TSTP Solution File: NUM826^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : NUM826^5 : TPTP v7.0.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n070.star.cs.uiowa.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory : 32218.625MB
% OS : Linux 3.10.0-693.2.2.el7.x86_64
% CPULimit : 300s
% DateTime : Mon Jan 8 13:11:51 EST 2018
% Result : Timeout 300.01s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----No solution output by system
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : NUM826^5 : TPTP v7.0.0. Released v4.0.0.
% 0.00/0.03 % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.03/0.23 % Computer : n070.star.cs.uiowa.edu
% 0.03/0.23 % Model : x86_64 x86_64
% 0.03/0.23 % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% 0.03/0.23 % Memory : 32218.625MB
% 0.03/0.23 % OS : Linux 3.10.0-693.2.2.el7.x86_64
% 0.03/0.23 % CPULimit : 300
% 0.03/0.23 % DateTime : Fri Jan 5 14:33:51 CST 2018
% 0.03/0.23 % CPUTime :
% 0.03/0.25 Python 2.7.13
% 0.07/0.52 Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% 0.07/0.52 FOF formula (<kernel.Constant object at 0x2ae2298f1200>, <kernel.DependentProduct object at 0x2ae2298f2050>) of role type named cG
% 0.07/0.52 Using role type
% 0.07/0.52 Declaring cG:(fofType->fofType)
% 0.07/0.52 FOF formula (<kernel.Constant object at 0x2ae228e725f0>, <kernel.DependentProduct object at 0x2ae2298f2758>) of role type named cQ
% 0.07/0.52 Using role type
% 0.07/0.52 Declaring cQ:(fofType->Prop)
% 0.07/0.52 FOF formula (<kernel.Constant object at 0x2ae228e725f0>, <kernel.DependentProduct object at 0x2ae2298f2440>) of role type named cP
% 0.07/0.52 Using role type
% 0.07/0.52 Declaring cP:(fofType->Prop)
% 0.07/0.52 FOF formula (<kernel.Constant object at 0x2ae2298f1950>, <kernel.DependentProduct object at 0x2ae2298f29e0>) of role type named cF
% 0.07/0.52 Using role type
% 0.07/0.52 Declaring cF:(fofType->(fofType->fofType))
% 0.07/0.52 FOF formula (<kernel.Constant object at 0x2ae2298f1200>, <kernel.Single object at 0x2ae2298f2440>) of role type named cA
% 0.07/0.52 Using role type
% 0.07/0.52 Declaring cA:fofType
% 0.07/0.52 FOF formula (<kernel.Constant object at 0x2ae2298f1950>, <kernel.Single object at 0x2ae2298f2758>) of role type named cB
% 0.07/0.52 Using role type
% 0.07/0.52 Declaring cB:fofType
% 0.07/0.52 FOF formula (((and ((and ((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (((eq fofType) (cG cB)) cA))) (forall (Xx:fofType) (Xy:fofType), (((eq fofType) (cG ((cF Xx) Xy))) ((cF (cG Xx)) (cG Xy)))))->(forall (Xx:fofType), ((cP Xx)->(cQ (cG Xx))))) of role conjecture named cTHM622_pme
% 0.07/0.52 Conjecture to prove = (((and ((and ((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (((eq fofType) (cG cB)) cA))) (forall (Xx:fofType) (Xy:fofType), (((eq fofType) (cG ((cF Xx) Xy))) ((cF (cG Xx)) (cG Xy)))))->(forall (Xx:fofType), ((cP Xx)->(cQ (cG Xx))))):Prop
% 0.07/0.52 We need to prove ['(((and ((and ((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (((eq fofType) (cG cB)) cA))) (forall (Xx:fofType) (Xy:fofType), (((eq fofType) (cG ((cF Xx) Xy))) ((cF (cG Xx)) (cG Xy)))))->(forall (Xx:fofType), ((cP Xx)->(cQ (cG Xx)))))']
% 0.07/0.52 Parameter fofType:Type.
% 0.07/0.52 Parameter cG:(fofType->fofType).
% 0.07/0.52 Parameter cQ:(fofType->Prop).
% 0.07/0.52 Parameter cP:(fofType->Prop).
% 0.07/0.52 Parameter cF:(fofType->(fofType->fofType)).
% 0.07/0.52 Parameter cA:fofType.
% 0.07/0.52 Parameter cB:fofType.
% 0.07/0.52 Trying to prove (((and ((and ((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (((eq fofType) (cG cB)) cA))) (forall (Xx:fofType) (Xy:fofType), (((eq fofType) (cG ((cF Xx) Xy))) ((cF (cG Xx)) (cG Xy)))))->(forall (Xx:fofType), ((cP Xx)->(cQ (cG Xx)))))
% 17.90/18.11 Found eq_ref00:=(eq_ref0 (cG Xx)):(((eq fofType) (cG Xx)) (cG Xx))
% 17.90/18.11 Found (eq_ref0 (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 17.90/18.11 Found ((eq_ref fofType) (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 17.90/18.11 Found ((eq_ref fofType) (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 17.90/18.11 Found ((eq_ref fofType) (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 17.90/18.11 Found eq_ref00:=(eq_ref0 (cG Xx)):(((eq fofType) (cG Xx)) (cG Xx))
% 17.90/18.11 Found (eq_ref0 (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 17.90/18.11 Found ((eq_ref fofType) (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 17.90/18.11 Found ((eq_ref fofType) (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 17.90/18.11 Found ((eq_ref fofType) (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 17.90/18.11 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 17.90/18.11 Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% 17.90/18.11 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 17.90/18.11 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 17.90/18.11 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 17.90/18.11 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 17.90/18.11 Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% 17.90/18.11 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 17.90/18.11 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 17.90/18.11 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 17.90/18.11 Found eq_ref00:=(eq_ref0 (cG Xx)):(((eq fofType) (cG Xx)) (cG Xx))
% 17.90/18.11 Found (eq_ref0 (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 17.90/18.11 Found ((eq_ref fofType) (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 17.90/18.11 Found ((eq_ref fofType) (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 17.90/18.11 Found ((eq_ref fofType) (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 17.90/18.11 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 17.90/18.11 Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% 17.90/18.11 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 17.90/18.11 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 17.90/18.11 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 17.90/18.11 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 17.90/18.11 Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% 17.90/18.11 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 17.90/18.11 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 17.90/18.11 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 17.90/18.11 Found eq_ref00:=(eq_ref0 (cG Xx)):(((eq fofType) (cG Xx)) (cG Xx))
% 17.90/18.11 Found (eq_ref0 (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 17.90/18.11 Found ((eq_ref fofType) (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 17.90/18.11 Found ((eq_ref fofType) (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 17.90/18.11 Found ((eq_ref fofType) (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 17.90/18.11 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 17.90/18.11 Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% 17.90/18.11 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 17.90/18.11 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 17.90/18.11 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 17.90/18.11 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 17.90/18.11 Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% 17.90/18.11 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 17.90/18.11 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 17.90/18.11 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 17.90/18.11 Found eq_ref00:=(eq_ref0 (cG Xx)):(((eq fofType) (cG Xx)) (cG Xx))
