 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 171.77/171.98 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 171.77/171.98 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 171.77/171.98 Found (((fun (x01:(((eq fofType) (x P0)) b0))=> ((eq_sym000 x01) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 171.77/171.98 Found (((fun (x01:(((eq fofType) (x P0)) x0))=> (((eq_sym00 x0) x01) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 171.77/171.98 Found (((fun (x01:(((eq fofType) (x P0)) x0))=> ((((eq_sym0 (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 171.77/171.98 Found (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 171.77/171.98 Found (fun (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of (P0 (x P0))
% 171.77/171.98 Found (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of ((forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))->(P0 (x P0)))
% 172.12/172.37 Found (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of ((P0 x0)->((forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))->(P0 (x P0))))
% 172.12/172.37 Found (and_rect00 (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 172.12/172.37 Found ((and_rect0 (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 172.12/172.37 Found (((fun (P1:Type) (x2:((P0 x0)->((forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))->P1)))=> (((((and_rect (P0 x0)) (forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))) P1) x2) x1)) (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 172.12/172.37 Found (fun (x1:(((unique fofType) P0) x0))=> (((fun (P1:Type) (x2:((P0 x0)->((forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))->P1)))=> (((((and_rect (P0 x0)) (forall (x':fofType), ((P0 x')->(((eq fofType) x0) x')))) P1) x2) x1)) (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (x':fofType), ((P0 x')->(((eq fofType) x0) x'))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10)))) as proof of (P0 (x P0))
% 172.12/172.37 Found eq_ref00:=(eq_ref0 (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))):(((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P))))))
% 172.12/172.37 Found (eq_ref0 (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))) b1)
% 172.12/172.37 Found ((eq_ref (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))) b1)
% 172.12/172.37 Found ((eq_ref (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))) b1)
% 172.12/172.37 Found ((eq_ref (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))) as proof of (((eq (((fofType->Prop)->fofType)->Prop)) (fun (D:((fofType->Prop)->fofType))=> (forall (P:(fofType->Prop)), (((ex fofType) ((unique fofType) P))->(P (D P)))))) b1)
% 172.12/172.37 Found x3:(P x1)
% 172.12/172.37 Instantiate: a:=x1:fofType
% 172.12/172.37 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a)
% 172.12/172.37 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a))
% 172.12/172.37 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a)))
% 172.12/172.37 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 172.42/172.62 Found ((and_rect0 (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 172.42/172.62 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 172.42/172.62 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 172.42/172.62 Found x3:(P x1)
% 172.42/172.62 Instantiate: a:=x1:fofType
% 172.42/172.62 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a)
% 172.42/172.62 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a))
% 172.42/172.62 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a)))
% 172.42/172.62 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 172.42/172.62 Found ((and_rect0 (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 172.42/172.62 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 172.42/172.62 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 172.42/172.62 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a)
% 172.42/172.62 Found x3:(P x1)
% 172.42/172.62 Instantiate: b0:=x1:fofType
% 172.42/172.62 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 172.42/172.62 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 172.42/172.62 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 172.42/172.62 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 172.42/172.62 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 172.42/172.62 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 172.42/172.62 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 172.42/172.62 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 175.33/175.58 Found x3:(P x1)
% 175.33/175.58 Instantiate: b0:=x1:fofType
% 175.33/175.58 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 175.33/175.58 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 175.33/175.58 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 175.33/175.58 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 175.33/175.58 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 175.33/175.58 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 175.33/175.59 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b0)
% 175.33/175.59 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 175.33/175.59 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 175.33/175.59 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 175.33/175.59 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 175.33/175.59 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 175.33/175.59 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 175.33/175.59 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 175.33/175.59 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 175.33/175.59 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 175.33/175.59 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 175.33/175.59 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 175.33/175.59 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 175.33/175.59 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 175.33/175.59 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 175.33/175.59 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 175.33/175.59 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 175.33/175.59 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 175.33/175.59 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 175.33/175.59 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 175.33/175.59 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 175.33/175.59 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 175.33/175.59 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 175.33/175.59 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 175.33/175.59 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 175.33/175.59 Found x3:(P x1)
% 175.33/175.59 Instantiate: b0:=x1:fofType
% 175.33/175.59 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 175.33/175.59 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 175.33/175.59 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 175.41/175.65 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 175.41/175.65 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 175.41/175.65 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 175.41/175.65 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 175.41/175.65 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 175.41/175.65 Found x3:(P x1)
% 175.41/175.65 Instantiate: b0:=x1:fofType
% 175.41/175.65 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 175.41/175.65 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 175.41/175.65 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 175.41/175.65 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 175.41/175.65 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 175.41/175.65 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 175.41/175.65 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b0)
% 175.41/175.65 Found x3:(P x1)
% 175.41/175.65 Instantiate: a:=x1:fofType
% 175.41/175.65 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a)
% 175.41/175.65 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a))
% 175.41/175.65 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a)))
% 175.41/175.65 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 175.41/175.65 Found ((and_rect0 (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 175.41/175.65 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 175.41/175.65 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 175.41/175.65 Found x3:(P x1)
% 175.41/175.67 Instantiate: a:=x1:fofType
% 175.41/175.67 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a)
% 175.41/175.67 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a))
% 175.41/175.67 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a)))
% 175.41/175.67 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 175.41/175.67 Found ((and_rect0 (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 175.41/175.67 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 175.41/175.67 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 175.41/175.67 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a)
% 175.41/175.67 Found x3:(P x1)
% 175.41/175.67 Instantiate: a:=x1:fofType
% 175.41/175.67 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a)
% 175.41/175.67 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a))
% 175.41/175.67 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a)))
% 175.41/175.67 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 175.41/175.67 Found ((and_rect0 (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 175.41/175.67 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 175.41/175.67 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 175.41/175.67 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a)
% 175.41/175.67 Found x3:(P x1)
% 175.41/175.67 Instantiate: a:=x1:fofType
% 175.41/175.67 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a)
% 175.41/175.67 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a))
% 175.41/175.67 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a)))
% 175.41/175.67 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 175.41/175.67 Found ((and_rect0 (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 175.41/175.67 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 175.60/175.82 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 175.60/175.82 Found x3:(P x1)
% 175.60/175.82 Instantiate: b0:=x1:fofType
% 175.60/175.82 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 175.60/175.82 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 175.60/175.82 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 175.60/175.82 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 175.60/175.82 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 175.60/175.82 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 175.60/175.82 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 175.60/175.82 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 175.60/175.82 Found x3:(P x1)
% 175.60/175.82 Instantiate: b0:=x1:fofType
% 175.60/175.82 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 175.60/175.82 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 175.60/175.82 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 175.60/175.82 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 175.60/175.82 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 175.60/175.82 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 175.60/175.82 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b0)
% 175.60/175.82 Found x3:(P x1)
% 175.60/175.82 Instantiate: b0:=x1:fofType
% 175.60/175.82 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 175.60/175.82 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 175.60/175.82 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 177.77/178.06 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 177.77/178.06 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 177.77/178.06 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 177.77/178.06 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 177.77/178.06 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 177.77/178.06 Found x3:(P x1)
% 177.77/178.06 Instantiate: b0:=x1:fofType
% 177.77/178.06 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 177.77/178.06 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 177.77/178.06 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 177.77/178.06 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 177.77/178.06 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 177.77/178.06 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 177.77/178.06 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b0)
% 177.77/178.06 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 177.77/178.06 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 177.77/178.06 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 177.77/178.06 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 177.77/178.06 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 177.77/178.06 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 177.77/178.06 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 177.77/178.06 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) a0)
% 177.77/178.06 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 177.77/178.06 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 177.77/178.06 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 177.77/178.06 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 177.77/178.06 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 177.77/178.06 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 177.77/178.06 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 177.77/178.06 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 177.77/178.06 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 179.59/179.87 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 179.59/179.87 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 179.59/179.87 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 179.59/179.87 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 179.59/179.87 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 179.59/179.87 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 179.59/179.87 Found x3:(P x1)
% 179.59/179.87 Instantiate: b0:=x1:fofType
% 179.59/179.87 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 179.59/179.87 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 179.59/179.87 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 179.59/179.87 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 179.59/179.87 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 179.59/179.87 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 179.59/179.87 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P b0)
% 179.59/179.87 Found x3:(P x1)
% 179.59/179.87 Instantiate: b0:=x1:fofType
% 179.59/179.87 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P b0)
% 179.59/179.87 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0))
% 179.59/179.87 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P b0)))