% 17.90/18.11 Found (eq_ref0 (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 17.90/18.11 Found ((eq_ref fofType) (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 17.90/18.11 Found ((eq_ref fofType) (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 17.90/18.11 Found ((eq_ref fofType) (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 17.90/18.11 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 17.90/18.11 Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% 17.90/18.11 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 72.94/73.17 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 72.94/73.17 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 72.94/73.17 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 72.94/73.17 Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% 72.94/73.17 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 72.94/73.17 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 72.94/73.17 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 72.94/73.17 Found x60:=(x6 (fun (x14:fofType)=> (cQ cB))):((cQ cB)->(cQ cB))
% 72.94/73.17 Found (x6 (fun (x14:fofType)=> (cQ cB))) as proof of (P cB)
% 72.94/73.17 Found (x6 (fun (x14:fofType)=> (cQ cB))) as proof of (P cB)
% 72.94/73.17 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 72.94/73.17 Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% 72.94/73.17 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 72.94/73.17 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 72.94/73.17 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 72.94/73.17 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 72.94/73.17 Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% 72.94/73.17 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 72.94/73.17 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 72.94/73.17 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 72.94/73.17 Found eq_ref00:=(eq_ref0 (cG Xx)):(((eq fofType) (cG Xx)) (cG Xx))
% 72.94/73.17 Found (eq_ref0 (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 72.94/73.17 Found ((eq_ref fofType) (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 72.94/73.17 Found ((eq_ref fofType) (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 72.94/73.17 Found ((eq_ref fofType) (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 72.94/73.17 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 72.94/73.17 Found (eq_ref0 b) as proof of (((eq fofType) b) (cG Xx))
% 72.94/73.17 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (cG Xx))
% 72.94/73.17 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (cG Xx))
% 72.94/73.17 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (cG Xx))
% 72.94/73.17 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 72.94/73.17 Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% 72.94/73.17 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 72.94/73.17 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 72.94/73.17 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 72.94/73.17 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 72.94/73.17 Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% 72.94/73.17 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 72.94/73.17 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 72.94/73.17 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 72.94/73.17 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 72.94/73.17 Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% 72.94/73.17 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 72.94/73.17 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 72.94/73.17 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 72.94/73.17 Found eq_ref00:=(eq_ref0 (cG Xx)):(((eq fofType) (cG Xx)) (cG Xx))
% 72.94/73.17 Found (eq_ref0 (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 72.94/73.17 Found ((eq_ref fofType) (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 72.94/73.17 Found ((eq_ref fofType) (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 72.94/73.17 Found ((eq_ref fofType) (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 72.94/73.17 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 72.94/73.17 Found (eq_ref0 b) as proof of (((eq fofType) b) (cG Xx))
% 72.94/73.17 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (cG Xx))
% 72.94/73.17 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (cG Xx))
% 72.94/73.17 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (cG Xx))
% 72.94/73.17 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 72.94/73.17 Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% 72.94/73.17 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 72.94/73.17 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 72.94/73.17 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% 72.94/73.17 Found x4:(((eq fofType) (cG cB)) cA)
% 72.94/73.17 Instantiate: a:=(cG cB):fofType;b:=cA:fofType
% 72.94/73.17 Found x4 as proof of (((eq fofType) a) b)
% 72.94/73.17 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 72.94/73.17 Found (eq_ref0 b) as proof of (((eq fofType) b) (cG Xx))
% 72.94/73.17 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (cG Xx))
% 72.94/73.17 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (cG Xx))
% 72.94/73.17 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (cG Xx))
% 72.94/73.17 Found x14:(cQ cB)
% 72.94/73.17 Instantiate: b:=cB:fofType
% 72.94/73.17 Found x14 as proof of (P b)
% 76.69/76.93 Found eq_ref00:=(eq_ref0 (cG Xx)):(((eq fofType) (cG Xx)) (cG Xx))
% 76.69/76.93 Found (eq_ref0 (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 76.69/76.93 Found ((eq_ref fofType) (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 76.69/76.93 Found ((eq_ref fofType) (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 76.69/76.93 Found ((eq_ref fofType) (cG Xx)) as proof of (((eq fofType) (cG Xx)) b)
% 76.69/76.93 Found x40:=(x4 (fun (x14:fofType)=> (cQ cB))):((cQ cB)->(cQ cB))
% 76.69/76.93 Found (x4 (fun (x14:fofType)=> (cQ cB))) as proof of ((cQ cB)->(P b))
% 76.69/76.93 Found (x4 (fun (x14:fofType)=> (cQ cB))) as proof of ((cQ cB)->(P b))
% 76.69/76.93 Found (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))) as proof of ((cQ cB)->(P b))
% 76.69/76.93 Found (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))) as proof of ((cP cA)->((cQ cB)->(P b)))
% 76.69/76.93 Found (and_rect60 (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))) as proof of (P b)
% 76.69/76.93 Found ((and_rect6 (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))) as proof of (P b)
% 76.69/76.93 Found (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))) as proof of (P b)
% 76.69/76.93 Found (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))) as proof of (P b)
% 76.69/76.93 Found x40:=(x4 (fun (x14:fofType)=> (cQ cB))):((cQ cB)->(cQ cB))
% 76.69/76.93 Found (x4 (fun (x14:fofType)=> (cQ cB))) as proof of ((cQ cB)->(P b))
% 76.69/76.93 Found (x4 (fun (x14:fofType)=> (cQ cB))) as proof of ((cQ cB)->(P b))
% 76.69/76.93 Found (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))) as proof of ((cQ cB)->(P b))