% 179.59/179.87 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 179.59/179.87 Found ((and_rect0 (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 179.59/179.87 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 179.59/179.87 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P b0)
% 179.59/179.87 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 179.59/179.87 Found x3:(P x1)
% 179.59/179.87 Instantiate: a:=x1:fofType
% 179.59/179.87 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a)
% 179.59/179.87 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a))
% 179.59/179.87 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a)))
% 180.89/181.12 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 180.89/181.12 Found ((and_rect0 (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 180.89/181.12 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 180.89/181.12 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a)
% 180.89/181.12 Found x3:(P x1)
% 180.89/181.12 Instantiate: a:=x1:fofType
% 180.89/181.12 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a)
% 180.89/181.12 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a))
% 180.89/181.12 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a)))
% 180.89/181.12 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 180.89/181.12 Found ((and_rect0 (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 180.89/181.12 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 180.89/181.12 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a)
% 180.89/181.12 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a)
% 180.89/181.12 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 180.89/181.12 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 180.89/181.12 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 180.89/181.12 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 180.89/181.12 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 180.89/181.12 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 180.89/181.12 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 180.89/181.12 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 180.89/181.12 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 180.89/181.12 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 180.89/181.12 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 180.89/181.12 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 180.89/181.12 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 180.89/181.12 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 180.89/181.12 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 180.89/181.12 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 180.89/181.12 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 184.18/184.40 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 184.18/184.40 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 184.18/184.40 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 184.18/184.40 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 184.18/184.40 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 184.18/184.40 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 184.18/184.40 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 184.18/184.40 Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% 184.18/184.40 Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) P0)
% 184.18/184.40 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P0)
% 184.18/184.40 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P0)
% 184.18/184.40 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P0)
% 184.18/184.40 Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% 184.18/184.40 Found (eq_ref0 b0) as proof of (((eq fofType) b0) (x P0))
% 184.18/184.40 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (x P0))
% 184.18/184.40 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (x P0))
% 184.18/184.40 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (x P0))
% 184.18/184.40 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 184.18/184.40 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 184.18/184.40 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 184.18/184.40 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 184.18/184.40 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 184.18/184.40 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 184.18/184.40 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 184.18/184.40 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 184.18/184.40 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 184.18/184.40 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 184.18/184.40 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 184.18/184.40 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 184.18/184.40 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 184.18/184.40 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 184.18/184.40 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 184.18/184.40 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 184.18/184.40 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 184.18/184.40 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 184.18/184.40 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 184.18/184.40 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 184.18/184.40 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 184.18/184.40 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 184.18/184.40 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 184.18/184.40 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 184.18/184.40 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 184.18/184.40 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 184.18/184.40 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 184.18/184.40 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 184.18/184.40 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 184.18/184.40 Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% 184.18/184.40 Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% 184.18/184.40 Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% 184.18/184.40 Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% 184.18/184.40 Found (fun (x:((fofType->Prop)->fofType))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% 184.18/184.40 Found (fun (x:((fofType->Prop)->fofType))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:((fofType->Prop)->fofType)), (((eq Prop) (f1 x)) (f0 x)))
% 184.18/184.40 Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% 187.29/187.54 Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% 187.29/187.54 Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% 187.29/187.54 Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% 187.29/187.54 Found (fun (x:((fofType->Prop)->fofType))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% 187.29/187.54 Found (fun (x:((fofType->Prop)->fofType))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:((fofType->Prop)->fofType)), (((eq Prop) (f1 x)) (f0 x)))
% 187.29/187.54 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 187.29/187.54 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) a0)
% 187.29/187.54 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 187.29/187.54 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 187.29/187.54 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 187.29/187.54 Found eq_ref00:=(eq_ref0 (x P0)):(((eq fofType) (x P0)) (x P0))
% 187.29/187.54 Found (eq_ref0 (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 187.29/187.54 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 187.29/187.54 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 187.29/187.54 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 187.29/187.54 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 187.29/187.54 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 187.29/187.54 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 187.29/187.54 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 187.29/187.54 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 187.29/187.54 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 187.29/187.54 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 187.29/187.54 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 187.29/187.54 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 187.29/187.54 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 187.29/187.54 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 187.29/187.54 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 187.29/187.54 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 187.29/187.54 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 187.29/187.54 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 187.29/187.54 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 187.29/187.54 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 187.29/187.54 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 187.29/187.54 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 187.29/187.54 Found (eq_ref0 b) as proof of (((eq fofType) b) (x P0))
% 187.29/187.54 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x P0))
% 187.29/187.54 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x P0))
% 187.29/187.54 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x P0))
% 187.29/187.54 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 187.29/187.54 Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% 187.29/187.54 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 187.29/187.54 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 187.29/187.54 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 187.29/187.54 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 187.29/187.54 Found (eq_ref0 b) as proof of (((eq fofType) b) (x P0))
% 187.29/187.54 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x P0))
% 187.29/187.54 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x P0))
% 187.29/187.54 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x P0))
% 187.29/187.54 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 187.29/187.54 Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% 187.29/187.54 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 187.29/187.54 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 187.29/187.54 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 187.29/187.54 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 191.87/192.10 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 191.87/192.10 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 191.87/192.10 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 191.87/192.10 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 191.87/192.10 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 191.87/192.10 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 191.87/192.10 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 191.87/192.10 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 191.87/192.10 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 191.87/192.10 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 191.87/192.10 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 191.87/192.10 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 191.87/192.10 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 191.87/192.10 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 191.87/192.10 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 191.87/192.10 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 191.87/192.10 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 191.87/192.10 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 191.87/192.10 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) a0)
% 191.87/192.10 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 191.87/192.10 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 191.87/192.10 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 191.87/192.10 Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 191.87/192.10 Found (eta_expansion00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 191.87/192.10 Found ((eta_expansion0 Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 191.87/192.10 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 191.87/192.10 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 191.87/192.10 Found (eq_sym000 (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 191.87/192.10 Found ((eq_sym00 a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 191.87/192.10 Found (((eq_sym0 a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 191.87/192.10 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 191.87/192.10 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 191.87/192.10 Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 191.87/192.10 Found (eta_expansion00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 191.87/192.10 Found ((eta_expansion0 Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 191.87/192.10 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 191.87/192.10 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 191.87/192.10 Found (eq_sym000 (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 191.87/192.10 Found ((eq_sym00 a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 191.87/192.10 Found (((eq_sym0 a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 191.87/192.10 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 191.87/192.10 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 191.87/192.10 Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% 191.87/192.10 Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% 191.87/192.10 Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% 191.87/192.10 Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% 191.87/192.10 Found (fun (x:((fofType->Prop)->fofType))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% 197.59/197.88 Found (fun (x:((fofType->Prop)->fofType))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:((fofType->Prop)->fofType)), (((eq Prop) (f1 x)) (f0 x)))
% 197.59/197.88 Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% 197.59/197.88 Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% 197.59/197.88 Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% 197.59/197.88 Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% 197.59/197.88 Found (fun (x:((fofType->Prop)->fofType))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% 197.59/197.88 Found (fun (x:((fofType->Prop)->fofType))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:((fofType->Prop)->fofType)), (((eq Prop) (f1 x)) (f0 x)))