% 76.69/76.93 Found (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))) as proof of ((cP cA)->((cQ cB)->(P b)))
% 76.69/76.93 Found (and_rect60 (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))) as proof of (P b)
% 76.69/76.93 Found ((and_rect6 (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))) as proof of (P b)
% 76.69/76.93 Found (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))) as proof of (P b)
% 76.69/76.93 Found (fun (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))) as proof of (P b)
% 76.69/76.93 Found (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))) as proof of ((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->(P b))
% 76.69/76.93 Found (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))) as proof of (((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->(P b)))
% 76.69/76.93 Found (and_rect50 (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))) as proof of (P b)
% 76.69/76.93 Found ((and_rect5 (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))) as proof of (P b)
% 76.69/76.93 Found (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))) as proof of (P b)
% 77.22/77.48 Found (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))) as proof of (P b)
% 77.22/77.48 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 77.22/77.48 Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% 77.22/77.48 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 77.22/77.48 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 77.22/77.48 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 77.22/77.48 Found x40:=(x4 (fun (x14:fofType)=> (cQ cB))):((cQ cB)->(cQ cB))
% 77.22/77.48 Found (x4 (fun (x14:fofType)=> (cQ cB))) as proof of ((cQ cB)->(P b))
% 77.22/77.48 Found (x4 (fun (x14:fofType)=> (cQ cB))) as proof of ((cQ cB)->(P b))
% 77.22/77.48 Found (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))) as proof of ((cQ cB)->(P b))
% 77.22/77.48 Found (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))) as proof of ((cP cA)->((cQ cB)->(P b)))
% 77.22/77.48 Found (and_rect60 (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))) as proof of (P b)
% 77.22/77.48 Found ((and_rect6 (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))) as proof of (P b)
% 77.22/77.48 Found (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))) as proof of (P b)
% 77.22/77.48 Found (fun (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))) as proof of (P b)
% 77.22/77.48 Found (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))) as proof of ((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->(P b))
% 77.22/77.48 Found (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))) as proof of (((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->(P b)))
% 77.22/77.48 Found (and_rect50 (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))) as proof of (P b)
% 77.22/77.48 Found ((and_rect5 (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))) as proof of (P b)
% 77.22/77.48 Found (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))) as proof of (P b)
% 77.22/77.48 Found (fun (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))) as proof of (P b)
% 77.22/77.48 Found (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))) as proof of ((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->(P b))
% 77.22/77.48 Found (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))) as proof of (((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->(P b)))
% 77.22/77.48 Found (and_rect40 (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))) as proof of (P b)
% 77.22/77.48 Found ((and_rect4 (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))) as proof of (P b)
% 77.73/78.01 Found (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))) as proof of (P b)
% 77.73/78.01 Found (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))) as proof of (P b)
% 77.73/78.01 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 77.73/78.01 Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% 77.73/78.01 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 77.73/78.01 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 77.73/78.01 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 77.73/78.01 Found x40:=(x4 (fun (x14:fofType)=> (cQ cB))):((cQ cB)->(cQ cB))
% 77.73/78.01 Found (x4 (fun (x14:fofType)=> (cQ cB))) as proof of ((cQ cB)->(P b))
% 77.73/78.01 Found (x4 (fun (x14:fofType)=> (cQ cB))) as proof of ((cQ cB)->(P b))
% 77.73/78.01 Found (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))) as proof of ((cQ cB)->(P b))
% 77.73/78.01 Found (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))) as proof of ((cP cA)->((cQ cB)->(P b)))
% 77.73/78.01 Found (and_rect60 (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))) as proof of (P b)
% 77.73/78.01 Found ((and_rect6 (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))) as proof of (P b)
% 77.73/78.01 Found (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))) as proof of (P b)
% 77.73/78.02 Found (fun (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))) as proof of (P b)
% 77.73/78.02 Found (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))) as proof of ((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->(P b))
% 77.73/78.02 Found (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))) as proof of (((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->(P b)))
% 77.73/78.02 Found (and_rect50 (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))) as proof of (P b)
% 77.73/78.02 Found ((and_rect5 (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))) as proof of (P b)
% 77.73/78.02 Found (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))) as proof of (P b)
% 77.73/78.02 Found (fun (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))) as proof of (P b)
% 77.73/78.02 Found (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))) as proof of ((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->(P b))
% 77.73/78.02 Found (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))) as proof of (((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->(P b)))
% 77.82/78.02 Found (and_rect40 (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))) as proof of (P b)
% 77.82/78.02 Found ((and_rect4 (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))) as proof of (P b)
% 77.82/78.02 Found (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))) as proof of (P b)
% 77.82/78.02 Found (fun (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))))) as proof of (P b)
% 77.82/78.03 Found (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))))) as proof of ((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->(P b))
% 77.82/78.03 Found (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))))) as proof of (((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->(P b)))
% 77.82/78.04 Found (and_rect30 (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))))) as proof of (P b)
% 77.82/78.04 Found ((and_rect3 (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))))) as proof of (P b)
% 77.82/78.04 Found (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))))) as proof of (P b)
% 78.26/78.50 Found (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))))) as proof of (P b)
% 78.26/78.50 Found x40:=(x4 (fun (x14:fofType)=> (cQ cB))):((cQ cB)->(cQ cB))