% 197.59/197.88 Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 197.59/197.88 Found (eta_expansion_dep00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 197.59/197.88 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 197.59/197.88 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 197.59/197.88 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 197.59/197.88 Found (eq_sym000 (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 197.59/197.88 Found ((eq_sym00 a) (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 197.59/197.88 Found (((eq_sym0 a0) a) (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 197.59/197.88 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 197.59/197.88 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 197.59/197.88 Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 197.59/197.88 Found (eta_expansion_dep00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 197.59/197.88 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 197.59/197.88 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 197.59/197.88 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 197.59/197.88 Found (eq_sym000 (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 197.59/197.88 Found ((eq_sym00 a) (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 197.59/197.88 Found (((eq_sym0 a0) a) (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 197.59/197.88 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 197.59/197.88 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 197.59/197.88 Found x3:(P x1)
% 197.59/197.88 Instantiate: a0:=x1:fofType
% 197.59/197.88 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a0)
% 197.59/197.88 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0))
% 197.59/197.88 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0)))
% 197.59/197.88 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 197.59/197.88 Found ((and_rect0 (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 197.59/197.88 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 197.59/197.88 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a0)
% 200.10/200.39 Found x3:(P x1)
% 200.10/200.39 Instantiate: a0:=x1:fofType
% 200.10/200.39 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a0)
% 200.10/200.39 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0))
% 200.10/200.39 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0)))
% 200.10/200.39 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 200.10/200.39 Found ((and_rect0 (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 200.10/200.39 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 200.10/200.39 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 200.10/200.39 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a0)
% 200.10/200.39 Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 200.10/200.39 Found (eta_expansion00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 200.10/200.39 Found ((eta_expansion0 Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 200.10/200.39 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 200.10/200.39 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 200.10/200.39 Found (eq_sym000 (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 200.10/200.39 Found ((eq_sym00 a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 200.10/200.39 Found (((eq_sym0 a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 200.10/200.39 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 200.10/200.39 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 200.10/200.39 Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 200.10/200.39 Found (eta_expansion00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 200.10/200.39 Found ((eta_expansion0 Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 200.10/200.39 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 200.10/200.39 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 200.10/200.39 Found (eq_sym000 (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 200.10/200.39 Found ((eq_sym00 a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 200.10/200.39 Found (((eq_sym0 a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 200.10/200.39 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 200.10/200.39 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 200.10/200.39 Found x3:(P x1)
% 200.10/200.39 Instantiate: a0:=x1:fofType
% 200.10/200.39 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a0)
% 200.10/200.39 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0))
% 200.10/200.39 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0)))
% 200.10/200.39 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 200.10/200.39 Found ((and_rect0 (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 200.10/200.39 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 200.10/200.39 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 200.10/200.39 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a0)
% 200.10/200.39 Found x3:(P x1)
% 200.10/200.39 Instantiate: a0:=x1:fofType
% 200.10/200.39 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a0)
% 200.10/200.39 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0))
% 200.10/200.39 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0)))
% 200.10/200.39 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 200.10/200.39 Found ((and_rect0 (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 200.10/200.39 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 200.10/200.39 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a0)
% 200.10/200.39 Found x3:(P x1)
% 200.10/200.39 Instantiate: a0:=x1:fofType
% 200.10/200.39 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a0)
% 200.10/200.39 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0))
% 200.10/200.39 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0)))
% 200.10/200.39 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 200.10/200.39 Found ((and_rect0 (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 200.10/200.39 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 200.10/200.39 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a0)
% 201.77/202.00 Found x3:(P x1)
% 201.77/202.00 Instantiate: a0:=x1:fofType
% 201.77/202.00 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a0)
% 201.77/202.00 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0))
% 201.77/202.00 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0)))
% 201.77/202.00 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 201.77/202.00 Found ((and_rect0 (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 201.77/202.00 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 201.77/202.00 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 201.77/202.00 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a0)
% 201.77/202.00 Found x3:(P x1)
% 201.77/202.00 Instantiate: a0:=x1:fofType
% 201.77/202.00 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a0)
% 201.77/202.00 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0))
% 201.77/202.00 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0)))
% 201.77/202.00 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 201.77/202.00 Found ((and_rect0 (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 201.77/202.00 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 201.77/202.00 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 201.77/202.00 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a0)
% 201.77/202.00 Found x3:(P x1)
% 201.77/202.00 Instantiate: a0:=x1:fofType
% 201.77/202.00 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P a0)
% 201.77/202.00 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0))
% 201.77/202.00 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P a0)))
% 201.77/202.00 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 202.41/202.70 Found ((and_rect0 (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 202.41/202.70 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P a0)
% 202.41/202.70 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P a0)
% 202.41/202.70 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 202.41/202.70 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 202.41/202.70 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 202.41/202.70 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 202.41/202.70 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 202.41/202.70 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 202.41/202.70 Found x10:(P0 x0)
% 202.41/202.70 Instantiate: b0:=x0:fofType
% 202.41/202.70 Found x10 as proof of (P1 b0)
% 202.41/202.70 Found eq_ref00:=(eq_ref0 (x P0)):(((eq fofType) (x P0)) (x P0))
% 202.41/202.70 Found (eq_ref0 (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 202.41/202.70 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 202.41/202.70 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 202.41/202.70 Found ((eq_ref fofType) (x P0)) as proof of (((eq fofType) (x P0)) b0)
% 202.41/202.70 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 202.41/202.70 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 202.41/202.70 Found (((fun (x01:(((eq fofType) (x P0)) b0))=> ((eq_sym000 x01) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 202.41/202.70 Found (((fun (x01:(((eq fofType) (x P0)) x0))=> (((eq_sym00 x0) x01) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 202.41/202.70 Found (((fun (x01:(((eq fofType) (x P0)) x0))=> ((((eq_sym0 (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 202.41/202.70 Found (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10) as proof of (P0 (x P0))
% 202.41/202.70 Found (fun (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of (P0 (x P0))
% 202.41/202.70 Found (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 (x P0)))
% 202.41/202.70 Found (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10)) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 (x P0))))
% 202.41/202.70 Found (and_rect00 (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 202.41/202.70 Found ((and_rect0 (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 202.41/202.70 Found (((fun (P1:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P1)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P1) x2) x1)) (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10))) as proof of (P0 (x P0))
% 203.24/203.47 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (P1:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P1)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P1) x2) x1)) (P0 (x P0))) (fun (x10:(P0 x0)) (x11:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) x10)))) as proof of (P0 (x P0))
% 203.24/203.47 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 203.24/203.47 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 203.24/203.47 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 203.24/203.47 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 203.24/203.47 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 203.24/203.47 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 203.24/203.47 Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% 203.24/203.47 Found (eq_ref0 b0) as proof of (((eq fofType) b0) (x P0))
% 203.24/203.47 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (x P0))
% 203.24/203.47 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (x P0))
% 203.24/203.47 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (x P0))
% 203.24/203.47 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 203.24/203.47 Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% 203.24/203.47 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 203.24/203.47 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 203.24/203.47 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% 203.24/203.47 Found x2:(P0 x0)
% 203.24/203.47 Instantiate: b0:=x0:fofType
% 203.24/203.47 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P1 b0)
% 203.24/203.47 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 b0))
% 203.24/203.47 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 b0)))
% 203.24/203.47 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b0)
% 203.24/203.47 Found ((and_rect0 (P1 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b0)
% 203.24/203.47 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b0)
% 203.24/203.47 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b0)
% 203.24/203.47 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 203.24/203.47 Found ((eq_sym0000 ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 203.24/203.47 Found (((fun (x01:(((eq fofType) (x P0)) b0))=> ((eq_sym000 x01) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 203.46/203.77 Found (((fun (x01:(((eq fofType) (x P0)) x0))=> (((eq_sym00 x0) x01) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 x0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 203.46/203.77 Found (((fun (x01:(((eq fofType) (x P0)) x0))=> ((((eq_sym0 (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 x0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 203.46/203.77 Found (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 x0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 (x P0))
% 203.46/203.77 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (x01:(((eq fofType) (x P0)) x0))=> (((((eq_sym fofType) (x P0)) x0) x01) P0)) ((eq_ref fofType) (x P0))) (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 x0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)))) as proof of (P0 (x P0))
% 203.46/203.77 Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% 203.46/203.77 Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) P0)
% 203.46/203.77 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P0)
% 203.46/203.77 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P0)
% 203.46/203.77 Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) P0)
% 203.46/203.77 Found x2:(P0 x0)
% 203.46/203.77 Instantiate: b0:=x0:fofType
% 203.46/203.77 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P1 b0)
% 203.46/203.77 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 b0))
% 203.46/203.77 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 b0)))
% 203.46/203.77 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b0)
% 203.46/203.77 Found ((and_rect0 (P1 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b0)
% 203.46/203.77 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b0)
% 203.46/203.77 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P1 b0)
% 203.46/203.77 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 203.46/203.77 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 203.46/203.77 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 203.46/203.77 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 208.67/208.97 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 208.67/208.97 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 208.67/208.97 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 208.67/208.97 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) a0)
% 208.67/208.97 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 208.67/208.97 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 208.67/208.97 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 208.67/208.97 Found x2:(P0 x0)
% 208.67/208.97 Instantiate: b0:=x0:fofType
% 208.67/208.97 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P1 b0)
% 208.67/208.97 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 b0))
% 208.67/208.97 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 b0)))
% 208.67/208.97 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b0)
% 208.67/208.97 Found ((and_rect0 (P1 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b0)
% 208.67/208.97 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 b0)
% 208.67/208.97 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P1 b0)
% 208.67/208.97 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 208.67/208.97 Found (eq_ref0 b) as proof of (((eq fofType) b) (x P0))
% 208.67/208.97 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x P0))
% 208.67/208.97 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x P0))
% 208.67/208.97 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x P0))
% 208.67/208.97 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 208.67/208.97 Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% 208.67/208.97 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 208.67/208.97 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 208.67/208.97 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 208.67/208.97 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 208.67/208.97 Found (eq_ref0 b) as proof of (((eq fofType) b) (x P0))
% 208.67/208.97 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x P0))
% 208.67/208.97 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x P0))
% 208.67/208.97 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (x P0))
% 208.67/208.97 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 208.67/208.97 Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% 208.67/208.97 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 208.67/208.97 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 208.67/208.97 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 208.67/208.97 Found x2:(P0 x00)
% 208.67/208.97 Instantiate: a:=x00:fofType
% 208.67/208.97 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2) as proof of (P1 a)
% 208.67/208.97 Found (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->(P1 a))
% 208.67/208.97 Found (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->(P1 a)))
% 208.67/208.97 Found (and_rect00 (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a)
% 208.67/208.97 Found ((and_rect0 (P1 a)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a)
% 208.67/208.97 Found (((fun (P2:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a)
% 210.38/210.68 Found (((fun (P2:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a)
% 210.38/210.68 Found x3:(P0 x1)
% 210.38/210.68 Instantiate: a:=x1:fofType
% 210.38/210.68 Found (fun (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a)
% 210.38/210.68 Found (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->(P1 a))
% 210.38/210.68 Found (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->(P1 a)))
% 210.38/210.68 Found (and_rect00 (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 210.38/210.68 Found ((and_rect0 (P1 a)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 210.38/210.68 Found (((fun (P2:Type) (x3:((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 210.38/210.68 Found (fun (x2:((and (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a)
% 210.38/210.68 Found x2:(P0 x00)
% 210.38/210.68 Instantiate: a:=x00:fofType
% 210.38/210.68 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2) as proof of (P1 a)
% 210.38/210.68 Found (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->(P1 a))
% 210.38/210.68 Found (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2) as proof of ((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->(P1 a)))
% 210.38/210.68 Found (and_rect00 (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a)
% 210.38/210.68 Found ((and_rect0 (P1 a)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a)
% 210.38/210.68 Found (((fun (P2:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a)
% 210.38/210.68 Found (((fun (P2:Type) (x2:((P0 x00)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))->P2)))=> (((((and_rect (P0 x00)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x00)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x00) Y))))=> x2)) as proof of (P1 a)
% 210.38/210.68 Found x3:(P0 x1)
% 210.38/210.68 Instantiate: a:=x1:fofType
% 210.38/210.68 Found (fun (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a)
% 210.38/210.68 Found (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->(P1 a))
% 210.38/210.68 Found (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->(P1 a)))
% 210.38/210.68 Found (and_rect00 (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 210.38/210.68 Found ((and_rect0 (P1 a)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 210.38/210.68 Found (((fun (P2:Type) (x3:((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a)
% 212.17/212.39 Found (fun (x2:((and (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a)
% 212.17/212.39 Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 212.17/212.39 Found (eta_expansion00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 212.17/212.39 Found ((eta_expansion0 Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 212.17/212.39 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 212.17/212.39 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 212.17/212.39 Found (eq_sym000 (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 212.17/212.39 Found ((eq_sym00 a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 212.17/212.39 Found (((eq_sym0 a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 212.17/212.39 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 212.17/212.39 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 212.17/212.39 Found x2:(P0 x0)
% 212.17/212.39 Instantiate: a:=x0:fofType
% 212.17/212.39 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P1 a)
% 212.17/212.39 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 a))
% 212.17/212.39 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 a)))
% 212.17/212.39 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 212.17/212.39 Found ((and_rect0 (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 212.17/212.39 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 212.17/212.39 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 212.17/212.39 Found x2:(P0 x0)
% 212.17/212.39 Instantiate: a:=x0:fofType
% 212.17/212.39 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P1 a)
% 212.17/212.39 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 a))
% 212.17/212.39 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 a)))
% 212.17/212.39 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 212.17/212.39 Found ((and_rect0 (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 212.17/212.39 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 212.17/212.39 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 212.68/212.90 Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 212.68/212.90 Found (eta_expansion00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 212.68/212.90 Found ((eta_expansion0 Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 212.68/212.90 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 212.68/212.90 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 212.68/212.90 Found (eq_sym000 (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 212.68/212.90 Found ((eq_sym00 a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 212.68/212.90 Found (((eq_sym0 a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 212.68/212.90 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 212.68/212.90 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 212.68/212.90 Found x2:(P0 x0)
% 212.68/212.90 Instantiate: a:=x0:fofType
% 212.68/212.90 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P1 a)
% 212.68/212.90 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 a))
% 212.68/212.90 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 a)))
% 212.68/212.90 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 212.68/212.90 Found ((and_rect0 (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 212.68/212.90 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 212.68/212.90 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P1 a)
% 212.68/212.90 Found x2:(P0 x0)
% 212.68/212.90 Instantiate: a:=x0:fofType
% 212.68/212.90 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P1 a)
% 212.68/212.90 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 a))
% 212.68/212.90 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 a)))
% 212.68/212.90 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 212.68/212.90 Found ((and_rect0 (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 212.68/212.90 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 212.68/212.90 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P1 a)
% 212.68/212.90 Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 212.68/212.90 Found (eta_expansion00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 212.68/212.90 Found ((eta_expansion0 Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 212.68/212.90 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 213.65/213.94 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 213.65/213.94 Found (eq_sym000 (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 213.65/213.94 Found ((eq_sym00 a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 213.65/213.94 Found (((eq_sym0 a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 213.65/213.94 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 213.65/213.94 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 213.65/213.94 Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 213.65/213.94 Found (eta_expansion00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 213.65/213.94 Found ((eta_expansion0 Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 213.65/213.94 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 213.65/213.94 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 213.65/213.94 Found (eq_sym000 (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 213.65/213.94 Found ((eq_sym00 a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 213.65/213.94 Found (((eq_sym0 a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 213.65/213.94 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 213.65/213.94 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 213.65/213.94 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 213.65/213.94 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) a0)
% 213.65/213.94 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 213.65/213.94 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 213.65/213.94 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 213.65/213.94 Found x2:(P0 x0)
% 213.65/213.94 Instantiate: a:=x0:fofType
% 213.65/213.94 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P1 a)
% 213.65/213.94 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 a))
% 213.65/213.94 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 a)))
% 213.65/213.94 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 213.65/213.94 Found ((and_rect0 (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 213.65/213.94 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 213.65/213.94 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 213.65/213.94 Found x2:(P0 x0)
% 213.65/213.94 Instantiate: a:=x0:fofType
% 213.65/213.94 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P1 a)
% 213.65/213.94 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 a))
% 213.65/213.94 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 a)))
% 213.65/213.94 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 213.65/213.94 Found ((and_rect0 (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 213.65/213.94 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 215.37/215.67 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 215.37/215.67 Found x2:(P0 x0)
% 215.37/215.67 Instantiate: a:=x0:fofType
% 215.37/215.67 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P1 a)
% 215.37/215.67 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 a))
% 215.37/215.67 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 a)))
% 215.37/215.67 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 215.37/215.67 Found ((and_rect0 (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 215.37/215.67 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 215.37/215.67 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P1 a)
% 215.37/215.67 Found x2:(P0 x0)
% 215.37/215.67 Instantiate: a:=x0:fofType
% 215.37/215.67 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P1 a)
% 215.37/215.67 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 a))
% 215.37/215.67 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P1 a)))
% 215.37/215.67 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 215.37/215.67 Found ((and_rect0 (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 215.37/215.67 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P1 a)
% 215.37/215.67 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P1 a)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P1 a)
% 215.37/215.67 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 215.37/215.67 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) a0)
% 215.37/215.67 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 215.37/215.67 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 215.37/215.67 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 215.37/215.67 Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 215.37/215.67 Found (eta_expansion_dep00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 215.37/215.67 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 215.37/215.67 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 216.18/216.45 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 216.18/216.45 Found (eq_sym000 (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 216.18/216.45 Found ((eq_sym00 a) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 216.18/216.45 Found (((eq_sym0 a0) a) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 216.18/216.45 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 216.18/216.45 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 216.18/216.45 Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 216.18/216.45 Found (eta_expansion_dep00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 216.18/216.45 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 216.18/216.45 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 216.18/216.45 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 216.18/216.45 Found (eq_sym000 (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 216.18/216.45 Found ((eq_sym00 a) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 216.18/216.45 Found (((eq_sym0 a0) a) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 216.18/216.45 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 216.18/216.45 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 216.18/216.45 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 216.18/216.45 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 216.18/216.45 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 216.18/216.45 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 216.18/216.45 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 216.18/216.45 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 216.18/216.45 Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 216.18/216.45 Found (eta_expansion_dep00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 216.18/216.45 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 216.18/216.45 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 216.18/216.45 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 216.18/216.45 Found (eq_sym000 (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 216.18/216.45 Found ((eq_sym00 a) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 216.18/216.45 Found (((eq_sym0 a0) a) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 216.18/216.45 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 216.18/216.45 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 216.18/216.45 Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 216.18/216.45 Found (eta_expansion_dep00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 216.18/216.45 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 216.18/216.45 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 218.87/219.09 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 218.87/219.09 Found (eq_sym000 (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 218.87/219.09 Found ((eq_sym00 a) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 218.87/219.09 Found (((eq_sym0 a0) a) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 218.87/219.09 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 218.87/219.09 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 218.87/219.09 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 218.87/219.09 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) a0)