% 78.26/78.50 Found (x4 (fun (x14:fofType)=> (cQ cB))) as proof of ((cQ cB)->(P b))
% 78.26/78.50 Found (x4 (fun (x14:fofType)=> (cQ cB))) as proof of ((cQ cB)->(P b))
% 78.26/78.51 Found (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))) as proof of ((cQ cB)->(P b))
% 78.26/78.51 Found (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))) as proof of ((cP cA)->((cQ cB)->(P b)))
% 78.26/78.51 Found (and_rect60 (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))) as proof of (P b)
% 78.26/78.51 Found ((and_rect6 (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))) as proof of (P b)
% 78.26/78.51 Found (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))) as proof of (P b)
% 78.26/78.51 Found (fun (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))) as proof of (P b)
% 78.26/78.51 Found (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))) as proof of ((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->(P b))
% 78.26/78.51 Found (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))) as proof of (((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->(P b)))
% 78.26/78.51 Found (and_rect50 (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))) as proof of (P b)
% 78.26/78.51 Found ((and_rect5 (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))) as proof of (P b)
% 78.26/78.51 Found (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))) as proof of (P b)
% 78.26/78.51 Found (fun (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))) as proof of (P b)
% 78.26/78.51 Found (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))) as proof of ((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->(P b))
% 78.26/78.51 Found (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))) as proof of (((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->(P b)))
% 78.26/78.51 Found (and_rect40 (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))) as proof of (P b)
% 78.26/78.51 Found ((and_rect4 (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))) as proof of (P b)
% 78.26/78.51 Found (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))) as proof of (P b)
% 78.26/78.52 Found (fun (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))))) as proof of (P b)
% 78.26/78.52 Found (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))))) as proof of ((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->(P b))
% 78.26/78.52 Found (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))))) as proof of (((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->(P b)))
% 78.26/78.52 Found (and_rect30 (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))))) as proof of (P b)
% 78.26/78.53 Found ((and_rect3 (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))))) as proof of (P b)
% 78.26/78.53 Found (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))))) as proof of (P b)
% 78.26/78.54 Found (fun (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))))))) as proof of (P b)
% 78.26/78.54 Found (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))))))) as proof of ((((eq fofType) (cG cA)) cB)->(P b))
% 78.26/78.55 Found (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))))))) as proof of (((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))->((((eq fofType) (cG cA)) cB)->(P b)))
% 78.26/78.56 Found (and_rect20 (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))))))) as proof of (P b)
% 78.26/78.56 Found ((and_rect2 (P b)) (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))))))) as proof of (P b)
% 78.35/78.56 Found (((fun (P0:Type) (x5:(((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))->((((eq fofType) (cG cA)) cB)->P0)))=> (((((and_rect ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (((eq fofType) (cG cA)) cB)) P0) x5) x3)) (P b)) (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))))))) as proof of (P b)
% 78.35/78.57 Found (((fun (P0:Type) (x5:(((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))->((((eq fofType) (cG cA)) cB)->P0)))=> (((((and_rect ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (((eq fofType) (cG cA)) cB)) P0) x5) x3)) (P b)) (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))))))) as proof of (P b)
% 78.74/78.96 Found eq_ref000:=(eq_ref00 cQ):((cQ cB)->(cQ cB))
% 78.74/78.96 Found (eq_ref00 cQ) as proof of ((cQ cB)->(P b))
% 78.74/78.96 Found ((eq_ref0 cB) cQ) as proof of ((cQ cB)->(P b))
% 78.74/78.96 Found (((eq_ref fofType) cB) cQ) as proof of ((cQ cB)->(P b))
% 78.74/78.96 Found (((eq_ref fofType) cB) cQ) as proof of ((cQ cB)->(P b))
% 78.74/78.96 Found (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ)) as proof of ((cQ cB)->(P b))
% 78.74/78.96 Found (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ)) as proof of ((cP cA)->((cQ cB)->(P b)))
% 78.74/78.96 Found (and_rect60 (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ))) as proof of (P b)
% 78.74/78.96 Found ((and_rect6 (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ))) as proof of (P b)
% 78.74/78.96 Found (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ))) as proof of (P b)
% 78.74/78.96 Found (fun (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ)))) as proof of (P b)
% 78.74/78.96 Found (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ)))) as proof of ((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->(P b))
% 78.74/78.96 Found (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ)))) as proof of (((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->(P b)))
% 78.74/78.96 Found (and_rect50 (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ))))) as proof of (P b)
% 78.74/78.96 Found ((and_rect5 (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ))))) as proof of (P b)
% 78.74/78.96 Found (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ))))) as proof of (P b)
% 78.74/78.97 Found (fun (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ)))))) as proof of (P b)
% 78.74/78.97 Found (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ)))))) as proof of ((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->(P b))
% 78.74/78.97 Found (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ)))))) as proof of (((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->(P b)))
% 78.74/78.97 Found (and_rect40 (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ))))))) as proof of (P b)
% 78.74/78.97 Found ((and_rect4 (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ))))))) as proof of (P b)
% 78.74/78.97 Found (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ))))))) as proof of (P b)
% 78.74/78.97 Found (fun (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ)))))))) as proof of (P b)
% 78.74/78.97 Found (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ)))))))) as proof of ((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->(P b))
% 78.74/78.98 Found (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ)))))))) as proof of (((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->(P b)))
% 78.74/78.98 Found (and_rect30 (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ))))))))) as proof of (P b)
% 78.74/78.98 Found ((and_rect3 (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ))))))))) as proof of (P b)
% 78.74/78.98 Found (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ))))))))) as proof of (P b)