% 218.87/219.09 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 218.87/219.09 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 218.87/219.09 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 218.87/219.09 Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 218.87/219.09 Found (eta_expansion00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 218.87/219.09 Found ((eta_expansion0 Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 218.87/219.09 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 218.87/219.09 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 218.87/219.09 Found (eq_sym000 (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 218.87/219.09 Found ((eq_sym00 a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 218.87/219.09 Found (((eq_sym0 a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 218.87/219.09 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 218.87/219.09 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 218.87/219.09 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 218.87/219.09 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 218.87/219.09 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 218.87/219.09 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 218.87/219.09 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 218.87/219.09 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 218.87/219.09 Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 218.87/219.09 Found (eta_expansion00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 218.87/219.09 Found ((eta_expansion0 Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 218.87/219.09 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 218.87/219.09 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 218.87/219.09 Found (eq_sym000 (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 218.87/219.09 Found ((eq_sym00 a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 218.87/219.09 Found (((eq_sym0 a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 218.87/219.09 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 218.87/219.09 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 218.87/219.09 Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 218.87/219.09 Found (eta_expansion00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 218.87/219.09 Found ((eta_expansion0 Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 218.87/219.09 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 218.87/219.09 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 218.87/219.09 Found (eq_sym000 (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found ((eq_sym00 a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found (((eq_sym0 a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 221.86/222.09 Found (eta_expansion00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 221.86/222.09 Found ((eta_expansion0 Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 221.86/222.09 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 221.86/222.09 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 221.86/222.09 Found (eq_sym000 (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found ((eq_sym00 a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found (((eq_sym0 a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 221.86/222.09 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 221.86/222.09 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 221.86/222.09 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 221.86/222.09 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 221.86/222.09 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 221.86/222.09 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 230.55/230.79 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 230.55/230.79 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 230.55/230.79 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 230.55/230.79 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 230.55/230.79 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 230.55/230.79 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 230.55/230.79 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 230.55/230.79 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 230.55/230.79 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 230.55/230.79 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 230.55/230.79 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 230.55/230.79 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 230.55/230.79 Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% 230.55/230.79 Found (eq_ref0 b0) as proof of (((eq fofType) b0) (x P))
% 230.55/230.79 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (x P))
% 230.55/230.79 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (x P))
% 230.55/230.79 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (x P))
% 230.55/230.79 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 230.55/230.79 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b0)
% 230.55/230.79 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 230.55/230.79 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 230.55/230.79 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 230.55/230.79 Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% 230.55/230.79 Found (eq_ref0 b0) as proof of (((eq fofType) b0) a)
% 230.55/230.79 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) a)
% 230.55/230.79 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) a)
% 230.55/230.79 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) a)
% 230.55/230.79 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 230.55/230.79 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b0)
% 230.55/230.79 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 230.55/230.79 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 230.55/230.79 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 230.55/230.79 Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% 230.55/230.79 Found (eq_ref0 b0) as proof of (((eq fofType) b0) a)
% 230.55/230.79 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) a)
% 230.55/230.79 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) a)
% 230.55/230.79 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) a)
% 230.55/230.79 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 230.55/230.79 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b0)
% 230.55/230.79 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 230.55/230.79 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 230.55/230.79 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 230.55/230.79 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 230.55/230.79 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) a0)
% 230.55/230.79 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 230.55/230.79 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 230.55/230.79 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 230.55/230.79 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 230.55/230.79 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 230.55/230.79 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 230.55/230.79 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 230.55/230.79 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 230.55/230.79 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 230.55/230.79 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 230.55/230.79 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 230.55/230.79 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 237.62/237.90 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 237.62/237.90 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 237.62/237.90 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% 237.62/237.90 Found (eq_ref0 b0) as proof of (((eq fofType) b0) b1)
% 237.62/237.90 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b1)
% 237.62/237.90 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b1)
% 237.62/237.90 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b1)
% 237.62/237.90 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 237.62/237.90 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 237.62/237.90 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 237.62/237.90 Found (eta_expansion_dep00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 237.62/237.90 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 237.62/237.90 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 237.62/237.90 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 237.62/237.90 Found (eq_sym000 (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found ((eq_sym00 a) (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 237.62/237.90 Found (((eq_sym0 a0) a) (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 247.24/247.50 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 247.24/247.50 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 247.24/247.50 Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 247.24/247.50 Found (eta_expansion_dep00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 247.24/247.50 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 247.24/247.50 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 247.24/247.50 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 247.24/247.50 Found (eq_sym000 (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 247.24/247.50 Found ((eq_sym00 a) (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 247.24/247.50 Found (((eq_sym0 a0) a) (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 247.24/247.50 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 247.24/247.50 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 247.24/247.50 Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% 247.24/247.50 Found (eq_ref0 b0) as proof of (((eq fofType) b0) (x P))
% 247.24/247.50 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (x P))
% 247.24/247.50 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (x P))
% 247.24/247.50 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (x P))
% 247.24/247.50 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 247.24/247.50 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b0)
% 247.24/247.50 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 247.24/247.50 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 247.24/247.50 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 247.24/247.50 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 247.24/247.50 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 247.24/247.50 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 247.24/247.50 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 247.24/247.50 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 247.24/247.50 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 247.24/247.50 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 247.24/247.50 Found (eq_ref0 b) as proof of (((eq fofType) b) b00)
% 247.24/247.50 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b00)
% 247.24/247.50 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b00)
% 247.24/247.50 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b00)
% 247.24/247.50 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 247.24/247.50 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 247.24/247.50 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 247.24/247.50 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 247.24/247.50 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 247.24/247.50 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 247.24/247.50 Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% 247.24/247.50 Found (eq_ref0 b0) as proof of (((eq fofType) b0) a)
% 247.24/247.50 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) a)
% 247.24/247.50 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) a)
% 247.24/247.50 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) a)
% 247.24/247.50 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 247.24/247.50 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b0)
% 247.24/247.50 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 247.24/247.50 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 247.24/247.50 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 247.24/247.50 Found x3:(P x1)
% 247.24/247.50 Instantiate: a0:=x1:fofType
% 247.24/247.50 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P00 a0)
% 247.24/247.50 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P00 a0))
% 250.68/250.99 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P00 a0)))
% 250.68/250.99 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 a0)
% 250.68/250.99 Found ((and_rect0 (P00 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 a0)
% 250.68/250.99 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P00 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 a0)
% 250.68/250.99 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P00 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 a0)
% 250.68/250.99 Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% 250.68/250.99 Found (eq_ref0 b0) as proof of (((eq fofType) b0) a)
% 250.68/250.99 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) a)
% 250.68/250.99 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) a)
% 250.68/250.99 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) a)
% 250.68/250.99 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 250.68/250.99 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b0)
% 250.68/250.99 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 250.68/250.99 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 250.68/250.99 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 250.68/250.99 Found x3:(P x1)
% 250.68/250.99 Instantiate: a0:=x1:fofType
% 250.68/250.99 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P00 a0)
% 250.68/250.99 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P00 a0))
% 250.68/250.99 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P00 a0)))
% 250.68/250.99 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 a0)
% 250.68/250.99 Found ((and_rect0 (P00 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 a0)
% 250.68/250.99 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P00 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 a0)
% 250.68/250.99 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P00 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P00 a0)
% 250.68/250.99 Found x3:(P x1)
% 250.68/250.99 Instantiate: a0:=x1:fofType
% 250.68/250.99 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P00 a0)
% 250.68/250.99 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P00 a0))
% 250.68/250.99 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P00 a0)))
% 250.68/250.99 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 a0)
% 250.68/250.99 Found ((and_rect0 (P00 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 a0)
% 250.68/250.99 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P00 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 a0)
% 257.79/258.12 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P00 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 a0)
% 257.79/258.12 Found x3:(P x1)
% 257.79/258.12 Instantiate: a0:=x1:fofType
% 257.79/258.12 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P00 a0)
% 257.79/258.12 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P00 a0))
% 257.79/258.12 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P00 a0)))
% 257.79/258.12 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 a0)
% 257.79/258.12 Found ((and_rect0 (P00 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 a0)
% 257.79/258.12 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P00 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P00 a0)
% 257.79/258.12 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P00 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P00 a0)
% 257.79/258.12 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 257.79/258.12 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) a0)
% 257.79/258.12 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 257.79/258.12 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 257.79/258.12 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 257.79/258.12 Found x2:(P x00)
% 257.79/258.12 Instantiate: a0:=x00:fofType
% 257.79/258.12 Found x2 as proof of (P00 a0)
% 257.79/258.12 Found x3:(P x1)
% 257.79/258.12 Instantiate: a0:=x1:fofType
% 257.79/258.12 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a0)
% 257.79/258.12 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0))
% 257.79/258.12 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0)))
% 257.79/258.12 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 257.79/258.12 Found ((and_rect0 (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 257.79/258.12 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 257.79/258.12 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 257.79/258.12 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 257.79/258.12 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) a0)
% 257.79/258.12 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 257.79/258.12 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 257.79/258.12 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 257.79/258.12 Found x3:(P x1)
% 257.79/258.12 Instantiate: a0:=x1:fofType
% 257.79/258.12 Found x3 as proof of (P00 a0)
% 257.79/258.12 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 257.79/258.12 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) a0)
% 257.79/258.12 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 260.24/260.50 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 260.24/260.50 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 260.24/260.50 Found x3:(P x1)
% 260.24/260.50 Instantiate: a0:=x1:fofType
% 260.24/260.50 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a0)
% 260.24/260.50 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0))
% 260.24/260.50 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0)))
% 260.24/260.50 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 260.24/260.50 Found ((and_rect0 (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 260.24/260.50 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 260.24/260.50 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a0)
% 260.24/260.50 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 260.24/260.50 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 260.24/260.50 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 260.24/260.50 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 260.24/260.50 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 260.24/260.50 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 260.24/260.50 Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% 260.24/260.50 Found (eq_ref0 b0) as proof of (((eq fofType) b0) (x P))
% 260.24/260.50 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (x P))
% 260.24/260.50 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (x P))
% 260.24/260.50 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (x P))
% 260.24/260.50 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 260.24/260.50 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b0)
% 260.24/260.50 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 260.24/260.50 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 260.24/260.50 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 260.24/260.50 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 260.24/260.50 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b0)
% 260.24/260.50 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 260.24/260.50 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 260.24/260.50 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 260.24/260.50 Found x3:(P x1)
% 260.24/260.50 Instantiate: a0:=x1:fofType
% 260.24/260.50 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a0)
% 260.24/260.50 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0))
% 260.24/260.50 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0)))
% 260.24/260.50 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 260.24/260.50 Found ((and_rect0 (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 260.24/260.50 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 262.83/263.13 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 262.83/263.13 Found fofType_DUMMY:fofType
% 262.83/263.13 Found fofType_DUMMY as proof of fofType
% 262.83/263.13 Found (choice_operator0 fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) b1)
% 262.83/263.13 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) b1)
% 262.83/263.13 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) b1)
% 262.83/263.13 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) b1)
% 262.83/263.13 Found ((choice_operator fofType) fofType_DUMMY) as proof of (P1 b1)
% 262.83/263.13 Found x3:(P x1)
% 262.83/263.13 Instantiate: a0:=x1:fofType
% 262.83/263.13 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a0)
% 262.83/263.13 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0))
% 262.83/263.13 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0)))
% 262.83/263.13 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 262.83/263.13 Found ((and_rect0 (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 262.83/263.13 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 262.83/263.13 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 262.83/263.13 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 262.83/263.13 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 262.83/263.13 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 262.83/263.13 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 262.83/263.13 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 262.83/263.13 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 262.83/263.13 Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% 262.83/263.13 Found (eq_ref0 b0) as proof of (((eq fofType) b0) (x P))
% 262.83/263.13 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (x P))
% 262.83/263.13 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (x P))
% 262.83/263.13 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (x P))
% 262.83/263.13 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 262.83/263.13 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b0)
% 262.83/263.13 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 262.83/263.13 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 262.83/263.13 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 262.83/263.13 Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% 262.83/263.13 Found (eq_ref0 a0) as proof of (((eq fofType) a0) b0)
% 262.83/263.13 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 262.83/263.13 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 262.83/263.13 Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b0)
% 262.83/263.13 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 262.83/263.13 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 262.83/263.13 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 262.83/263.13 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 262.83/263.13 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 262.83/263.13 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 262.83/263.13 Found x3:(P x1)
% 262.83/263.13 Instantiate: a0:=x1:fofType
% 264.28/264.57 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a0)
% 264.28/264.57 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0))
% 264.28/264.57 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0)))
% 264.28/264.57 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 264.28/264.57 Found ((and_rect0 (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 264.28/264.57 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 264.28/264.57 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a0)
% 264.28/264.57 Found x3:(P x1)
% 264.28/264.57 Instantiate: a0:=x1:fofType
% 264.28/264.57 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a0)
% 264.28/264.57 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0))
% 264.28/264.57 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0)))
% 264.28/264.57 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 264.28/264.57 Found ((and_rect0 (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 264.28/264.57 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 264.28/264.57 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a0)
% 264.28/264.57 Found x3:(P x1)
% 264.28/264.57 Instantiate: a0:=x1:fofType
% 264.28/264.57 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a0)
% 264.28/264.57 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0))
% 264.28/264.57 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0)))
% 264.28/264.57 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 264.28/264.57 Found ((and_rect0 (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 264.28/264.57 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 264.28/264.57 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 264.28/264.57 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 267.64/267.92 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) a0)
% 267.64/267.92 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 267.64/267.92 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 267.64/267.92 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 267.64/267.92 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 267.64/267.92 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 267.64/267.92 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 267.64/267.92 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 267.64/267.92 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 267.64/267.92 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 267.64/267.92 Found x3:(P x1)
% 267.64/267.92 Instantiate: a0:=x1:fofType
% 267.64/267.92 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P1 a0)
% 267.64/267.92 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0))
% 267.64/267.92 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P1 a0)))
% 267.64/267.92 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 267.64/267.92 Found ((and_rect0 (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 267.64/267.92 Found (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P1 a0)
% 267.64/267.92 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P1 a0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a0)
% 267.64/267.92 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 267.64/267.92 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 267.64/267.92 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 267.64/267.92 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 267.64/267.92 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 267.64/267.92 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 267.64/267.92 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 267.64/267.92 Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% 267.64/267.92 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% 267.64/267.92 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% 267.64/267.92 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% 267.64/267.92 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 267.64/267.92 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 267.64/267.92 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 267.64/267.92 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 267.64/267.92 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 267.64/267.92 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 267.64/267.92 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 267.64/267.92 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 267.64/267.92 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 267.64/267.92 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 274.61/274.88 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 274.61/274.88 Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% 274.61/274.88 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% 274.61/274.88 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% 274.61/274.88 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% 274.61/274.88 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 274.61/274.88 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found fofType_DUMMY:fofType