% 78.74/78.99 Found (fun (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ)))))))))) as proof of (P b)
% 78.74/78.99 Found (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ)))))))))) as proof of ((((eq fofType) (cG cA)) cB)->(P b))
% 78.80/79.00 Found (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ)))))))))) as proof of (((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))->((((eq fofType) (cG cA)) cB)->(P b)))
% 78.80/79.00 Found (and_rect20 (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ))))))))))) as proof of (P b)
% 78.80/79.01 Found ((and_rect2 (P b)) (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ))))))))))) as proof of (P b)
% 78.80/79.02 Found (((fun (P0:Type) (x5:(((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))->((((eq fofType) (cG cA)) cB)->P0)))=> (((((and_rect ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (((eq fofType) (cG cA)) cB)) P0) x5) x3)) (P b)) (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ))))))))))) as proof of (P b)
% 78.80/79.02 Found (fun (x4:(((eq fofType) (cG cB)) cA))=> (((fun (P0:Type) (x5:(((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))->((((eq fofType) (cG cA)) cB)->P0)))=> (((((and_rect ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (((eq fofType) (cG cA)) cB)) P0) x5) x3)) (P b)) (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ)))))))))))) as proof of (P b)
% 78.80/79.03 Found (fun (x3:((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (((eq fofType) (cG cA)) cB))) (x4:(((eq fofType) (cG cB)) cA))=> (((fun (P0:Type) (x5:(((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))->((((eq fofType) (cG cA)) cB)->P0)))=> (((((and_rect ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (((eq fofType) (cG cA)) cB)) P0) x5) x3)) (P b)) (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ)))))))))))) as proof of ((((eq fofType) (cG cB)) cA)->(P b))
% 78.80/79.04 Found (fun (x3:((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (((eq fofType) (cG cA)) cB))) (x4:(((eq fofType) (cG cB)) cA))=> (((fun (P0:Type) (x5:(((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))->((((eq fofType) (cG cA)) cB)->P0)))=> (((((and_rect ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (((eq fofType) (cG cA)) cB)) P0) x5) x3)) (P b)) (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ)))))))))))) as proof of (((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (((eq fofType) (cG cA)) cB))->((((eq fofType) (cG cB)) cA)->(P b)))
% 78.80/79.05 Found (and_rect10 (fun (x3:((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (((eq fofType) (cG cA)) cB))) (x4:(((eq fofType) (cG cB)) cA))=> (((fun (P0:Type) (x5:(((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))->((((eq fofType) (cG cA)) cB)->P0)))=> (((((and_rect ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (((eq fofType) (cG cA)) cB)) P0) x5) x3)) (P b)) (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ))))))))))))) as proof of (P b)
% 78.80/79.06 Found ((and_rect1 (P b)) (fun (x3:((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (((eq fofType) (cG cA)) cB))) (x4:(((eq fofType) (cG cB)) cA))=> (((fun (P0:Type) (x5:(((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))->((((eq fofType) (cG cA)) cB)->P0)))=> (((((and_rect ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (((eq fofType) (cG cA)) cB)) P0) x5) x3)) (P b)) (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ))))))))))))) as proof of (P b)
% 78.86/79.07 Found (((fun (P0:Type) (x3:(((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (((eq fofType) (cG cA)) cB))->((((eq fofType) (cG cB)) cA)->P0)))=> (((((and_rect ((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (((eq fofType) (cG cA)) cB))) (((eq fofType) (cG cB)) cA)) P0) x3) x1)) (P b)) (fun (x3:((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (((eq fofType) (cG cA)) cB))) (x4:(((eq fofType) (cG cB)) cA))=> (((fun (P0:Type) (x5:(((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))->((((eq fofType) (cG cA)) cB)->P0)))=> (((((and_rect ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (((eq fofType) (cG cA)) cB)) P0) x5) x3)) (P b)) (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ))))))))))))) as proof of (P b)
% 78.86/79.08 Found (((fun (P0:Type) (x3:(((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (((eq fofType) (cG cA)) cB))->((((eq fofType) (cG cB)) cA)->P0)))=> (((((and_rect ((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (((eq fofType) (cG cA)) cB))) (((eq fofType) (cG cB)) cA)) P0) x3) x1)) (P b)) (fun (x3:((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (((eq fofType) (cG cA)) cB))) (x4:(((eq fofType) (cG cB)) cA))=> (((fun (P0:Type) (x5:(((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))->((((eq fofType) (cG cA)) cB)->P0)))=> (((((and_rect ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (((eq fofType) (cG cA)) cB)) P0) x5) x3)) (P b)) (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xp Xx0)) (Xq Xy))->(Xq ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (Xq Xx0)) (Xp Xy))->(Xp ((cF Xx0) Xy)))))->((and (forall (Xx0:fofType), ((cP Xx0)->(Xp Xx0)))) (forall (Xx0:fofType), ((cQ Xx0)->(Xq Xx0)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))->((forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))) (x10:(forall (Xx0:fofType) (Xy:fofType), (((and (cQ Xx0)) (cP Xy))->(cP ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx0:fofType) (Xy:fofType), (((and (cP Xx0)) (cQ Xy))->(cQ ((cF Xx0) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (((eq_ref fofType) cB) cQ))))))))))))) as proof of (P b)
% 78.99/79.19 Found x40:=(x4 (fun (x14:fofType)=> (cQ cB))):((cQ cB)->(cQ cB))
% 78.99/79.19 Found (x4 (fun (x14:fofType)=> (cQ cB))) as proof of ((cQ cB)->(P b))
% 78.99/79.19 Found (x4 (fun (x14:fofType)=> (cQ cB))) as proof of ((cQ cB)->(P b))
% 78.99/79.19 Found (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))) as proof of ((cQ cB)->(P b))
% 78.99/79.19 Found (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))) as proof of ((cP cA)->((cQ cB)->(P b)))
% 78.99/79.19 Found (and_rect60 (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))) as proof of (P b)
% 78.99/79.19 Found ((and_rect6 (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))) as proof of (P b)
% 78.99/79.19 Found (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))) as proof of (P b)
% 78.99/79.19 Found (fun (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))) as proof of (P b)
% 78.99/79.19 Found (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))) as proof of ((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->(P b))
% 78.99/79.20 Found (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))) as proof of (((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->(P b)))
% 78.99/79.20 Found (and_rect50 (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))) as proof of (P b)
% 78.99/79.20 Found ((and_rect5 (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))) as proof of (P b)
% 78.99/79.20 Found (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))) as proof of (P b)
% 78.99/79.20 Found (fun (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))) as proof of (P b)
% 78.99/79.20 Found (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))) as proof of ((forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))->(P b))
% 78.99/79.20 Found (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))) as proof of (((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))->((forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))->(P b)))
% 78.99/79.20 Found (and_rect40 (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))) as proof of (P b)
% 78.99/79.20 Found ((and_rect4 (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))) as proof of (P b)
% 78.99/79.20 Found (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))->((forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))) as proof of (P b)
% 78.99/79.20 Found (fun (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))->((forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))))) as proof of (P b)
% 78.99/79.21 Found (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))->((forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))))) as proof of ((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))->(P b))
% 78.99/79.21 Found (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))->((forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))))) as proof of (((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))->(P b)))
% 78.99/79.22 Found (and_rect30 (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))->((forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))))) as proof of (P b)
% 78.99/79.22 Found ((and_rect3 (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))->((forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))))) as proof of (P b)
% 78.99/79.22 Found (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))->((forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))))) as proof of (P b)
% 78.99/79.22 Found (fun (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))->((forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))))))) as proof of (P b)
% 78.99/79.23 Found (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))->((forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))))))) as proof of ((((eq fofType) (cG cA)) cB)->(P b))
% 79.03/79.23 Found (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))->((forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))))))) as proof of (((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))->((((eq fofType) (cG cA)) cB)->(P b)))
% 79.04/79.24 Found (and_rect20 (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))->((forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))))))) as proof of (P b)
% 79.04/79.24 Found ((and_rect2 (P b)) (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))->((forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))))))) as proof of (P b)
% 79.04/79.25 Found (((fun (P0:Type) (x5:(((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))->((((eq fofType) (cG cA)) cB)->P0)))=> (((((and_rect ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB)) P0) x5) x3)) (P b)) (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))->((forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))))))) as proof of (P b)
% 79.04/79.26 Found (fun (x4:(((eq fofType) (cG cB)) cA))=> (((fun (P0:Type) (x5:(((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))->((((eq fofType) (cG cA)) cB)->P0)))=> (((((and_rect ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB)) P0) x5) x3)) (P b)) (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))->((forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))))))))) as proof of (P b)
% 79.04/79.27 Found (fun (x3:((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (x4:(((eq fofType) (cG cB)) cA))=> (((fun (P0:Type) (x5:(((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))->((((eq fofType) (cG cA)) cB)->P0)))=> (((((and_rect ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB)) P0) x5) x3)) (P b)) (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))->((forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))))))))) as proof of ((((eq fofType) (cG cB)) cA)->(P b))
% 79.04/79.28 Found (fun (x3:((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (x4:(((eq fofType) (cG cB)) cA))=> (((fun (P0:Type) (x5:(((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))->((((eq fofType) (cG cA)) cB)->P0)))=> (((((and_rect ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB)) P0) x5) x3)) (P b)) (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))->((forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))))))))) as proof of (((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))->((((eq fofType) (cG cB)) cA)->(P b)))
% 79.04/79.28 Found (and_rect10 (fun (x3:((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (x4:(((eq fofType) (cG cB)) cA))=> (((fun (P0:Type) (x5:(((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))->((((eq fofType) (cG cA)) cB)->P0)))=> (((((and_rect ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB)) P0) x5) x3)) (P b)) (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))->((forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))))))))) as proof of (P b)
% 79.04/79.29 Found ((and_rect1 (P b)) (fun (x3:((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (x4:(((eq fofType) (cG cB)) cA))=> (((fun (P0:Type) (x5:(((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))->((((eq fofType) (cG cA)) cB)->P0)))=> (((((and_rect ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB)) P0) x5) x3)) (P b)) (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))->((forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))))))))) as proof of (P b)
% 79.10/79.30 Found (((fun (P0:Type) (x3:(((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))->((((eq fofType) (cG cB)) cA)->P0)))=> (((((and_rect ((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (((eq fofType) (cG cB)) cA)) P0) x3) x1)) (P b)) (fun (x3:((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (x4:(((eq fofType) (cG cB)) cA))=> (((fun (P0:Type) (x5:(((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))->((((eq fofType) (cG cA)) cB)->P0)))=> (((((and_rect ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB)) P0) x5) x3)) (P b)) (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))->((forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))))))))) as proof of (P b)
% 79.10/79.31 Found (fun (x2:(forall (Xx:fofType) (Xy:fofType), (((eq fofType) (cG ((cF Xx) Xy))) ((cF (cG Xx)) (cG Xy)))))=> (((fun (P0:Type) (x3:(((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))->((((eq fofType) (cG cB)) cA)->P0)))=> (((((and_rect ((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (((eq fofType) (cG cB)) cA)) P0) x3) x1)) (P b)) (fun (x3:((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (x4:(((eq fofType) (cG cB)) cA))=> (((fun (P0:Type) (x5:(((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))->((((eq fofType) (cG cA)) cB)->P0)))=> (((((and_rect ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB)) P0) x5) x3)) (P b)) (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))->((forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))))))))))) as proof of (P b)