% 274.61/274.88 Found fofType_DUMMY as proof of fofType
% 274.61/274.88 Found (choice_operator0 fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) b1)
% 274.61/274.88 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) b1)
% 274.61/274.88 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) b1)
% 274.61/274.88 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) b1)
% 274.61/274.88 Found ((choice_operator fofType) fofType_DUMMY) as proof of (P1 b1)
% 274.61/274.88 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 274.61/274.88 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 274.61/274.88 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 274.61/274.88 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 274.61/274.88 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 274.61/274.88 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 281.40/281.76 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 281.40/281.76 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 281.40/281.76 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 281.40/281.76 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 281.40/281.76 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 281.40/281.76 Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 281.40/281.76 Found (eta_expansion00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 281.40/281.76 Found ((eta_expansion0 Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 281.40/281.76 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 281.40/281.76 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 281.40/281.76 Found (eq_sym000 (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 281.40/281.76 Found ((eq_sym00 a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 281.40/281.76 Found (((eq_sym0 a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 281.40/281.76 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 281.40/281.76 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 281.40/281.76 Found eq_ref00:=(eq_ref0 (x a0)):(((eq fofType) (x a0)) (x a0))
% 281.40/281.76 Found (eq_ref0 (x a0)) as proof of (((eq fofType) (x a0)) b)
% 281.40/281.76 Found ((eq_ref fofType) (x a0)) as proof of (((eq fofType) (x a0)) b)
% 281.40/281.76 Found ((eq_ref fofType) (x a0)) as proof of (((eq fofType) (x a0)) b)
% 281.40/281.76 Found ((eq_ref fofType) (x a0)) as proof of (((eq fofType) (x a0)) b)
% 281.40/281.76 Found fofType_DUMMY:fofType
% 281.40/281.76 Found fofType_DUMMY as proof of fofType
% 281.40/281.76 Found (choice_operator0 fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f1)
% 281.40/281.76 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f1)
% 281.40/281.76 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f1)
% 281.40/281.76 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f1)
% 281.40/281.76 Found ((choice_operator fofType) fofType_DUMMY) as proof of (P1 f1)
% 281.40/281.76 Found fofType_DUMMY:fofType
% 281.40/281.76 Found fofType_DUMMY as proof of fofType
% 281.40/281.76 Found (choice_operator0 fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f1)
% 281.40/281.76 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f1)
% 281.40/281.76 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f1)
% 281.40/281.76 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f1)
% 281.40/281.76 Found ((choice_operator fofType) fofType_DUMMY) as proof of (P1 f1)
% 281.40/281.76 Found eq_ref00:=(eq_ref0 (x a0)):(((eq fofType) (x a0)) (x a0))
% 281.40/281.76 Found (eq_ref0 (x a0)) as proof of (((eq fofType) (x a0)) b)
% 281.40/281.76 Found ((eq_ref fofType) (x a0)) as proof of (((eq fofType) (x a0)) b)
% 281.40/281.76 Found ((eq_ref fofType) (x a0)) as proof of (((eq fofType) (x a0)) b)
% 281.40/281.76 Found ((eq_ref fofType) (x a0)) as proof of (((eq fofType) (x a0)) b)
% 281.40/281.76 Found x3:(P x1)
% 281.40/281.76 Instantiate: b1:=x1:fofType
% 281.40/281.76 Found x3 as proof of (P01 b1)
% 281.40/281.76 Found eq_ref00:=(eq_ref0 (x P)):(((eq fofType) (x P)) (x P))
% 281.40/281.76 Found (eq_ref0 (x P)) as proof of (((eq fofType) (x P)) b1)
% 281.40/281.76 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b1)
% 281.40/281.76 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b1)
% 281.40/281.76 Found ((eq_ref fofType) (x P)) as proof of (((eq fofType) (x P)) b1)
% 281.40/281.76 Found eq_ref00:=(eq_ref0 (x a0)):(((eq fofType) (x a0)) (x a0))
% 281.40/281.76 Found (eq_ref0 (x a0)) as proof of (((eq fofType) (x a0)) b)
% 281.40/281.76 Found ((eq_ref fofType) (x a0)) as proof of (((eq fofType) (x a0)) b)
% 281.40/281.76 Found ((eq_ref fofType) (x a0)) as proof of (((eq fofType) (x a0)) b)
% 281.40/281.76 Found ((eq_ref fofType) (x a0)) as proof of (((eq fofType) (x a0)) b)
% 281.40/281.76 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 281.40/281.76 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 281.40/281.76 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 285.28/285.61 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 285.28/285.61 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 285.28/285.61 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 285.28/285.61 Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 285.28/285.61 Found (eta_expansion_dep00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 285.28/285.61 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 285.28/285.61 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 285.28/285.61 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 285.28/285.61 Found (eq_sym000 (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 285.28/285.61 Found ((eq_sym00 a) (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 285.28/285.61 Found (((eq_sym0 a0) a) (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 285.28/285.61 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 285.28/285.61 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 285.28/285.61 Found eq_ref00:=(eq_ref0 (x a0)):(((eq fofType) (x a0)) (x a0))
% 285.28/285.61 Found (eq_ref0 (x a0)) as proof of (((eq fofType) (x a0)) b)
% 285.28/285.61 Found ((eq_ref fofType) (x a0)) as proof of (((eq fofType) (x a0)) b)
% 285.28/285.61 Found ((eq_ref fofType) (x a0)) as proof of (((eq fofType) (x a0)) b)
% 285.28/285.61 Found ((eq_ref fofType) (x a0)) as proof of (((eq fofType) (x a0)) b)
% 285.28/285.61 Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% 285.28/285.61 Found (eq_ref0 b0) as proof of (((eq fofType) b0) b1)
% 285.28/285.61 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b1)
% 285.28/285.61 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b1)
% 285.28/285.61 Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b1)
% 285.28/285.61 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 285.28/285.61 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 285.28/285.61 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 285.28/285.61 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 285.28/285.61 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 285.28/285.61 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 285.28/285.61 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 285.28/285.61 Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% 285.28/285.61 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% 285.28/285.61 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% 285.28/285.61 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% 285.28/285.61 Found x00:(P02 (x P))
% 285.28/285.61 Instantiate: b:=x1:fofType
% 285.28/285.61 Found (fun (x00:(P02 (x P)))=> x00) as proof of (P02 b1)
% 285.28/285.61 Found (fun (P02:(fofType->Prop)) (x00:(P02 (x P)))=> x00) as proof of ((P02 (x P))->(P02 b1))
% 285.28/285.61 Found (fun (P02:(fofType->Prop)) (x00:(P02 (x P)))=> x00) as proof of (((eq fofType) (x P)) b1)
% 285.28/285.61 Found ((eq_sym0020 (fun (P02:(fofType->Prop)) (x00:(P02 (x P)))=> x00)) x3) as proof of (P00 b0)
% 285.28/285.61 Found ((eq_sym0020 (fun (P02:(fofType->Prop)) (x00:(P02 (x P)))=> x00)) x3) as proof of (P00 b0)
% 285.28/285.61 Found (((fun (x00:(((eq fofType) (x P)) b1))=> ((eq_sym002 x00) P00)) (fun (P02:(fofType->Prop)) (x00:(P02 (x P)))=> x00)) x3) as proof of (P00 b0)
% 285.28/285.61 Found (((fun (x00:(((eq fofType) (x P)) x1))=> (((eq_sym00 x1) x00) P00)) (fun (P02:(fofType->Prop)) (x00:(P02 (x P)))=> x00)) x3) as proof of (P00 b0)
% 285.28/285.61 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> (((fun (x00:(((eq fofType) (x P)) x1))=> (((eq_sym00 x1) x00) P00)) (fun (P02:(fofType->Prop)) (x00:(P02 (x P)))=> x00)) x3)) as proof of (P00 b0)
% 285.28/285.61 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> (((fun (x00:(((eq fofType) (x P)) x1))=> (((eq_sym00 x1) x00) P00)) (fun (P02:(fofType->Prop)) (x00:(P02 (x P)))=> x00)) x3)) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P00 b0))
% 285.28/285.62 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> (((fun (x00:(((eq fofType) (x P)) x1))=> (((eq_sym00 x1) x00) P00)) (fun (P02:(fofType->Prop)) (x00:(P02 (x P)))=> x00)) x3)) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P00 b0)))
% 285.28/285.62 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> (((fun (x00:(((eq fofType) (x P)) x1))=> (((eq_sym00 x1) x00) P00)) (fun (P02:(fofType->Prop)) (x00:(P02 (x P)))=> x00)) x3))) as proof of (P00 b0)
% 285.28/285.62 Found ((and_rect0 (P00 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> (((fun (x00:(((eq fofType) (x P)) x1))=> (((eq_sym00 x1) x00) P00)) (fun (P02:(fofType->Prop)) (x00:(P02 (x P)))=> x00)) x3))) as proof of (P00 b0)
% 285.28/285.62 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P00 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> (((fun (x00:(((eq fofType) (x P)) x1))=> (((eq_sym00 x1) x00) P00)) (fun (P02:(fofType->Prop)) (x00:(P02 (x P)))=> x00)) x3))) as proof of (P00 b0)
% 285.28/285.62 Found (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P00 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> (((fun (x00:(((eq fofType) (x P)) x1))=> (((eq_sym00 x1) x00) P00)) (fun (P02:(fofType->Prop)) (x00:(P02 (x P)))=> x00)) x3))) as proof of (P00 b0)
% 285.28/285.62 Found ((eq_sym0010 ((eq_ref fofType) (x P))) (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P00 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> (((fun (x00:(((eq fofType) (x P)) x1))=> (((eq_sym00 x1) x00) P00)) (fun (P02:(fofType->Prop)) (x00:(P02 (x P)))=> x00)) x3)))) as proof of (P0 b)
% 285.28/285.62 Found ((eq_sym0010 ((eq_ref fofType) (x P))) (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P00 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> (((fun (x00:(((eq fofType) (x P)) x1))=> (((eq_sym00 x1) x00) P00)) (fun (P02:(fofType->Prop)) (x00:(P02 (x P)))=> x00)) x3)))) as proof of (P0 b)
% 285.28/285.62 Found (((fun (x00:(((eq fofType) (x P)) b0))=> ((eq_sym001 x00) P0)) ((eq_ref fofType) (x P))) (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P0 b0)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> (((fun (x00:(((eq fofType) (x P)) x1))=> (((eq_sym00 x1) x00) P0)) (fun (P02:(fofType->Prop)) (x00:(P02 (x P)))=> x00)) x3)))) as proof of (P0 b)
% 285.28/285.62 Found (((fun (x00:(((eq fofType) (x P)) b))=> (((eq_sym00 b) x00) P0)) ((eq_ref fofType) (x P))) (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P0 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> (((fun (x00:(((eq fofType) (x P)) x1))=> (((eq_sym00 x1) x00) P0)) (fun (P02:(fofType->Prop)) (x00:(P02 (x P)))=> x00)) x3)))) as proof of (P0 b)
% 285.28/285.62 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (x00:(((eq fofType) (x P)) b))=> (((eq_sym00 b) x00) P0)) ((eq_ref fofType) (x P))) (((fun (P1:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P1)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P1) x3) x2)) (P0 b)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> (((fun (x00:(((eq fofType) (x P)) x1))=> (((eq_sym00 x1) x00) P0)) (fun (P02:(fofType->Prop)) (x00:(P02 (x P)))=> x00)) x3))))) as proof of (P0 b)
% 288.76/289.05 Found fofType_DUMMY:fofType
% 288.76/289.05 Found fofType_DUMMY as proof of fofType
% 288.76/289.05 Found (choice_operator0 fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f1)
% 288.76/289.05 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f1)
% 288.76/289.05 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f1)
% 288.76/289.05 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f1)
% 288.76/289.05 Found ((choice_operator fofType) fofType_DUMMY) as proof of (P1 f1)
% 288.76/289.05 Found fofType_DUMMY:fofType
% 288.76/289.05 Found fofType_DUMMY as proof of fofType
% 288.76/289.05 Found (choice_operator0 fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f1)
% 288.76/289.05 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f1)
% 288.76/289.05 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f1)
% 288.76/289.05 Found ((choice_operator fofType) fofType_DUMMY) as proof of ((ex ((fofType->Prop)->fofType)) f1)
% 288.76/289.05 Found ((choice_operator fofType) fofType_DUMMY) as proof of (P1 f1)
% 288.76/289.05 Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 288.76/289.05 Found (eta_expansion00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 288.76/289.05 Found ((eta_expansion0 Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 288.76/289.05 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 288.76/289.05 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 288.76/289.05 Found (eq_sym000 (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 288.76/289.05 Found ((eq_sym00 a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 288.76/289.05 Found (((eq_sym0 a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 288.76/289.05 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 288.76/289.05 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 288.76/289.05 Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 288.76/289.05 Found (eta_expansion00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 288.76/289.05 Found ((eta_expansion0 Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 288.76/289.05 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 288.76/289.05 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 288.76/289.05 Found (eq_sym000 (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 288.76/289.05 Found ((eq_sym00 a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 288.76/289.05 Found (((eq_sym0 a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 288.76/289.05 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 288.76/289.05 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 288.76/289.05 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 288.76/289.05 Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% 288.76/289.05 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% 288.76/289.05 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% 288.76/289.05 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% 288.76/289.05 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 288.76/289.05 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 288.76/289.05 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 288.76/289.05 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 288.76/289.05 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 288.76/289.05 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) a0)