% 79.10/79.32 Found (fun (x1:((and ((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (((eq fofType) (cG cB)) cA))) (x2:(forall (Xx:fofType) (Xy:fofType), (((eq fofType) (cG ((cF Xx) Xy))) ((cF (cG Xx)) (cG Xy)))))=> (((fun (P0:Type) (x3:(((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))->((((eq fofType) (cG cB)) cA)->P0)))=> (((((and_rect ((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (((eq fofType) (cG cB)) cA)) P0) x3) x1)) (P b)) (fun (x3:((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (x4:(((eq fofType) (cG cB)) cA))=> (((fun (P0:Type) (x5:(((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))->((((eq fofType) (cG cA)) cB)->P0)))=> (((((and_rect ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB)) P0) x5) x3)) (P b)) (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))->((forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))))))))))) as proof of ((forall (Xx:fofType) (Xy:fofType), (((eq fofType) (cG ((cF Xx) Xy))) ((cF (cG Xx)) (cG Xy))))->(P b))
% 79.10/79.33 Found (fun (x1:((and ((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (((eq fofType) (cG cB)) cA))) (x2:(forall (Xx:fofType) (Xy:fofType), (((eq fofType) (cG ((cF Xx) Xy))) ((cF (cG Xx)) (cG Xy)))))=> (((fun (P0:Type) (x3:(((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))->((((eq fofType) (cG cB)) cA)->P0)))=> (((((and_rect ((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (((eq fofType) (cG cB)) cA)) P0) x3) x1)) (P b)) (fun (x3:((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (x4:(((eq fofType) (cG cB)) cA))=> (((fun (P0:Type) (x5:(((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))->((((eq fofType) (cG cA)) cB)->P0)))=> (((((and_rect ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB)) P0) x5) x3)) (P b)) (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))->((forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB)))))))))))))))) as proof of (((and ((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (((eq fofType) (cG cB)) cA))->((forall (Xx:fofType) (Xy:fofType), (((eq fofType) (cG ((cF Xx) Xy))) ((cF (cG Xx)) (cG Xy))))->(P b)))
% 79.10/79.34 Found (and_rect00 (fun (x1:((and ((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (((eq fofType) (cG cB)) cA))) (x2:(forall (Xx:fofType) (Xy:fofType), (((eq fofType) (cG ((cF Xx) Xy))) ((cF (cG Xx)) (cG Xy)))))=> (((fun (P0:Type) (x3:(((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))->((((eq fofType) (cG cB)) cA)->P0)))=> (((((and_rect ((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (((eq fofType) (cG cB)) cA)) P0) x3) x1)) (P b)) (fun (x3:((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (x4:(((eq fofType) (cG cB)) cA))=> (((fun (P0:Type) (x5:(((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))->((((eq fofType) (cG cA)) cB)->P0)))=> (((((and_rect ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB)) P0) x5) x3)) (P b)) (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))->((forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))))))))))) as proof of (P b)
% 79.10/79.35 Found ((and_rect0 (P b)) (fun (x1:((and ((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (((eq fofType) (cG cB)) cA))) (x2:(forall (Xx:fofType) (Xy:fofType), (((eq fofType) (cG ((cF Xx) Xy))) ((cF (cG Xx)) (cG Xy)))))=> (((fun (P0:Type) (x3:(((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))->((((eq fofType) (cG cB)) cA)->P0)))=> (((((and_rect ((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (((eq fofType) (cG cB)) cA)) P0) x3) x1)) (P b)) (fun (x3:((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (x4:(((eq fofType) (cG cB)) cA))=> (((fun (P0:Type) (x5:(((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))->((((eq fofType) (cG cA)) cB)->P0)))=> (((((and_rect ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB)) P0) x5) x3)) (P b)) (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))->((forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))))))))))) as proof of (P b)
% 79.16/79.37 Found (((fun (P0:Type) (x1:(((and ((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (((eq fofType) (cG cB)) cA))->((forall (Xx:fofType) (Xy:fofType), (((eq fofType) (cG ((cF Xx) Xy))) ((cF (cG Xx)) (cG Xy))))->P0)))=> (((((and_rect ((and ((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (((eq fofType) (cG cB)) cA))) (forall (Xx:fofType) (Xy:fofType), (((eq fofType) (cG ((cF Xx) Xy))) ((cF (cG Xx)) (cG Xy))))) P0) x1) x)) (P b)) (fun (x1:((and ((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (((eq fofType) (cG cB)) cA))) (x2:(forall (Xx:fofType) (Xy:fofType), (((eq fofType) (cG ((cF Xx) Xy))) ((cF (cG Xx)) (cG Xy)))))=> (((fun (P0:Type) (x3:(((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))->((((eq fofType) (cG cB)) cA)->P0)))=> (((((and_rect ((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (((eq fofType) (cG cB)) cA)) P0) x3) x1)) (P b)) (fun (x3:((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (x4:(((eq fofType) (cG cB)) cA))=> (((fun (P0:Type) (x5:(((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))->((((eq fofType) (cG cA)) cB)->P0)))=> (((((and_rect ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB)) P0) x5) x3)) (P b)) (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))->((forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))))))))))) as proof of (P b)
% 79.16/79.38 Found (((fun (P0:Type) (x1:(((and ((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (((eq fofType) (cG cB)) cA))->((forall (Xx:fofType) (Xy:fofType), (((eq fofType) (cG ((cF Xx) Xy))) ((cF (cG Xx)) (cG Xy))))->P0)))=> (((((and_rect ((and ((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (((eq fofType) (cG cB)) cA))) (forall (Xx:fofType) (Xy:fofType), (((eq fofType) (cG ((cF Xx) Xy))) ((cF (cG Xx)) (cG Xy))))) P0) x1) x)) (P b)) (fun (x1:((and ((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (((eq fofType) (cG cB)) cA))) (x2:(forall (Xx:fofType) (Xy:fofType), (((eq fofType) (cG ((cF Xx) Xy))) ((cF (cG Xx)) (cG Xy)))))=> (((fun (P0:Type) (x3:(((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))->((((eq fofType) (cG cB)) cA)->P0)))=> (((((and_rect ((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (((eq fofType) (cG cB)) cA)) P0) x3) x1)) (P b)) (fun (x3:((and ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB))) (x4:(((eq fofType) (cG cB)) cA))=> (((fun (P0:Type) (x5:(((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))->((((eq fofType) (cG cA)) cB)->P0)))=> (((((and_rect ((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (((eq fofType) (cG cA)) cB)) P0) x5) x3)) (P b)) (fun (x5:((and ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))) (x6:(((eq fofType) (cG cA)) cB))=> (((fun (P0:Type) (x7:(((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))->((forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))->P0)))=> (((((and_rect ((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx))))))) P0) x7) x5)) (P b)) (fun (x7:((and ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))) (x8:(forall (Xp:(fofType->Prop)) (Xq:(fofType->Prop)), (((and ((and ((and (Xp cA)) (Xq cB))) (forall (Xx:fofType) (Xy:fofType), (((and (Xp Xx)) (Xq Xy))->(Xq ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (Xq Xx)) (Xp Xy))->(Xp ((cF Xx) Xy)))))->((and (forall (Xx:fofType), ((cP Xx)->(Xp Xx)))) (forall (Xx:fofType), ((cQ Xx)->(Xq Xx)))))))=> (((fun (P0:Type) (x9:(((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))->((forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy))))) P0) x9) x7)) (P b)) (fun (x9:((and ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))) (x10:(forall (Xx:fofType) (Xy:fofType), (((and (cQ Xx)) (cP Xy))->(cP ((cF Xx) Xy)))))=> (((fun (P0:Type) (x11:(((and (cP cA)) (cQ cB))->((forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))->P0)))=> (((((and_rect ((and (cP cA)) (cQ cB))) (forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy))))) P0) x11) x9)) (P b)) (fun (x11:((and (cP cA)) (cQ cB))) (x12:(forall (Xx:fofType) (Xy:fofType), (((and (cP Xx)) (cQ Xy))->(cQ ((cF Xx) Xy)))))=> (((fun (P0:Type) (x13:((cP cA)->((cQ cB)->P0)))=> (((((and_rect (cP cA)) (cQ cB)) P0) x13) x11)) (P b)) (fun (x13:(cP cA))=> (x4 (fun (x14:fofType)=> (cQ cB))))))))))))))))) as proof of (P b)
% 100.34/100.60 Found x6:(((eq fofType) (cG cA)) cB)
% 100.34/100.60 Found x6 as proof of (((eq fofType) (cG cA)) cB)
% 100.34/100.60 Found x6:(((eq fofType) (cG cA)) cB)
% 100.34/100.60 Found x6 as proof of (((eq fofType) (cG cA)) cB)
% 100.34/100.60 Found x4:(((eq fofType) (cG cB)) cA)
% 100.34/100.60 Instantiate: a:=(cG cB):fofType;b:=cA:fofType
% 100.34/100.60 Found x4 as proof of (((eq fofType) a) b)
% 100.34/100.60 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 100.34/100.60 Found (eq_ref0 b) as proof of (((eq fofType) b) (cG Xx))
% 100.34/100.60 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (cG Xx))
% 100.34/100.60 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (cG Xx))
% 100.34/100.60 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (cG Xx))
% 100.34/100.60 Found x40:=(x4 (fun (x14:fofType)=> (cQ cB))):((cQ cB)->(cQ cB))
% 100.34/100.60 Found (x4 (fun (x14:fofType)=> (cQ cB))) as proof of (P cB)
% 100.34/100.60 Found (x4 (fun (x14:fofType)=> (cQ cB))) as proof of (P cB)
% 100.34/100.60 Found eq_ref00:=(eq_ref0 cB):(((eq fofType) cB) cB)
% 100.34/100.60 Found (eq_ref0 cB) as proof of (((eq fofType) cB) b)
% 100.34/100.60 Found ((eq_ref fofType) cB) as proof of (((eq fofType) cB) b)
% 100.34/100.60 Found ((eq_ref fofType) cB) as proof of (((eq fofType) cB) b)
% 100.34/100.60 Found ((eq_ref fofType) cB) as proof of (((eq fofType) cB) b)
% 100.34/100.60 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 100.34/100.60 Found (eq_ref0 b) as proof of (((eq fofType) b) (cG Xx))
% 100.34/100.60 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (cG Xx))
% 264.55/264.85 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (cG Xx))
% 264.55/264.85 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (cG Xx))
% 264.55/264.85 Found eq_ref00:=(eq_ref0 cB):(((eq fofType) cB) cB)
% 264.55/264.85 Found (eq_ref0 cB) as proof of (((eq fofType) cB) b)
% 264.55/264.85 Found ((eq_ref fofType) cB) as proof of (((eq fofType) cB) b)
% 264.55/264.85 Found ((eq_ref fofType) cB) as proof of (((eq fofType) cB) b)
% 264.55/264.85 Found ((eq_ref fofType) cB) as proof of (((eq fofType) cB) b)
% 264.55/264.85 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 264.55/264.85 Found (eq_ref0 b) as proof of (((eq fofType) b) (cG Xx))
% 264.55/264.85 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (cG Xx))
% 264.55/264.85 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (cG Xx))
% 264.55/264.85 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (cG Xx))
% 264.55/264.85 Found x6:(((eq fofType) (cG cA)) cB)
% 264.55/264.85 Instantiate: a:=(cG cA):fofType;b:=cB:fofType
% 264.55/264.85 Found x6 as proof of (((eq fofType) a) b)
% 264.55/264.85 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 264.55/264.85 Found (eq_ref0 b) as proof of (((eq fofType) b) (cG Xx))
% 264.55/264.85 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (cG Xx))
% 264.55/264.85 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (cG Xx))
% 264.55/264.85 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (cG Xx))
% 264.55/264.85 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 264.55/264.85 Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% 264.55/264.85 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 264.55/264.85 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 264.55/264.85 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 264.55/264.85 Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% 264.55/264.85 Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% 264.55/264.85 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 264.55/264.85 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 264.55/264.85 Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% 264.55/264.85 Found eq_ref000:=(eq_ref00 P0):((P0 (cG Xx))->(P0 (cG Xx)))
% 264.55/264.85 Found (eq_ref00 P0) as proof of (P1 (cG Xx))
% 264.55/264.85 Found ((eq_ref0 (cG Xx)) P0) as proof of (P1 (cG Xx))
% 264.55/264.85 Found (((eq_ref fofType) (cG Xx)) P0) as proof of (P1 (cG Xx))
% 264.55/264.85 Found (((eq_ref fofType) (cG Xx)) P0) as proof of (P1 (cG Xx))
% 264.55/264.85 Found eq_ref000:=(eq_ref00 P0):((P0 (cG Xx))->(P0 (cG Xx)))
% 264.55/264.85 Found (eq_ref00 P0) as proof of (P1 (cG Xx))
% 264.55/264.85 Found ((eq_ref0 (cG Xx)) P0) as proof of (P1 (cG Xx))
% 264.55/264.85 Found (((eq_ref fofType) (cG Xx)) P0) as proof of (P1 (cG Xx))
% 264.55/264.85 Found (((eq_ref fofType) (cG Xx)) P0) as proof of (P1 (cG Xx))
% 264.55/264.85 Found x6:(((eq fofType) (cG cA)) cB)
% 264.55/264.85 Instantiate: a:=(cG cA):fofType;b:=cB:fofType
% 264.55/264.85 Found x6 as proof of (((eq fofType) a) b)
% 264.55/264.85 Found eq_ref000:=(eq_ref00 P0):((P0 (cG Xx))->(P0 (cG Xx)))
% 264.55/264.85 Found (eq_ref00 P0) as proof of (P1 (cG Xx))
% 264.55/264.85 Found ((eq_ref0 (cG Xx)) P0) as proof of (P1 (cG Xx))
% 264.55/264.85 Found (((eq_ref fofType) (cG Xx)) P0) as proof of (P1 (cG Xx))
% 264.55/264.85 Found (((eq_ref fofType) (cG Xx)) P0) as proof of (P1 (cG Xx))
% 264.55/264.85 Found eq_ref000:=(eq_ref00 P0):((P0 (cG Xx))->(P0 (cG Xx)))
% 264.55/264.85 Found (eq_ref00 P0) as proof of (P1 (cG Xx))
% 264.55/264.85 Found ((eq_ref0 (cG Xx)) P0) as proof of (P1 (cG Xx))
% 264.55/264.85 Found (((eq_ref fofType) (cG Xx)) P0) as proof of (P1 (cG Xx))
% 264.55/264.85 Found (((eq_ref fofType) (cG Xx)) P0) as proof of (P1 (cG Xx))
% 264.55/264.85 Found x40:=(x4 (fun (x14:fofType)=> (P0 (cG Xx)))):((P0 (cG Xx))->(P0 (cG Xx)))
% 264.55/264.85 Found (x4 (fun (x14:fofType)=> (P0 (cG Xx)))) as proof of (P1 (cG Xx))
% 264.55/264.85 Found (x4 (fun (x14:fofType)=> (P0 (cG Xx)))) as proof of (P1 (cG Xx))
% 264.55/264.85 Found x40:=(x4 (fun (x14:fofType)=> (P0 (cG Xx)))):((P0 (cG Xx))->(P0 (cG Xx)))
% 264.55/264.85 Found (x4 (fun (x14:fofType)=> (P0 (cG Xx)))) as proof of (P1 (cG Xx))
% 264.55/264.85 Found (x4 (fun (x14:fofType)=> (P0 (cG Xx)))) as proof of (P1 (cG Xx))
% 264.55/264.85 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 264.55/264.85 Found (eq_ref0 b) as proof of (((eq fofType) b) (cG Xx))
% 264.55/264.85 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (cG Xx))
% 264.55/264.85 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (cG Xx))
% 264.55/264.85 Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (cG Xx))
% 264.55/264.85 Found x6:(((eq fofType) (cG cA)) cB)
% 264.55/264.85 Found x6 as proof of (((eq fofType) (cG cA)) cB)
% 264.55/264.85 Found x6:(((eq fofType) (cG cA)) cB)
% 264.55/264.85 Found x6 as proof of (((eq fofType) (cG cA)) cB)
% 264.55/264.85 Found x6:(((eq fofType) (cG cA)) cB)
% 264.55/264.85 Found x6 as proof of (((eq fofType) (cG cA)) b)
% 264.55/264.85 Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% 264.55/264.85 Found (eq_ref0 b) as proof of
%------------------------------------------------------------------------------