% 288.76/289.05 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 290.55/290.88 Found (eq_ref0 b) as proof of (((eq fofType) b) b00)
% 290.55/290.88 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b00)
% 290.55/290.88 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b00)
% 290.55/290.88 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b00)
% 290.55/290.88 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 290.55/290.88 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) a0)
% 290.55/290.88 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 290.55/290.88 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 290.55/290.88 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 290.55/290.88 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) a0)
% 290.55/290.88 Found x2:(P0 x0)
% 290.55/290.88 Instantiate: b0:=x0:fofType
% 290.55/290.88 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P0 b0)
% 290.55/290.88 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 b0))
% 290.55/290.88 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 b0)))
% 290.55/290.88 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b0)
% 290.55/290.88 Found ((and_rect0 (P0 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b0)
% 290.55/290.88 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b0)
% 290.55/290.88 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b0)
% 290.55/290.88 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P1 b0)
% 290.55/290.88 Found x2:(P0 x0)
% 290.55/290.88 Instantiate: b0:=x0:fofType
% 290.55/290.88 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P0 b0)
% 290.55/290.88 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 b0))
% 290.55/290.88 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 b0)))
% 290.55/290.88 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b0)
% 290.55/290.89 Found ((and_rect0 (P0 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b0)
% 290.55/290.89 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b0)
% 290.55/290.89 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 b0)
% 290.55/290.89 Found x2:(P0 x0)
% 290.55/290.89 Instantiate: b0:=x0:fofType
% 290.55/290.89 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P0 b0)
% 290.55/290.89 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 b0))
% 294.56/294.86 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 b0)))
% 294.56/294.86 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b0)
% 294.56/294.86 Found ((and_rect0 (P0 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b0)
% 294.56/294.86 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b0)
% 294.56/294.86 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P0 b0)
% 294.56/294.86 Found x2:(P0 x0)
% 294.56/294.86 Instantiate: b0:=x0:fofType
% 294.56/294.86 Found (fun (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of (P0 b0)
% 294.56/294.86 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 b0))
% 294.56/294.86 Found (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2) as proof of ((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->(P0 b0)))
% 294.56/294.86 Found (and_rect00 (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b0)
% 294.56/294.86 Found ((and_rect0 (P0 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b0)
% 294.56/294.86 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b0)
% 294.56/294.86 Found (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2)) as proof of (P0 b0)
% 294.56/294.86 Found (fun (x1:((and (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))))=> (((fun (P2:Type) (x2:((P0 x0)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))->P2)))=> (((((and_rect (P0 x0)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y)))) P2) x2) x1)) (P0 b0)) (fun (x2:(P0 x0)) (x3:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x0) Y))))=> x2))) as proof of (P1 b0)
% 294.56/294.86 Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% 294.56/294.86 Found (eta_expansion00 a0) as proof of (((eq (fofType->Prop)) a0) a)
% 294.56/294.86 Found ((eta_expansion0 Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 294.56/294.86 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 294.56/294.86 Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) a)
% 294.56/294.86 Found (eq_sym000 (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 294.56/294.86 Found ((eq_sym00 a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 294.56/294.86 Found (((eq_sym0 a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 294.56/294.86 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 294.56/294.86 Found ((((eq_sym (fofType->Prop)) a0) a) (((eta_expansion fofType) Prop) a0)) as proof of (((eq (fofType->Prop)) a) a0)
% 294.56/294.86 Found eq_ref000:=(eq_ref00 P2):((P2 a)->(P2 a))
% 294.56/294.86 Found (eq_ref00 P2) as proof of (P3 a)
% 294.56/294.86 Found ((eq_ref0 a) P2) as proof of (P3 a)
% 294.56/294.86 Found (((eq_ref fofType) a) P2) as proof of (P3 a)
% 294.56/294.86 Found (((eq_ref fofType) a) P2) as proof of (P3 a)
% 297.79/298.10 Found eq_ref000:=(eq_ref00 P2):((P2 a)->(P2 a))
% 297.79/298.10 Found (eq_ref00 P2) as proof of (P3 a)
% 297.79/298.10 Found ((eq_ref0 a) P2) as proof of (P3 a)
% 297.79/298.10 Found (((eq_ref fofType) a) P2) as proof of (P3 a)
% 297.79/298.10 Found (((eq_ref fofType) a) P2) as proof of (P3 a)
% 297.79/298.10 Found x3:(P x1)
% 297.79/298.10 Instantiate: b1:=x1:fofType
% 297.79/298.10 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P2 b1)
% 297.79/298.10 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P2 b1))
% 297.79/298.10 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P2 b1)))
% 297.79/298.10 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P2 b1)
% 297.79/298.10 Found ((and_rect0 (P2 b1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P2 b1)
% 297.79/298.10 Found (((fun (P3:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P3)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P3) x3) x2)) (P2 b1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P2 b1)
% 297.79/298.10 Found (((fun (P3:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P3)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P3) x3) x2)) (P2 b1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P2 b1)
% 297.79/298.10 Found ((eq_sym0200 ((eq_ref fofType) b0)) (((fun (P3:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P3)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P3) x3) x2)) (P2 b1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 297.79/298.10 Found ((eq_sym0200 ((eq_ref fofType) b0)) (((fun (P3:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P3)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P3) x3) x2)) (P2 b1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 297.79/298.10 Found (((fun (x3:(((eq fofType) b0) b1))=> ((eq_sym020 x3) P1)) ((eq_ref fofType) b0)) (((fun (P3:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P3)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P3) x3) x2)) (P1 b1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 297.79/298.10 Found (((fun (x3:(((eq fofType) b0) x1))=> (((eq_sym02 x1) x3) P1)) ((eq_ref fofType) b0)) (((fun (P3:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P3)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P3) x3) x2)) (P1 x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 297.79/298.10 Found (((fun (x3:(((eq fofType) b0) x1))=> ((((eq_sym0 b0) x1) x3) P1)) ((eq_ref fofType) b0)) (((fun (P3:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P3)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P3) x3) x2)) (P1 x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 b0)
% 297.79/298.10 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (x3:(((eq fofType) b0) x1))=> ((((eq_sym0 b0) x1) x3) P1)) ((eq_ref fofType) b0)) (((fun (P3:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P3)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P3) x3) x2)) (P1 x1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)))) as proof of (P1 b0)
% 297.79/298.10 Found x3:(P x1)
% 297.79/298.10 Instantiate: b1:=x1:fofType
% 297.79/298.10 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P2 b1)
% 297.79/298.10 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P2 b1))
% 297.95/298.30 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P2 b1)))
% 297.95/298.30 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P2 b1)
% 297.95/298.30 Found ((and_rect0 (P2 b1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P2 b1)
% 297.95/298.30 Found (((fun (P3:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P3)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P3) x3) x2)) (P2 b1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P2 b1)
% 297.95/298.30 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P3:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P3)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P3) x3) x2)) (P2 b1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P2 b1)
% 297.95/298.30 Found x3:(P0 x1)
% 297.95/298.30 Instantiate: a:=x1:fofType
% 297.95/298.30 Found (fun (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P0 a)
% 297.95/298.30 Found (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->(P0 a))
% 297.95/298.30 Found (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->(P0 a)))
% 297.95/298.30 Found (and_rect00 (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 a)
% 297.95/298.30 Found ((and_rect0 (P0 a)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 a)
% 297.95/298.30 Found (((fun (P2:Type) (x3:((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P0 a)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 a)
% 297.95/298.30 Found (((fun (P2:Type) (x3:((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P0 a)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 a)
% 297.95/298.30 Found (fun (x2:((and (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P0 a)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P1 a)
% 297.95/298.30 Found x3:(P0 x1)
% 297.95/298.30 Instantiate: a:=x1:fofType
% 297.95/298.30 Found (fun (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P0 a)
% 297.95/298.30 Found (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->(P0 a))
% 297.95/298.30 Found (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->(P0 a)))
% 297.95/298.30 Found (and_rect00 (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 a)
% 297.95/298.30 Found ((and_rect0 (P0 a)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 a)
% 297.95/298.30 Found (((fun (P2:Type) (x3:((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P0 a)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P0 a)
% 297.95/298.30 Found (fun (x2:((and (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))))=> (((fun (P2:Type) (x3:((P0 x1)->((forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))->P2)))=> (((((and_rect (P0 x1)) (forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y)))) P2) x3) x2)) (P0 a)) (fun (x3:(P0 x1)) (x4:(forall (Y:fofType), ((P0 Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P0 a)
% 299.82/300.12 Found x3:(P x1)
% 299.82/300.12 Instantiate: b1:=x1:fofType
% 299.82/300.12 Found (fun (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of (P2 b1)
% 299.82/300.12 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P2 b1))
% 299.82/300.12 Found (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3) as proof of ((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->(P2 b1)))
% 299.82/300.12 Found (and_rect00 (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P2 b1)
% 299.82/300.12 Found ((and_rect0 (P2 b1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P2 b1)
% 299.82/300.12 Found (((fun (P3:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P3)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P3) x3) x2)) (P2 b1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3)) as proof of (P2 b1)
% 299.82/300.12 Found (fun (x2:((and (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))))=> (((fun (P3:Type) (x3:((P x1)->((forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))->P3)))=> (((((and_rect (P x1)) (forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y)))) P3) x3) x2)) (P2 b1)) (fun (x3:(P x1)) (x4:(forall (Y:fofType), ((P Y)->(((eq fofType) x1) Y))))=> x3))) as proof of (P2 b1)
% 299.82/300.12 Found x01:(P x00)
% 299.82/300.12 Instantiate: b:=x00:fofType
% 299.82/300.12 Found x01 as proof of (P2 b)
% 299.82/300.12 Found eq_ref00:=(eq_ref0 (x a0)):(((eq fofType) (x a0)) (x a0))
% 299.82/300.12 Found (eq_ref0 (x a0)) as proof of (((eq fofType) (x a0)) b)
% 299.82/300.12 Found ((eq_ref fofType) (x a0)) as proof of (((eq fofType) (x a0)) b)
% 299.82/300.12 Found ((eq_ref fofType) (x a0)) as proof of (((eq fofType) (x a0)) b)
% 299.82/300.12 Found ((eq_ref fofType) (x a0)) as proof of (((eq fofType) (x a0)) b)
% 299.82/300.12 Found ((eq_sym0000 ((eq_ref fofType) (x a0))) x01) as proof of (P1 (x a0))
% 299.82/300.12 Found ((eq_sym0000 ((eq_ref fofType) (x a0))) x01) as proof of (P1 (x a0))
% 299.82/300.12 Found (((fun (x03:(((eq fofType) (x a0)) b))=> ((eq_sym000 x03) P1)) ((eq_ref fofType) (x a0))) x01) as proof of (P1 (x a0))
% 299.82/300.12 Found (((fun (x03:(((eq fofType) (x a0)) x00))=> (((eq_sym00 x00) x03) P1)) ((eq_ref fofType) (x a0))) x01) as proof of (P1 (x a0))
% 299.82/300.12 Found (((fun (x03:(((eq fofType) (x a0)) x00))=> ((((eq_sym0 (x a0)) x00) x03) P1)) ((eq_ref fofType) (x a0))) x01) as proof of (P1 (x a0))
% 299.82/300.12 Found (((fun (x03:(((eq fofType) (x a0)) x00))=> (((((eq_sym fofType) (x a0)) x00) x03) P1)) ((eq_ref fofType) (x a0))) x01) as proof of (P1 (x a0))
% 299.82/300.12 Found (fun (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a0)) x00))=> (((((eq_sym fofType) (x a0)) x00) x03) P1)) ((eq_ref fofType) (x a0))) x01)) as proof of (P1 (x a0))
% 299.82/300.12 Found (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a0)) x00))=> (((((eq_sym fofType) (x a0)) x00) x03) P1)) ((eq_ref fofType) (x a0))) x01)) as proof of ((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 (x a0)))
% 299.82/300.12 Found (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a0)) x00))=> (((((eq_sym fofType) (x a0)) x00) x03) P1)) ((eq_ref fofType) (x a0))) x01)) as proof of ((P x00)->((forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y)))->(P1 (x a0))))
% 299.82/300.12 Found (and_rect00 (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a0)) x00))=> (((((eq_sym fofType) (x a0)) x00) x03) P1)) ((eq_ref fofType) (x a0))) x01))) as proof of (P1 (x a0))
% 299.82/300.12 Found ((and_rect0 (P1 (x a0))) (fun (x01:(P x00)) (x02:(forall (Y:fofType), ((P Y)->(((eq fofType) x00) Y))))=> (((fun (x03:(((eq fofType) (x a0)) x00))=> (((((eq_sym fofType) (x a0)) x00) x03) P1)) ((eq_ref fofType) (x a0))) x01))) as proof of (P1 (x a0))
% 299.82/300.12 Found (((fun (P2:Type) (x2:((P
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